mean shift 图像分割_算法_其他

<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/6244dcfd6cd7507199bf81c5/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6244dcfd6cd7507199bf81c5/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Mean<span class="_ _0"> </span>Shift:<span class="_ _0"> </span>A<span class="_ _0"> </span>Robust<span class="_ _0"> </span>Approach</div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">Toward<span class="_ _0"> </span>Feature<span class="_ _0"> </span>Space<span class="_ _0"> </span>Analysis</div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls1 ws0">Dorin<span class="_ _1"> </span>Comaniciu,<span class="_ _1"> </span><span class="ff2 ls2">Member</span><span class="ls3">,<span class="_ _1"> </span><span class="ff2 ls4">IEEE</span><span class="ls5">,<span class="_ _1"> </span>and<span class="_ _1"> </span>Peter<span class="_ _1"> </span>Meer,<span class="_ _1"> </span><span class="ff2 ls6">Senior<span class="_ _1"> </span>Member</span></span>,<span class="_ _1"> </span><span class="ff2 ls5">IEEE</span></span></div><div class="t m0 x4 h4 y4 ff3 fs2 fc0 sc0 ls3 ws0">Abstract<span class="ff1">ÐA<span class="_ _2"> </span>general<span class="_ _2"> </span>nonparametric<span class="_ _2"> </span>technique<span class="_ _2"> </span>is<span class="_ _2"> </span>proposed<span class="_ _2"> </span>for<span class="_ _2"> </span>the<span class="_ _2"> </span>analysis<span class="_ _2"> </span>of<span class="_ _2"> </span>a<span class="_ _3"> </span>complex<span class="_ _2"> </span>multimodal<span class="_ _2"> </span>feature<span class="_ _2"> </span>space<span class="_ _2"> </span>and<span class="_ _2"> </span>to<span class="_ _2"> </span>delineate</span></div><div class="t m0 x4 h4 y5 ff1 fs2 fc0 sc0 ls3 ws0">arbitrarily<span class="_ _2"> </span>shaped<span class="_ _2"> </span>clusters<span class="_ _2"> </span>in<span class="_ _2"> </span>it.<span class="_ _2"> </span>The<span class="_ _3"> </span>basic<span class="_ _2"> </span>computational<span class="_ _2"> </span>module<span class="_ _2"> </span>of<span class="_ _2"> </span>the<span class="_ _2"> </span>technique<span class="_ _2"> </span>is<span class="_ _2"> </span>an<span class="_ _2"> </span>old<span class="_ _3"> </span>pattern<span class="_ _2"> </span>recognition<span class="_ _2"> </span>procedure,<span class="_ _2"> </span>the<span class="_ _2"> </span>mean</div><div class="t m0 x4 h4 y6 ff1 fs2 fc0 sc0 ls3 ws0">shift.<span class="_ _4"> </span>We<span class="_ _4"> </span>prove<span class="_ _4"> </span>for<span class="_ _4"> </span>discrete<span class="_ _4"> </span>data<span class="_ _4"> </span>the<span class="_ _4"> </span>convergence<span class="_ _4"> </span>of<span class="_ _4"> </span>a<span class="_ _4"> </span>recursive<span class="_ _4"> </span>mean<span class="_ _4"> </span>shift<span class="_ _4"> </span>procedure<span class="_ _4"> </span>to<span class="_ _4"> </span>the<span class="_ _4"> </span>nearest<span class="_ _4"> </span>stationary<span class="_ _4"> </span>point<span class="_ _4"> </span>of<span class="_ _4"> </span>the<span class="_ _2"> </span>underlying</div><div class="t m0 x4 h4 y7 ff1 fs2 fc0 sc0 ls3 ws1">density<span class="_ _2"> </span>function<span class="_ _4"> </span>and,<span class="_ _2"> </span>thus,<span class="_ _2"> </span>its<span class="_ _4"> </span>utility<span class="_ _2"> </span>in<span class="_ _2"> </span>detecting<span class="_ _4"> </span>the<span class="_ _2"> </span>modes<span class="_ _2"> </span>of<span class="_ _2"> </span>the<span class="_ _4"> </span>density.<span class="_ _2"> </span>The<span class="_ _2"> </span>relation<span class="_ _4"> </span>of<span class="_ _2"> </span>the<span class="_ _2"> </span>mean<span class="_ _4"> </span>shift<span class="_ _2"> </span>procedure<span class="_ _2"> </span>to<span class="_ _2"> </span>the<span class="_ _4"> </span>Nadaraya-</div><div class="t m0 x4 h4 y8 ff1 fs2 fc0 sc0 ls3 ws0">Watson<span class="_ _5"> </span>estimator<span class="_ _5"> </span>from<span class="_ _5"> </span>kernel<span class="_ _5"> </span>regression<span class="_ _5"> </span>and<span class="_ _5"> </span>the<span class="_ _5"> </span>robust<span class="_ _5"> </span>M-estimators<span class="_ _5"> </span>of<span class="_ _5"> </span>location<span class="_ _5"> </span>is<span class="_ _5"> </span>also<span class="_ _5"> </span>established.<span class="_ _5"> </span>Algorithms<span class="_ _5"> </span>for<span class="_ _5"> </span>two<span class="_ _5"> </span>low-level<span class="_ _5"> </span>vision</div><div class="t m0 x4 h4 y9 ff1 fs2 fc0 sc0 ls3 ws0">tasks,<span class="_ _2"> </span>discontinuity<span class="_ _4"> </span>preserving<span class="_ _2"> </span>smoothing<span class="_ _4"> </span>and<span class="_ _2"> </span>image<span class="_ _2"> </span>segmentation,<span class="_ _4"> </span>are<span class="_ _2"> </span>described<span class="_ _2"> </span>as<span class="_ _2"> </span>applications.<span class="_ _4"> </span>In<span class="_ _2"> </span>these<span class="_ _2"> </span>algorithms,<span class="_ _4"> </span>the<span class="_ _2"> </span>only<span class="_ _2"> </span>user</div><div class="t m0 x4 h4 ya ff1 fs2 fc0 sc0 ls3 ws0">set<span class="_ _2"> </span>parameter<span class="_ _2"> </span>is<span class="_ _2"> </span>the<span class="_ _2"> </span>resolution<span class="_ _2"> </span>of<span class="_ _3"> </span>the<span class="_ _2"> </span>analysis<span class="_ _2"> </span>and<span class="_ _2"> </span>either<span class="_ _2"> </span>gray<span class="_ _2"> </span>level<span class="_ _2"> </span>or<span class="_ _3"> </span>color<span class="_ _2"> </span>images<span class="_ _2"> </span>are<span class="_ _2"> </span>accepted<span class="_ _2"> </span>as<span class="_ _2"> </span>input.<span class="_ _2"> </span>Extensive<span class="_ _2"> </span>experimental</div><div class="t m0 x4 h4 yb ff1 fs2 fc0 sc0 ls3 ws0">results<span class="_ _2"> </span>illustrate<span class="_ _2"> </span>their<span class="_ _2"> </span>excellent<span class="_ _2"> </span>performance.</div><div class="t m0 x4 h4 yc ff3 fs2 fc0 sc0 ls3 ws0">Index<span class="_ _2"> </span>Terms<span class="ff1">ÐMean<span class="_ _2"> </span>shift,<span class="_ _2"> </span>clustering,<span class="_ _2"> </span>image<span class="_ _2"> </span>segmentation,<span class="_ _2"> </span>image<span class="_ _2"> </span>smoothin<span class="_ _6"></span>g,<span class="_ _2"> </span>feature<span class="_ _2"> </span>space,<span class="_ _2"> </span>low-level<span class="_ _2"> </span>vision.</span></div><div class="t m0 x5 h5 yd ff4 fs3 fc0 sc0 ls3 ws0">æ</div><div class="t m0 x6 h6 ye ff3 fs4 fc0 sc0 ls7 ws0">1I<span class="_ _7"></span><span class="fs5 ls3">NTRODUCTION</span></div><div class="t m0 x6 h7 yf ff5 fs6 fc0 sc0 ls3 ws0">L</div><div class="t m0 x7 h8 y10 ff5 fs7 fc0 sc0 ls8 ws0">OW-LEVEL<span class="_ _3"> </span><span class="fs8 ls3">computer<span class="_ _8"> </span>vision<span class="_ _1"> </span>tasks<span class="_ _8"> </span>are<span class="_ _1"> </span>misleadingly<span class="_ _8"> </span>diffi-</span></div><div class="t m0 x7 h8 y11 ff5 fs8 fc0 sc0 ls3 ws0">cult.<span class="_ _8"> </span>Incorrect<span class="_ _8"> </span>results<span class="_ _8"> </span>can<span class="_ _8"> </span>be<span class="_ _8"> </span>easily<span class="_ _1"> </span>obtained<span class="_ _8"> </span>since<span class="_ _9"> </span>the</div><div class="t m0 x6 h8 y12 ff5 fs8 fc0 sc0 ls3 ws0">employed<span class="_ _a"> </span>techniques<span class="_ _a"> </span>often<span class="_ _a"> </span>rely<span class="_ _a"> </span>upon<span class="_ _a"> </span>the<span class="_ _a"> </span>user<span class="_ _a"> </span>correctly</div><div class="t m0 x6 h8 y13 ff5 fs8 fc0 sc0 ls3 ws0">guessing<span class="_ _3"> </span>the<span class="_ _b"> </span>values<span class="_ _3"> </span>for<span class="_ _b"> </span>the<span class="_ _3"> </span>tuning<span class="_ _b"> </span>parameters.<span class="_ _b"> </span>To<span class="_ _3"> </span>improve</div><div class="t m0 x6 h8 y14 ff5 fs8 fc0 sc0 ls3 ws2">performance,<span class="_ _4"> </span>the<span class="_ _4"> </span>execution<span class="_ _2"> </span>of<span class="_ _5"> </span>low-level<span class="_ _2"> </span>tasks<span class="_ _4"> </span>should<span class="_ _4"> </span>be<span class="_ _4"> </span>task</div><div class="t m0 x6 h8 y15 ff5 fs8 fc0 sc0 ls3 ws3">driven,<span class="_ _b"> </span>i.e.,<span class="_ _1"> </span>supported<span class="_ _b"> </span>by<span class="_ _1"> </span>independent<span class="_ _b"> </span>high-level<span class="_ _1"> </span>informa-</div><div class="t m0 x6 h8 y16 ff5 fs8 fc0 sc0 ls3 ws4">tion.<span class="_ _1"> </span>This<span class="_ _1"> </span>approach,<span class="_ _b"> </span>however,<span class="_ _1"> </span>requires<span class="_ _1"> </span>that,<span class="_ _1"> </span>first,<span class="_ _1"> </span>the<span class="_ _1"> </span>low-</div><div class="t m0 x6 h8 y17 ff5 fs8 fc0 sc0 ls3 ws0">level<span class="_ _3"> </span>stage<span class="_ _3"> </span>provides<span class="_ _3"> </span>a<span class="_ _3"> </span>reliable<span class="_ _3"> </span>enough<span class="_ _3"> </span>representation<span class="_ _3"> </span>of<span class="_ _2"> </span>the</div><div class="t m0 x6 h8 y18 ff5 fs8 fc0 sc0 ls3 ws0">input<span class="_ _b"> </span>and<span class="_ _1"> </span>that<span class="_ _b"> </span>the<span class="_ _1"> </span>feature<span class="_ _b"> </span>extraction<span class="_ _1"> </span>process<span class="_ _b"> </span>be<span class="_ _1"> </span>controlled</div><div class="t m0 x6 h8 y19 ff5 fs8 fc0 sc0 ls3 ws0">only<span class="_ _c"> </span>by<span class="_ _c"> </span>very<span class="_ _c"> </span>few<span class="_ _c"> </span>tuning<span class="_ _c"> </span>parameters<span class="_ _c"> </span>corresponding<span class="_ _c"> </span>to</div><div class="t m0 x6 h8 y1a ff5 fs8 fc0 sc0 ls3 ws0">intuitive<span class="_ _b"> </span>measures<span class="_ _3"> </span>in<span class="_ _b"> </span>the<span class="_ _b"> </span>input<span class="_ _3"> </span>domain.</div><div class="t m0 x8 h8 y1b ff5 fs8 fc0 sc0 ls3 ws0">Feature<span class="_ _d"> </span>space-based<span class="_ _d"> </span>analysis<span class="_ _d"> </span>of<span class="_ _d"> </span>images<span class="_ _d"> </span>is<span class="_ _d"> </span>a<span class="_ _d"> </span>paradigm</div><div class="t m0 x6 h8 y1c ff5 fs8 fc0 sc0 ls3 ws0">which<span class="_"> </span>can<span class="_"> </span>achieve<span class="_"> </span>the<span class="_"> </span>above-stated<span class="_"> </span>goals.<span class="_"> </span>A<span class="_"> </span>feature<span class="_"> </span>space<span class="_ _2"> </span>is</div><div class="t m0 x6 h8 y1d ff5 fs8 fc0 sc0 ls3 ws0">a<span class="_ _3"> </span>mapping<span class="_ _3"> </span>of<span class="_ _b"> </span>the<span class="_ _3"> </span>input<span class="_ _3"> </span>obtained<span class="_ _b"> </span>through<span class="_ _3"> </span>the<span class="_ _3"> </span>processing<span class="_ _b"> </span>of</div><div class="t m0 x6 h8 y1e ff5 fs8 fc0 sc0 ls3 ws5">the<span class="_ _a"> </span>data<span class="_ _a"> </span>in<span class="_ _a"> </span>small<span class="_ _a"> </span>subsets<span class="_ _a"> </span>at<span class="_ _a"> </span>a<span class="_ _e"> </span>time.<span class="_ _a"> </span>For<span class="_ _a"> </span>each<span class="_ _a"> </span>subset,<span class="_ _a"> </span>a</div><div class="t m0 x6 h8 y1f ff5 fs8 fc0 sc0 ls3 ws0">para<span class="_ _f"></span>metri<span class="_ _f"></span>c<span class="_ _10"> </span>repre<span class="_ _f"></span>sent<span class="_ _f"></span>ation<span class="_ _10"> </span>o<span class="_ _f"></span>f<span class="_ _c"> </span>t<span class="_ _f"></span>he<span class="_ _10"> </span>feat<span class="_ _f"></span>ure<span class="_ _10"> </span>of<span class="_ _10"> </span>int<span class="_ _f"></span>ere<span class="_ _f"></span>st<span class="_ _c"> </span>i<span class="_ _f"></span>s</div><div class="t m0 x6 h8 y20 ff5 fs8 fc0 sc0 ls3 ws0">obtained<span class="_ _e"> </span>and<span class="_ _e"> </span>the<span class="_ _e"> </span>result<span class="_ _e"> </span>is<span class="_ _e"> </span>mapped<span class="_ _e"> </span>into<span class="_ _11"> </span>a<span class="_ _e"> </span>point<span class="_ _e"> </span>in<span class="_ _e"> </span>the</div><div class="t m0 x6 h8 y21 ff5 fs8 fc0 sc0 ls3 ws0">multidimensional<span class="_ _8"> </span>space<span class="_ _1"> </span>of<span class="_ _8"> </span>the<span class="_ _1"> </span>parameter.<span class="_ _8"> </span>After<span class="_ _1"> </span>the<span class="_ _8"> </span>entire</div><div class="t m0 x6 h8 y22 ff5 fs8 fc0 sc0 ls3 ws6">input<span class="_ _4"> </span>is<span class="_ _2"> </span>processed,<span class="_ _4"> </span>significant<span class="_ _4"> </span>features<span class="_ _2"> </span>correspond<span class="_ _4"> </span>to<span class="_ _2"> </span>denser</div><div class="t m0 x6 h8 y23 ff5 fs8 fc0 sc0 ls3 ws7">regions<span class="_ _3"> </span>in<span class="_ _2"> </span>the<span class="_ _3"> </span>feature<span class="_ _3"> </span>space,<span class="_ _3"> </span>i.e.,<span class="_ _3"> </span>to<span class="_ _2"> </span>clusters,<span class="_ _3"> </span>and<span class="_ _3"> </span>the<span class="_ _3"> </span>goal<span class="_ _3"> </span>of</div><div class="t m0 x6 h8 y24 ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _b"> </span>analysis<span class="_ _3"> </span>is<span class="_ _b"> </span>the<span class="_ _3"> </span>delineation<span class="_ _b"> </span>of<span class="_ _b"> </span>these<span class="_ _3"> </span>clusters.</div><div class="t m0 x8 h8 y25 ff5 fs8 fc0 sc0 ls3 ws0">The<span class="_"> </span>nature<span class="_ _2"> </span>of<span class="_ _2"> </span>the<span class="_ _2"> </span>feature<span class="_ _2"> </span>space<span class="_ _2"> </span>is<span class="_ _2"> </span>application<span class="_ _2"> </span>dependent.</div><div class="t m0 x6 h8 y26 ff5 fs8 fc0 sc0 ls3 ws0">The<span class="_ _e"> </span>subsets<span class="_ _e"> </span>employed<span class="_ _11"> </span>in<span class="_ _e"> </span>the<span class="_ _e"> </span>mapping<span class="_ _11"> </span>can<span class="_ _e"> </span>range<span class="_ _11"> </span>from</div><div class="t m0 x6 h8 y27 ff5 fs8 fc0 sc0 ls3 ws8">individual<span class="_ _3"> </span>pixels,<span class="_ _3"> </span>as<span class="_ _3"> </span>in<span class="_ _3"> </span>the<span class="_ _3"> </span>color<span class="_ _3"> </span>space<span class="_ _b"> </span>representation<span class="_ _3"> </span>of<span class="_ _3"> </span>an</div><div class="t m0 x6 h8 y28 ff5 fs8 fc0 sc0 ls3 ws9">image,<span class="_ _3"> </span>to<span class="_ _b"> </span>a<span class="_ _3"> </span>set<span class="_ _3"> </span>of<span class="_ _b"> </span>quasi-randomly<span class="_ _3"> </span>chosen<span class="_ _3"> </span>data<span class="_ _b"> </span>points,<span class="_ _3"> </span>as<span class="_ _3"> </span>in</div><div class="t m0 x6 h8 y29 ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _3"> </span>probabilistic<span class="_ _3"> </span>Hough<span class="_ _3"> </span>transform.<span class="_ _3"> </span>Both<span class="_ _3"> </span>the<span class="_ _2"> </span>advantage<span class="_ _3"> </span>and</div><div class="t m0 x6 h8 y2a ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _b"> </span>disadvantage<span class="_ _b"> </span>of<span class="_ _b"> </span>the<span class="_ _b"> </span>feature<span class="_ _b"> </span>space<span class="_ _b"> </span>paradigm<span class="_ _b"> </span>arise<span class="_ _b"> </span>from</div><div class="t m0 x6 h8 y2b ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _2"> </span>global<span class="_ _2"> </span>nature<span class="_ _2"> </span>of<span class="_ _2"> </span>the<span class="_ _3"> </span>derived<span class="_"> </span>representation<span class="_ _3"> </span>of<span class="_ _2"> </span>the<span class="_ _2"> </span>input.</div><div class="t m0 x6 h8 y2c ff5 fs8 fc0 sc0 ls3 wsa">On<span class="_ _c"> </span>one<span class="_ _10"> </span>hand,<span class="_ _c"> </span>all<span class="_ _10"> </span>the<span class="_ _c"> </span>evidence<span class="_ _10"> </span>for<span class="_ _c"> </span>the<span class="_ _10"> </span>presence<span class="_ _10"> </span>of<span class="_ _c"> </span>a</div><div class="t m0 x9 h8 y2d ff5 fs8 fc0 sc0 ls3 wsb">significant<span class="_ _9"> </span>feature<span class="_ _9"> </span>is<span class="_ _9"> </span>pooled<span class="_ _9"> </span>together,<span class="_ _d"> </span>providing<span class="_ _9"> </span>excellent</div><div class="t m0 x9 h8 y2e ff5 fs8 fc0 sc0 ls3 ws0">tolerance<span class="_ _3"> </span>to<span class="_ _b"> </span>a<span class="_ _3"> </span>noise<span class="_ _3"> </span>level<span class="_ _b"> </span>which<span class="_ _3"> </span>may<span class="_ _b"> </span>render<span class="_ _3"> </span>local<span class="_ _3"> </span>decisions</div><div class="t m0 x9 h8 y2f ff5 fs8 fc0 sc0 ls3 wsc">unreliable.<span class="_ _3"> </span>On<span class="_ _b"> </span>the<span class="_ _3"> </span>other<span class="_ _b"> </span>hand,<span class="_ _b"> </span>features<span class="_ _3"> </span>with<span class="_ _b"> </span>lesser<span class="_ _3"> </span>support</div><div class="t m0 x9 h8 y30 ff5 fs8 fc0 sc0 ls3 ws0">in<span class="_ _1"> </span>the<span class="_ _b"> </span>feature<span class="_ _1"> </span>space<span class="_ _1"> </span>may<span class="_ _1"> </span>not<span class="_ _1"> </span>be<span class="_ _b"> </span>detected<span class="_ _1"> </span>in<span class="_ _1"> </span>spite<span class="_ _1"> </span>of<span class="_ _1"> </span>being</div><div class="t m0 x9 h8 y31 ff5 fs8 fc0 sc0 ls3 ws0">salient<span class="_ _11"> </span>for<span class="_ _c"> </span>the<span class="_ _e"> </span>task<span class="_ _c"> </span>to<span class="_ _11"> </span>be<span class="_ _c"> </span>executed.<span class="_ _e"> </span>This<span class="_ _c"> </span>disadvantage,</div><div class="t m0 x9 h8 y32 ff5 fs8 fc0 sc0 ls3 wsd">however,<span class="_ _b"> </span>can<span class="_ _b"> </span>be<span class="_ _b"> </span>largely<span class="_ _b"> </span>avoided<span class="_ _b"> </span>by<span class="_ _b"> </span>either<span class="_ _b"> </span>augmenting<span class="_ _b"> </span>the</div><div class="t m0 x9 h8 y33 ff5 fs8 fc0 sc0 ls3 ws0">feature<span class="_ _3"> </span>space<span class="_ _b"> </span>with<span class="_ _3"> </span>additional<span class="_ _3"> </span>(spatial)<span class="_ _b"> </span>parameters<span class="_ _3"> </span>from<span class="_ _3"> </span>the</div><div class="t m0 x9 h8 y34 ff5 fs8 fc0 sc0 ls3 ws0">input<span class="_ _a"> </span>domain<span class="_ _e"> </span>or<span class="_ _a"> </span>by<span class="_ _e"> </span>robust<span class="_ _a"> </span>postprocessing<span class="_ _e"> </span>of<span class="_ _a"> </span>the<span class="_ _e"> </span>input</div><div class="t m0 x9 h8 y35 ff5 fs8 fc0 sc0 ls3 ws0">domain<span class="_ _3"> </span>guided<span class="_ _3"> </span>by<span class="_ _b"> </span>the<span class="_ _3"> </span>results<span class="_ _3"> </span>of<span class="_ _b"> </span>the<span class="_ _3"> </span>feature<span class="_ _3"> </span>space<span class="_ _3"> </span>analysis.</div><div class="t m0 xa h8 y36 ff5 fs8 fc0 sc0 ls3 ws0">Analysis<span class="_ _2"> </span>of<span class="_ _2"> </span>the<span class="_ _3"> </span>feature<span class="_ _2"> </span>space<span class="_ _3"> </span>is<span class="_ _2"> </span>application<span class="_ _2"> </span>independent.</div><div class="t m0 x9 h8 y37 ff5 fs8 fc0 sc0 ls3 ws0">While<span class="_ _1"> </span>there<span class="_ _8"> </span>are<span class="_ _1"> </span>a<span class="_ _8"> </span>plethora<span class="_ _8"> </span>of<span class="_ _1"> </span>published<span class="_ _8"> </span>clustering<span class="_ _1"> </span>techni-</div><div class="t m0 x9 h8 y38 ff5 fs8 fc0 sc0 ls3 wse">ques,<span class="_ _d"> </span>most<span class="_ _a"> </span>of<span class="_ _d"> </span>them<span class="_ _12"> </span>are<span class="_ _d"> </span>not<span class="_ _12"> </span>adequate<span class="_ _12"> </span>to<span class="_ _12"> </span>analyze<span class="_ _12"> </span>feature</div><div class="t m0 x9 h8 y39 ff5 fs8 fc0 sc0 ls3 ws0">spaces<span class="_ _8"> </span>derived<span class="_ _8"> </span>from<span class="_ _9"> </span>real<span class="_ _8"> </span>data.<span class="_ _9"> </span>Methods<span class="_ _8"> </span>which<span class="_ _9"> </span>rely<span class="_ _8"> </span>upon</div><div class="t m0 x9 h8 y3a ff5 fs8 fc0 sc0 ls3 ws0">a<span class="_ _c"> </span>priori<span class="_ _c"> </span>knowledge<span class="_ _c"> </span>of<span class="_ _c"> </span>the<span class="_ _c"> </span>number<span class="_ _c"> </span>of<span class="_ _c"> </span>clusters<span class="_ _c"> </span>present</div><div class="t m0 x9 h8 y3b ff5 fs8 fc0 sc0 ls9 ws0">(including<span class="_ _10"> </span>those<span class="_ _10"> </span>which<span class="_ _10"> </span>use<span class="_ _10"> </span>optimization<span class="_ _10"> </span>of<span class="_ _10"> </span>a<span class="_ _10"> </span>glo<span class="_ _f"></span>bal</div><div class="t m0 x9 h8 y3c ff5 fs8 fc0 sc0 ls3 wsf">criterion<span class="_ _8"> </span>to<span class="_ _d"> </span>find<span class="_ _8"> </span>this<span class="_ _9"> </span>number),<span class="_ _9"> </span>as<span class="_ _9"> </span>well<span class="_ _9"> </span>as<span class="_ _9"> </span>methods<span class="_ _9"> </span>which</div><div class="t m0 x9 h8 y3d ff5 fs8 fc0 sc0 ls3 ws0">implicitly<span class="_ _3"> </span>assume<span class="_ _3"> </span>the<span class="_ _3"> </span>same<span class="_ _3"> </span>shape<span class="_ _3"> </span>(most<span class="_ _3"> </span>often<span class="_ _3"> </span>elliptical)<span class="_ _b"> </span>for</div><div class="t m0 x9 h8 y3e ff5 fs8 fc0 sc0 ls3 ws10">all<span class="_ _12"> </span>the<span class="_ _12"> </span>clusters<span class="_ _12"> </span>in<span class="_ _12"> </span>the<span class="_ _12"> </span>space,<span class="_ _12"> </span>are<span class="_ _12"> </span>not<span class="_ _a"> </span>able<span class="_ _12"> </span>to<span class="_ _12"> </span>handle<span class="_ _12"> </span>the</div><div class="t m0 x9 h8 y3f ff5 fs8 fc0 sc0 ls3 ws0">complexity<span class="_ _8"> </span>of<span class="_ _1"> </span>a<span class="_ _8"> </span>real<span class="_ _8"> </span>feature<span class="_ _1"> </span>space.<span class="_ _8"> </span>For<span class="_ _8"> </span>a<span class="_ _1"> </span>recent<span class="_ _8"> </span>survey<span class="_ _8"> </span>of</div><div class="t m0 x9 h8 y40 ff5 fs8 fc0 sc0 ls3 ws11">such<span class="_ _3"> </span>methods,<span class="_ _b"> </span>see<span class="_ _b"> </span>[29,<span class="_ _3"> </span>Section<span class="_ _b"> </span>8].</div><div class="t m0 xa h8 y41 ff5 fs8 fc0 sc0 ls3 ws12">In<span class="_ _2"> </span>Fig.<span class="_ _3"> </span>1,<span class="_ _3"> </span>a<span class="_ _2"> </span>typical<span class="_ _3"> </span>example<span class="_ _3"> </span>is<span class="_ _3"> </span>shown.<span class="_ _2"> </span>The<span class="_ _3"> </span>color<span class="_ _3"> </span>image<span class="_ _2"> </span>in</div><div class="t m0 x9 h8 y42 ff5 fs8 fc0 sc0 ls3 ws0">Fig.<span class="_ _1"> </span>1a<span class="_ _1"> </span>is<span class="_ _1"> </span>mapped<span class="_ _1"> </span>into<span class="_ _1"> </span>the<span class="_ _1"> </span>three-dimensional<span class="_ _1"> </span>L*u*v*<span class="_ _1"> </span>color</div><div class="t m0 x9 h8 y43 ff5 fs8 fc0 sc0 ls3 ws0">space<span class="_ _b"> </span>(to<span class="_ _1"> </span>be<span class="_ _b"> </span>discussed<span class="_ _1"> </span>in<span class="_ _b"> </span>Section<span class="_ _1"> </span>4).<span class="_ _b"> </span>There<span class="_ _b"> </span>is<span class="_ _1"> </span>a<span class="_ _b"> </span>continuous</div><div class="t m0 x9 h8 y44 ff5 fs8 fc0 sc0 ls3 ws0">transition<span class="_ _1"> </span>between<span class="_ _8"> </span>the<span class="_ _1"> </span>clusters<span class="_ _8"> </span>arising<span class="_ _8"> </span>from<span class="_ _1"> </span>the<span class="_ _8"> </span>dominant</div><div class="t m0 x9 h8 y45 ff5 fs8 fc0 sc0 ls3 ws0">colors<span class="_ _b"> </span>and<span class="_ _3"> </span>a<span class="_ _b"> </span>decomposition<span class="_ _b"> </span>of<span class="_ _b"> </span>the<span class="_ _b"> </span>space<span class="_ _3"> </span>into<span class="_ _b"> </span>elliptical<span class="_ _b"> </span>tiles</div><div class="t m0 x9 h8 y46 ff5 fs8 fc0 sc0 lsa ws0">will<span class="_ _10"> </span>introduce<span class="_ _10"> </span>severe<span class="_ _c"> </span>artifacts.<span class="_ _10"> </span>Enforcing<span class="_ _10"> </span>a<span class="_ _10"> </span>Gaussian</div><div class="t m0 x9 h8 y47 ff5 fs8 fc0 sc0 ls3 ws13">mixture<span class="_ _b"> </span>model<span class="_ _1"> </span>over<span class="_ _b"> </span>such<span class="_ _b"> </span>data<span class="_ _1"> </span>is<span class="_ _b"> </span>doomed<span class="_ _b"> </span>to<span class="_ _1"> </span>fail,<span class="_ _b"> </span>e.g.,<span class="_ _1"> </span>[49],<span class="_ _b"> </span></div><div class="t m0 x9 h8 y48 ff5 fs8 fc0 sc0 ls3 ws0">and<span class="_ _b"> </span>even<span class="_ _1"> </span>the<span class="_ _1"> </span>use<span class="_ _b"> </span>of<span class="_ _1"> </span>a<span class="_ _b"> </span>robust<span class="_ _1"> </span>approach<span class="_ _b"> </span>with<span class="_ _1"> </span>contaminated</div><div class="t m0 x9 h8 y49 ff5 fs8 fc0 sc0 ls3 ws0">Gaussian<span class="_ _c"> </span>densities<span class="_ _11"> </span>[67]<span class="_ _c"> </span>cannot<span class="_ _11"> </span>be<span class="_ _c"> </span>satisfactory<span class="_ _c"> </span>for<span class="_ _11"> </span>such</div><div class="t m0 x9 h8 y4a ff5 fs8 fc0 sc0 ls3 ws0">complex<span class="_ _1"> </span>cases.<span class="_ _8"> </span>Note<span class="_ _1"> </span>also<span class="_ _1"> </span>that<span class="_ _8"> </span>the<span class="_ _1"> </span>mixture<span class="_ _8"> </span>models<span class="_ _1"> </span>require</div><div class="t m0 x9 h8 y4b ff5 fs8 fc0 sc0 ls3 ws14">the<span class="_ _2"> </span>number<span class="_ _2"> </span>of<span class="_ _3"> </span>clusters<span class="_ _2"> </span>as<span class="_ _3"> </span>a<span class="_ _2"> </span>parameter,<span class="_ _2"> </span>which<span class="_ _3"> </span>raises<span class="_ _2"> </span>its<span class="_ _3"> </span>own</div><div class="t m0 x9 h8 y4c ff5 fs8 fc0 sc0 ls3 ws15">challenges.<span class="_ _c"> </span>For<span class="_ _11"> </span>example,<span class="_ _c"> </span>the<span class="_ _c"> </span>method<span class="_ _c"> </span>described<span class="_ _11"> </span>in<span class="_ _c"> </span>[45]</div><div class="t m0 x9 h8 y4d ff5 fs8 fc0 sc0 ls3 ws0">proposes<span class="_ _3"> </span>several<span class="_ _b"> </span>different<span class="_ _b"> </span>ways<span class="_ _3"> </span>to<span class="_ _b"> </span>determine<span class="_ _b"> </span>this<span class="_ _3"> </span>number.</div><div class="t m0 xa h8 y4e ff5 fs8 fc0 sc0 ls3 ws0">Arbitrarily<span class="_ _a"> </span>structured<span class="_ _a"> </span>feature<span class="_ _a"> </span>spaces<span class="_ _a"> </span>can<span class="_ _a"> </span>be<span class="_ _a"> </span>analyzed</div><div class="t m0 x9 h8 y4f ff5 fs8 fc0 sc0 ls3 ws0">only<span class="_"> </span>by<span class="_"> </span>nonparametric<span class="_ _2"> </span>methods<span class="_ _2"> </span>since<span class="_"> </span>these<span class="_ _2"> </span>methods<span class="_"> </span>do<span class="_ _2"> </span>not</div><div class="t m0 x9 h8 y50 ff5 fs8 fc0 sc0 ls3 ws0">have<span class="_ _c"> </span>embedded<span class="_ _11"> </span>assumptions.<span class="_ _c"> </span>Numerous<span class="_ _11"> </span>nonparametric</div><div class="t m0 x9 h8 y51 ff5 fs8 fc0 sc0 ls3 ws0">clustering<span class="_ _12"> </span>methods<span class="_ _a"> </span>were<span class="_ _12"> </span>described<span class="_ _a"> </span>in<span class="_ _12"> </span>the<span class="_ _a"> </span>literature<span class="_ _a"> </span>and</div><div class="t m0 x9 h8 y52 ff5 fs8 fc0 sc0 ls3 ws0">they<span class="_ _9"> </span>can<span class="_ _d"> </span>be<span class="_ _d"> </span>classified<span class="_ _9"> </span>into<span class="_ _d"> </span>two<span class="_ _d"> </span>large<span class="_ _d"> </span>classes:<span class="_ _9"> </span>hierarchical</div><div class="t m0 x9 h8 y53 ff5 fs8 fc0 sc0 ls3 ws0">clustering<span class="_ _d"> </span>and<span class="_ _9"> </span>density<span class="_ _d"> </span>estimation.<span class="_ _d"> </span>Hierarchical<span class="_ _d"> </span>clustering</div><div class="t m0 x9 h8 y54 ff5 fs8 fc0 sc0 ls3 ws0">techniques<span class="_ _d"> </span>either<span class="_ _12"> </span>aggregate<span class="_ _d"> </span>or<span class="_ _12"> </span>divide<span class="_ _d"> </span>the<span class="_ _12"> </span>data<span class="_ _12"> </span>based<span class="_ _d"> </span>on</div><div class="t m0 xb h9 y55 ff1 fs7 fc0 sc0 ls8 ws0">IEEE<span class="_ _4"> </span>TRANSAC<span class="_ _f"></span>TIONS<span class="_ _4"> </span>ON<span class="_ _2"> </span>PATTERN<span class="_ _4"> </span>ANALYSI<span class="_ _f"></span>S<span class="_ _4"> </span>AND<span class="_ _2"> </span>MACHINE<span class="_ _4"> </span>INTELLIGEN<span class="_ _f"></span>CE,<span class="_ _13"> </span>VOL.<span class="_ _2"> </span>24,<span class="_ _13"> </span>NO.<span class="_ _4"> </span>5,<span class="_ _13"> </span>MAY<span class="_ _2"> </span>2002<span class="_ _14"> </span>603</div><div class="t m0 x6 ha y56 ff6 fs2 fc0 sc0 ls3 ws0">.<span class="_ _c"> </span><span class="ff7">D.<span class="_ _5"> </span>Comanici<span class="_ _6"></span>u<span class="_ _5"> </span>is<span class="_ _5"> </span>with<span class="_ _5"> </span>the<span class="_ _5"> </span>Imaging<span class="_ _5"> </span>and<span class="_ _5"> </span>Visualization<span class="_ _5"> </span>Department,<span class="_ _1"> </span>Siemens</span></div><div class="t m0 xc ha y57 ff7 fs2 fc0 sc0 ls3 ws16">Corporate<span class="_ _2"> </span>Research,<span class="_ _2"> </span>755<span class="_ _2"> </span>College<span class="_ _2"> </span>Road<span class="_ _2"> </span>East,<span class="_ _2"> </span>Princeton,<span class="_ _2"> </span>NJ<span class="_ _2"> </span>08540.</div><div class="t m0 xc ha y58 ff7 fs2 fc0 sc0 ls3 ws0">E-mail:<span class="_"> </span>comanici@scr.siemens.com.</div><div class="t m0 x6 ha y59 ff6 fs2 fc0 sc0 ls3 ws0">.<span class="_ _c"> </span><span class="ff7">P.<span class="_ _b"> </span>Meer<span class="_ _1"> </span>is<span class="_ _1"> </span>with<span class="_ _1"> </span>the<span class="_ _1"> </span>Electrical<span class="_ _1"> </span>and<span class="_ _b"> </span>Computer<span class="_ _1"> </span>Engineering<span class="_ _1"> </span>Department,</span></div><div class="t m0 xc ha y5a ff7 fs2 fc0 sc0 ls3 ws16">Rutgers<span class="_ _2"> </span>University,<span class="_ _2"> </span>94<span class="_ _2"> </span>Brett<span class="_ _2"> </span>Road,<span class="_ _2"> </span>Piscataway,<span class="_ _2"> </span>NJ<span class="_ _2"> </span>08854-8058.</div><div class="t m0 xc ha y5b ff7 fs2 fc0 sc0 ls3 ws0">E-mail:<span class="_"> </span>meer@caip.rutgers.edu.</div><div class="t m0 x6 ha y5c ff7 fs2 fc0 sc0 ls3 ws0">Manuscript<span class="_ _b"> </span>received<span class="_ _b"> </span>17<span class="_ _b"> </span>Jan.<span class="_ _b"> </span>2001;<span class="_ _b"> </span>revised<span class="_ _b"> </span>16<span class="_ _b"> </span>July<span class="_ _1"> </span>2001;<span class="_ _b"> </span>accepted<span class="_ _b"> </span>21<span class="_ _b"> </span>Nov.</div><div class="t m0 x6 ha y5d ff7 fs2 fc0 sc0 ls3 ws0">2001.</div><div class="t m0 x6 ha y5e ff7 fs2 fc0 sc0 ls3 ws0">Recommended<span class="_"> </span>for<span class="_"> </span>acceptance<span class="_"> </span>by<span class="_"> </span>V.<span class="_"> </span>Solo.</div><div class="t m0 x6 ha y5f ff7 fs2 fc0 sc0 ls3 ws0">For<span class="_ _b"> </span>information<span class="_ _b"> </span>on<span class="_ _b"> </span>obtaining<span class="_ _b"> </span>reprints<span class="_ _b"> </span>of<span class="_ _b"> </span>this<span class="_ _b"> </span>article,<span class="_ _b"> </span>please<span class="_ _b"> </span>send<span class="_ _b"> </span>e-mail<span class="_ _b"> </span>to:</div><div class="t m0 x6 ha y60 ff7 fs2 fc0 sc0 ls3 ws0">tpami@computer.org,<span class="_"> </span>and<span class="_"> </span>reference<span class="_"> </span>IEEECS<span class="_ _4"> </span>Log<span class="_"> </span>Number<span class="_"> </span>113483.</div><div class="t m0 xd hb y61 ff1 fs9 fc0 sc0 lsb ws0">0162-8828/02/$17.00<span class="_ _5"> </span><span class="ff8 ls3">ß<span class="_ _5"> </span></span>2002<span class="_ _4"> </span>IEEE</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div> </body> </html>

<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6244dcfd6cd7507199bf81c5/bg2.jpg"><div class="t m1 x6 h8 y62 ff5 fs8 fc0 sc0 ls3 ws17">some<span class="_ _b"> </span>proximity<span class="_ _b"> </span>measure.<span class="_ _b"> </span>See<span class="_ _b"> </span>[28,<span class="_ _b"> </span>Section<span class="_ _1"> </span>3.2]<span class="_ _3"> </span>for<span class="_ _1"> </span>a<span class="_ _b"> </span>survey</div><div class="t m1 x6 h8 y63 ff5 fs8 fc0 sc0 ls3 ws0">of<span class="_ _1"> </span>hierarchical<span class="_ _8"> </span>clustering<span class="_ _8"> </span>methods.<span class="_ _1"> </span>The<span class="_ _8"> </span>hierarchical<span class="_ _8"> </span>meth-</div><div class="t m1 x6 h8 y64 ff5 fs8 fc0 sc0 ls3 ws0">ods<span class="_"> </span>tend<span class="_"> </span>to<span class="_"> </span>be<span class="_ _4"> </span>computationally<span class="_"> </span>expensive<span class="_"> </span>and<span class="_ _4"> </span>the<span class="_"> </span>definition</div><div class="t m1 x6 h8 y65 ff5 fs8 fc0 sc0 ls3 ws0">of<span class="_ _10"> </span>a<span class="_ _10"> </span>mean<span class="_ _f"></span>ingfu<span class="_ _f"></span>l<span class="_ _c"> </span>st<span class="_ _f"></span>oppi<span class="_ _f"></span>ng<span class="_ _c"> </span>c<span class="_ _f"></span>rite<span class="_ _f"></span>rion<span class="_ _10"> </span>for<span class="_ _13"> </span>the<span class="_ _c"> </span>fusi<span class="_ _f"></span>on<span class="_ _10"> </span>(or</div><div class="t m1 x6 h8 y66 ff5 fs8 fc0 sc0 ls3 ws0">division)<span class="_ _b"> </span>of<span class="_ _3"> </span>the<span class="_ _b"> </span>data<span class="_ _b"> </span>is<span class="_ _3"> </span>not<span class="_ _b"> </span>straightforward.</div><div class="t m1 x8 h8 y67 ff5 fs8 fc0 sc0 ls3 ws0">The<span class="_ _1"> </span>rationale<span class="_ _1"> </span>behind<span class="_ _1"> </span>the<span class="_ _1"> </span>density<span class="_ _1"> </span>estimation-based<span class="_ _1"> </span>non-</div><div class="t m1 x6 h8 y68 ff5 fs8 fc0 sc0 ls3 ws0">parametric<span class="_ _2"> </span>clustering<span class="_ _2"> </span>approach<span class="_ _2"> </span>is<span class="_ _2"> </span>that<span class="_ _2"> </span>the<span class="_ _2"> </span>feature<span class="_ _2"> </span>space<span class="_ _2"> </span>can</div><div class="t m1 x6 h8 y69 ff5 fs8 fc0 sc0 ls3 ws0">be<span class="_ _8"> </span>regarded<span class="_ _9"> </span>as<span class="_ _9"> </span>the<span class="_ _8"> </span>empirical<span class="_ _9"> </span>probability<span class="_ _9"> </span>density<span class="_ _9"> </span>function</div><div class="t m1 x6 h8 y6a ff5 fs8 fc0 sc0 ls3 ws0">(p.d.f.)<span class="_ _1"> </span>of<span class="_ _b"> </span>the<span class="_ _1"> </span>represented<span class="_ _b"> </span>parameter.<span class="_ _1"> </span>Dense<span class="_ _b"> </span>regions<span class="_ _1"> </span>in<span class="_ _1"> </span>the</div><div class="t m1 x6 h8 y6b ff5 fs8 fc0 sc0 ls3 ws0">feature<span class="_ _2"> </span>space<span class="_ _3"> </span>thus<span class="_ _3"> </span>correspond<span class="_ _2"> </span>to<span class="_ _3"> </span>local<span class="_ _3"> </span>maxima<span class="_ _2"> </span>of<span class="_ _3"> </span>the<span class="_ _3"> </span>p.d.f.,</div><div class="t m1 x6 hc y6c ff5 fs8 fc0 sc0 ls3 ws18">that<span class="_ _12"> </span>is,<span class="_ _a"> </span>to<span class="_ _12"> </span>the<span class="_ _a"> </span><span class="ff7 ws0">modes<span class="_ _12"> </span><span class="ff5">of<span class="_ _a"> </span>the<span class="_ _12"> </span>unknown<span class="_ _a"> </span>density.<span class="_ _12"> </span>Once<span class="_ _a"> </span>the</span></span></div><div class="t m1 x6 h8 y6d ff5 fs8 fc0 sc0 ls3 ws19">location<span class="_ _d"> </span>of<span class="_ _12"> </span>a<span class="_ _12"> </span>mode<span class="_ _12"> </span>is<span class="_ _12"> </span>determined,<span class="_ _d"> </span>the<span class="_ _12"> </span>cluster<span class="_ _12"> </span>associated</div><div class="t m1 x6 h8 y6e ff5 fs8 fc0 sc0 ls3 ws0">with<span class="_ _d"> </span>it<span class="_ _12"> </span>is<span class="_ _12"> </span>delineated<span class="_ _d"> </span>based<span class="_ _12"> </span>on<span class="_ _12"> </span>the<span class="_ _12"> </span>local<span class="_ _d"> </span>structure<span class="_ _12"> </span>of<span class="_ _12"> </span>the</div><div class="t m1 x6 h8 y6f ff5 fs8 fc0 sc0 ls3 ws1a">feature<span class="_ _b"> </span>space<span class="_ _3"> </span>[25],<span class="_ _b"> </span>[60],<span class="_ _3"> </span>[63].</div><div class="t m1 x8 h8 y70 ff5 fs8 fc0 sc0 ls3 ws0">Our<span class="_ _15"> </span>approach<span class="_ _15"> </span>to<span class="_ _15"> </span>mode<span class="_ _15"> </span>detection<span class="_ _15"> </span>and<span class="_ _15"> </span>clustering<span class="_ _15"> </span>is<span class="_ _15"> </span>based<span class="_ _15"> </span>on</div><div class="t m1 x6 h8 y71 ff5 fs8 fc0 sc0 ls3 ws1b">the<span class="_ _5"> </span>mean<span class="_ _15"> </span>shift<span class="_ _5"> </span>procedure,<span class="_ _5"> </span>proposed<span class="_ _15"> </span>in<span class="_ _5"> </span>1975<span class="_ _5"> </span>by<span class="_ _15"> </span>Fukunaga<span class="_ _5"> </span>and</div><div class="t m1 x6 h8 y72 ff5 fs8 fc0 sc0 ls3 ws0">Hostetler<span class="_ _3"> </span>[21]<span class="_ _3"> </span>and<span class="_ _3"> </span>largely<span class="_ _3"> </span>forgotten<span class="_ _3"> </span>until<span class="_ _3"> </span>Cheng's<span class="_ _b"> </span>paper<span class="_ _2"> </span>[7]</div><div class="t m1 x6 h8 y73 ff5 fs8 fc0 sc0 ls3 ws1c">rekindled<span class="_ _2"> </span>interest<span class="_ _3"> </span>in<span class="_ _3"> </span>it.<span class="_ _2"> </span>In<span class="_ _3"> </span>spite<span class="_ _2"> </span>of<span class="_ _3"> </span>its<span class="_ _3"> </span>excellent<span class="_ _2"> </span>qualities,<span class="_ _3"> </span>the</div><div class="t m1 x6 h8 y74 ff5 fs8 fc0 sc0 ls3 ws0">mean<span class="_ _5"> </span>shift<span class="_ _15"> </span>procedure<span class="_ _5"> </span>does<span class="_ _5"> </span>not<span class="_ _15"> </span>seem<span class="_ _5"> </span>to<span class="_ _5"> </span>be<span class="_ _15"> </span>known<span class="_ _5"> </span>in<span class="_ _5"> </span>statistical</div><div class="t m1 x6 h8 y75 ff5 fs8 fc0 sc0 ls3 ws1d">literature.<span class="_ _15"> </span>While<span class="_ _15"> </span>the<span class="_ _5"> </span>book<span class="_ _15"> </span>[54,<span class="_ _15"> </span>Section<span class="_ _5"> </span>6.2.2]<span class="_ _15"> </span>discusses<span class="_ _15"> </span>[21],<span class="_ _15"> </span>the</div><div class="t m1 x6 h8 y76 ff5 fs8 fc0 sc0 ls3 ws0">advantages<span class="_ _8"> </span>of<span class="_ _9"> </span>employing<span class="_ _9"> </span>a<span class="_ _8"> </span>mean<span class="_ _9"> </span>shift<span class="_ _9"> </span>type<span class="_ _8"> </span>procedure<span class="_ _9"> </span>in</div><div class="t m1 x6 h8 y77 ff5 fs8 fc0 sc0 ls3 ws0">density<span class="_"> </span>estimation<span class="_"> </span>were<span class="_ _4"> </span>only<span class="_"> </span>recently<span class="_"> </span>rediscovered<span class="_"> </span>[8].</div><div class="t m1 x8 h8 y78 ff5 fs8 fc0 sc0 ls3 ws1e">As<span class="_ _2"> </span>will<span class="_ _2"> </span>be<span class="_ _2"> </span>proven<span class="_ _2"> </span>in<span class="_ _2"> </span>the<span class="_ _2"> </span>sequel,<span class="_ _2"> </span>a<span class="_ _2"> </span>computational<span class="_ _2"> </span>module</div><div class="t m1 x6 h8 y79 ff5 fs8 fc0 sc0 ls3 ws0">based<span class="_ _2"> </span>on<span class="_ _2"> </span>the<span class="_ _2"> </span>mean<span class="_ _3"> </span>shift<span class="_"> </span>procedure<span class="_ _3"> </span>is<span class="_ _2"> </span>an<span class="_ _2"> </span>extremely<span class="_ _2"> </span>versatile</div><div class="t m1 x6 h8 y7a ff5 fs8 fc0 sc0 ls3 ws0">tool<span class="_ _a"> </span>for<span class="_ _a"> </span>feature<span class="_ _a"> </span>space<span class="_ _a"> </span>analysis<span class="_ _a"> </span>and<span class="_ _a"> </span>can<span class="_ _a"> </span>provide<span class="_ _a"> </span>reliable</div><div class="t m1 x6 h8 y7b ff5 fs8 fc0 sc0 ls3 ws1f">solutions<span class="_ _2"> </span>for<span class="_ _3"> </span>many<span class="_ _2"> </span>vision<span class="_ _3"> </span>tasks.<span class="_ _2"> </span>In<span class="_ _2"> </span>Section<span class="_ _3"> </span>2,<span class="_ _2"> </span>the<span class="_ _3"> </span>mean<span class="_ _2"> </span>shift</div><div class="t m1 x6 h8 y7c ff5 fs8 fc0 sc0 ls3 ws0">procedure<span class="_ _d"> </span>is<span class="_ _d"> </span>defined<span class="_ _12"> </span>and<span class="_ _d"> </span>its<span class="_ _12"> </span>properties<span class="_ _d"> </span>are<span class="_ _12"> </span>analyzed.<span class="_ _d"> </span>In</div><div class="t m1 x6 h8 y7d ff5 fs8 fc0 sc0 ls3 ws20">Section<span class="_ _c"> </span>3,<span class="_ _11"> </span>the<span class="_ _c"> </span>procedure<span class="_ _11"> </span>is<span class="_ _c"> </span>used<span class="_ _11"> </span>as<span class="_ _c"> </span>the<span class="_ _11"> </span>computational</div><div class="t m1 x6 h8 y7e ff5 fs8 fc0 sc0 ls3 ws0">module<span class="_ _b"> </span>for<span class="_ _b"> </span>robust<span class="_ _1"> </span>feature<span class="_ _b"> </span>space<span class="_ _b"> </span>analysis<span class="_ _1"> </span>and<span class="_ _b"> </span>implementa-</div><div class="t m1 x6 h8 y7f ff5 fs8 fc0 sc0 ls3 ws21">tional<span class="_ _1"> </span>issues<span class="_ _1"> </span>are<span class="_ _8"> </span>discussed.<span class="_ _1"> </span>In<span class="_ _1"> </span>Section<span class="_ _1"> </span>4,<span class="_ _8"> </span>the<span class="_ _1"> </span>feature<span class="_ _1"> </span>space</div><div class="t m1 x6 h8 y80 ff5 fs8 fc0 sc0 ls3 ws0">analysis<span class="_ _3"> </span>technique<span class="_ _b"> </span>is<span class="_ _b"> </span>applied<span class="_ _3"> </span>to<span class="_ _b"> </span>two<span class="_ _b"> </span>low-level<span class="_ _3"> </span>vision<span class="_ _b"> </span>tasks:</div><div class="t m1 x6 h8 y81 ff5 fs8 fc0 sc0 ls3 ws0">discontinuity<span class="_ _b"> </span>preserving<span class="_ _b"> </span>filtering<span class="_ _b"> </span>and<span class="_ _b"> </span>image<span class="_ _b"> </span>segmentation.</div><div class="t m1 x6 h8 y82 ff5 fs8 fc0 sc0 ls3 ws0">Both<span class="_ _2"> </span>algorithms<span class="_ _3"> </span>can<span class="_ _2"> </span>have<span class="_ _3"> </span>as<span class="_ _2"> </span>input<span class="_ _2"> </span>either<span class="_ _3"> </span>gray<span class="_ _2"> </span>level<span class="_ _3"> </span>or<span class="_ _2"> </span>color</div><div class="t m1 x6 h8 y83 ff5 fs8 fc0 sc0 ls3 ws0">images<span class="_ _1"> </span>and<span class="_ _b"> </span>the<span class="_ _1"> </span>only<span class="_ _1"> </span>parameter<span class="_ _1"> </span>to<span class="_ _1"> </span>be<span class="_ _b"> </span>tuned<span class="_ _1"> </span>by<span class="_ _1"> </span>the<span class="_ _1"> </span>user<span class="_ _b"> </span>is</div><div class="t m1 x6 h8 y84 ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _2"> </span>resolution<span class="_ _3"> </span>of<span class="_ _3"> </span>the<span class="_ _2"> </span>analysis.<span class="_ _3"> </span>The<span class="_ _3"> </span>applicability<span class="_ _2"> </span>of<span class="_ _3"> </span>the<span class="_ _3"> </span>mean</div><div class="t m1 x6 h8 y85 ff5 fs8 fc0 sc0 ls3 ws0">shift<span class="_ _3"> </span>procedure<span class="_ _b"> </span>is<span class="_ _3"> </span>not<span class="_ _b"> </span>restricted<span class="_ _3"> </span>to<span class="_ _b"> </span>the<span class="_ _3"> </span>presented<span class="_ _b"> </span>examples.</div><div class="t m1 x6 h8 y86 ff5 fs8 fc0 sc0 ls3 ws22">In<span class="_ _e"> </span>Section<span class="_ _11"> </span>5,<span class="_ _e"> </span>other<span class="_ _e"> </span>applications<span class="_ _11"> </span>are<span class="_ _11"> </span>mentioned<span class="_ _e"> </span>and<span class="_ _11"> </span>the</div><div class="t m1 x6 h8 y87 ff5 fs8 fc0 sc0 ls3 ws0">procedure<span class="_ _b"> </span>is<span class="_ _3"> </span>put<span class="_ _b"> </span>into<span class="_ _b"> </span>a<span class="_ _3"> </span>more<span class="_ _b"> </span>general<span class="_ _b"> </span>context.</div><div class="t m1 x6 h6 y88 ff3 fs4 fc0 sc0 ls7 ws0">2T<span class="_ _7"></span><span class="fs5 ls3">HE<span class="_ _1"> </span><span class="fs4">M</span>EAN<span class="_ _8"> </span><span class="fs4">S</span>HIFT<span class="_ _1"> </span><span class="fs4">P</span>ROCEDURE</span></div><div class="t m1 x6 h8 y89 ff5 fs8 fc0 sc0 ls3 ws0">Kernel<span class="_ _9"> </span>density<span class="_ _9"> </span>estimation<span class="_ _9"> </span>(known<span class="_ _9"> </span>as<span class="_ _9"> </span>the<span class="_ _9"> </span>Parzen<span class="_ _9"> </span>window</div><div class="t m1 x6 h8 y8a ff5 fs8 fc0 sc0 ls3 ws23">technique<span class="_ _5"> </span>in<span class="_ _5"> </span>pattern<span class="_ _5"> </span>recognition<span class="_ _4"> </span>literature<span class="_ _5"> </span>[17,<span class="_ _5"> </span>Section<span class="_ _5"> </span>4.3])<span class="_ _5"> </span>is</div><div class="t m1 x6 hd y8b ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _3"> </span>most<span class="_ _b"> </span>popular<span class="_ _b"> </span>density<span class="_ _3"> </span>estimation<span class="_ _b"> </span>method.<span class="_ _3"> </span>Given<span class="_ _b"> </span><span class="ff9">n<span class="_ _3"> </span></span>data</div><div class="t m1 x9 h8 y8c ff5 fs8 fc0 sc0 ls3 ws0">points<span class="_ _12"> </span><span class="ffa">x</span></div><div class="t m1 xe he y8d ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 xf hf y62 ff5 fs8 fc0 sc0 ls3 ws0">,<span class="_ _12"> </span><span class="ff9">i<span class="_ _2"> </span><span class="ffb"><span class="_ _2"> </span><span class="ffc">1</span></span>;<span class="_ _5"> </span><span class="ffc lsc">...<span class="ff9">;n<span class="_ _b"> </span></span></span></span>in<span class="_ _12"> </span>the<span class="_ _12"> </span><span class="ff9">d<span class="ffc">-dimensional<span class="_ _12"> </span></span></span>space<span class="_ _a"> </span><span class="ff9">R</span></div><div class="t m1 x10 he y8e ff9 fsa fc0 sc0 ls3 ws0">d</div><div class="t m1 x11 h8 y62 ff5 fs8 fc0 sc0 ls3 ws24">,<span class="_ _12"> </span>the</div><div class="t m1 x9 hf y8f ff7 fs8 fc0 sc0 ls3 ws0">multivariate<span class="_ _1"> </span>kernel<span class="_ _1"> </span>density<span class="_ _1"> </span>estimator<span class="_ _1"> </span><span class="ff5">with<span class="_ _1"> </span>kernel<span class="_ _1"> </span><span class="ff9">K<span class="_ _f"></span><span class="ffb"><span class="ffa">x</span><span class="_ _1"> </span></span></span>and<span class="_ _1"> </span>a</span></div><div class="t m1 x9 hf y90 ff5 fs8 fc0 sc0 lsd ws0">symmetric<span class="_ _a"> </span>positive<span class="_ _a"> </span>definite<span class="_ _a"> </span><span class="ff9 ls3">d<span class="_ _4"> </span><span class="ffb"><span class="_ _4"> </span></span>d<span class="_ _a"> </span><span class="ff5">b<span class="_ _f"></span>and<span class="_ _f"></span>widt<span class="_ _f"></span>h<span class="_ _a"> </span>m<span class="_ _f"></span>atri<span class="_ _f"></span>x<span class="_ _e"> </span><span class="ffa">H</span>,</span></span></div><div class="t m1 x9 h8 y91 ff5 fs8 fc0 sc0 ls3 ws0">computed<span class="_ _5"> </span>in<span class="_"> </span>the<span class="_ _5"> </span>point<span class="_"> </span><span class="ffa">x<span class="_ _5"> </span></span>is<span class="_"> </span>given<span class="_ _5"> </span>by</div><div class="t m1 x12 h10 y92 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x13 hf y93 ff9 fs8 fc0 sc0 ls3 ws0">f<span class="_ _16"></span><span class="ffb"><span class="ffa">x</span><span class="lse"></span></span></div><div class="t m1 x14 h10 y94 ffc fs8 fc0 sc0 ls3 ws0">1</div><div class="t m1 x14 hd y95 ff9 fs8 fc0 sc0 ls3 ws0">n</div><div class="t m1 x15 h11 y96 ffd fs8 fc0 sc0 ls3 ws0">X</div><div class="t m1 x16 he y97 ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 x17 h12 y98 ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x18 hd y99 ff9 fs8 fc0 sc0 ls3 ws0">K</div><div class="t m1 x19 h13 y9a ffa fsa fc0 sc0 ls3 ws0">H</div><div class="t m1 x1a hf y99 ffa fs8 fc0 sc0 ls3 ws0">x<span class="_ _4"> </span><span class="ffb"><span class="_ _4"> </span></span>x</div><div class="t m1 x1b he y9a ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 x1c hf y99 ffb fs8 fc0 sc0 lsf ws0"><span class="_ _17"></span><span class="ff9 ls3">;<span class="_ _18"> </span><span class="ffb"><span class="ffc">1</span></span></span></div><div class="t m1 x9 h8 y9b ff5 fs8 fc0 sc0 ls3 ws0">where</div><div class="t m1 x1d hd y9c ff9 fs8 fc0 sc0 ls3 ws0">K</div><div class="t m1 x1e h13 y9d ffa fsa fc0 sc0 ls3 ws0">H</div><div class="t m1 x1f hf y9e ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="lse"><span class="_ _19"></span>j<span class="ffa ls3">H<span class="_ _2"> </span><span class="ffb">j</span></span></span></div><div class="t m1 x20 h12 y9f ffb fsa fc0 sc0 ls3 ws0"><span class="ffc">1<span class="ff9">=</span>2</span></div><div class="t m1 x21 hf y9e ff9 fs8 fc0 sc0 ls3 ws0">K<span class="_ _f"></span><span class="ffb"><span class="ffa">H</span></span></div><div class="t m1 x22 h12 y9f ffb fsa fc0 sc0 ls3 ws0"><span class="ffc">1<span class="ff9">=</span>2</span></div><div class="t m1 x23 hf y9e ffa fs8 fc0 sc0 ls3 ws0">x<span class="ffb"><span class="ff9">:<span class="_ _1a"> </span></span><span class="ffc">2</span></span></div><div class="t m1 x9 hf ya0 ff5 fs8 fc0 sc0 ls3 ws0">The<span class="_ _11"> </span><span class="ff9">d<span class="ffc">-variate<span class="_ _c"> </span></span></span>kernel<span class="_ _11"> </span><span class="ff9">K<span class="_ _16"></span><span class="ffb"><span class="ffa">x</span><span class="_ _11"> </span></span></span>is<span class="_ _c"> </span>a<span class="_ _11"> </span>bounded<span class="_ _c"> </span>function<span class="_ _11"> </span>with</div><div class="t m1 x9 h8 ya1 ff5 fs8 fc0 sc0 ls3 ws25">compact<span class="_ _3"> </span>support<span class="_ _b"> </span>satisfying<span class="_ _b"> </span>[62,<span class="_ _3"> </span>p.<span class="_ _b"> </span>95]</div><div class="t m1 x24 h11 ya2 ffd fs8 fc0 sc0 ls3 ws0">Z</div><div class="t m1 x25 he ya3 ff9 fsa fc0 sc0 ls3 ws0">R</div><div class="t m1 x26 h14 ya4 ff9 fsb fc0 sc0 ls3 ws0">d</div><div class="t m1 x27 hf ya5 ff9 fs8 fc0 sc0 ls3 ws0">K<span class="_ _f"></span><span class="ffb"><span class="ffa">x</span><span class="ffc">d<span class="ffa">x<span class="_ _2"> </span></span></span><span class="_ _2"> </span><span class="ffc">1<span class="_ _1b"> </span>lim</span></span></div><div class="t m1 x28 h12 ya4 ffb fsa fc0 sc0 ls3 ws0">k<span class="ffa">x</span><span class="ls10">k!1</span></div><div class="t m1 x29 hf ya5 ffb fs8 fc0 sc0 ls3 ws0">k<span class="ffa">x</span>k</div><div class="t m1 x2a he ya6 ff9 fsa fc0 sc0 ls3 ws0">d</div><div class="t m1 x2b hf ya5 ff9 fs8 fc0 sc0 ls3 ws0">K<span class="_ _f"></span><span class="ffb"><span class="ffa">x</span><span class="ls11"></span><span class="ffc">0</span></span></div><div class="t m1 x2c h11 ya7 ffd fs8 fc0 sc0 ls3 ws0">Z</div><div class="t m1 x24 he ya8 ff9 fsa fc0 sc0 ls3 ws0">R</div><div class="t m1 x2d h14 ya9 ff9 fsb fc0 sc0 ls3 ws0">d</div><div class="t m1 x2e hf yaa ffa fs8 fc0 sc0 ls3 ws0">x<span class="ff9">K<span class="_ _f"></span><span class="ffb"></span></span>x<span class="ffb"><span class="ffc">d</span></span>x<span class="_ _2"> </span><span class="ffb"><span class="_ _2"> </span><span class="ffc">0</span></span></div><div class="t m1 x28 h11 yab ffd fs8 fc0 sc0 ls3 ws0">Z</div><div class="t m1 x2f he ya8 ff9 fsa fc0 sc0 ls3 ws0">R</div><div class="t m1 x19 h14 ya9 ff9 fsb fc0 sc0 ls3 ws0">d</div><div class="t m1 x21 h15 yaa ffa fs8 fc0 sc0 ls3 ws0">xx</div><div class="t m1 x30 h12 yac ffb fsa fc0 sc0 ls3 ws0">></div><div class="t m1 x31 hf yaa ff9 fs8 fc0 sc0 ls3 ws0">K<span class="_ _f"></span><span class="ffb"><span class="ffa">x</span><span class="ffc">d<span class="ffa">x<span class="_ _2"> </span></span></span><span class="_ _2"> </span></span>c</div><div class="t m1 x32 he yad ff9 fsa fc0 sc0 ls3 ws0">K</div><div class="t m1 x33 hd yaa ffa fs8 fc0 sc0 ls3 ws0">I<span class="ff9">;</span></div><div class="t m1 x34 hf yae ffb fs8 fc0 sc0 ls3 ws0"><span class="ffc">3</span></div><div class="t m1 x9 hd yaf ff5 fs8 fc0 sc0 ls3 ws0">where<span class="_ _a"> </span><span class="ff9">c</span></div><div class="t m1 x35 he yb0 ff9 fsa fc0 sc0 ls3 ws0">K</div><div class="t m1 x36 h8 yb1 ff5 fs8 fc0 sc0 ls3 ws0">is<span class="_ _a"> </span>a<span class="_ _a"> </span>constant.<span class="_ _a"> </span>The<span class="_ _a"> </span>multivariate<span class="_ _a"> </span>kernel<span class="_ _e"> </span>can<span class="_ _a"> </span>be</div><div class="t m1 x9 hd yb2 ff5 fs8 fc0 sc0 ls3 ws0">generated<span class="_ _2"> </span>from<span class="_ _3"> </span>a<span class="_ _2"> </span>symmetric<span class="_ _3"> </span>univariate<span class="_ _2"> </span>kernel<span class="_ _3"> </span><span class="ff9">K</span></div><div class="t m1 x32 h16 yb3 ffc fsa fc0 sc0 ls3 ws0">1</div><div class="t m1 x37 hf yb4 ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">x</span><span class="_ _2"> </span><span class="ff5">in<span class="_ _3"> </span>two</span></div><div class="t m1 x9 h8 yb5 ff5 fs8 fc0 sc0 ls3 ws0">different<span class="_ _3"> </span>ways</div><div class="t m1 x38 hd yb6 ff9 fs8 fc0 sc0 ls3 ws0">K</div><div class="t m1 x39 he yb7 ff9 fsa fc0 sc0 ls3 ws0">P</div><div class="t m1 x3a hf yb8 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="lse"></span></div><div class="t m1 x27 h11 yb9 ffd fs8 fc0 sc0 ls3 ws0">Y</div><div class="t m1 x3b he yba ff9 fsa fc0 sc0 ls3 ws0">d</div><div class="t m1 x3c h12 ybb ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x3d hd yb8 ff9 fs8 fc0 sc0 ls3 ws0">K</div><div class="t m1 x3e h16 ybc ffc fsa fc0 sc0 ls3 ws0">1</div><div class="t m1 x3f hf yb8 ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">x</span></div><div class="t m1 x40 he ybc ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 x41 hf yb8 ffb fs8 fc0 sc0 ls3 ws0"><span class="_ _1c"> </span><span class="ff9">K</span></div><div class="t m1 x42 he yb7 ff9 fsa fc0 sc0 ls3 ws0">S</div><div class="t m1 x43 hf yb8 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="ls11"></span><span class="ff9">a</span></div><div class="t m1 x44 he ybc ff9 fsa fc0 sc0 ls3 ws0">k;d</div><div class="t m1 x45 hd yb8 ff9 fs8 fc0 sc0 ls3 ws0">K</div><div class="t m1 x46 h16 ybc ffc fsa fc0 sc0 ls3 ws0">1</div><div class="t m1 x47 hf yb8 ffb fs8 fc0 sc0 ls3 ws0">k<span class="ffa">x</span>k<span class="ff9">;<span class="_ _1d"> </span></span><span class="ffc">4</span></div><div class="t m1 x9 hd ybd ff5 fs8 fc0 sc0 ls3 ws0">where<span class="_"> </span><span class="ff9">K</span></div><div class="t m1 xe he ybe ff9 fsa fc0 sc0 ls3 ws0">P</div><div class="t m1 x2d hf ybf ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="_ _2"> </span><span class="ff5">is<span class="_"> </span>obtained<span class="_ _2"> </span>from<span class="_ _2"> </span>the<span class="_ _2"> </span>product<span class="_ _2"> </span>of<span class="_ _2"> </span>the<span class="_ _2"> </span>univariate</span></div><div class="t m1 x9 hd yc0 ff5 fs8 fc0 sc0 ls3 ws0">kernels<span class="_ _3"> </span>and<span class="_ _b"> </span><span class="ff9">K</span></div><div class="t m1 x48 he yc1 ff9 fsa fc0 sc0 ls3 ws0">S</div><div class="t m1 x49 hf yc2 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="_ _3"> </span><span class="ff5">from<span class="_ _b"> </span>rotating<span class="_ _b"> </span><span class="ff9">K</span></span></div><div class="t m1 x29 h16 yc3 ffc fsa fc0 sc0 ls3 ws0">1</div><div class="t m1 x4a hf yc2 ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">x</span><span class="_ _3"> </span><span class="ff5">in<span class="_ _b"> </span><span class="ff9">R</span></span></div><div class="t m1 x4b he yc1 ff9 fsa fc0 sc0 ls3 ws0">d</div><div class="t m1 x4c hd yc2 ff5 fs8 fc0 sc0 ls3 ws26">,<span class="_ _3"> </span>i.e.,<span class="_ _b"> </span><span class="ff9 ws0">K</span></div><div class="t m1 x4d he yc1 ff9 fsa fc0 sc0 ls3 ws0">S</div><div class="t m1 x4e hf yc2 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="_ _3"> </span><span class="ff5">is</span></div><div class="t m1 x9 hd yc4 ff5 fs8 fc0 sc0 ls12 ws0">radially<span class="_ _10"> </span>symmetric.<span class="_ _10"> </span>The<span class="_ _13"> </span>constant<span class="_ _10"> </span><span class="ff9 ls3">a</span></div><div class="t m1 x4f h12 yc5 ffb fsa fc0 sc0 ls3 ws0"><span class="ffc">1</span></div><div class="t m1 x4f he yc6 ff9 fsa fc0 sc0 ls3 ws0">k;d</div><div class="t m1 x50 hf yc7 ffb fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x4c h11 yc8 ffd fs8 fc0 sc0 ls3 ws0">R</div><div class="t m1 x51 he yc9 ff9 fsa fc0 sc0 ls3 ws0">R</div><div class="t m1 x52 h14 yca ff9 fsb fc0 sc0 ls3 ws0">d</div><div class="t m1 x53 hd yc7 ff9 fs8 fc0 sc0 ls3 ws0">K</div><div class="t m1 x54 h16 ycb ffc fsa fc0 sc0 ls3 ws0">1</div><div class="t m1 x55 hf yc7 ffb fs8 fc0 sc0 ls3 ws0">k<span class="ffa">x</span>k<span class="ffc">d<span class="ffa">x</span></span></div><div class="t m1 x9 hd ycc ff5 fs8 fc0 sc0 ls3 ws0">assures<span class="_ _3"> </span>that<span class="_ _b"> </span><span class="ff9">K</span></div><div class="t m1 x56 he ycd ff9 fsa fc0 sc0 ls3 ws0">S</div><div class="t m1 x57 hf yce ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="_ _3"> </span><span class="ff5 ws27">integrates<span class="_ _b"> </span>to<span class="_ _b"> </span>one,<span class="_ _b"> </span>though<span class="_ _3"> </span>this<span class="_ _b"> </span>condition</span></div><div class="t m1 x9 h8 ycf ff5 fs8 fc0 sc0 ls3 ws0">can<span class="_ _8"> </span>be<span class="_ _8"> </span>relaxed<span class="_ _8"> </span>in<span class="_ _8"> </span>our<span class="_ _8"> </span>context.<span class="_ _8"> </span>Either<span class="_ _8"> </span>type<span class="_ _8"> </span>of<span class="_ _8"> </span>multivariate</div><div class="t m1 x9 h8 yd0 ff5 fs8 fc0 sc0 ls13 ws28">kernel<span class="_ _10"> </span>obeys<span class="_ _10"> </span>(3),<span class="_ _10"> </span>but,<span class="_ _10"> </span>for<span class="_ _10"> </span>our<span class="_ _10"> </span>purp<span class="_ _f"></span>oses,<span class="_ _10"> </span>the<span class="_ _10"> </span>radially</div><div class="t m1 x9 h8 yd1 ff5 fs8 fc0 sc0 ls3 ws0">symmetric<span class="_ _3"> </span>kernels<span class="_ _b"> </span>are<span class="_ _b"> </span>often<span class="_ _3"> </span>more<span class="_ _b"> </span>suitable.</div><div class="t m1 xa h8 yd2 ff5 fs8 fc0 sc0 ls3 ws0">We<span class="_ _e"> </span>are<span class="_ _e"> </span>interested<span class="_ _e"> </span>only<span class="_ _e"> </span>in<span class="_ _11"> </span>a<span class="_ _e"> </span>special<span class="_ _e"> </span>class<span class="_ _e"> </span>of<span class="_ _e"> </span>radially</div><div class="t m1 x9 h8 yd3 ff5 fs8 fc0 sc0 ls3 ws0">symmetric<span class="_ _3"> </span>kernels<span class="_ _b"> </span>satisfying</div><div class="t m1 x58 hf yd4 ff9 fs8 fc0 sc0 ls3 ws0">K<span class="_ _f"></span><span class="ffb"><span class="ffa">x</span><span class="ls11"></span></span>c</div><div class="t m1 x59 he yd5 ff9 fsa fc0 sc0 ls3 ws0">k;d</div><div class="t m1 x5a hf yd6 ff9 fs8 fc0 sc0 ls3 ws0">k<span class="ffb">k<span class="ffa">x</span>k</span></div><div class="t m1 x5b h16 yd7 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x22 hf yd6 ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">;<span class="_ _1e"> </span></span><span class="ffc">5</span></div><div class="t m1 x9 hf yd8 ff5 fs8 fc0 sc0 ls3 ws0">in<span class="_ _1"> </span>which<span class="_ _1"> </span>case<span class="_ _1"> </span>it<span class="_ _1"> </span>suffices<span class="_ _1"> </span>to<span class="_ _1"> </span>define<span class="_ _1"> </span>the<span class="_ _1"> </span>function<span class="_ _1"> </span><span class="ff9">k<span class="ffb"></span>x<span class="ffb"><span class="_ _1"> </span></span></span>called</div><div class="t m1 x9 hf yd9 ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _1"> </span><span class="ff7">profile<span class="_ _1"> </span></span><span class="ws29">of<span class="_ _1"> </span>the<span class="_ _1"> </span>kernel,<span class="_ _1"> </span>only<span class="_ _1"> </span>for<span class="_ _1"> </span></span><span class="ff9">x<span class="_ _2"> </span><span class="ffb"><span class="_ _2"> </span><span class="ffc">0</span></span></span>.<span class="_ _1"> </span>The<span class="_ _1"> </span>normalization</div><div class="t m1 x9 hd yda ff5 fs8 fc0 sc0 ls14 ws0">constant<span class="_ _c"> </span><span class="ff9 ls3">c</span></div><div class="t m1 x36 he ydb ff9 fsa fc0 sc0 ls3 ws0">k;d</div><div class="t m1 x5c hf ydc ff5 fs8 fc0 sc0 ls15 ws2a">,<span class="_ _c"> </span>which<span class="_ _13"> </span>makes<span class="_ _c"> </span><span class="ff9 ls3 ws0">K<span class="_ _f"></span><span class="ffb"><span class="ffa">x</span><span class="_ _10"> </span></span></span><span class="ls16 ws2b">integrate<span class="_ _10"> </span>to<span class="_ _10"> </span>one,<span class="_ _10"> </span>is</span></div><div class="t m1 x9 h8 ydd ff5 fs8 fc0 sc0 ls3 ws0">assumed<span class="_ _3"> </span>strictly<span class="_ _b"> </span>positive.</div><div class="t m1 xa h8 yde ff5 fs8 fc0 sc0 ls3 ws0">Using<span class="_ _b"> </span>a<span class="_ _b"> </span>fully<span class="_ _3"> </span>parameterized<span class="_ _b"> </span><span class="ffa">H<span class="_ _b"> </span></span>increases<span class="_ _b"> </span>the<span class="_ _3"> </span>complexity</div><div class="t m1 x9 h8 ydf ff5 fs8 fc0 sc0 ls3 ws2c">of<span class="_ _4"> </span>the<span class="_ _2"> </span>estimation<span class="_ _2"> </span>[62,<span class="_ _4"> </span>p.<span class="_ _2"> </span>106]<span class="_ _4"> </span>and,<span class="_ _2"> </span>in<span class="_ _4"> </span>practice,<span class="_ _2"> </span>the<span class="_ _4"> </span>bandwidth</div><div class="t m1 x9 hf ye0 ff5 fs8 fc0 sc0 ls3 ws0">matrix<span class="_ _3"> </span><span class="ffa">H<span class="_ _b"> </span></span>is<span class="_ _b"> </span>chosen<span class="_ _b"> </span>either<span class="_ _b"> </span>as<span class="_ _3"> </span>diagonal<span class="_ _b"> </span><span class="ffa">H<span class="_ _2"> </span><span class="ffb"><span class="_ _2"> </span></span>diag<span class="ffb"><span class="ff9">h</span></span></span></div><div class="t m1 x5d h16 ye1 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x5d h16 ye2 ffc fsa fc0 sc0 ls3 ws0">1</div><div class="t m1 x5e hd ye3 ff9 fs8 fc0 sc0 ls3 ws0">;<span class="_ _15"> </span><span class="ffc lsc">...<span class="ff9">;h</span></span></div><div class="t m1 x5f h16 ye1 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x5f he ye2 ff9 fsa fc0 sc0 ls3 ws0">d</div><div class="t m1 x60 hf ye3 ffb fs8 fc0 sc0 ls3 ws0"><span class="ff5">,</span></div><div class="t m1 xb h9 y55 ff1 fs7 fc0 sc0 ls8 ws0">604<span class="_ _14"> </span>IEEE<span class="_ _2"> </span>TRANSACTIONS<span class="_ _4"> </span>ON<span class="_ _2"> </span>PATTERN<span class="_ _4"> </span>ANALYSI<span class="_ _f"></span>S<span class="_ _4"> </span>AND<span class="_ _2"> </span>MACHINE<span class="_ _4"> </span>INTELLIGEN<span class="_ _f"></span>CE,<span class="_ _13"> </span>VOL.<span class="_ _4"> </span>24,<span class="_ _1f"> </span>NO.<span class="_ _4"> </span>5,<span class="_ _1f"> </span>MAY<span class="_ _4"> </span>2002</div><div class="t m1 x6 h4 ye4 ff1 fs2 fc0 sc0 ls3 ws0">Fig.<span class="_ _2"> </span>1.<span class="_ _2"> </span>Example<span class="_ _2"> </span>of<span class="_ _2"> </span>a<span class="_ _2"> </span>feature<span class="_ _2"> </span>space.<span class="_ _2"> </span>(a)<span class="_ _2"> </span>A<span class="_ _2"> </span><span class="ffc">400<span class="_ _5"> </span><span class="ffb"><span class="_ _5"> </span></span>276<span class="_ _2"> </span></span>color<span class="_ _2"> </span>image.<span class="_ _2"> </span>(b)<span class="_ _2"> </span>Corresponding<span class="_ _2"> </span>L*u*v*<span class="_ _2"> </span>color<span class="_ _2"> </span>space<span class="_ _2"> </span>with<span class="_ _2"> </span><span class="ffc">110<span class="ff9">;<span class="_ _15"> </span></span>400<span class="_ _2"> </span></span>data<span class="_ _2"> </span>points.</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6244dcfd6cd7507199bf81c5/bg3.jpg"><div class="t m1 x6 hf ye5 ff5 fs8 fc0 sc0 ls3 ws0">or<span class="_ _9"> </span>proportional<span class="_ _d"> </span>to<span class="_ _8"> </span>the<span class="_ _d"> </span>identity<span class="_ _9"> </span>matrix<span class="_ _9"> </span><span class="ffa">H<span class="_ _2"> </span><span class="ffb"><span class="_ _3"> </span><span class="ff9">h</span></span></span></div><div class="t m1 x61 h16 ye6 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x62 h8 ye5 ffa fs8 fc0 sc0 ls3 ws0">I<span class="ff5">.<span class="_ _9"> </span>The<span class="_ _9"> </span>clear</span></div><div class="t m1 x6 h8 ye7 ff5 fs8 fc0 sc0 ls3 ws0">advantage<span class="_ _d"> </span>of<span class="_ _12"> </span>the<span class="_ _12"> </span>latter<span class="_ _d"> </span>case<span class="_ _12"> </span>is<span class="_ _d"> </span>that<span class="_ _12"> </span>only<span class="_ _12"> </span>one<span class="_ _12"> </span>bandwidth</div><div class="t m1 x6 hd ye8 ff5 fs8 fc0 sc0 ls3 ws0">parameter<span class="_"> </span><span class="ff9 ls17">h></span><span class="ffc">0<span class="_ _2"> </span></span><span class="ws2d">must<span class="_ _2"> </span>be<span class="_ _2"> </span>provided;<span class="_ _4"> </span>however,<span class="_ _2"> </span>as<span class="_ _2"> </span>can<span class="_ _2"> </span>be<span class="_ _4"> </span>seen</span></div><div class="t m1 x6 h8 ye9 ff5 fs8 fc0 sc0 ls3 ws2e">from<span class="_ _8"> </span>(2),<span class="_ _8"> </span>then<span class="_ _8"> </span>the<span class="_ _9"> </span>validity<span class="_ _8"> </span>of<span class="_ _8"> </span>an<span class="_ _9"> </span>Euclidean<span class="_ _8"> </span>metric<span class="_ _8"> </span>for<span class="_ _9"> </span>the</div><div class="t m1 x6 h8 yea ff5 fs8 fc0 sc0 ls3 ws0">feature<span class="_ _d"> </span>space<span class="_ _d"> </span>should<span class="_ _9"> </span>be<span class="_ _d"> </span>confirmed<span class="_ _d"> </span>first.<span class="_ _d"> </span>Employing<span class="_ _d"> </span>only</div><div class="t m1 x6 h8 yeb ff5 fs8 fc0 sc0 ls3 ws2f">one<span class="_ _b"> </span>bandwidth<span class="_ _b"> </span>parameter,<span class="_ _b"> </span>the<span class="_ _b"> </span>kernel<span class="_ _b"> </span>density<span class="_ _1"> </span>estimator<span class="_ _b"> </span>(1)</div><div class="t m1 x6 h8 yec ff5 fs8 fc0 sc0 ls3 ws0">becomes<span class="_ _b"> </span>the<span class="_ _3"> </span>well-known<span class="_ _b"> </span>expression</div><div class="t m1 x3 h10 yed ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x63 hf yee ff9 fs8 fc0 sc0 ls3 ws0">f<span class="_ _16"></span><span class="ffb"><span class="ffa">x</span><span class="ls11"></span></span></div><div class="t m1 x64 h10 yef ffc fs8 fc0 sc0 ls3 ws0">1</div><div class="t m1 x65 hd yf0 ff9 fs8 fc0 sc0 ls3 ws0">nh</div><div class="t m1 x66 he yf1 ff9 fsa fc0 sc0 ls3 ws0">d</div><div class="t m1 x67 h11 yf2 ffd fs8 fc0 sc0 ls3 ws0">X</div><div class="t m1 x68 he yf3 ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 x69 h12 yf4 ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x6a hd yf5 ff9 fs8 fc0 sc0 ls3 ws0">K</div><div class="t m1 x6b hf yef ffa fs8 fc0 sc0 ls3 ws0">x<span class="_ _4"> </span><span class="ffb"><span class="_ _4"> </span></span>x</div><div class="t m1 x6c he yf6 ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 x6d hd yf7 ff9 fs8 fc0 sc0 ls3 ws0">h</div><div class="t m1 x6e h11 yf8 ffd fs8 fc0 sc0 ls18 ws0"></div><div class="t m1 x6f hf yf9 ff9 fs8 fc0 sc0 ls3 ws0">:<span class="_ _20"> </span><span class="ffb"><span class="ffc">6</span></span></div><div class="t m1 x8 h8 yfa ff5 fs8 fc0 sc0 ls3 ws0">The<span class="_ _2"> </span>quality<span class="_ _3"> </span>of<span class="_ _3"> </span>a<span class="_ _2"> </span>kernel<span class="_ _3"> </span>density<span class="_ _3"> </span>estimator<span class="_ _3"> </span>is<span class="_ _2"> </span>measured<span class="_ _3"> </span>by</div><div class="t m1 x6 h8 yfb ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _8"> </span>mean<span class="_ _1"> </span>of<span class="_ _8"> </span>the<span class="_ _8"> </span>square<span class="_ _1"> </span>error<span class="_ _8"> </span>between<span class="_ _8"> </span>the<span class="_ _8"> </span>density<span class="_ _1"> </span>and<span class="_ _8"> </span>its</div><div class="t m1 x6 h8 yfc ff5 fs8 fc0 sc0 ls3 ws30">estimate,<span class="_ _15"> </span>integrated<span class="_ _15"> </span>over<span class="_ _15"> </span>the<span class="_ _15"> </span>domain<span class="_ _15"></span>of<span class="_ _21"></span>definition.<span class="_ _15"> </span>In<span class="_ _21"> </span>practice,<span class="_ _15"> </span></div><div class="t m1 x6 h8 yfd ff5 fs8 fc0 sc0 ls3 ws31">however,<span class="_ _4"> </span>only<span class="_ _4"> </span>an<span class="_ _4"> </span>asymptotic<span class="_ _4"> </span>approximation<span class="_ _2"> </span>of<span class="_ _5"> </span>this<span class="_ _4"> </span>measure</div><div class="t m1 x6 h8 yfe ff5 fs8 fc0 sc0 ls3 ws0">(denoted<span class="_ _2"> </span>as<span class="_ _3"> </span>AMISE)<span class="_ _3"> </span>can<span class="_ _3"> </span>be<span class="_ _3"> </span>computed.<span class="_ _2"> </span>Under<span class="_ _3"> </span>the<span class="_ _3"> </span>asympto-</div><div class="t m1 x6 hf yff ff5 fs8 fc0 sc0 ls3 ws32">tics,<span class="_ _2"> </span>the<span class="_ _2"> </span>number<span class="_ _2"> </span>of<span class="_ _2"> </span>data<span class="_ _2"> </span>points<span class="_ _2"> </span><span class="ff9 ws0">n<span class="_ _21"> </span><span class="ffb ls19">!1<span class="_ _22"></span><span class="ff5 ls3 ws33">,<span class="_ _2"> </span>while<span class="_ _2"> </span>the<span class="_ _2"> </span>bandwidth</span></span></span></div><div class="t m1 x6 hf y100 ff9 fs8 fc0 sc0 ls3 ws0">h<span class="_ _21"></span><span class="ffb">!<span class="_ _16"></span><span class="ffc">0<span class="_ _4"> </span><span class="ff5">at<span class="_ _5"> </span>a<span class="_ _4"> </span>rate<span class="_ _5"> </span>slower<span class="_ _5"> </span>than<span class="_"> </span></span></span></span>n</div><div class="t m1 x70 h12 y101 ffb fsa fc0 sc0 ls3 ws0"><span class="ffc">1</span></div><div class="t m1 x71 h8 y102 ff5 fs8 fc0 sc0 ls3 ws0">.<span class="_ _5"> </span>For<span class="_ _5"> </span>both<span class="_"> </span>types<span class="_ _5"> </span>of<span class="_ _5"> </span>multivariate</div><div class="t m1 x6 h8 y103 ff5 fs8 fc0 sc0 ls3 ws34">kernels,<span class="_ _5"> </span>the<span class="_ _4"> </span>AMISE<span class="_ _5"> </span>measure<span class="_ _4"> </span>is<span class="_ _5"> </span>minimized<span class="_ _4"> </span>by<span class="_ _5"> </span>the<span class="_ _4"> </span>Epanechni-</div><div class="t m1 x6 h8 y104 ff5 fs8 fc0 sc0 ls3 ws35">kov<span class="_ _4"> </span>kernel<span class="_ _2"> </span>[51,<span class="_ _5"> </span>p.<span class="_ _2"> </span>139],<span class="_ _4"> </span>[62,<span class="_ _4"> </span>p.<span class="_ _2"> </span>104]<span class="_ _5"> </span>having<span class="_ _2"> </span>the<span class="_ _4"> </span>profile</div><div class="t m1 x72 hd y105 ff9 fs8 fc0 sc0 ls3 ws0">k</div><div class="t m1 x3 he y106 ff9 fsa fc0 sc0 ls3 ws0">E</div><div class="t m1 x73 hf y107 ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">x</span><span class="ls11"></span></div><div class="t m1 x74 hf y108 ffc fs8 fc0 sc0 ls3 ws0">1<span class="_ _5"> </span><span class="ffb"><span class="_ _4"> </span><span class="ff9">x<span class="_ _23"> </span></span></span>0<span class="_ _2"> </span><span class="ffb"><span class="_ _2"> </span><span class="ff9">x<span class="_ _2"> </span></span><span class="_ _2"> </span></span>1</div><div class="t m1 x74 hd y109 ffc fs8 fc0 sc0 ls3 ws0">0<span class="_ _1b"> </span><span class="ff9 ls1a">x></span>1<span class="ff9">;</span></div><div class="t m1 x75 h11 y10a ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x76 hf y107 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffc">7</span></div><div class="t m1 x6 h8 y10b ff5 fs8 fc0 sc0 ls3 ws0">which<span class="_ _b"> </span>yields<span class="_ _3"> </span>the<span class="_ _b"> </span>radially<span class="_ _b"> </span>symmetric<span class="_ _3"> </span>kernel</div><div class="t m1 x77 hd y10c ff9 fs8 fc0 sc0 ls3 ws0">K</div><div class="t m1 x78 he y10d ff9 fsa fc0 sc0 ls3 ws0">E</div><div class="t m1 x79 hf y10e ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="lse"></span></div><div class="t m1 x7a h16 y10f ffc fsa fc0 sc0 ls3 ws0">1</div><div class="t m1 x7a h16 y110 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x1 hd y111 ff9 fs8 fc0 sc0 ls3 ws0">c</div><div class="t m1 x7b h12 y112 ffb fsa fc0 sc0 ls3 ws0"><span class="ffc">1</span></div><div class="t m1 x7b he y113 ff9 fsa fc0 sc0 ls3 ws0">d</div><div class="t m1 x7c hf y111 ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">d<span class="_ _5"> </span></span><span class="_ _4"> </span><span class="ffc">2</span><span class="ffc">1<span class="_ _4"> </span></span><span class="ls1b">k<span class="_ _24"></span><span class="ffa ls3">x<span class="ffb">k</span></span></span></div><div class="t m1 x7d h16 y114 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x7e hf y111 ffb fs8 fc0 sc0 ls1c ws0">k<span class="_ _25"></span><span class="ffa ls3">x<span class="ffb ls1d">k</span><span class="ffc">1</span></span></div><div class="t m1 x7f hd y115 ffc fs8 fc0 sc0 ls3 ws0">0<span class="_ _26"> </span>otherwise<span class="ff9">;</span></div><div class="t m1 x63 h11 y116 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x76 hf y117 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffc">8</span></div><div class="t m1 x6 hd y118 ff5 fs8 fc0 sc0 ls3 ws0">where<span class="_ _9"> </span><span class="ff9">c</span></div><div class="t m1 x80 he y119 ff9 fsa fc0 sc0 ls3 ws0">d</div><div class="t m1 x81 hd y11a ff5 fs8 fc0 sc0 ls3 ws0">is<span class="_ _9"> </span>the<span class="_ _d"> </span>volume<span class="_ _9"> </span>of<span class="_ _d"> </span>the<span class="_ _9"> </span>unit<span class="_ _d"> </span><span class="ff9">d<span class="ffc">-dimensional<span class="_"> </span></span></span>sphere.</div><div class="t m1 x6 h8 y11b ff5 fs8 fc0 sc0 ls3 ws0">Note<span class="_ _1"> </span>that<span class="_ _b"> </span>the<span class="_ _1"> </span>Epanechnikov<span class="_ _1"> </span>profile<span class="_ _1"> </span>is<span class="_ _1"> </span>not<span class="_ _b"> </span>differentiable<span class="_ _1"> </span>at</div><div class="t m1 x6 h8 y11c ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _b"> </span>boundary.<span class="_ _3"> </span>The<span class="_ _b"> </span>profile</div><div class="t m1 x82 hd y11d ff9 fs8 fc0 sc0 ls3 ws0">k</div><div class="t m1 x83 he y11e ff9 fsa fc0 sc0 ls3 ws0">N</div><div class="t m1 x84 hf y11f ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">x</span><span class="lse"></span><span class="ffc">exp<span class="_ _13"> </span></span></div><div class="t m1 x69 h10 y120 ffc fs8 fc0 sc0 ls3 ws0">1</div><div class="t m1 x69 h10 y121 ffc fs8 fc0 sc0 ls3 ws0">2</div><div class="t m1 x85 hd y122 ff9 fs8 fc0 sc0 ls3 ws0">x</div><div class="t m1 x86 h11 y123 ffd fs8 fc0 sc0 ls1e ws0"></div><div class="t m1 x87 hf y122 ff9 fs8 fc0 sc0 ls3 ws0">x<span class="_ _2"> </span><span class="ffb"><span class="_ _2"> </span><span class="ffc">0<span class="_ _27"> </span></span><span class="ffc">9</span></span></div><div class="t m1 x6 h8 y124 ff5 fs8 fc0 sc0 ls3 ws0">yields<span class="_ _b"> </span>the<span class="_ _3"> </span>multivariate<span class="_ _b"> </span>normal<span class="_ _b"> </span>kernel</div><div class="t m1 x83 hd y125 ff9 fs8 fc0 sc0 ls3 ws0">K</div><div class="t m1 x88 he y126 ff9 fsa fc0 sc0 ls3 ws0">N</div><div class="t m1 x3 hf y127 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="ls1f"><span class="_ _19"></span><span class="ffc ls3">2<span class="ff9"><span class="ffb"></span></span></span></span></div><div class="t m1 x89 h12 y128 ffb fsa fc0 sc0 ls3 ws0"><span class="ff9">d=<span class="ffc">2</span></span></div><div class="t m1 x8a hf y127 ffc fs8 fc0 sc0 ls3 ws0">exp<span class="_ _13"> </span><span class="ffb"></span></div><div class="t m1 x7d h10 y129 ffc fs8 fc0 sc0 ls3 ws0">1</div><div class="t m1 x7d h10 y12a ffc fs8 fc0 sc0 ls3 ws0">2</div><div class="t m1 x8b hf y12b ffb fs8 fc0 sc0 ls3 ws0">k<span class="ffa">x</span>k</div><div class="t m1 x8c h16 y128 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x8d h11 y12c ffd fs8 fc0 sc0 ls20 ws0"></div><div class="t m1 x8e hf y12d ffb fs8 fc0 sc0 ls3 ws0"><span class="ffc">10</span></div><div class="t m1 x6 h8 y12e ff5 fs8 fc0 sc0 ls3 ws0">for<span class="_"> </span>both<span class="_"> </span>types<span class="_"> </span>of<span class="_"> </span>composition<span class="_"> </span>(4).<span class="_"> </span>The<span class="_"> </span>normal<span class="_"> </span>kernel<span class="_"> </span>is<span class="_"> </span>often</div><div class="t m1 x6 h8 y12f ff5 fs8 fc0 sc0 ls3 ws0">symmetrically<span class="_ _5"> </span>truncated<span class="_ _5"> </span>to<span class="_ _4"> </span>have<span class="_ _5"> </span>a<span class="_ _5"> </span>kernel<span class="_ _5"> </span>with<span class="_ _5"> </span>finite<span class="_ _5"> </span>support.</div><div class="t m1 x8 h8 y130 ff5 fs8 fc0 sc0 ls3 ws0">While<span class="_"> </span>these<span class="_"> </span>two<span class="_"> </span>kernels<span class="_"> </span>will<span class="_ _4"> </span>suffice<span class="_"> </span>for<span class="_"> </span>most<span class="_"> </span>applications</div><div class="t m1 x6 h8 y131 ff5 fs8 fc0 sc0 ls3 ws36">we<span class="_ _4"> </span>are<span class="_ _5"> </span>interested<span class="_ _4"> </span>in,<span class="_ _4"> </span>all<span class="_ _4"> </span>the<span class="_ _4"> </span>results<span class="_ _5"> </span>presented<span class="_ _4"> </span>below<span class="_ _4"> </span>are<span class="_ _4"> </span>valid</div><div class="t m1 x6 h8 y132 ff5 fs8 fc0 sc0 ls3 ws0">for<span class="_ _a"> </span>arbitrary<span class="_ _a"> </span>kernels<span class="_ _a"> </span>within<span class="_ _a"> </span>the<span class="_ _a"> </span>conditions<span class="_ _a"> </span>to<span class="_ _a"> </span>be<span class="_ _a"> </span>stated.</div><div class="t m1 x6 h8 y133 ff5 fs8 fc0 sc0 ls3 ws37">Employing<span class="_ _4"> </span>the<span class="_ _4"> </span>profile<span class="_ _4"> </span>notation,<span class="_ _4"> </span>the<span class="_ _4"> </span>density<span class="_ _4"> </span>estimator<span class="_ _2"> </span>(6)<span class="_ _5"> </span>can</div><div class="t m1 x6 h8 y134 ff5 fs8 fc0 sc0 ls3 ws0">be<span class="_"> </span>rewritten<span class="_"> </span>as</div><div class="t m1 x8f h10 y135 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x83 hd y136 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 x90 he y137 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 x91 hf y138 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="lse"></span></div><div class="t m1 x65 hd y139 ff9 fs8 fc0 sc0 ls3 ws0">c</div><div class="t m1 x92 he y13a ff9 fsa fc0 sc0 ls3 ws0">k;d</div><div class="t m1 x93 hd y13b ff9 fs8 fc0 sc0 ls3 ws0">nh</div><div class="t m1 x74 he y13c ff9 fsa fc0 sc0 ls3 ws0">d</div><div class="t m1 x94 h11 y13d ffd fs8 fc0 sc0 ls3 ws0">X</div><div class="t m1 x7f he y13e ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 x67 h12 y13f ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x95 hd y138 ff9 fs8 fc0 sc0 ls3 ws0">k</div><div class="t m1 x96 hf y139 ffa fs8 fc0 sc0 ls3 ws0">x<span class="_ _5"> </span><span class="ffb"><span class="_ _4"> </span></span>x</div><div class="t m1 x97 he y13a ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 x98 hd y13b ff9 fs8 fc0 sc0 ls3 ws0">h</div><div class="t m1 x99 h11 y140 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x99 h11 y141 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x99 h11 y142 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x9a h11 y140 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x9a h11 y141 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x9a h11 y142 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x9b h16 y143 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x9c h11 y144 ffd fs8 fc0 sc0 ls21 ws0"></div><div class="t m1 x9d hf y145 ff9 fs8 fc0 sc0 ls3 ws0">:<span class="_ _28"> </span><span class="ffb"><span class="ffc">11</span></span></div><div class="t m1 x6 h8 y146 ff5 fs8 fc0 sc0 ls3 ws0">The<span class="_ _9"> </span>first<span class="_ _9"> </span>step<span class="_ _d"> </span>in<span class="_ _9"> </span>the<span class="_ _9"> </span>analysis<span class="_ _9"> </span>of<span class="_ _9"> </span>a<span class="_ _d"> </span>feature<span class="_ _9"> </span>space<span class="_ _9"> </span>with<span class="_ _9"> </span>the</div><div class="t m1 x6 hf y147 ff5 fs8 fc0 sc0 ls3 ws0">underlying<span class="_"> </span>density<span class="_ _2"> </span><span class="ff9">f<span class="_ _16"></span><span class="ffb"><span class="ffa">x</span><span class="_ _2"> </span></span></span>is<span class="_"> </span>to<span class="_"> </span>find<span class="_ _2"> </span>the<span class="_"> </span>modes<span class="_ _2"> </span>of<span class="_"> </span>this<span class="_ _2"> </span>density.</div><div class="t m1 x6 h8 y148 ff5 fs8 fc0 sc0 ls3 ws0">The<span class="_ _d"> </span>modes<span class="_ _12"> </span>are<span class="_ _d"> </span>located<span class="_ _12"> </span>among<span class="_ _d"> </span>the<span class="_ _12"> </span>zeros<span class="_ _d"> </span>of<span class="_ _12"> </span>the<span class="_ _d"> </span>gradient</div><div class="t m1 x6 hf y149 ffb fs8 fc0 sc0 ls3 ws0">r<span class="ff9">f<span class="_ _16"></span></span><span class="ffa">x</span><span class="ls11"></span><span class="ffa">0<span class="_ _3"> </span><span class="ff5">and<span class="_ _b"> </span>the<span class="_ _3"> </span>mean<span class="_ _b"> </span>shift<span class="_ _3"> </span>procedure<span class="_ _b"> </span>is<span class="_ _3"> </span>an<span class="_ _3"> </span>elegant<span class="_ _b"> </span>way</span></span></div><div class="t m1 x6 hc y14a ff5 fs8 fc0 sc0 ls3 ws0">to<span class="_ _b"> </span>locate<span class="_ _3"> </span>these<span class="_ _b"> </span>zeros<span class="_ _b"> </span><span class="ff7">without<span class="_"> </span></span>estimating<span class="_ _3"> </span>the<span class="_ _b"> </span>density.</div><div class="t m1 x6 h17 y14b ff3 fsc fc0 sc0 ls3 ws0">2.1<span class="_ _29"> </span>Density<span class="_ _b"> </span>Gradient<span class="_ _1"> </span>Estimation</div><div class="t m1 x6 h8 y14c ff5 fs8 fc0 sc0 ls3 ws0">The<span class="_"> </span>density<span class="_"> </span>gradient<span class="_"> </span>estimator<span class="_ _5"> </span>is<span class="_"> </span>obtained<span class="_"> </span>as<span class="_"> </span>the<span class="_"> </span>gradient<span class="_ _4"> </span>of</div><div class="t m1 x6 h8 y14d ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _b"> </span>density<span class="_ _3"> </span>estimator<span class="_ _b"> </span>by<span class="_ _b"> </span>exploiting<span class="_ _3"> </span>the<span class="_ _b"> </span>linearity<span class="_ _b"> </span>of<span class="_ _3"> </span>(11)</div><div class="t m1 xc h10 y14e ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x9e hf y14f ffb fs8 fc0 sc0 ls3 ws0">r<span class="ff9">f</span></div><div class="t m1 x9f he y150 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 xa0 hf y151 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="ls1f">r</span></div><div class="t m1 x72 h10 y152 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x88 hd y151 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 x63 he y150 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 x2 hf y151 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="ls11"></span></div><div class="t m1 xa1 hd y153 ffc fs8 fc0 sc0 ls3 ws0">2<span class="ff9">c</span></div><div class="t m1 x69 he y154 ff9 fsa fc0 sc0 ls3 ws0">k;d</div><div class="t m1 xa2 hd y155 ff9 fs8 fc0 sc0 ls3 ws0">nh</div><div class="t m1 x69 h12 y156 ff9 fsa fc0 sc0 ls3 ws0">d<span class="ffb"><span class="ffc">2</span></span></div><div class="t m1 x6a h11 y157 ffd fs8 fc0 sc0 ls3 ws0">X</div><div class="t m1 xa3 he y158 ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 xa4 h12 y159 ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 xa5 hf y151 ffa fs8 fc0 sc0 ls3 ws0">x<span class="_ _4"> </span><span class="ffb"><span class="_ _5"> </span></span>x</div><div class="t m1 xa6 he y150 ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 x6b hf y151 ffb fs8 fc0 sc0 ls22 ws0"><span class="_ _17"></span><span class="ff9 ls3">k</span></div><div class="t m1 xa7 h12 y15a ffb fsa fc0 sc0 ls3 ws0">0</div><div class="t m1 xa8 hf y153 ffa fs8 fc0 sc0 ls3 ws0">x<span class="_ _4"> </span><span class="ffb"><span class="_ _5"> </span></span>x</div><div class="t m1 xa9 he y154 ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 xd hd y155 ff9 fs8 fc0 sc0 ls3 ws0">h</div><div class="t m1 xaa h11 y15b ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xaa h11 y15c ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xaa h11 y15d ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xab h11 y15b ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xab h11 y15c ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xab h11 y15d ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xac h16 y88 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 xad h11 y15e ffd fs8 fc0 sc0 ls21 ws0"></div><div class="t m1 xae hd y15f ff9 fs8 fc0 sc0 ls3 ws0">:</div><div class="t m1 x8e hf y160 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffc">12</span></div><div class="t m1 x9 h8 y161 ff5 fs8 fc0 sc0 ls3 ws0">We<span class="_ _3"> </span>define<span class="_ _b"> </span>the<span class="_ _b"> </span>function</div><div class="t m1 xaf hf y162 ff9 fs8 fc0 sc0 ls3 ws0">g<span class="ffb"></span>x<span class="ffb ls1f"><span class="_ _19"></span><span class="ff9 ls3">k</span></span></div><div class="t m1 xb0 h12 y163 ffb fsa fc0 sc0 ls3 ws0">0</div><div class="t m1 x21 hf y164 ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">x</span><span class="ff9">;<span class="_ _2a"> </span></span><span class="ffc">13</span></div><div class="t m1 x9 hd y165 ff5 fs8 fc0 sc0 ls3 ws0">assuming<span class="_"> </span>that<span class="_"> </span>the<span class="_"> </span>derivative<span class="_ _4"> </span>of<span class="_"> </span>the<span class="_"> </span>kernel<span class="_ _4"> </span>profile<span class="_"> </span><span class="ff9">k<span class="_ _4"> </span></span>exists<span class="_"> </span>for</div><div class="t m1 x9 hf y166 ff5 fs8 fc0 sc0 ls3 ws0">all<span class="_ _1"> </span><span class="ff9">x<span class="_ _2"> </span><span class="ffb ls23">2<span class="_ _19"></span><span class="ffc ls3">0<span class="ff9">;<span class="_ _15"> </span><span class="ffb">1<span class="ff5 ws38">,<span class="_ _1"> </span>except<span class="_ _1"> </span>for<span class="_ _1"> </span>a<span class="_ _1"> </span>finite<span class="_ _1"> </span>set<span class="_ _1"> </span>of<span class="_ _1"> </span>points.<span class="_ _1"> </span>Now,<span class="_ _1"> </span>using</span></span></span></span></span></span></div><div class="t m1 x9 hf y167 ff9 fs8 fc0 sc0 ls3 ws0">g<span class="ffb"></span>x<span class="ffb"><span class="_ _3"> </span><span class="ff5 ws39">for<span class="_ _b"> </span>profile,<span class="_ _b"> </span>the<span class="_ _3"> </span>kernel<span class="_ _b"> </span></span></span>G<span class="ffb"><span class="ffa">x</span><span class="_ _b"> </span><span class="ff5">is<span class="_ _b"> </span>defined<span class="_ _3"> </span>as</span></span></div><div class="t m1 xb1 hf y168 ff9 fs8 fc0 sc0 ls3 ws0">G<span class="ffb"><span class="ffa">x</span><span class="lse"></span></span>c</div><div class="t m1 x20 he y169 ff9 fsa fc0 sc0 ls3 ws0">g;d</div><div class="t m1 x2f hf y16a ff9 fs8 fc0 sc0 ls3 ws0">g<span class="_ _e"> </span><span class="ffb">k<span class="ffa">x</span>k</span></div><div class="t m1 x4a h16 y16b ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 xb2 h11 y16c ffd fs8 fc0 sc0 ls24 ws0"></div><div class="t m1 xb3 hf y16d ff9 fs8 fc0 sc0 ls3 ws0">;<span class="_ _2b"> </span><span class="ffb"><span class="ffc">14</span></span></div><div class="t m1 x9 hd y16e ff5 fs8 fc0 sc0 ls3 ws0">where<span class="_ _2"> </span><span class="ff9">c</span></div><div class="t m1 x2c he y16f ff9 fsa fc0 sc0 ls3 ws0">g;d</div><div class="t m1 x25 h8 y170 ff5 fs8 fc0 sc0 ls3 ws0">is<span class="_ _2"> </span>the<span class="_ _2"> </span>corresponding<span class="_ _2"> </span>normalization<span class="_ _2"> </span>constant.<span class="_ _3"> </span>The</div><div class="t m1 x9 hf y171 ff5 fs8 fc0 sc0 ls3 ws0">kernel<span class="_ _5"> </span><span class="ff9">K<span class="_ _16"></span><span class="ffb"><span class="ffa">x</span><span class="_ _5"> </span></span></span>was<span class="_ _5"> </span>called<span class="_ _5"> </span>the<span class="_ _5"> </span>shadow<span class="_"> </span>of<span class="_ _5"> </span><span class="ff9">G<span class="ffb"><span class="ffa">x</span><span class="_ _5"> </span></span></span>in<span class="_ _5"> </span>[7]<span class="_"> </span>in<span class="_ _5"> </span>a<span class="_ _5"> </span>slightly</div><div class="t m1 x9 h8 y172 ff5 fs8 fc0 sc0 ls3 ws0">different<span class="_ _b"> </span>context.<span class="_ _3"> </span>Note<span class="_ _b"> </span>that<span class="_ _b"> </span>the<span class="_ _b"> </span>Epanechnikov<span class="_ _b"> </span>kernel<span class="_ _b"> </span>is<span class="_ _b"> </span>the</div><div class="t m1 x9 hd y173 ff5 fs8 fc0 sc0 ls3 ws3a">shadow<span class="_ _8"> </span>of<span class="_ _8"> </span>the<span class="_ _8"> </span>uniform<span class="_ _9"> </span>kernel,<span class="_ _8"> </span>i.e.,<span class="_ _8"> </span>the<span class="_ _8"> </span><span class="ff9 ws0">d<span class="ffc">-dimensional<span class="_ _9"> </span><span class="ff5">unit</span></span></span></div><div class="t m1 x9 h8 y174 ff5 fs8 fc0 sc0 ls3 ws3b">sphere,<span class="_ _21"> </span>while<span class="_ _15"> </span>the<span class="_ _15"> </span>normal<span class="_ _15"> </span>kernel<span class="_ _15"> </span>and<span class="_ _21"> </span>its<span class="_ _15"> </span>shadow<span class="_ _15"> </span>have<span class="_ _15"> </span>the<span class="_ _15"> </span>same</div><div class="t m1 x9 h8 y175 ff5 fs8 fc0 sc0 ls3 ws0">expression.</div><div class="t m1 xa hf y176 ff5 fs8 fc0 sc0 ls3 ws0">Introducing<span class="_ _b"> </span><span class="ff9">g<span class="ffb"></span>x<span class="ffb"><span class="_ _3"> </span></span></span>into<span class="_ _b"> </span>(12)<span class="_ _b"> </span>yields,</div><div class="t m1 xa h10 y177 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 xb4 hf y178 ffb fs8 fc0 sc0 ls3 ws0">r<span class="ff9">f</span></div><div class="t m1 x39 he y179 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 x35 hf y17a ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span></div><div class="t m1 xb5 hf y17b ffb fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xb6 hd y17c ffc fs8 fc0 sc0 ls3 ws0">2<span class="ff9">c</span></div><div class="t m1 xb7 he y17d ff9 fsa fc0 sc0 ls3 ws0">k;d</div><div class="t m1 x39 hd y17e ff9 fs8 fc0 sc0 ls3 ws0">nh</div><div class="t m1 xb7 h12 y17f ff9 fsa fc0 sc0 ls3 ws0">d<span class="ffb"><span class="ffc">2</span></span></div><div class="t m1 x2e h11 y180 ffd fs8 fc0 sc0 ls3 ws0">X</div><div class="t m1 xb8 he y181 ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 x26 h12 y182 ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 xb9 h15 y183 ffa fs8 fc0 sc0 ls3 ws0">x</div><div class="t m1 xba he y184 ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 x3e hf y183 ffb fs8 fc0 sc0 ls3 ws0"><span class="_ _4"> </span><span class="ffa">x<span class="_ _2c"></span><span class="ffb ls22"><span class="_ _17"></span><span class="ff9 ls3">g</span></span></span></div><div class="t m1 x15 hf y17c ffa fs8 fc0 sc0 ls3 ws0">x<span class="_ _4"> </span><span class="ffb"><span class="_ _4"> </span></span>x</div><div class="t m1 xb2 he y17d ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 x20 hd y17e ff9 fs8 fc0 sc0 ls3 ws0">h</div><div class="t m1 xbb h11 y185 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xbb h11 y186 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xbb h11 y187 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xb0 h11 y185 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xb0 h11 y186 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xb0 h11 y187 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x42 h16 y188 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 xbc h11 y189 ffd fs8 fc0 sc0 ls25 ws0"></div><div class="t m1 xb5 hf y18a ffb fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xb6 hd y18b ffc fs8 fc0 sc0 ls3 ws0">2<span class="ff9">c</span></div><div class="t m1 xb7 he y18c ff9 fsa fc0 sc0 ls3 ws0">k;d</div><div class="t m1 x39 hd y18d ff9 fs8 fc0 sc0 ls3 ws0">nh</div><div class="t m1 xb7 h12 y18e ff9 fsa fc0 sc0 ls3 ws0">d<span class="ffb"><span class="ffc">2</span></span></div><div class="t m1 xb8 h11 y18f ffd fs8 fc0 sc0 ls3 ws0">X</div><div class="t m1 x3b he y190 ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 x3c h12 y191 ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x3d hd y192 ff9 fs8 fc0 sc0 ls3 ws0">g</div><div class="t m1 x58 hf y18b ffa fs8 fc0 sc0 ls3 ws0">x<span class="_ _4"> </span><span class="ffb"><span class="_ _4"> </span></span>x</div><div class="t m1 xbd he y18c ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 xaf hd y18d ff9 fs8 fc0 sc0 ls3 ws0">h</div><div class="t m1 x3f h11 y193 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x3f h11 y194 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x3f h11 y195 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x15 h11 y193 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x15 h11 y194 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x15 h11 y195 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x16 h16 y196 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x1e h11 y197 ffd fs8 fc0 sc0 ls21 ws0"></div><div class="t m1 x2e h11 y198 ffd fs8 fc0 sc0 ls26 ws0">"#</div><div class="t m1 x21 h11 y199 ffd fs8 fc0 sc0 ls3 ws0">P</div><div class="t m1 xbe he y19a ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 xbe h12 y19b ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 xbf h15 y19c ffa fs8 fc0 sc0 ls3 ws0">x</div><div class="t m1 x1b he y196 ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 x4f hd y19c ff9 fs8 fc0 sc0 ls3 ws0">g</div><div class="t m1 xc0 h12 y19d ffa fsa fc0 sc0 ls3 ws0">x<span class="ffb"></span>x</div><div class="t m1 x47 h14 y19e ff9 fsb fc0 sc0 ls3 ws0">i</div><div class="t m1 xc1 he y19f ff9 fsa fc0 sc0 ls3 ws0">h</div><div class="t m1 xc2 h11 y1a0 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc2 h11 y1a1 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc3 h11 y1a0 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc3 h11 y1a1 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x32 h16 y1a2 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 xc4 h11 y1a3 ffd fs8 fc0 sc0 ls27 ws0"></div><div class="t m1 xc5 h11 y1a4 ffd fs8 fc0 sc0 ls3 ws0">P</div><div class="t m1 xc6 he y1a5 ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 xc6 h12 y1a6 ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x2a hd y1a7 ff9 fs8 fc0 sc0 ls3 ws0">g</div><div class="t m1 xc7 h12 y1a8 ffa fsa fc0 sc0 ls3 ws0">x<span class="ffb"></span>x</div><div class="t m1 x46 h14 y1a9 ff9 fsb fc0 sc0 ls3 ws0">i</div><div class="t m1 x45 he y1aa ff9 fsa fc0 sc0 ls3 ws0">h</div><div class="t m1 x23 h11 y97 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x23 h11 y1ab ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x51 h11 y97 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x51 h11 y1ab ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x52 h16 y1ac ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x4f h11 y1ad ffd fs8 fc0 sc0 ls27 ws0"></div><div class="t m1 x5e hf y1ae ffb fs8 fc0 sc0 ls3 ws0"><span class="_ _4"> </span><span class="ffa">x</span></div><div class="t m1 x5a h11 y1af ffd fs8 fc0 sc0 ls3 ws0">2</div><div class="t m1 x5a h11 y1b0 ffd fs8 fc0 sc0 ls3 ws0">4</div><div class="t m1 xc8 h11 y1af ffd fs8 fc0 sc0 ls3 ws0">3</div><div class="t m1 xc8 h11 y1b0 ffd fs8 fc0 sc0 ls3 ws0">5</div><div class="t m1 xc9 hd y1ae ff9 fs8 fc0 sc0 ls3 ws0">;</div><div class="t m1 xca hf y1b1 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffc">15</span></div><div class="t m1 x9 h8 y1b2 ff5 fs8 fc0 sc0 ls3 ws0">where</div><div class="t m1 x3a h11 y1b3 ffd fs8 fc0 sc0 ls3 ws0">P</div><div class="t m1 x24 he y1b4 ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 x24 h12 y1b5 ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 xcb hd y1b6 ff9 fs8 fc0 sc0 ls3 ws0">g</div><div class="t m1 x57 h12 y1b7 ffa fsa fc0 sc0 ls3 ws0">x<span class="ffb"></span>x</div><div class="t m1 x3f h14 y1b8 ff9 fsb fc0 sc0 ls3 ws0">i</div><div class="t m1 x1e he y1b9 ff9 fsa fc0 sc0 ls3 ws0">h</div><div class="t m1 x48 h11 y1ba ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x48 h11 y1bb ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xcc h11 y1ba ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xcc h11 y1bb ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xcd h16 y1bc ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x5c h11 y1bd ffd fs8 fc0 sc0 ls27 ws0"></div><div class="t m1 xbc h8 y1be ff5 fs8 fc0 sc0 ls3 ws0">is<span class="_ _2"> </span>assumed<span class="_ _2"> </span>to<span class="_ _2"> </span>be<span class="_ _3"> </span>a<span class="_ _2"> </span>positive<span class="_ _2"> </span>number.</div><div class="t m1 x9 h8 y1bf ff5 fs8 fc0 sc0 ls3 ws0">This<span class="_ _1"> </span>condition<span class="_ _1"> </span>is<span class="_ _1"> </span>easy<span class="_ _1"> </span>to<span class="_ _8"> </span>satisfy<span class="_ _1"> </span>for<span class="_ _1"> </span>all<span class="_ _1"> </span>the<span class="_ _1"> </span>profiles<span class="_ _1"> </span>met<span class="_ _8"> </span>in</div><div class="t m1 x9 h8 y1c0 ff5 fs8 fc0 sc0 ls3 ws0">practice.<span class="_ _12"> </span>Both<span class="_ _12"> </span>terms<span class="_ _12"> </span>of<span class="_ _12"> </span>the<span class="_ _a"> </span>product<span class="_ _12"> </span>in<span class="_ _12"> </span>(15)<span class="_ _12"> </span>have<span class="_ _12"> </span>special</div><div class="t m1 x9 h8 y1c1 ff5 fs8 fc0 sc0 ls3 ws3c">significance.<span class="_ _3"> </span>From<span class="_ _b"> </span>(11),<span class="_ _3"> </span>the<span class="_ _b"> </span>first<span class="_ _3"> </span>term<span class="_ _b"> </span>is<span class="_ _3"> </span>proportional<span class="_ _b"> </span>to<span class="_ _b"> </span>the</div><div class="t m1 x9 hd y1c2 ff5 fs8 fc0 sc0 ls3 ws0">density<span class="_ _3"> </span>estimate<span class="_ _b"> </span>at<span class="_ _b"> </span><span class="ffa">x<span class="_ _3"> </span></span>computed<span class="_ _b"> </span>with<span class="_ _b"> </span>the<span class="_ _b"> </span>kernel<span class="_ _3"> </span><span class="ff9">G</span></div><div class="t m1 xce h10 y1c3 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x3b hd y1c4 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 xcf he y1c5 ff9 fsa fc0 sc0 ls3 ws0">h;G</div><div class="t m1 x12 hf y1c6 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="ls11"></span></div><div class="t m1 xd0 hd y1c7 ff9 fs8 fc0 sc0 ls3 ws0">c</div><div class="t m1 x14 he y1c8 ff9 fsa fc0 sc0 ls3 ws0">g;d</div><div class="t m1 xd1 hd y1c9 ff9 fs8 fc0 sc0 ls3 ws0">nh</div><div class="t m1 x15 he y1ca ff9 fsa fc0 sc0 ls3 ws0">d</div><div class="t m1 x16 h11 y1cb ffd fs8 fc0 sc0 ls3 ws0">X</div><div class="t m1 xd2 he y1cc ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 xd3 h12 y1cd ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x5a hd y1c6 ff9 fs8 fc0 sc0 ls3 ws0">g</div><div class="t m1 xbe hf y1c7 ffa fs8 fc0 sc0 ls3 ws0">x<span class="_ _4"> </span><span class="ffb"><span class="_ _5"> </span></span>x</div><div class="t m1 xd4 he y1c8 ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 xd5 hd y1c9 ff9 fs8 fc0 sc0 ls3 ws0">h</div><div class="t m1 xc5 h11 y1ce ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc5 h11 y1cf ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc5 h11 y1d0 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc4 h11 y1ce ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc4 h11 y1cf ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc4 h11 y1d0 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc2 h16 y1d1 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 xd6 h11 y1d2 ffd fs8 fc0 sc0 ls21 ws0"></div><div class="t m1 xc1 hf y1d3 ff9 fs8 fc0 sc0 ls3 ws0">:<span class="_ _2d"> </span><span class="ffb"><span class="ffc">16</span></span></div><div class="t m1 x9 hc y1d4 ff5 fs8 fc0 sc0 ls3 ws0">The<span class="_ _3"> </span>second<span class="_ _b"> </span>term<span class="_ _b"> </span>is<span class="_ _3"> </span>the<span class="_ _b"> </span><span class="ff7">mean<span class="_"> </span>shift</span></div><div class="t m1 x27 h15 y1d5 ffa fs8 fc0 sc0 ls3 ws0">m</div><div class="t m1 xcf he y1d6 ff9 fsa fc0 sc0 ls3 ws0">h;G</div><div class="t m1 x12 hf y1d7 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="ls11"></span></div><div class="t m1 xd1 h11 y1d8 ffd fs8 fc0 sc0 ls3 ws0">P</div><div class="t m1 xd7 he y1d9 ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 xd7 h12 y1da ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x59 h15 y1db ffa fs8 fc0 sc0 ls3 ws0">x</div><div class="t m1 xd8 he y1dc ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 xd9 hd y1db ff9 fs8 fc0 sc0 ls3 ws0">g</div><div class="t m1 x43 h12 y1dd ffa fsa fc0 sc0 ls3 ws0">x<span class="ffb"></span>x</div><div class="t m1 x22 h14 y1de ff9 fsb fc0 sc0 ls3 ws0">i</div><div class="t m1 xda he y1df ff9 fsa fc0 sc0 ls3 ws0">h</div><div class="t m1 x1c h11 y1e0 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x1c h11 y1e1 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xbf h11 y1e0 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xbf h11 y1e1 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xdb h16 y1e2 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 xdc h11 y1e3 ffd fs8 fc0 sc0 ls27 ws0"></div><div class="t m1 x14 h11 y1e4 ffd fs8 fc0 sc0 ls3 ws0">P</div><div class="t m1 xdd he y1e5 ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 xdd h12 y1e6 ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x18 hd y1e7 ff9 fs8 fc0 sc0 ls3 ws0">g</div><div class="t m1 x42 h12 y1e8 ffa fsa fc0 sc0 ls3 ws0">x<span class="ffb"></span>x</div><div class="t m1 x5b h14 y1e9 ff9 fsb fc0 sc0 ls3 ws0">i</div><div class="t m1 xde he y1ea ff9 fsa fc0 sc0 ls3 ws0">h</div><div class="t m1 xd6 h11 y1eb ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xd6 h11 y1ec ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xdf h11 y1eb ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xdf h11 y1ec ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xe0 h16 y1ed ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 xd9 h11 y1ee ffd fs8 fc0 sc0 ls28 ws0"></div><div class="t m1 xe1 hf y1ef ffb fs8 fc0 sc0 ls3 ws0"><span class="_ _5"> </span><span class="ffa">x<span class="ff9">;<span class="_ _2e"> </span></span></span><span class="ffc">17</span></div><div class="t m1 x9 h8 y1f0 ff5 fs8 fc0 sc0 ls3 ws3d">i.e.,<span class="_ _8"> </span>the<span class="_ _8"> </span>difference<span class="_ _8"> </span>between<span class="_ _1"> </span>the<span class="_ _8"> </span>weighted<span class="_ _8"> </span>mean,<span class="_ _8"> </span>using<span class="_ _8"> </span>the</div><div class="t m1 x9 hd y1f1 ff5 fs8 fc0 sc0 ls3 ws0">kernel<span class="_ _11"> </span><span class="ff9">G<span class="_ _c"> </span></span><span class="ws3e">for<span class="_ _11"> </span>weights,<span class="_ _c"> </span>and<span class="_ _11"> </span></span><span class="ffa">x</span><span class="ws3f">,<span class="_ _c"> </span>the<span class="_ _11"> </span>center<span class="_ _c"> </span>of<span class="_ _11"> </span>the<span class="_ _c"> </span>kernel</span></div><div class="t m1 x9 h8 y1f2 ff5 fs8 fc0 sc0 ls3 ws3c">(window).<span class="_ _3"> </span>From<span class="_ _b"> </span>(16)<span class="_ _b"> </span>and<span class="_ _3"> </span>(17),<span class="_ _b"> </span>(15)<span class="_ _b"> </span>becomes</div><div class="t m1 x5c h10 y1f3 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x3c hf y1f4 ffb fs8 fc0 sc0 ls3 ws0">r<span class="ff9">f</span></div><div class="t m1 x57 he y1f5 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 x1f hf y1f6 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="ls11"></span></div><div class="t m1 xbb h10 y1f7 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x14 hd y1f6 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 xd7 he y1f5 ff9 fsa fc0 sc0 ls3 ws0">h;G</div><div class="t m1 x20 hf y1f6 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span></div><div class="t m1 xb0 hd y1f8 ffc fs8 fc0 sc0 ls3 ws0">2<span class="ff9">c</span></div><div class="t m1 x43 he y1f9 ff9 fsa fc0 sc0 ls3 ws0">k;d</div><div class="t m1 xdc hd y1fa ff9 fs8 fc0 sc0 ls3 ws0">h</div><div class="t m1 x1c h16 y1fb ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x1a hd y1fa ff9 fs8 fc0 sc0 ls3 ws0">c</div><div class="t m1 xbe he y1fc ff9 fsa fc0 sc0 ls3 ws0">g;d</div><div class="t m1 xd5 h15 y1f6 ffa fs8 fc0 sc0 ls3 ws0">m</div><div class="t m1 x1b he y1f5 ff9 fsa fc0 sc0 ls3 ws0">h;G</div><div class="t m1 xc2 hf y1f6 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="ff9">;<span class="_ _2f"> </span></span><span class="ffc">18</span></div><div class="t m1 x9 h8 y1fd ff5 fs8 fc0 sc0 ls3 ws0">yielding</div><div class="t m1 xba h15 y1fe ffa fs8 fc0 sc0 ls3 ws0">m</div><div class="t m1 x3f he y1ff ff9 fsa fc0 sc0 ls3 ws0">h;G</div><div class="t m1 x41 hf y200 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="lse"></span></div><div class="t m1 xe2 h10 y201 ffc fs8 fc0 sc0 ls3 ws0">1</div><div class="t m1 xe2 h10 y202 ffc fs8 fc0 sc0 ls3 ws0">2</div><div class="t m1 x28 hd y203 ff9 fs8 fc0 sc0 ls3 ws0">h</div><div class="t m1 x2f h16 y204 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 xe3 hd y200 ff9 fs8 fc0 sc0 ls3 ws0">c</div><div class="t m1 xe4 h10 y205 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x21 hf y206 ffb fs8 fc0 sc0 ls3 ws0">r<span class="ff9">f</span></div><div class="t m1 x30 he y207 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 xe0 hf y201 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span></div><div class="t m1 x1a h10 y208 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 xc5 hd y209 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 xde he y20a ff9 fsa fc0 sc0 ls3 ws0">h;G</div><div class="t m1 xd5 hf y20b ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span></div><div class="t m1 x44 hf y20c ff9 fs8 fc0 sc0 ls3 ws0">:<span class="_ _30"> </span><span class="ffb"><span class="ffc">19</span></span></div><div class="t m1 x9 h8 y20d ff5 fs8 fc0 sc0 ls3 ws40">The<span class="_ _2"> </span>expression<span class="_ _2"> </span>(19)<span class="_ _4"> </span>shows<span class="_ _2"> </span>that,<span class="_ _2"> </span>at<span class="_ _2"> </span>location<span class="_ _2"> </span><span class="ffa ws0">x</span><span class="ws41">,<span class="_ _2"> </span>the<span class="_ _4"> </span>mean<span class="_ _2"> </span>shift</span></div><div class="t m1 x9 hd y20e ff5 fs8 fc0 sc0 ls3 ws0">vector<span class="_ _5"> </span>computed<span class="_"> </span>with<span class="_ _5"> </span>kernel<span class="_ _5"> </span><span class="ff9">G<span class="_ _5"> </span></span>is<span class="_"> </span>proportional<span class="_ _5"> </span>to<span class="_ _5"> </span>the<span class="_"> </span><span class="ff7">normal-</span></div><div class="t m1 x9 hd y20f ff7 fs8 fc0 sc0 ls3 ws0">ized<span class="_"> </span><span class="ff5">density<span class="_ _b"> </span>gradient<span class="_ _3"> </span>estimate<span class="_ _b"> </span>obtained<span class="_ _b"> </span>with<span class="_ _b"> </span>kernel<span class="_ _3"> </span><span class="ff9">K<span class="_ _16"></span></span>.<span class="_ _3"> </span>The</span></div><div class="t m1 x9 h8 y210 ff5 fs8 fc0 sc0 ls3 ws0">normalization<span class="_"> </span>is<span class="_ _5"> </span>by<span class="_"> </span>the<span class="_"> </span>density<span class="_"> </span>estimate<span class="_ _5"> </span>in<span class="_"> </span><span class="ffa">x<span class="_ _4"> </span></span>computed<span class="_"> </span>with</div><div class="t m1 x9 hd y211 ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _15"> </span>kernel<span class="_ _15"> </span><span class="ff9">G</span>.<span class="_ _21"> </span>The<span class="_ _5"> </span>mean<span class="_ _21"> </span>shift<span class="_ _15"> </span>vector<span class="_ _15"> </span>thus<span class="_ _15"> </span>always<span class="_ _15"> </span>points<span class="_ _15"> </span>toward</div><div class="t m1 x9 h8 y212 ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _3"> </span>direction<span class="_ _b"> </span>of<span class="_ _3"> </span>maximum<span class="_ _3"> </span>increase<span class="_ _b"> </span>in<span class="_ _3"> </span>the<span class="_ _3"> </span>density.<span class="_ _b"> </span>This<span class="_ _3"> </span>is<span class="_ _b"> </span>a</div><div class="t m1 x9 h8 y213 ff5 fs8 fc0 sc0 ls3 ws0">more<span class="_ _2"> </span>general<span class="_ _3"> </span>formulation<span class="_ _2"> </span>of<span class="_ _3"> </span>the<span class="_ _2"> </span>property<span class="_ _3"> </span>first<span class="_ _2"> </span>remarked<span class="_ _3"> </span>by</div><div class="t m1 x9 h8 y214 ff5 fs8 fc0 sc0 ls3 ws42">Fukunaga<span class="_ _15"> </span>and<span class="_ _15"> </span>Hostetler<span class="_ _15"> </span>[20,<span class="_ _15"> </span>p.<span class="_ _15"> </span>535],<span class="_ _15"> </span>[21],<span class="_ _5"> </span>and<span class="_ _21"> </span>discussed<span class="_ _5"> </span>in<span class="_ _21"> </span>[7].</div><div class="t m1 xa h8 y215 ff5 fs8 fc0 sc0 ls3 ws43">The<span class="_ _4"> </span>relation<span class="_ _4"> </span>captured<span class="_ _2"> </span>in<span class="_ _5"> </span>(19)<span class="_ _4"> </span>is<span class="_ _2"> </span>intuitive,<span class="_ _4"> </span>the<span class="_ _4"> </span>local<span class="_ _4"> </span>mean<span class="_ _4"> </span>is</div><div class="t m1 x9 h8 y216 ff5 fs8 fc0 sc0 ls3 ws0">shifted<span class="_ _a"> </span>toward<span class="_ _a"> </span>the<span class="_ _e"> </span>region<span class="_ _e"> </span>in<span class="_ _a"> </span>which<span class="_ _a"> </span>the<span class="_ _e"> </span>majority<span class="_ _e"> </span>of<span class="_ _a"> </span>the</div><div class="t m1 xb h9 y55 ff1 fs7 fc0 sc0 ls8 ws0">COMANICIU<span class="_ _2"> </span>AND<span class="_ _4"> </span>MEER:<span class="_ _1"> </span>MEAN<span class="_ _4"> </span>SHIFT:<span class="_ _2"> </span>A<span class="_ _4"> </span>ROBUST<span class="_ _2"> </span>APPROACH<span class="_ _4"> </span>TOWARD<span class="_ _2"> </span>FEATURE<span class="_ _4"> </span>SPACE<span class="_ _2"> </span>ANALYSIS<span class="_ _31"> </span>605</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>

<div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6244dcfd6cd7507199bf81c5/bg4.jpg"><div class="t m1 x6 h8 ye5 ff5 fs8 fc0 sc0 ls3 ws0">points<span class="_"> </span>reside.<span class="_"> </span>Since<span class="_"> </span>the<span class="_"> </span>mean<span class="_"> </span>shift<span class="_"> </span>vector<span class="_"> </span>is<span class="_"> </span>aligned<span class="_ _2"> </span>with<span class="_"> </span>the</div><div class="t m1 x6 h8 y217 ff5 fs8 fc0 sc0 ls3 ws44">local<span class="_ _9"> </span>gradient<span class="_ _d"> </span>estimate,<span class="_ _9"> </span>it<span class="_ _9"> </span>can<span class="_ _d"> </span>define<span class="_ _9"> </span>a<span class="_ _9"> </span>path<span class="_ _d"> </span>leading<span class="_ _9"> </span>to<span class="_ _9"> </span>a</div><div class="t m1 x6 hc y218 ff5 fs8 fc0 sc0 ls3 ws0">stationary<span class="_ _b"> </span>point<span class="_ _1"> </span>of<span class="_ _b"> </span>the<span class="_ _b"> </span><span class="ff7">estimated<span class="_ _1"> </span></span>density.<span class="_ _b"> </span>The<span class="_ _1"> </span>modes<span class="_ _b"> </span>of<span class="_ _b"> </span>the</div><div class="t m1 x6 hc y219 ff5 fs8 fc0 sc0 ls3 ws0">density<span class="_ _3"> </span>are<span class="_ _3"> </span>such<span class="_ _b"> </span>stationary<span class="_ _3"> </span>points.<span class="_ _3"> </span>The<span class="_ _b"> </span><span class="ff7">mean<span class="_"> </span>shift<span class="_ _2"> </span>procedure</span>,</div><div class="t m1 x6 h8 y21a ff5 fs8 fc0 sc0 ls3 ws0">obtained<span class="_ _b"> </span>by<span class="_ _3"> </span>successive</div><div class="t m1 x8 h8 y21b ff6 fs8 fc0 sc0 ls3 ws0">.<span class="_ _32"> </span><span class="ff5">computation<span class="_ _3"> </span>of<span class="_ _b"> </span>the<span class="_ _b"> </span>mean<span class="_ _3"> </span>shift<span class="_ _b"> </span>vector<span class="_ _b"> </span><span class="ffa">m</span></span></div><div class="t m1 x62 he y21c ff9 fsa fc0 sc0 ls3 ws0">h;G</div><div class="t m1 xe5 hf y21d ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="ff5">,</span></div><div class="t m1 x8 hf y21e ff6 fs8 fc0 sc0 ls3 ws0">.<span class="_ _32"> </span><span class="ff5">translation<span class="_ _3"> </span>of<span class="_ _3"> </span>the<span class="_ _3"> </span>kernel<span class="_ _3"> </span>(window)<span class="_ _3"> </span><span class="ff9">G<span class="_ _f"></span><span class="ffb"><span class="ffa">x</span><span class="_ _3"> </span></span></span>by<span class="_ _3"> </span><span class="ffa">m</span></span></div><div class="t m1 xab he y21f ff9 fsa fc0 sc0 ls3 ws0">h;G</div><div class="t m1 xe6 hf y220 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">x</span><span class="ff5">,</span></div><div class="t m1 x6 h8 y221 ff5 fs8 fc0 sc0 ls3 ws0">is<span class="_ _21"> </span>guaranteed<span class="_ _21"> </span>to<span class="_ _15"> </span>converge<span class="_ _21"> </span>at<span class="_ _21"> </span>a<span class="_ _15"> </span>nearby<span class="_ _21"> </span>point<span class="_ _21"> </span>where<span class="_ _15"> </span>the<span class="_ _21"> </span>estimate</div><div class="t m1 x6 h8 y222 ff5 fs8 fc0 sc0 ls3 ws45">(11)<span class="_ _21"> </span>has<span class="_ _15"> </span>zero<span class="_ _21"> </span>gradient,<span class="_ _15"> </span>as<span class="_ _21"> </span>will<span class="_ _15"> </span>be<span class="_ _15"> </span>shown<span class="_ _21"> </span>in<span class="_ _15"> </span>the<span class="_ _21"> </span>next<span class="_ _15"> </span>section.<span class="_ _15"> </span>The</div><div class="t m1 x6 h8 y223 ff5 fs8 fc0 sc0 ls3 ws0">presence<span class="_ _1"> </span>of<span class="_ _1"> </span>the<span class="_ _1"> </span>normalization<span class="_ _1"> </span>by<span class="_ _1"> </span>the<span class="_ _b"> </span>density<span class="_ _1"> </span>estimate<span class="_ _1"> </span>is<span class="_ _1"> </span>a</div><div class="t m1 x6 h8 y224 ff5 fs8 fc0 sc0 ls3 ws0">desirable<span class="_ _5"> </span>feature.<span class="_ _5"> </span>The<span class="_ _4"> </span>regions<span class="_ _5"> </span>of<span class="_ _5"> </span>low-density<span class="_ _5"> </span>values<span class="_ _4"> </span>are<span class="_ _5"> </span>of<span class="_ _5"> </span>no</div><div class="t m1 x6 h8 y225 ff5 fs8 fc0 sc0 ls3 ws46">interest<span class="_ _15"> </span>for<span class="_ _5"> </span>the<span class="_ _15"> </span>feature<span class="_ _15"> </span>space<span class="_ _5"> </span>analysis<span class="_ _15"> </span>and,<span class="_ _5"> </span>in<span class="_ _21"> </span>such<span class="_ _5"> </span>regions,<span class="_ _15"> </span>the</div><div class="t m1 x6 h8 y226 ff5 fs8 fc0 sc0 ls3 ws47">mean<span class="_ _3"> </span>shift<span class="_ _3"> </span>steps<span class="_ _3"> </span>are<span class="_ _3"> </span>large.<span class="_ _3"> </span>Similarly,<span class="_ _3"> </span>near<span class="_ _2"> </span>local<span class="_ _3"> </span>maxima<span class="_ _b"> </span>the</div><div class="t m1 x6 h8 y227 ff5 fs8 fc0 sc0 ls3 ws0">steps<span class="_ _5"> </span>are<span class="_ _5"> </span>small<span class="_"> </span>and<span class="_ _5"> </span>the<span class="_ _5"> </span>analysis<span class="_ _5"> </span>more<span class="_ _5"> </span>refined.<span class="_"> </span>The<span class="_ _5"> </span>mean<span class="_ _5"> </span>shift</div><div class="t m1 x6 h8 y228 ff5 fs8 fc0 sc0 ls3 ws0">procedure<span class="_ _5"> </span>thus<span class="_"> </span>is<span class="_ _5"> </span>an<span class="_ _5"> </span>adaptive<span class="_"> </span>gradient<span class="_ _5"> </span>ascent<span class="_ _5"> </span>method.</div><div class="t m1 x6 h17 y229 ff3 fsc fc0 sc0 ls3 ws0">2.2<span class="_ _29"> </span>Sufficient<span class="_ _b"> </span>Condi<span class="_ _f"></span>tion<span class="_ _b"> </span>for<span class="_ _b"> </span>Convergence</div><div class="t m1 x6 hf y22a ff5 fs8 fc0 sc0 ls3 ws0">Denote<span class="_ _2"> </span>by<span class="_ _2"> </span><span class="ffb">f<span class="ffa">y</span></span></div><div class="t m1 xe7 he y22b ff9 fsa fc0 sc0 ls3 ws0">j</div><div class="t m1 x8f hf y22a ffb fs8 fc0 sc0 ls3 ws0">g</div><div class="t m1 x84 h12 y22c ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span>;<span class="ffc">2...</span></div><div class="t m1 xe8 h8 y22a ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _2"> </span>sequence<span class="_ _2"> </span>of<span class="_ _2"> </span>successive<span class="_ _2"> </span>locations<span class="_ _3"> </span>of</div><div class="t m1 x6 hd y22d ff5 fs8 fc0 sc0 ls3 ws0">the<span class="_ _b"> </span>kernel<span class="_ _3"> </span><span class="ff9">G</span><span class="ws48">,<span class="_ _b"> </span>where,<span class="_ _3"> </span>from<span class="_ _b"> </span>(17),<span class="_ _b"> </span></span></div><div class="t m1 x78 h15 y22e ffa fs8 fc0 sc0 ls3 ws0">y</div><div class="t m1 x79 h12 y22f ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x8f hf y230 ffb fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xe9 h11 y231 ffd fs8 fc0 sc0 ls3 ws0">P</div><div class="t m1 x7a he y232 ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 x7a h12 y233 ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 xea h15 y234 ffa fs8 fc0 sc0 ls3 ws0">x</div><div class="t m1 x7c he y235 ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 x93 hd y234 ff9 fs8 fc0 sc0 ls3 ws0">g</div><div class="t m1 x89 h12 y236 ffa fsa fc0 sc0 ls3 ws0">x<span class="ffb"></span>x</div><div class="t m1 xeb h14 y237 ff9 fsb fc0 sc0 ls3 ws0">i</div><div class="t m1 x7f he y238 ff9 fsa fc0 sc0 ls3 ws0">h</div><div class="t m1 xec h11 y239 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xec h11 y23a ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x8a h11 y239 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x8a h11 y23a ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xed h16 y23b ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 xee h11 y23c ffd fs8 fc0 sc0 ls27 ws0"></div><div class="t m1 xef h11 y23d ffd fs8 fc0 sc0 ls3 ws0">P</div><div class="t m1 x2 he y23e ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 x2 h12 y23f ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 xf0 hd y240 ff9 fs8 fc0 sc0 ls3 ws0">g</div><div class="t m1 xa1 h12 y241 ffa fsa fc0 sc0 ls3 ws0">x<span class="ffb"></span>x</div><div class="t m1 x71 h14 y242 ff9 fsb fc0 sc0 ls3 ws0">i</div><div class="t m1 x94 he y243 ff9 fsa fc0 sc0 ls3 ws0">h</div><div class="t m1 x92 h11 y244 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x92 h11 y245 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xf1 h11 y244 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xf1 h11 y245 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x8a h16 y246 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x93 h11 y247 ffd fs8 fc0 sc0 ls27 ws0"></div><div class="t m1 xf2 hf y248 ff9 fs8 fc0 sc0 ls3 ws0">j<span class="_ _2"> </span><span class="ffb"><span class="_ _2"> </span><span class="ffc">1</span></span>;<span class="_ _15"> </span><span class="ffc">2</span>;<span class="_ _5"> </span><span class="ffc lsc">...<span class="_ _33"> </span></span><span class="ffb"><span class="ffc">20</span></span></div><div class="t m1 x6 h8 y249 ff5 fs8 fc0 sc0 ls3 ws0">is<span class="_ _2"> </span>the<span class="_ _3"> </span>weighted<span class="_ _3"> </span>mean<span class="_ _3"> </span>at<span class="_ _2"> </span><span class="ffa">y</span></div><div class="t m1 xec he y24a ff9 fsa fc0 sc0 ls3 ws0">j</div><div class="t m1 xf3 hd y24b ff5 fs8 fc0 sc0 ls3 ws0">computed<span class="_ _3"> </span>with<span class="_ _2"> </span>kernel<span class="_ _3"> </span><span class="ff9">G<span class="_ _3"> </span></span>and<span class="_ _2"> </span><span class="ffa">y</span></div><div class="t m1 xf4 h16 y24a ffc fsa fc0 sc0 ls3 ws0">1</div><div class="t m1 x6 h8 y24c ff5 fs8 fc0 sc0 ls3 ws0">is<span class="_ _c"> </span>the<span class="_ _11"> </span>center<span class="_ _c"> </span>of<span class="_ _11"> </span>the<span class="_ _c"> </span>initial<span class="_ _11"> </span>position<span class="_ _c"> </span>of<span class="_ _11"> </span>the<span class="_ _c"> </span>kernel.<span class="_ _c"> </span>The</div><div class="t m1 x6 h8 y24d ff5 fs8 fc0 sc0 ls3 ws0">corresponding<span class="_ _c"> </span>sequence<span class="_ _c"> </span>of<span class="_ _c"> </span>density<span class="_ _11"> </span>estimates<span class="_ _c"> </span>computed</div><div class="t m1 x6 hf y24e ff5 fs8 fc0 sc0 ls3 ws0">with<span class="_ _b"> </span>kernel<span class="_ _3"> </span><span class="ff9">K<span class="_ _16"></span></span>,<span class="_ _3"> </span><span class="ffb">f</span></div><div class="t m1 x3 h10 y24f ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x63 hd y24e ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 x91 he y250 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 xf5 hf y251 ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">j</span>g</div><div class="t m1 x75 h12 y252 ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span>;<span class="ffc">2...</span></div><div class="t m1 x85 h8 y251 ff5 fs8 fc0 sc0 ls3 ws26">,<span class="_ _3"> </span>is<span class="_ _b"> </span>given<span class="_ _b"> </span>by</div><div class="t m1 x83 h10 y253 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 xe7 hd y254 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 xf6 he y255 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 xef hf y256 ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">j</span><span class="ls11"></span></div><div class="t m1 xf7 h10 y257 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x7c hd y256 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 x65 he y255 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 xf8 hf y256 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">y</span></div><div class="t m1 x68 he y258 ff9 fsa fc0 sc0 ls3 ws0">j</div><div class="t m1 x85 hf y256 ffb fs8 fc0 sc0 ls3 ws0"><span class="_ _34"> </span><span class="ff9">j<span class="_ _2"> </span></span><span class="_ _2"> </span><span class="ffc">1<span class="ff9">;<span class="_ _15"> </span></span><span class="ls29">2...</span><span class="ff9">:<span class="_ _2f"> </span></span></span><span class="ffc">21</span></div><div class="t m1 x6 hd y259 ff5 fs8 fc0 sc0 ls3 ws49">As<span class="_ _b"> </span>stated<span class="_ _b"> </span>by<span class="_ _b"> </span>the<span class="_ _b"> </span>following<span class="_ _1"> </span>theorem,<span class="_ _b"> </span>a<span class="_ _b"> </span>kernel<span class="_ _b"> </span><span class="ff9 ws0">K<span class="_ _1"> </span><span class="ff5">that<span class="_ _b"> </span>obeys</span></span></div><div class="t m1 x6 h8 y25a ff5 fs8 fc0 sc0 ls3 ws0">some<span class="_ _9"> </span>mild<span class="_ _9"> </span>conditions<span class="_ _9"> </span>suffices<span class="_ _d"> </span>for<span class="_ _9"> </span>the<span class="_ _9"> </span>convergence<span class="_ _9"> </span>of<span class="_ _9"> </span>the</div><div class="t m1 x6 hf y25b ff5 fs8 fc0 sc0 ls3 ws0">sequences<span class="_ _b"> </span><span class="ffb">f<span class="ffa">y</span></span></div><div class="t m1 xe7 he y25c ff9 fsa fc0 sc0 ls3 ws0">j</div><div class="t m1 x8f hf y25d ffb fs8 fc0 sc0 ls3 ws0">g</div><div class="t m1 x84 h12 y25e ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span>;<span class="ffc">2...</span></div><div class="t m1 xe8 hf y25d ff5 fs8 fc0 sc0 ls3 ws0">and<span class="_ _3"> </span><span class="ffb">f</span></div><div class="t m1 x70 h10 y25f ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 xf8 hd y25d ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 x67 he y260 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 xf9 hf y25d ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">j</span>g</div><div class="t m1 x8d h12 y25e ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span>;<span class="ffc">2...</span></div><div class="t m1 x8b h8 y25d ff5 fs8 fc0 sc0 ls3 ws0">.</div><div class="t m1 x6 hd y261 ffe fs8 fc0 sc0 ls3 ws0">Theorem<span class="_ _b"> </span>1.<span class="_ _b"> </span><span class="ff7">If<span class="_ _d"> </span>the<span class="_ _d"> </span>kernel<span class="_ _12"> </span><span class="ff9">K<span class="_ _a"> </span></span>has<span class="_ _d"> </span>a<span class="_ _d"> </span>convex<span class="_ _d"> </span>and<span class="_ _12"> </span>monotonically</span></div><div class="t m1 x8 hc y262 ff7 fs8 fc0 sc0 ls2a ws0">decreasing<span class="_ _35"> </span>profile,<span class="_ _35"> </span>the<span class="_ _35"> </span>sequences<span class="_ _36"> </span><span class="ffa ls3">y</span></div><div class="t m1 x61 he y263 ff9 fsa fc0 sc0 ls3 ws0">j</div><div class="t m1 xfa h11 y264 ffd fs8 fc0 sc0 ls2b ws0"></div><div class="t m1 xfb h12 y265 ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span>;<span class="ffc">2...</span></div><div class="t m1 xe6 hc y266 ff7 fs8 fc0 sc0 ls2c ws0">and</div><div class="t m1 x8 hf y267 ffb fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 xfc h10 y268 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x7 hd y267 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 xfd he yb1 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 xfe hf y269 ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">j</span>g</div><div class="t m1 xff h12 y26a ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span>;<span class="ffc">2...</span></div><div class="t m1 x100 hf y269 ff7 fs8 fc0 sc0 ls3 ws0">converge<span class="_ _13"> </span>and<span class="_ _10"> </span><span class="ffb">f</span></div><div class="t m1 x101 h10 y26b ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x9c hd y269 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 x6e he yb1 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 x102 hf y269 ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">j</span>g</div><div class="t m1 x6c h12 y26a ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span>;<span class="ffc">2...</span></div><div class="t m1 xa8 hc y269 ff7 fs8 fc0 sc0 ls3 ws0">is<span class="_ _13"> </span>monotoni-</div><div class="t m1 x8 hc y26c ff7 fs8 fc0 sc0 ls3 ws0">cally<span class="_ _d"> </span>increasing.</div><div class="t m1 x8 h8 y26d ff5 fs8 fc0 sc0 ls2d ws0">The<span class="_ _e"> </span>proof<span class="_ _11"> </span>is<span class="_ _11"> </span>given<span class="_ _11"> </span>in<span class="_ _e"> </span>the<span class="_ _11"> </span>Appendix.<span class="_ _11"> </span>The<span class="_ _e"> </span>theorem</div><div class="t m1 x6 hd y26e ff5 fs8 fc0 sc0 ls3 ws4a">generalizes<span class="_ _1"> </span>the<span class="_ _1"> </span>result<span class="_ _1"> </span>derived<span class="_ _1"> </span>differently<span class="_ _1"> </span>in<span class="_ _1"> </span>[13],<span class="_ _b"> </span>where<span class="_ _1"> </span><span class="ff9 ws0">K</span></div><div class="t m1 x6 hd y26f ff5 fs8 fc0 sc0 ls3 ws0">was<span class="_"> </span>the<span class="_ _5"> </span>Epanechnikov<span class="_"> </span>kernel<span class="_"> </span>and<span class="_ _5"> </span><span class="ff9">G<span class="_ _4"> </span></span>the<span class="_"> </span>uniform<span class="_"> </span>kernel.<span class="_ _5"> </span>The</div><div class="t m1 x6 h8 y270 ff5 fs8 fc0 sc0 ls3 ws0">theorem<span class="_"> </span>remains<span class="_ _4"> </span>valid<span class="_ _4"> </span>when<span class="_"> </span>each<span class="_ _5"> </span>data<span class="_"> </span>point<span class="_"> </span><span class="ffa">x</span></div><div class="t m1 xaa he y271 ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 xa8 h8 y272 ff5 fs8 fc0 sc0 ls3 ws0">is<span class="_"> </span>associated</div><div class="t m1 x6 hd y273 ff5 fs8 fc0 sc0 ls3 ws0">with<span class="_ _1"> </span>a<span class="_ _1"> </span>nonnegative<span class="_ _1"> </span>weight<span class="_ _8"> </span><span class="ff9">w</span></div><div class="t m1 xf1 he y274 ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 xf9 h8 y275 ff5 fs8 fc0 sc0 ls3 ws0">.<span class="_ _1"> </span>An<span class="_ _1"> </span>example<span class="_ _1"> </span>of<span class="_ _1"> </span>nonconver-</div><div class="t m1 x6 hd y276 ff5 fs8 fc0 sc0 ls3 ws0">gence<span class="_ _5"> </span>when<span class="_ _5"> </span>the<span class="_ _5"> </span>kernel<span class="_ _5"> </span><span class="ff9">K<span class="_ _2"> </span></span><span class="ws23">is<span class="_ _5"> </span>not<span class="_ _5"> </span>convex<span class="_ _5"> </span>is<span class="_ _5"> </span>shown<span class="_ _5"> </span>in<span class="_ _4"> </span>[10,<span class="_ _5"> </span>p.<span class="_ _5"> </span>16].</span></div><div class="t m1 x8 h8 y277 ff5 fs8 fc0 sc0 ls3 ws0">The<span class="_ _d"> </span>convergence<span class="_ _d"> </span>property<span class="_ _d"> </span>of<span class="_ _d"> </span>the<span class="_ _12"> </span>mean<span class="_ _d"> </span>shift<span class="_ _d"> </span>was<span class="_ _d"> </span>also</div><div class="t m1 x6 h8 y278 ff5 fs8 fc0 sc0 ls3 ws4b">discussed<span class="_ _15"> </span>in<span class="_ _15"> </span>[7,<span class="_ _5"> </span>Section<span class="_ _21"> </span>iv].<span class="_ _5"> </span>(Note,<span class="_ _21"> </span>however,<span class="_ _5"> </span>that<span class="_ _15"> </span>almost<span class="_ _15"> </span>all<span class="_ _15"> </span>the</div><div class="t m1 x6 h8 y279 ff5 fs8 fc0 sc0 ls3 ws0">discussion<span class="_"> </span>there<span class="_"> </span>is<span class="_"> </span>concerned<span class="_ _2"> </span>with<span class="_"> </span>the<span class="_"> </span>ªblurringº<span class="_"> </span>process<span class="_ _2"> </span>in</div><div class="t m1 x6 h8 y27a ff5 fs8 fc0 sc0 ls3 ws0">which<span class="_ _5"> </span>the<span class="_"> </span>input<span class="_ _5"> </span>is<span class="_ _5"> </span>recursively<span class="_ _5"> </span>modified<span class="_"> </span>after<span class="_ _5"> </span>each<span class="_ _5"> </span>mean<span class="_ _5"> </span>shift</div><div class="t m1 x6 h8 y27b ff5 fs8 fc0 sc0 ls3 ws0">step.)<span class="_ _b"> </span>The<span class="_ _1"> </span>convergence<span class="_ _b"> </span>of<span class="_ _b"> </span>the<span class="_ _1"> </span>procedure<span class="_ _b"> </span>as<span class="_ _b"> </span>defined<span class="_ _1"> </span>in<span class="_ _b"> </span>this</div><div class="t m1 x6 h8 y27c ff5 fs8 fc0 sc0 ls3 ws0">paper<span class="_ _21"></span>was<span class="_ _21"></span>attributed<span class="_ _21"></span>in<span class="_ _21"> </span>[7]<span class="_ _21"></span>to<span class="_ _21"> </span>the<span class="_ _21"> </span>gradient<span class="_ _21"> </span>ascent<span class="_ _21"></span>nature<span class="_ _21"> </span>of<span class="_ _21"> </span>(19).</div><div class="t m1 x6 h8 y27d ff5 fs8 fc0 sc0 ls3 ws4c">However,<span class="_ _15"> </span>as<span class="_ _5"> </span>shown<span class="_ _15"> </span>in<span class="_ _15"> </span>[4,<span class="_ _5"> </span>Section<span class="_ _15"> </span>1.2],<span class="_ _15"> </span>moving<span class="_ _5"> </span>in<span class="_ _15"> </span>the<span class="_ _5"> </span>direction</div><div class="t m1 x6 h8 y27e ff5 fs8 fc0 sc0 ls2e ws0">of<span class="_ _e"> </span>the<span class="_ _e"> </span>local<span class="_ _e"> </span>gradient<span class="_ _e"> </span>guarantees<span class="_ _e"> </span>co<span class="_ _f"></span>nvergence<span class="_ _e"> </span>only<span class="_ _e"> </span>for</div><div class="t m1 x6 h8 y27f ff5 fs8 fc0 sc0 ls3 ws0">infinitesimal<span class="_ _1"> </span>steps.<span class="_ _b"> </span>The<span class="_ _1"> </span>step<span class="_ _1"> </span>size<span class="_ _b"> </span>of<span class="_ _1"> </span>a<span class="_ _1"> </span>gradient-based<span class="_ _b"> </span>algo-</div><div class="t m1 x6 h8 y280 ff5 fs8 fc0 sc0 ls3 ws0">rithm<span class="_"> </span>is<span class="_ _5"> </span>crucial<span class="_"> </span>for<span class="_ _5"> </span>the<span class="_"> </span>overall<span class="_ _5"> </span>performance.<span class="_"> </span>If<span class="_ _5"> </span>the<span class="_"> </span>step<span class="_ _5"> </span>size<span class="_"> </span>is</div><div class="t m1 x6 h8 y281 ff5 fs8 fc0 sc0 ls3 ws4d">too<span class="_ _21"></span>large,<span class="_ _21"></span>the<span class="_ _21"></span>algorithm<span class="_ _21"></span>will<span class="_ _21"></span>diverge,<span class="_ _21"></span>while<span class="_ _21"> </span>if<span class="_ _21"></span>the<span class="_ _21"></span>step<span class="_ _21"></span>size<span class="_ _21"></span>is<span class="_ _21"></span>too</div><div class="t m1 x6 h8 y282 ff5 fs8 fc0 sc0 ls3 ws4e">small,<span class="_ _5"> </span>the<span class="_ _15"> </span>rate<span class="_ _5"> </span>of<span class="_ _15"> </span>convergence<span class="_ _5"> </span>may<span class="_ _5"> </span>be<span class="_ _15"> </span>very<span class="_ _5"> </span>slow.<span class="_ _15"> </span>A<span class="_ _5"> </span>number<span class="_ _5"> </span>of</div><div class="t m1 x6 h8 y283 ff5 fs8 fc0 sc0 ls3 ws0">costly<span class="_ _5"> </span>procedures<span class="_ _15"> </span>have<span class="_ _5"> </span>been<span class="_ _5"> </span>developed<span class="_ _15"> </span>for<span class="_ _5"> </span>step<span class="_ _15"> </span>size<span class="_ _5"> </span>selection</div><div class="t m1 x6 h8 y284 ff5 fs8 fc0 sc0 ls2f ws4f">[4,<span class="_ _a"> </span>p.<span class="_ _e"> </span>24].<span class="_ _e"> </span>The<span class="_ _e"> </span>guaranteed<span class="_ _a"> </span>convergence<span class="_ _e"> </span>(as<span class="_ _e"> </span>shown<span class="_ _e"> </span>by</div><div class="t m1 x6 h8 y285 ff5 fs8 fc0 sc0 ls3 ws0">Theorem<span class="_ _b"> </span>1)<span class="_ _1"> </span>is<span class="_ _b"> </span>due<span class="_ _1"> </span>to<span class="_ _b"> </span>the<span class="_ _b"> </span>adaptive<span class="_ _1"> </span>magnitude<span class="_ _b"> </span>of<span class="_ _1"> </span>the<span class="_ _b"> </span>mean</div><div class="t m1 x6 h8 y286 ff5 fs8 fc0 sc0 ls3 ws50">shift<span class="_ _1"> </span>vector,<span class="_ _1"> </span>which<span class="_ _1"> </span>also<span class="_ _8"> </span>eliminates<span class="_ _1"> </span>the<span class="_ _1"> </span>need<span class="_ _1"> </span>for<span class="_ _1"> </span>additional</div><div class="t m1 x9 h8 y287 ff5 fs8 fc0 sc0 ls3 ws0">procedures<span class="_ _2"> </span>to<span class="_ _2"> </span>chose<span class="_ _3"> </span>the<span class="_ _2"> </span>adequate<span class="_ _2"> </span>step<span class="_ _3"> </span>sizes.<span class="_ _2"> </span>This<span class="_ _2"> </span>is<span class="_ _3"> </span>a<span class="_ _2"> </span>major</div><div class="t m1 x9 h8 y288 ff5 fs8 fc0 sc0 ls3 ws0">advantage<span class="_ _5"> </span>over<span class="_"> </span>the<span class="_ _5"> </span>traditional<span class="_"> </span>gradient-based<span class="_ _5"> </span>methods.</div><div class="t m1 xa h8 y289 ff5 fs8 fc0 sc0 ls3 wsb">For<span class="_ _9"> </span>discrete<span class="_ _d"> </span>data,<span class="_ _9"> </span>the<span class="_ _9"> </span>number<span class="_ _9"> </span>of<span class="_ _d"> </span>steps<span class="_ _9"> </span>to<span class="_ _9"> </span>convergence</div><div class="t m1 x9 hd y28a ff5 fs8 fc0 sc0 ls3 ws0">depends<span class="_ _1"> </span>on<span class="_ _1"> </span>the<span class="_ _1"> </span>employed<span class="_ _8"> </span>kernel.<span class="_ _1"> </span>When<span class="_ _1"> </span><span class="ff9">G<span class="_ _1"> </span></span>is<span class="_ _8"> </span>the<span class="_ _1"> </span>uniform</div><div class="t m1 x9 h8 y28b ff5 fs8 fc0 sc0 ls3 ws51">kernel,<span class="_ _3"> </span>convergence<span class="_ _b"> </span>is<span class="_ _b"> </span>achieved<span class="_ _3"> </span>in<span class="_ _b"> </span>a<span class="_ _3"> </span>finite<span class="_ _b"> </span>number<span class="_ _3"> </span>of<span class="_ _b"> </span>steps</div><div class="t m1 x9 h8 y28c ff5 fs8 fc0 sc0 ls3 ws0">since<span class="_ _e"> </span>the<span class="_ _e"> </span>number<span class="_ _11"> </span>of<span class="_ _e"> </span>locations<span class="_ _11"> </span>generating<span class="_ _e"> </span>distinct<span class="_ _11"> </span>mean</div><div class="t m1 x9 hd y28d ff5 fs8 fc0 sc0 ls3 ws52">values<span class="_ _d"> </span>is<span class="_ _d"> </span>finite.<span class="_ _d"> </span>However,<span class="_ _d"> </span>when<span class="_ _d"> </span>the<span class="_ _d"> </span>kernel<span class="_ _d"> </span><span class="ff9 ws0">G<span class="_ _12"> </span><span class="ff5">imposes<span class="_ _d"> </span>a</span></span></div><div class="t m1 x9 h8 y28e ff5 fs8 fc0 sc0 ls3 ws0">weighting<span class="_ _12"> </span>on<span class="_ _12"> </span>the<span class="_ _12"> </span>data<span class="_ _12"> </span>points<span class="_ _a"> </span>(according<span class="_ _12"> </span>to<span class="_ _12"> </span>the<span class="_ _12"> </span>distance</div><div class="t m1 x9 h8 y28f ff5 fs8 fc0 sc0 ls3 ws53">from<span class="_ _11"> </span>its<span class="_ _c"> </span>center),<span class="_ _e"> </span>the<span class="_ _c"> </span>mean<span class="_ _11"> </span>shift<span class="_ _11"> </span>procedure<span class="_ _c"> </span>is<span class="_ _e"> </span>infinitely</div><div class="t m1 x9 h8 y290 ff5 fs8 fc0 sc0 ls3 ws0">convergent.<span class="_ _2"> </span>The<span class="_ _2"> </span>practical<span class="_ _3"> </span>way<span class="_ _2"> </span>to<span class="_ _2"> </span>stop<span class="_ _3"> </span>the<span class="_ _2"> </span>iterations<span class="_ _3"> </span>is<span class="_ _2"> </span>to<span class="_ _2"> </span>set</div><div class="t m1 x9 h8 y291 ff5 fs8 fc0 sc0 ls3 ws0">a<span class="_ _3"> </span>lower<span class="_ _b"> </span>bound<span class="_ _b"> </span>for<span class="_ _3"> </span>the<span class="_ _b"> </span>magnitude<span class="_ _b"> </span>of<span class="_ _3"> </span>the<span class="_ _b"> </span>mean<span class="_ _b"> </span>shift<span class="_ _3"> </span>vector.</div><div class="t m1 x9 h17 y292 ff3 fsc fc0 sc0 ls3 ws0">2.3<span class="_ _29"> </span>Mean<span class="_ _b"> </span>Shift-Based<span class="_ _b"> </span>Mod<span class="_ _f"></span>e<span class="_ _b"> </span>Detection</div><div class="t m1 x9 h8 y293 ff5 fs8 fc0 sc0 ls3 ws0">Let<span class="_ _9"> </span>us<span class="_ _d"> </span>denote<span class="_ _d"> </span>by<span class="_ _d"> </span><span class="ffa">y</span></div><div class="t m1 x103 he y294 ff9 fsa fc0 sc0 ls3 ws0">c</div><div class="t m1 x104 h8 y293 ff5 fs8 fc0 sc0 ls3 ws0">and</div><div class="t m1 x16 h10 y295 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x105 hd y293 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 x106 he y296 ff9 fsa fc0 sc0 ls3 ws0">c</div><div class="t m1 x107 he y297 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 xb2 hf y293 ffb fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x1a h10 y295 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x108 hd y293 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 xde he y298 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 x109 hf y293 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">y</span></div><div class="t m1 x4f he y294 ff9 fsa fc0 sc0 ls3 ws0">c</div><div class="t m1 x10a hf y293 ffb fs8 fc0 sc0 ls3 ws0"><span class="_ _9"> </span><span class="ff5">the<span class="_ _d"> </span>convergence</span></div><div class="t m1 x9 hf y299 ff5 fs8 fc0 sc0 ls30 ws0">points<span class="_ _11"> </span>of<span class="_ _11"> </span>the<span class="_ _11"> </span>s<span class="_ _f"></span>equences<span class="_ _11"> </span><span class="ffb ls3">f<span class="ffa">y</span></span></div><div class="t m1 xd9 he y29a ff9 fsa fc0 sc0 ls3 ws0">j</div><div class="t m1 xb2 hf y29b ffb fs8 fc0 sc0 ls3 ws0">g</div><div class="t m1 x21 h12 y29c ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span>;<span class="ffc">2...</span></div><div class="t m1 xdb hf y29b ff5 fs8 fc0 sc0 ls31 ws0">and<span class="_ _11"> </span><span class="ffb ls3">f</span></div><div class="t m1 x51 h10 y29d ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x46 hd y29b ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 x10b he y29e ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 x10c hf y29b ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">j</span>g</div><div class="t m1 x10 h12 y29c ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span>;<span class="ffc">2...</span></div><div class="t m1 x10d h8 y29b ff5 fs8 fc0 sc0 ls3 ws0">,</div><div class="t m1 x9 h8 y29f ff5 fs8 fc0 sc0 ls3 ws0">respectively.<span class="_ _15"> </span>The<span class="_ _21"> </span>implications<span class="_ _5"> </span>of<span class="_ _21"> </span>Theorem<span class="_ _15"> </span>1<span class="_ _15"> </span>are<span class="_ _15"> </span>the<span class="_ _15"> </span>following.</div><div class="t m1 xa h8 y2a0 ff5 fs8 fc0 sc0 ls3 ws54">First,<span class="_ _4"> </span>the<span class="_ _4"> </span>magnitude<span class="_ _4"> </span>of<span class="_ _4"> </span>the<span class="_ _4"> </span>mean<span class="_ _4"> </span>shift<span class="_ _2"> </span>vector<span class="_ _5"> </span>converges<span class="_ _4"> </span>to</div><div class="t m1 x9 hd y2a1 ff5 fs8 fc0 sc0 ls3 ws55">zero.<span class="_ _4"> </span>Indeed,<span class="_ _4"> </span>from<span class="_ _2"> </span>(17)<span class="_ _4"> </span>and<span class="_ _4"> </span>(20)<span class="_ _4"> </span>the<span class="_ _2"> </span><span class="ff9 ws0">j<span class="ffc">th<span class="_ _4"> </span><span class="ff5">mean<span class="_"> </span>shift<span class="_"> </span>vector<span class="_"> </span>is</span></span></span></div><div class="t m1 xcc h15 y2a2 ffa fs8 fc0 sc0 ls3 ws0">m</div><div class="t m1 x41 he y2a3 ff9 fsa fc0 sc0 ls3 ws0">h;G</div><div class="t m1 x10e hf y2a4 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">y</span></div><div class="t m1 x15 he y2a5 ff9 fsa fc0 sc0 ls3 ws0">j</div><div class="t m1 x105 hf y2a4 ffb fs8 fc0 sc0 ls11 ws0"><span class="ffa ls3">y</span></div><div class="t m1 xb0 h12 y2a5 ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x30 hf y2a4 ffb fs8 fc0 sc0 ls3 ws0"><span class="_ _5"> </span><span class="ffa">y</span></div><div class="t m1 x10f he y2a5 ff9 fsa fc0 sc0 ls3 ws0">j</div><div class="t m1 xca hf y2a4 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffc">22</span></div><div class="t m1 x9 h8 y2a6 ff5 fs8 fc0 sc0 ls3 ws56">and,<span class="_ _4"> </span>at<span class="_ _4"> </span>the<span class="_ _2"> </span>limit,<span class="_ _4"> </span><span class="ffa ws0">m</span></div><div class="t m1 x1f he y2a7 ff9 fsa fc0 sc0 ls3 ws0">h;G</div><div class="t m1 x110 hf y2a8 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">y</span></div><div class="t m1 xd1 he y2a9 ff9 fsa fc0 sc0 ls3 ws0">c</div><div class="t m1 x111 hf y2a8 ffb fs8 fc0 sc0 ls11 ws0"><span class="ffa ls3">y</span></div><div class="t m1 x112 he y2a9 ff9 fsa fc0 sc0 ls3 ws0">c</div><div class="t m1 xd9 hf y2a8 ffb fs8 fc0 sc0 ls3 ws0"><span class="_ _4"> </span><span class="ffa">y</span></div><div class="t m1 x43 he y2a9 ff9 fsa fc0 sc0 ls3 ws0">c</div><div class="t m1 xda hf y2a8 ffb fs8 fc0 sc0 ls3 ws0"><span class="_ _2"> </span><span class="ffa">0<span class="ff5 ws57">.<span class="_ _4"> </span>In<span class="_ _4"> </span>other<span class="_ _2"> </span>words,<span class="_ _4"> </span>the</span></span></div><div class="t m1 x9 h8 y2aa ff5 fs8 fc0 sc0 ls3 ws0">gradient<span class="_ _2"> </span>of<span class="_ _3"> </span>the<span class="_ _3"> </span>density<span class="_ _2"> </span>estimate<span class="_ _3"> </span>(11)<span class="_ _3"> </span>computed<span class="_ _3"> </span>at<span class="_ _2"> </span><span class="ffa">y</span></div><div class="t m1 x55 he y2ab ff9 fsa fc0 sc0 ls3 ws0">c</div><div class="t m1 x4d h8 y2ac ff5 fs8 fc0 sc0 ls3 ws0">is<span class="_ _2"> </span>zero</div><div class="t m1 x104 hf y2ad ffb fs8 fc0 sc0 ls3 ws0">r</div><div class="t m1 x111 h10 y2ae ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x10e hd y2ad ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 x113 he y2af ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 x107 hf y2b0 ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">y</span></div><div class="t m1 xd8 he y2b1 ff9 fsa fc0 sc0 ls3 ws0">c</div><div class="t m1 x5a hf y2b0 ffb fs8 fc0 sc0 ls11 ws0"><span class="ffc ls3">0<span class="ff9">;<span class="_ _37"> </span><span class="ffb"></span></span>23<span class="ffb"></span></span></div><div class="t m1 x9 h8 y2b2 ff5 fs8 fc0 sc0 ls3 ws0">due<span class="_ _3"> </span>to<span class="_ _b"> </span>(19).<span class="_ _b"> </span>Hence,<span class="_ _b"> </span><span class="ffa">y</span></div><div class="t m1 x110 he y2b3 ff9 fsa fc0 sc0 ls3 ws0">c</div><div class="t m1 x114 h8 y2b4 ff5 fs8 fc0 sc0 ls3 ws0">is<span class="_ _b"> </span>a<span class="_ _3"> </span>stationary<span class="_ _b"> </span>point<span class="_ _b"> </span>of</div><div class="t m1 x47 h10 y2b5 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x51 hd y2b4 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 x52 he y2b6 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 x5d h8 y2b4 ff5 fs8 fc0 sc0 ls3 ws0">.<span class="_ _b"> </span>Second,</div><div class="t m1 x9 hf y2b7 ff5 fs8 fc0 sc0 ls3 ws0">since<span class="_ _1"> </span><span class="ffb">f</span></div><div class="t m1 x115 h10 y2b8 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x116 hd y2b7 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 xb7 he y2b9 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 x117 hf y2ba ffb fs8 fc0 sc0 ls3 ws0"><span class="ff9">j</span>g</div><div class="t m1 x56 h12 y2bb ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span>;<span class="ffc">2...</span></div><div class="t m1 xcd h8 y2ba ff5 fs8 fc0 sc0 ls3 ws58">is<span class="_ _1"> </span>monotonically<span class="_ _1"> </span>increasing,<span class="_ _1"> </span>the<span class="_ _1"> </span>mean</div><div class="t m1 x9 hc y2bc ff5 fs8 fc0 sc0 ls3 ws0">shift<span class="_"> </span>iterations<span class="_"> </span>satisfy<span class="_"> </span>the<span class="_"> </span>conditions<span class="_"> </span>required<span class="_"> </span>by<span class="_ _5"> </span>the<span class="_"> </span><span class="ff7">Capture</span></div><div class="t m1 x9 hc y2bd ff7 fs8 fc0 sc0 ls3 ws0">Theorem<span class="_"> </span><span class="ff5 ws59">[4,<span class="_ _b"> </span>p.<span class="_ _b"> </span>45],<span class="_ _b"> </span>which<span class="_ _3"> </span>states<span class="_ _b"> </span>that<span class="_ _b"> </span>the<span class="_ _b"> </span>trajectories<span class="_ _b"> </span>of<span class="_ _b"> </span>such</span></div><div class="t m1 x9 h8 y2be ff5 fs8 fc0 sc0 ls3 ws0">gradient<span class="_ _3"> </span>methods<span class="_ _b"> </span>are<span class="_ _3"> </span>attracted<span class="_ _b"> </span>by<span class="_ _b"> </span>local<span class="_ _3"> </span>maxima<span class="_ _b"> </span>if<span class="_ _3"> </span>they<span class="_ _b"> </span>are</div><div class="t m1 x9 h8 y2bf ff5 fs8 fc0 sc0 ls3 ws0">unique<span class="_ _12"> </span>(within<span class="_ _a"> </span>a<span class="_ _12"> </span>small<span class="_ _a"> </span>neighborhood)<span class="_ _a"> </span>stationary<span class="_ _12"> </span>points.</div><div class="t m1 x9 h8 y2c0 ff5 fs8 fc0 sc0 ls3 ws5a">That<span class="_ _b"> </span>is,<span class="_ _b"> </span>once<span class="_ _b"> </span><span class="ffa ws0">y</span></div><div class="t m1 xcf he y2c1 ff9 fsa fc0 sc0 ls3 ws0">j</div><div class="t m1 x118 h8 y2c2 ff5 fs8 fc0 sc0 ls3 ws0">gets<span class="_ _b"> </span>sufficiently<span class="_ _b"> </span>close<span class="_ _b"> </span>to<span class="_ _1"> </span>a<span class="_ _b"> </span>mode<span class="_ _b"> </span>of</div><div class="t m1 x119 h10 y2c3 ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x4d hd y2c2 ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 x4e he y2c4 ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 x34 h8 y2c2 ff5 fs8 fc0 sc0 ls3 ws5b">,<span class="_ _b"> </span>it</div><div class="t m1 x9 h8 y2c5 ff5 fs8 fc0 sc0 ls3 ws0">converges<span class="_ _3"> </span>to<span class="_ _3"> </span>it.<span class="_ _b"> </span>The<span class="_ _3"> </span>set<span class="_ _3"> </span>of<span class="_ _b"> </span>all<span class="_ _3"> </span>locations<span class="_ _3"> </span>that<span class="_ _b"> </span>converge<span class="_ _3"> </span>to<span class="_ _3"> </span>the</div><div class="t m1 x9 hc y2c6 ff5 fs8 fc0 sc0 ls3 ws0">same<span class="_ _3"> </span>mode<span class="_ _b"> </span>defines<span class="_ _b"> </span>the<span class="_ _3"> </span><span class="ff7">basin<span class="_"> </span>of<span class="_"> </span>attraction<span class="_ _b"> </span></span>of<span class="_ _b"> </span>that<span class="_ _b"> </span>mode.</div><div class="t m1 xa h8 y2c7 ff5 fs8 fc0 sc0 ls32 ws0">The<span class="_ _c"> </span>theoretical<span class="_ _10"> </span>observations<span class="_ _c"> </span>from<span class="_ _10"> </span>above<span class="_ _10"> </span>suggest<span class="_ _10"> </span>a</div><div class="t m1 x9 h8 y2c8 ff5 fs8 fc0 sc0 ls3 ws0">practical<span class="_ _b"> </span>algorithm<span class="_ _3"> </span>for<span class="_ _b"> </span>mode<span class="_ _3"> </span>detection:</div><div class="t m1 xa h8 y2c9 ff6 fs8 fc0 sc0 ls3 ws0">.<span class="_ _32"> </span><span class="ff5">Run<span class="_ _3"> </span>the<span class="_ _b"> </span>mean<span class="_ _b"> </span>shift<span class="_ _3"> </span>procedure<span class="_ _b"> </span>to<span class="_ _b"> </span>find<span class="_ _3"> </span>the<span class="_ _b"> </span>stationary</span></div><div class="t m1 x116 h8 y2ca ff5 fs8 fc0 sc0 ls3 ws0">points<span class="_ _b"> </span>of</div><div class="t m1 x1e h10 y2cb ffc fs8 fc0 sc0 ls3 ws0">^</div><div class="t m1 x118 hd y2ca ff9 fs8 fc0 sc0 ls3 ws0">f</div><div class="t m1 x11a he y1df ff9 fsa fc0 sc0 ls3 ws0">h;K</div><div class="t m1 xcd h8 y2cc ff5 fs8 fc0 sc0 ls3 ws0">,</div><div class="t m1 xa h8 y2cd ff6 fs8 fc0 sc0 ls3 ws0">.<span class="_ _32"> </span><span class="ff5 ls33">Prune<span class="_ _10"> </span>these<span class="_ _10"> </span>points<span class="_ _10"> </span>by<span class="_ _c"> </span>ret<span class="_ _f"></span>aining<span class="_ _c"> </span>only<span class="_ _10"> </span>the<span class="_ _10"> </span>local</span></div><div class="t m1 x116 h8 y2ce ff5 fs8 fc0 sc0 ls3 ws0">maxima.</div><div class="t m1 x9 h8 y2cf ff5 fs8 fc0 sc0 ls3 ws5c">The<span class="_ _e"> </span>local<span class="_ _11"> </span>maxima<span class="_ _11"> </span>points<span class="_ _11"> </span>are<span class="_ _11"> </span>defined,<span class="_ _e"> </span>according<span class="_ _11"> </span>to<span class="_ _11"> </span>the</div><div class="t m1 x9 h8 y2d0 ff5 fs8 fc0 sc0 ls3 ws5d">Capture<span class="_ _3"> </span>Theorem,<span class="_ _3"> </span>as<span class="_ _3"> </span>unique<span class="_ _3"> </span>stationary<span class="_ _b"> </span>points<span class="_ _3"> </span>within<span class="_ _3"> </span>some</div><div class="t m1 x9 h8 y2d1 ff5 fs8 fc0 sc0 ls34 ws0">small<span class="_ _10"> </span>open<span class="_ _10"> </span>spher<span class="_ _f"></span>e.<span class="_ _10"> </span>This<span class="_ _10"> </span>prope<span class="_ _f"></span>rty<span class="_ _10"> </span>can<span class="_ _10"> </span>be<span class="_ _13"> </span>tested<span class="_ _c"> </span>by</div><div class="t m1 x9 h8 y2d2 ff5 fs8 fc0 sc0 ls3 ws0">perturbing<span class="_ _12"> </span>each<span class="_ _12"> </span>stationary<span class="_ _12"> </span>point<span class="_ _a"> </span>by<span class="_ _12"> </span>a<span class="_ _12"> </span>random<span class="_ _12"> </span>vector<span class="_ _a"> </span>of</div><div class="t m1 x9 h8 y2d3 ff5 fs8 fc0 sc0 ls3 ws0">small<span class="_ _b"> </span>norm<span class="_ _b"> </span>and<span class="_ _b"> </span>letting<span class="_ _b"> </span>the<span class="_ _b"> </span>mean<span class="_ _b"> </span>shift<span class="_ _b"> </span>procedure<span class="_ _1"> </span>converge</div><div class="t m1 x9 h8 y2d4 ff5 fs8 fc0 sc0 ls3 ws0">again.<span class="_"> </span>Should<span class="_"> </span>the<span class="_"> </span>point<span class="_ _5"> </span>of<span class="_"> </span>convergence<span class="_"> </span>be<span class="_"> </span>unchanged<span class="_"> </span>(up<span class="_"> </span>to</div><div class="t m1 x9 h8 y2d5 ff5 fs8 fc0 sc0 ls3 ws5e">a<span class="_ _3"> </span>tolerance),<span class="_ _b"> </span>the<span class="_ _b"> </span>point<span class="_ _b"> </span>is<span class="_ _3"> </span>a<span class="_ _b"> </span>local<span class="_ _b"> </span>maximum.</div><div class="t m1 x9 h17 y2d6 ff3 fsc fc0 sc0 ls3 ws0">2.4<span class="_ _29"> </span>Smooth<span class="_ _b"> </span>Trajectory<span class="_ _b"> </span>Pro<span class="_ _f"></span>perty</div><div class="t m1 x9 h8 y2d7 ff5 fs8 fc0 sc0 ls3 ws0">The<span class="_ _1"> </span>mean<span class="_ _1"> </span>shift<span class="_ _1"> </span>procedure<span class="_ _1"> </span>employing<span class="_ _1"> </span>a<span class="_ _1"> </span>normal<span class="_ _1"> </span>kernel<span class="_ _1"> </span>has</div><div class="t m1 x9 h8 y2d8 ff5 fs8 fc0 sc0 ls3 ws0">an<span class="_ _2"> </span>interesting<span class="_ _3"> </span>property.<span class="_ _2"> </span>Its<span class="_ _3"> </span>path<span class="_ _2"> </span>toward<span class="_ _2"> </span>the<span class="_ _3"> </span>mode<span class="_ _2"> </span>follows<span class="_ _3"> </span>a</div><div class="t m1 x9 h8 y2d9 ff5 fs8 fc0 sc0 ls3 ws5f">smooth<span class="_ _2"> </span>trajectory,<span class="_ _2"> </span>the<span class="_ _2"> </span>angle<span class="_ _2"> </span>between<span class="_ _2"> </span>two<span class="_ _2"> </span>consecutive<span class="_ _3"> </span>mean</div><div class="t m1 x9 h8 y2da ff5 fs8 fc0 sc0 ls3 ws0">shift<span class="_ _3"> </span>vectors<span class="_ _b"> </span>being<span class="_ _b"> </span>always<span class="_ _3"> </span>less<span class="_ _b"> </span>than<span class="_ _b"> </span>90<span class="_ _3"> </span>degrees.</div><div class="t m1 xa hd y2db ff5 fs8 fc0 sc0 ls3 ws60">Using<span class="_ _2"> </span>the<span class="_ _2"> </span>normal<span class="_ _2"> </span>kernel<span class="_ _2"> </span>(10),<span class="_ _2"> </span>the<span class="_ _2"> </span><span class="ff9 ws0">j<span class="ffc">th<span class="_ _2"> </span><span class="ff5">mean<span class="_ _2"> </span>shift<span class="_ _2"> </span>vector<span class="_ _2"> </span>is</span></span></span></div><div class="t m1 x9 h8 y2dc ff5 fs8 fc0 sc0 ls3 ws0">given<span class="_ _3"> </span>by</div><div class="t m1 xa h15 y2dd ffa fs8 fc0 sc0 ls3 ws0">m</div><div class="t m1 x11b he y2de ff9 fsa fc0 sc0 ls3 ws0">h;N</div><div class="t m1 x2c hf y2df ffb fs8 fc0 sc0 ls3 ws0"><span class="ffa">y</span></div><div class="t m1 x11c he y2e0 ff9 fsa fc0 sc0 ls3 ws0">j</div><div class="t m1 x11d hf y2df ffb fs8 fc0 sc0 ls11 ws0"><span class="ffa ls3">y</span></div><div class="t m1 x57 h12 y2e0 ff9 fsa fc0 sc0 ls3 ws0">j<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x11e hf y2df ffb fs8 fc0 sc0 ls3 ws0"><span class="_ _5"> </span><span class="ffa">y</span></div><div class="t m1 x114 he y2e0 ff9 fsa fc0 sc0 ls3 ws0">j</div><div class="t m1 x10e hf y2df ffb fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x15 h11 y2e1 ffd fs8 fc0 sc0 ls3 ws0">P</div><div class="t m1 xd2 he y2e2 ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 xd2 h12 y2e3 ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 x19 h15 y2e4 ffa fs8 fc0 sc0 ls3 ws0">x</div><div class="t m1 xe4 he y2e5 ff9 fsa fc0 sc0 ls3 ws0">i</div><div class="t m1 x1a h10 y2e4 ffc fs8 fc0 sc0 ls3 ws0">exp</div><div class="t m1 x4f h12 y2e6 ffa fsa fc0 sc0 ls3 ws0">x<span class="ffb"></span>x</div><div class="t m1 x11f h14 y2e7 ff9 fsb fc0 sc0 ls3 ws0">i</div><div class="t m1 x23 he y2e8 ff9 fsa fc0 sc0 ls3 ws0">h</div><div class="t m1 x2a h11 y2e9 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x2a h11 y2ea ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc0 h11 y2e9 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc0 h11 y2ea ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc1 h16 y2eb ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 xd5 h11 y2ec ffd fs8 fc0 sc0 ls27 ws0"></div><div class="t m1 xe2 h11 y2ed ffd fs8 fc0 sc0 ls3 ws0">P</div><div class="t m1 x18 he y2ee ff9 fsa fc0 sc0 ls3 ws0">n</div><div class="t m1 x18 h12 y2ef ff9 fsa fc0 sc0 ls3 ws0">i<span class="ffb"><span class="ffc">1</span></span></div><div class="t m1 xe4 h10 y2f0 ffc fs8 fc0 sc0 ls3 ws0">exp</div><div class="t m1 x2a h12 y2f1 ffa fsa fc0 sc0 ls3 ws0">x<span class="ffb"></span>x</div><div class="t m1 xe1 h14 y2f2 ff9 fsb fc0 sc0 ls3 ws0">i</div><div class="t m1 x4f he y2f3 ff9 fsa fc0 sc0 ls3 ws0">h</div><div class="t m1 x22 h11 y158 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 x22 h11 y2f4 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc2 h11 y158 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc2 h11 y2f4 ffd fs8 fc0 sc0 ls3 ws0"></div><div class="t m1 xc0 h16 y88 ffc fsa fc0 sc0 ls3 ws0">2</div><div class="t m1 x4a h11 y2f5 ffd fs8 fc0 sc0 ls27 ws0"></div><div class="t m1 x120 hf y2f6 ffb fs8 fc0 sc0 ls3 ws0"><span class="_ _4"> </span><span class="ffa">y</span></div><div class="t m1 x5e he y2e0 ff9 fsa fc0 sc0 ls3 ws0">j</div><div class="t m1 x121 hf y2df ff9 fs8 fc0 sc0 ls3 ws0">:<span class="_ _38"> </span><span class="ffb"><span class="ffc">24</span></span></div><div class="t m1 x9 hf y2f7 ff5 fs8 fc0 sc0 ls35 ws0">The<span class="_ _c"> </span>following<span class="_ _10"> </span>theorem<span class="_ _10"> </span>holds<span class="_ _10"> </span>true<span class="_ _10"> </span>for<span class="_ _10"> </span>all<span class="_ _10"> </span><span class="ff9 ls3">j<span class="_ _2"> </span><span class="ffb"><span class="_ _2"> </span><span class="ffc">1</span></span>;<span class="_ _5"> </span><span class="ffc">2</span>;<span class="_ _15"> </span><span class="ffc lsc">...</span><span class="ff5">,</span></span></div><div class="t m1 x9 h8 y2f8 ff5 fs8 fc0 sc0 ls3 ws0">according<span class="_ _3"> </span>to<span class="_ _b"> </span>the<span class="_ _b"> </span>proof<span class="_ _b"> </span>given<span class="_ _3"> </span>in<span class="_ _b"> </span>the<span class="_ _b"> </span>Appendix.</div><div class="t m1 xb h9 y55 ff1 fs7 fc0 sc0 ls8 ws0">606<span class="_ _14"> </span>IEEE<span class="_ _2"> </span>TRANSACTIONS<span class="_ _4"> </span>ON<span class="_ _2"> </span>PATTERN<span class="_ _4"> </span>ANALYSI<span class="_ _f"></span>S<span class="_ _4"> </span>AND<span class="_ _2"> </span>MACHINE<span class="_ _4"> </span>INTELLIGEN<span class="_ _f"></span>CE,<span class="_ _13"> </span>VOL.<span class="_ _4"> </span>24,<span class="_ _1f"> </span>NO.<span class="_ _4"> </span>5,<span class="_ _1f"> </span>MAY<span class="_ _4"> </span>2002</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>