自适应小波阈值算法matlab源码

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自适应小波阈值算法matlab源码
自适应小波阈值算法matlab源码
  • adaptive wavelet thresholding
  • mainbays.m
    826B
  • bayes.m
    253B
  • barbara.png
    173.7KB
  • MSE.m
    220B
  • sthresh.m
    277B
  • Chang.adaptive wavelet thresholding for image denoising and compression.pdf
    424.7KB
  • lena512.bmp
    257.1KB
  • PSNR.m
    444B
  • lena.png
    147.7KB
  • BaysShrink.m
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内容介绍
<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/625a4c3c92dc900e62ea6970/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/625a4c3c92dc900e62ea6970/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">1532<span class="_ _0"> </span>IEEE<span class="_"> </span>TRANSACTIONS<span class="_"> </span>ON<span class="_ _1"> </span>IMA<span class="_ _2"></span>GE<span class="_"> </span>PROCESSING,<span class="_"> </span>VOL.<span class="_"> </span>9,<span class="_"> </span>NO.<span class="_ _1"> </span>9,<span class="_"> </span>SEPTEMBER<span class="_ _1"> </span>2000</div><div class="t m0 x2 h3 y2 ff1 fs1 fc0 sc0 ls0 ws0">Adapti<span class="_ _3"></span>v<span class="_ _3"></span>e<span class="_ _4"> </span>W<span class="_ _5"></span>a<span class="_ _3"></span>velet<span class="_"> </span>Thresholding<span class="_"> </span>for<span class="_"> </span>Image<span class="_ _4"> </span>Denoising</div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls0 ws0">and<span class="_ _6"> </span>Compression</div><div class="t m0 x4 h4 y4 ff1 fs2 fc0 sc0 ls0 ws0">S.<span class="_"> </span>Grace<span class="_"> </span>Chang<span class="ff2">,<span class="_"> </span>Student<span class="_"> </span>Member<span class="_ _7"></span>,<span class="_"> </span>IEEE<span class="ff1">,<span class="_"> </span>Bin<span class="_"> </span>Y<span class="_ _7"></span>u<span class="ff2">,<span class="_"> </span>Senior<span class="_"> </span>Member<span class="_ _7"></span>,<span class="_"> </span>IEEE<span class="ff1">,<span class="_"> </span>and<span class="_"> </span>Martin<span class="_"> </span>V<span class="_ _7"></span>etterli<span class="ff2">,<span class="_"> </span>F<span class="_ _7"></span>ellow<span class="_ _3"></span>,<span class="_"> </span>IEEE</span></span></span></span></span></div><div class="t m0 x5 h5 y5 ff3 fs3 fc0 sc0 ls0 ws0">Abstract&#8212;<span class="ff4">The<span class="_ _8"> </span>first<span class="_ _9"> </span>part<span class="_ _9"> </span>of<span class="_ _8"> </span>this<span class="_ _8"> </span>paper<span class="_ _8"> </span>proposes<span class="_ _8"> </span>an<span class="_ _8"> </span>adaptive,</span></div><div class="t m0 x1 h5 y6 ff4 fs3 fc0 sc0 ls0 ws0">data-driven<span class="_"> </span>thr<span class="_ _3"></span>eshold<span class="_"> </span>for<span class="_"> </span>image<span class="_"> </span>denoising<span class="_"> </span>via<span class="_"> </span>wa<span class="_ _3"></span>velet<span class="_"> </span>soft-thresh-</div><div class="t m0 x1 h5 y7 ff4 fs3 fc0 sc0 ls0 ws0">olding.<span class="_"> </span>The<span class="_ _1"> </span>threshold<span class="_ _1"> </span>is<span class="_"> </span>deriv<span class="_ _2"></span>ed<span class="_"> </span>in<span class="_"> </span>a<span class="_ _1"> </span>Bayesian<span class="_ _1"> </span>framework,<span class="_ _1"> </span>and<span class="_"> </span>the</div><div class="t m0 x1 h5 y8 ff4 fs3 fc0 sc0 ls0 ws0">prior<span class="_ _a"> </span>used<span class="_ _a"> </span>on<span class="_ _a"> </span>the<span class="_ _a"> </span>wa<span class="_ _3"></span>velet<span class="_ _a"> </span>coefficients<span class="_ _a"> </span>is<span class="_ _a"> </span>the<span class="_"> </span>generalized<span class="_ _a"> </span>Gaussian</div><div class="t m0 x1 h5 y9 ff4 fs3 fc0 sc0 ls0 ws0">distribution<span class="_"> </span>(GGD)<span class="_ _a"> </span>widely<span class="_ _a"> </span>used<span class="_"> </span>in<span class="_ _a"> </span>image<span class="_ _a"> </span>processing<span class="_"> </span>applications.</div><div class="t m0 x1 h5 ya ff4 fs3 fc0 sc0 ls0 ws0">The<span class="_"> </span>proposed<span class="_"> </span>threshold<span class="_ _a"> </span>is<span class="_"> </span>simple<span class="_"> </span>and<span class="_ _a"> </span>closed-f<span class="_ _3"></span>orm,<span class="_ _a"> </span>and<span class="_"> </span>it<span class="_ _a"> </span>is<span class="_"> </span>adap-</div><div class="t m0 x1 h5 yb ff4 fs3 fc0 sc0 ls0 ws0">tive<span class="_"> </span>to<span class="_ _a"> </span>each<span class="_ _a"> </span>subband<span class="_"> </span>because<span class="_ _a"> </span>it<span class="_ _a"> </span>depends<span class="_ _a"> </span>on<span class="_ _a"> </span>data-driv<span class="_ _2"></span>en<span class="_ _a"> </span>estimates</div><div class="t m0 x1 h5 yc ff4 fs3 fc0 sc0 ls0 ws0">of<span class="_ _9"> </span>the<span class="_ _b"> </span>parameters.<span class="_ _b"> </span>Experimental<span class="_ _9"> </span>results<span class="_ _b"> </span>show<span class="_ _b"> </span>that<span class="_ _b"> </span>the<span class="_ _9"> </span>proposed</div><div class="t m0 x1 h5 yd ff4 fs3 fc0 sc0 ls0 ws0">method,<span class="_ _9"> </span>called<span class="_ _8"> </span><span class="ff3">BayesShrink</span>,<span class="_ _9"> </span>is<span class="_ _8"> </span>typically<span class="_ _9"> </span>within<span class="_ _8"> </span>5%<span class="_ _8"> </span>of<span class="_ _9"> </span>the<span class="_ _8"> </span>MSE</div><div class="t m0 x1 h5 ye ff4 fs3 fc0 sc0 ls0 ws0">of<span class="_ _a"> </span>the<span class="_ _b"> </span>best<span class="_ _a"> </span>soft-thresholding<span class="_ _a"> </span>benchmark<span class="_ _a"> </span>with<span class="_ _a"> </span>the<span class="_ _b"> </span>image<span class="_ _a"> </span>assumed</div><div class="t m0 x1 h5 yf ff4 fs3 fc0 sc0 ls0 ws0">known.<span class="_ _b"> </span>It<span class="_ _b"> </span>also<span class="_ _a"> </span>outperforms<span class="_ _b"> </span>Donoho<span class="_ _b"> </span>and<span class="_ _b"> </span>J<span class="_ _2"></span>ohnstone&#8217;s<span class="_ _b"> </span><span class="ff3">SureShrink</span></div><div class="t m0 x1 h5 y10 ff4 fs3 fc0 sc0 ls0 ws0">most<span class="_ _a"> </span>of<span class="_ _a"> </span>the<span class="_ _a"> </span>time.</div><div class="t m0 x6 h5 y11 ff4 fs3 fc0 sc0 ls0 ws0">The<span class="_ _c"> </span>second<span class="_ _c"> </span>part<span class="_ _c"> </span>of<span class="_ _c"> </span>the<span class="_ _c"> </span>paper<span class="_ _c"> </span>attempts<span class="_ _c"> </span>to<span class="_ _c"> </span>further<span class="_ _d"> </span>validate</div><div class="t m0 x1 h5 y12 ff4 fs3 fc0 sc0 ls0 ws0">recent<span class="_ _9"> </span>claims<span class="_ _9"> </span>that<span class="_ _9"> </span>lossy<span class="_ _9"> </span>compression<span class="_ _9"> </span>can<span class="_ _9"> </span>be<span class="_ _8"> </span>used<span class="_ _9"> </span>for<span class="_ _9"> </span>denoising.</div><div class="t m0 x1 h5 y13 ff4 fs3 fc0 sc0 ls0 ws0">The<span class="_ _e"> </span><span class="ff3">BayesShrink<span class="_ _e"> </span></span>threshold<span class="_ _e"> </span>can<span class="_ _e"> </span>aid<span class="_ _e"> </span>in<span class="_ _e"> </span>the<span class="_ _e"> </span>parameter<span class="_ _e"> </span>selection</div><div class="t m0 x1 h5 y14 ff4 fs3 fc0 sc0 ls0 ws0">of<span class="_ _e"> </span>a<span class="_ _e"> </span>coder<span class="_ _c"> </span>designed<span class="_ _e"> </span>with<span class="_ _e"> </span>the<span class="_ _e"> </span>intention<span class="_ _e"> </span>of<span class="_ _e"> </span>denoising,<span class="_ _c"> </span>and<span class="_ _e"> </span>thus</div><div class="t m0 x1 h5 y15 ff4 fs3 fc0 sc0 ls0 ws0">achieving<span class="_ _9"> </span>simultaneous<span class="_ _8"> </span>denoising<span class="_ _8"> </span>and<span class="_ _8"> </span>compression.<span class="_ _9"> </span>Specifically<span class="_ _3"></span>,</div><div class="t m0 x1 h5 y16 ff4 fs3 fc0 sc0 ls0 ws0">the<span class="_"> </span>zero-zone<span class="_"> </span>in<span class="_ _1"> </span>the<span class="_"> </span>quantization<span class="_"> </span>step<span class="_"> </span>of<span class="_ _1"> </span>compression<span class="_"> </span>is<span class="_ _1"> </span>analogous</div><div class="t m0 x1 h5 y17 ff4 fs3 fc0 sc0 ls0 ws0">to<span class="_"> </span>the<span class="_"> </span>thr<span class="_ _3"></span>eshold<span class="_"> </span>value<span class="_"> </span>in<span class="_"> </span>the<span class="_"> </span>thr<span class="_ _3"></span>esholding<span class="_"> </span>function.<span class="_"> </span>The<span class="_"> </span>r<span class="_ _2"></span>emaining</div><div class="t m0 x1 h5 y18 ff4 fs3 fc0 sc0 ls0 ws0">coder<span class="_ _a"> </span>design<span class="_ _b"> </span>parameters<span class="_ _a"> </span>ar<span class="_ _2"></span>e<span class="_ _a"> </span>chosen<span class="_ _b"> </span>based<span class="_ _a"> </span>on<span class="_ _a"> </span>a<span class="_ _a"> </span>criterion<span class="_ _b"> </span>deriv<span class="_ _2"></span>ed</div><div class="t m0 x1 h5 y19 ff4 fs3 fc0 sc0 ls0 ws0">from<span class="_ _8"> </span>Rissanen&#8217;s<span class="_ _e"> </span>minimum<span class="_ _8"> </span>description<span class="_ _f"> </span>length<span class="_ _f"> </span>(MDL)<span class="_ _8"> </span>principle.</div><div class="t m0 x1 h5 y1a ff4 fs3 fc0 sc0 ls0 ws0">Experiments<span class="_ _b"> </span>show<span class="_ _a"> </span>that<span class="_ _b"> </span>this<span class="_ _a"> </span>compression<span class="_ _b"> </span>method<span class="_ _a"> </span>does<span class="_ _b"> </span>indeed<span class="_ _b"> </span>r<span class="_ _2"></span>e-</div><div class="t m0 x1 h5 y1b ff4 fs3 fc0 sc0 ls0 ws0">move<span class="_ _1"> </span>noise<span class="_ _10"> </span>significantly<span class="_ _3"></span>,<span class="_ _10"> </span>especially<span class="_"> </span>f<span class="_ _3"></span>or<span class="_"> </span>large<span class="_ _10"> </span>noise<span class="_"> </span>po<span class="_ _2"></span>wer<span class="_ _7"></span>.<span class="_"> </span>Ho<span class="_ _2"></span>wever<span class="_ _7"></span>,</div><div class="t m0 x1 h5 y1c ff4 fs3 fc0 sc0 ls0 ws0">it<span class="_"> </span>introduces<span class="_"> </span>quantization<span class="_"> </span>noise<span class="_ _a"> </span>and<span class="_"> </span>should<span class="_"> </span>be<span class="_"> </span>used<span class="_ _a"> </span>only<span class="_"> </span>if<span class="_"> </span>bitrate</div><div class="t m0 x1 h5 y1d ff4 fs3 fc0 sc0 ls0 ws0">were<span class="_"> </span>an<span class="_ _a"> </span>additional<span class="_"> </span>concern<span class="_"> </span>to<span class="_ _a"> </span>denoising<span class="_ _2"></span>.</div><div class="t m0 x6 h5 y1e ff3 fs3 fc0 sc0 ls0 ws0">Index<span class="_ _a"> </span>T<span class="_ _7"></span>erms&#8212;<span class="ff4">Adaptive<span class="_ _a"> </span>method,<span class="_"> </span>image<span class="_ _a"> </span>compression,<span class="_ _a"> </span>image<span class="_"> </span>de-</span></div><div class="t m0 x1 h5 y1f ff4 fs3 fc0 sc0 ls0 ws0">noising,<span class="_"> </span>image<span class="_ _a"> </span>r<span class="_ _2"></span>estoration,<span class="_"> </span>wavelet<span class="_"> </span>thresholding.</div><div class="t m0 x7 h6 y20 ff1 fs4 fc0 sc0 ls0 ws0">I.<span class="_ _c"> </span>I<span class="fs5">NTRODUCTION</span></div><div class="t m0 x1 h7 y21 ff4 fs6 fc0 sc0 ls0 ws0">A</div><div class="t m0 x8 h6 y22 ff1 fs4 fc0 sc0 ls0 ws0">N<span class="_"> </span>IMA<span class="_ _3"></span>GE<span class="_"> </span>is<span class="_ _1"> </span>often<span class="_"> </span>corrupted<span class="_"> </span>by<span class="_ _1"> </span>noise<span class="_"> </span>in<span class="_"> </span>its<span class="_ _1"> </span>acquisition<span class="_"> </span>or</div><div class="t m0 x8 h6 y23 ff1 fs4 fc0 sc0 ls0 ws0">transmission.<span class="_ _1"> </span>The<span class="_ _1"> </span>goal<span class="_ _1"> </span>of<span class="_ _1"> </span>denoising<span class="_ _1"> </span>is<span class="_ _1"> </span>to<span class="_"> </span>remo<span class="_ _3"></span>ve<span class="_ _1"> </span>the<span class="_ _1"> </span>noise</div><div class="t m0 x1 h6 y24 ff1 fs4 fc0 sc0 ls0 ws0">while<span class="_ _9"> </span>retaining<span class="_ _9"> </span>as<span class="_ _8"> </span>much<span class="_ _9"> </span>as<span class="_ _9"> </span>possible<span class="_ _8"> </span>the<span class="_ _9"> </span>important<span class="_ _8"> </span>signal<span class="_ _9"> </span>fea-</div><div class="t m0 x1 h6 y25 ff1 fs4 fc0 sc0 ls0 ws0">tures.<span class="_ _b"> </span>Traditionally<span class="_ _7"></span>,<span class="_ _9"> </span>this<span class="_ _b"> </span>is<span class="_ _9"> </span>achie<span class="_ _2"></span>ved<span class="_ _b"> </span>by<span class="_ _b"> </span>linear<span class="_ _9"> </span>processing<span class="_ _b"> </span>such</div><div class="t m0 x1 h6 y26 ff1 fs4 fc0 sc0 ls0 ws0">as<span class="_ _9"> </span>W<span class="_ _3"></span>iener<span class="_ _9"> </span>filtering.<span class="_ _9"> </span>A<span class="_ _9"> </span>vast<span class="_ _b"> </span>literature<span class="_ _9"> </span>has<span class="_ _9"> </span>emerged<span class="_ _9"> </span>recently<span class="_ _b"> </span>on</div><div class="t m0 x9 h8 y27 ff1 fs5 fc0 sc0 ls0 ws0">Manuscript<span class="_ _b"> </span>received<span class="_ _b"> </span>January<span class="_ _9"> </span>22,<span class="_ _b"> </span>1998;<span class="_ _9"> </span>revised<span class="_ _b"> </span>April<span class="_ _b"> </span>7,<span class="_ _9"> </span>2000.<span class="_ _9"> </span>This<span class="_ _b"> </span>work</div><div class="t m0 x1 h8 y28 ff1 fs5 fc0 sc0 ls0 ws0">was<span class="_ _f"> </span>supported<span class="_ _e"> </span>in<span class="_ _f"> </span>part<span class="_ _e"> </span>by<span class="_ _e"> </span>the<span class="_ _f"> </span>NSF<span class="_ _e"> </span>Graduate<span class="_ _f"> </span>Fellowship<span class="_ _f"> </span>and<span class="_ _e"> </span>the<span class="_ _e"> </span>Univ<span class="_ _2"></span>er-</div><div class="t m0 x1 h8 y29 ff1 fs5 fc0 sc0 ls0 ws0">sity<span class="_ _c"> </span>of<span class="_ _c"> </span>California<span class="_ _c"> </span>Dissertation<span class="_ _c"> </span>Fellowship<span class="_ _c"> </span>to<span class="_ _c"> </span>S.<span class="_ _c"> </span>G.<span class="_ _c"> </span>Chang;<span class="_ _c"> </span>ARO<span class="_ _11"> </span>Grant</div><div class="t m0 x1 h8 y2a ff1 fs5 fc0 sc0 ls0 ws0">D<span class="_ _2"></span>AAH04-94-G-0232<span class="_"> </span>and<span class="_ _12"> </span>NSF<span class="_"> </span>Grant<span class="_ _12"> </span>DMS-9322817<span class="_"> </span>to<span class="_ _12"> </span>B.<span class="_"> </span>Y<span class="_ _3"></span>u;<span class="_"> </span>and<span class="_ _12"> </span>NSF<span class="_"> </span>Grant</div><div class="t m0 x1 h8 y2b ff1 fs5 fc0 sc0 ls0 ws0">MIP-93-213002<span class="_ _12"> </span>and<span class="_"> </span>Swiss<span class="_ _12"> </span>NSF<span class="_ _1"> </span>Grant<span class="_ _12"> </span>20-52347.97<span class="_ _12"> </span>to<span class="_ _12"> </span>M.<span class="_"> </span>V<span class="_ _3"></span>etterli.<span class="_"> </span>Part<span class="_"> </span>of<span class="_ _12"> </span>this</div><div class="t m0 x1 h8 y2c ff1 fs5 fc0 sc0 ls0 ws0">work<span class="_ _10"> </span>was<span class="_ _10"> </span>presented<span class="_ _10"> </span>at<span class="_ _10"> </span>the<span class="_ _10"> </span>IEEE<span class="_"> </span>International<span class="_ _10"> </span>Conference<span class="_ _10"> </span>on<span class="_ _10"> </span>Image<span class="_ _10"> </span>Processing,</div><div class="t m0 x1 h8 y2d ff1 fs5 fc0 sc0 ls0 ws0">Santa<span class="_ _e"> </span>Barbara,<span class="_ _e"> </span>CA,<span class="_ _11"> </span>October<span class="_ _e"> </span>1997.<span class="_ _e"> </span>The<span class="_ _11"> </span>associate<span class="_ _e"> </span>editor<span class="_ _e"> </span>coordinating<span class="_ _11"> </span>the</div><div class="t m0 x1 h8 y2e ff1 fs5 fc0 sc0 ls0 ws0">revie<span class="_ _3"></span>w<span class="_ _12"> </span>of<span class="_"> </span>this<span class="_ _1"> </span>manuscript<span class="_ _12"> </span>and<span class="_"> </span>approving<span class="_"> </span>it<span class="_ _12"> </span>for<span class="_"> </span>publication<span class="_"> </span>was<span class="_ _1"> </span>Prof.<span class="_ _12"> </span>Patrick<span class="_"> </span>L.</div><div class="t m0 x1 h8 y2f ff1 fs5 fc0 sc0 ls0 ws0">Combettes.</div><div class="t m0 x9 h8 y30 ff1 fs5 fc0 sc0 ls0 ws0">S.<span class="_ _13"> </span>G.<span class="_ _13"> </span>Chang<span class="_ _13"> </span>was<span class="_ _13"> </span>with<span class="_ _13"> </span>the<span class="_ _13"> </span>Department<span class="_ _13"> </span>of<span class="_ _13"> </span>Electrical<span class="_ _13"> </span>Engineering<span class="_ _13"> </span>and<span class="_ _13"> </span>Computer</div><div class="t m0 x1 h8 y31 ff1 fs5 fc0 sc0 ls0 ws0">Sciences,<span class="_"> </span>University<span class="_"> </span>of<span class="_"> </span>California,<span class="_"> </span>Berkeley<span class="_ _3"></span>,<span class="_"> </span>CA<span class="_"> </span>94720<span class="_"> </span>USA.<span class="_ _12"> </span>She<span class="_"> </span>is<span class="_"> </span>now<span class="_"> </span>with</div><div class="t m0 x1 h8 y32 ff1 fs5 fc0 sc0 ls0 ws0">Hewlett-P<span class="_ _2"></span>ackard<span class="_"> </span>Company<span class="_ _3"></span>,<span class="_"> </span>Grenoble,<span class="_"> </span>France<span class="_"> </span>(e-mail:<span class="_"> </span>grchang@yahoo.com).</div><div class="t m0 x9 h8 y33 ff1 fs5 fc0 sc0 ls0 ws0">B.<span class="_ _10"> </span>Y<span class="_ _7"></span>u<span class="_"> </span>is<span class="_ _13"> </span>with<span class="_"> </span>the<span class="_ _13"> </span>Department<span class="_"> </span>of<span class="_ _13"> </span>Statistics,<span class="_"> </span>Uni<span class="_ _3"></span>versity<span class="_"> </span>of<span class="_ _13"> </span>California,<span class="_"> </span>Berkele<span class="_ _2"></span>y<span class="_ _3"></span>,</div><div class="t m0 x1 h8 y34 ff1 fs5 fc0 sc0 ls0 ws0">CA<span class="_"> </span>94720<span class="_ _12"> </span>USA<span class="_"> </span>(e-mail:<span class="_ _12"> </span>binyu@stat.berkele<span class="_ _2"></span>y<span class="_ _3"></span>.edu)</div><div class="t m0 x9 h8 y35 ff1 fs5 fc0 sc0 ls0 ws0">M.<span class="_ _a"> </span>V<span class="_ _7"></span>etterli<span class="_ _b"> </span>is<span class="_ _a"> </span>with<span class="_ _a"> </span>the<span class="_ _b"> </span>Laboratory<span class="_ _a"> </span>of<span class="_ _a"> </span>Audiovisual<span class="_ _a"> </span>Communications,<span class="_ _a"> </span>Swiss</div><div class="t m0 x1 h8 y36 ff1 fs5 fc0 sc0 ls0 ws0">Federal<span class="_ _12"> </span>Institute<span class="_ _12"> </span>of<span class="_ _12"> </span>T<span class="_ _2"></span>echnology<span class="_ _12"> </span>(EPFL),<span class="_ _12"> </span>Lausanne,<span class="_ _12"> </span>Switzerland<span class="_ _a"> </span>and<span class="_ _12"> </span>also<span class="_ _12"> </span>with</div><div class="t m0 x1 h8 y37 ff1 fs5 fc0 sc0 ls0 ws0">the<span class="_ _10"> </span>Department<span class="_ _13"> </span>of<span class="_"> </span>Electrical<span class="_ _13"> </span>Engineering<span class="_ _10"> </span>and<span class="_ _13"> </span>Computer<span class="_"> </span>Sciences,<span class="_ _13"> </span>University<span class="_ _13"> </span>of</div><div class="t m0 x1 h8 y38 ff1 fs5 fc0 sc0 ls0 ws0">California,<span class="_ _12"> </span>Berkeley<span class="_ _7"></span>,<span class="_ _12"> </span>CA<span class="_ _12"> </span>94720<span class="_"> </span>USA.</div><div class="t m0 x9 h8 y39 ff1 fs5 fc0 sc0 ls0 ws0">Publisher<span class="_"> </span>Item<span class="_ _12"> </span>Identifier<span class="_"> </span>S<span class="_"> </span>1057-7149(00)06914-1.</div><div class="t m0 xa h6 y5 ff1 fs4 fc0 sc0 ls0 ws0">signal<span class="_ _9"> </span>denoising<span class="_ _8"> </span>using<span class="_ _8"> </span>nonlinear<span class="_ _8"> </span>techniques,<span class="_ _8"> </span>in<span class="_ _8"> </span>the<span class="_ _8"> </span>setting<span class="_ _8"> </span>of</div><div class="t m0 xa h6 y3a ff1 fs4 fc0 sc0 ls0 ws0">additiv<span class="_ _3"></span>e<span class="_ _a"> </span>white<span class="_ _a"> </span>Gaussian<span class="_"> </span>noise.<span class="_ _a"> </span>The<span class="_ _a"> </span>seminal<span class="_ _a"> </span>work<span class="_"> </span>on<span class="_ _a"> </span>signal<span class="_ _a"> </span>de-</div><div class="t m0 xa h6 y3b ff1 fs4 fc0 sc0 ls0 ws0">noising<span class="_ _b"> </span>via<span class="_ _9"> </span><span class="ff2">wavelet<span class="_ _9"> </span>thr<span class="_ _2"></span>esholding<span class="_ _9"> </span><span class="ff1">or<span class="_ _b"> </span></span>shrinkage<span class="_ _b"> </span><span class="ff1">of<span class="_ _9"> </span>Donoho<span class="_ _9"> </span>and</span></span></div><div class="t m0 xa h6 y3c ff1 fs4 fc0 sc0 ls0 ws0">Johnstone<span class="_ _a"> </span>([13]&#8211;[16])<span class="_ _b"> </span>ha<span class="_ _2"></span>ve<span class="_"> </span>shown<span class="_ _a"> </span>that<span class="_ _a"> </span>various<span class="_ _a"> </span>wav<span class="_ _2"></span>elet<span class="_ _a"> </span>thresh-</div><div class="t m0 xa h6 y3d ff1 fs4 fc0 sc0 ls0 ws0">olding<span class="_ _9"> </span>schemes<span class="_ _9"> </span>for<span class="_ _9"> </span>denoising<span class="_ _8"> </span>hav<span class="_ _2"></span>e<span class="_ _9"> </span>near-optimal<span class="_ _9"> </span>properties<span class="_ _9"> </span>in</div><div class="t m0 xa h6 y3e ff1 fs4 fc0 sc0 ls0 ws0">the<span class="_ _9"> </span>minimax<span class="_ _8"> </span>sense<span class="_ _9"> </span>and<span class="_ _8"> </span>perform<span class="_ _8"> </span>well<span class="_ _9"> </span>in<span class="_ _8"> </span>simulation<span class="_ _9"> </span>studies<span class="_ _8"> </span>of</div><div class="t m0 xa h6 y3f ff1 fs4 fc0 sc0 ls0 ws0">one-dimensional<span class="_ _9"> </span>curve<span class="_ _9"> </span>estimation.<span class="_ _9"> </span>It<span class="_ _8"> </span>has<span class="_ _9"> </span>been<span class="_ _8"> </span>shown<span class="_ _9"> </span>to<span class="_ _9"> </span>hav<span class="_ _2"></span>e</div><div class="t m0 xa h6 y40 ff1 fs4 fc0 sc0 ls0 ws0">better<span class="_ _f"> </span>rates<span class="_ _f"> </span>of<span class="_ _f"> </span>conv<span class="_ _2"></span>ergence<span class="_ _8"> </span>than<span class="_ _e"> </span>linear<span class="_ _f"> </span>methods<span class="_ _f"> </span>for<span class="_ _f"> </span>approxi-</div><div class="t m0 xa h6 y41 ff1 fs4 fc0 sc0 ls0 ws0">mating<span class="_ _a"> </span>functions<span class="_ _b"> </span>in<span class="_ _a"> </span>Besov<span class="_ _a"> </span>spaces<span class="_ _b"> </span>([13],<span class="_ _a"> </span>[14]).<span class="_ _b"> </span>Thresholding<span class="_ _a"> </span>is</div><div class="t m0 xa h6 y42 ff1 fs4 fc0 sc0 ls0 ws0">a<span class="_ _a"> </span>nonlinear<span class="_ _b"> </span>technique,<span class="_ _a"> </span>yet<span class="_ _b"> </span>it<span class="_ _a"> </span>is<span class="_ _b"> </span>very<span class="_"> </span>simple<span class="_ _b"> </span>because<span class="_ _a"> </span>it<span class="_ _b"> </span>operates</div><div class="t m0 xa h6 y43 ff1 fs4 fc0 sc0 ls0 ws0">on<span class="_ _a"> </span>one<span class="_ _a"> </span>wavelet<span class="_"> </span>coeff<span class="_ _2"></span>icient<span class="_"> </span>at<span class="_ _b"> </span>a<span class="_"> </span>time.<span class="_ _b"> </span>Alternati<span class="_ _3"></span>ve<span class="_ _a"> </span>approaches<span class="_ _b"> </span>to</div><div class="t m0 xa h6 y44 ff1 fs4 fc0 sc0 ls0 ws0">nonlinear<span class="_ _10"> </span>wa<span class="_ _2"></span>velet-based<span class="_ _13"></span>denoising<span class="_ _1"> </span>can<span class="_ _10"> </span>be<span class="_ _10"> </span>found<span class="_ _10"> </span>in,<span class="_ _10"> </span>for<span class="_ _10"> </span>example,</div><div class="t m0 xa h6 y45 ff1 fs4 fc0 sc0 ls0 ws0">[1],<span class="_ _9"> </span>[4],<span class="_ _9"> </span>[8]&#8211;[10],<span class="_ _8"> </span>[12],<span class="_ _9"> </span>[18],<span class="_ _8"> </span>[19],<span class="_ _9"> </span>[24],<span class="_ _9"> </span>[27]&#8211;[29],<span class="_ _8"> </span>[32],<span class="_ _9"> </span>[33],</div><div class="t m0 xa h6 y46 ff1 fs4 fc0 sc0 ls0 ws0">[35],<span class="_ _a"> </span>and<span class="_ _a"> </span>references<span class="_ _b"> </span>therein.</div><div class="t m0 xb h6 y47 ff1 fs4 fc0 sc0 ls0 ws0">On<span class="_"> </span>a<span class="_ _a"> </span>seemingly<span class="_ _b"> </span>unrelated<span class="_"> </span>front,<span class="_"> </span>lossy<span class="_ _b"> </span>compression<span class="_"> </span>has<span class="_ _a"> </span>been</div><div class="t m0 xa h6 y48 ff1 fs4 fc0 sc0 ls0 ws0">proposed<span class="_ _f"> </span>for<span class="_ _e"> </span>denoising<span class="_ _e"> </span>in<span class="_ _f"> </span>several<span class="_ _f"> </span>works<span class="_ _f"> </span>[6],<span class="_ _e"> </span>[5],<span class="_ _f"> </span>[21],<span class="_ _e"> </span>[25],</div><div class="t m0 xa h6 y49 ff1 fs4 fc0 sc0 ls0 ws0">[28].<span class="_ _b"> </span>Concerns<span class="_ _b"> </span>regarding<span class="_ _b"> </span>the<span class="_ _b"> </span>compression<span class="_ _b"> </span>rate<span class="_ _b"> </span>were<span class="_ _b"> </span>explicitly</div><div class="t m0 xa h6 y4a ff1 fs4 fc0 sc0 ls0 ws0">addressed.<span class="_ _b"> </span>This<span class="_ _b"> </span>is<span class="_ _b"> </span>important<span class="_ _b"> </span>because<span class="_ _a"> </span>any<span class="_ _b"> </span>practical<span class="_ _b"> </span>coder<span class="_ _b"> </span>must</div><div class="t m0 xa h6 y4b ff1 fs4 fc0 sc0 ls0 ws0">assume<span class="_ _1"> </span>a<span class="_ _1"> </span>limited<span class="_"> </span>resource<span class="_ _10"> </span>(such<span class="_"> </span>as<span class="_ _10"> </span>bits)<span class="_"> </span>at<span class="_ _10"> </span>its<span class="_"> </span>disposal<span class="_ _10"> </span>for<span class="_"> </span>repre-</div><div class="t m0 xa h6 y4c ff1 fs4 fc0 sc0 ls0 ws0">senting<span class="_"> </span>the<span class="_ _a"> </span>data.<span class="_ _a"> </span>Other<span class="_ _a"> </span>works<span class="_ _a"> </span>[4],<span class="_ _a"> </span>[12]&#8211;[16]<span class="_ _a"> </span>also<span class="_ _a"> </span>addressed<span class="_ _a"> </span>the</div><div class="t m0 xa h6 y4d ff1 fs4 fc0 sc0 ls0 ws0">connection<span class="_ _10"> </span>between<span class="_ _10"> </span>compression<span class="_ _10"> </span>and<span class="_ _1"> </span>denoising,<span class="_ _10"> </span>especially<span class="_ _10"> </span>with</div><div class="t m0 xa h6 y4e ff1 fs4 fc0 sc0 ls0 ws0">nonlinear<span class="_ _b"> </span>algorithms<span class="_ _a"> </span>such<span class="_ _b"> </span>as<span class="_ _b"> </span>wa<span class="_ _3"></span>velet<span class="_ _b"> </span>thresholding<span class="_ _b"> </span>in<span class="_ _a"> </span>a<span class="_ _b"> </span>mathe-</div><div class="t m0 xa h6 y4f ff1 fs4 fc0 sc0 ls0 ws0">matical<span class="_ _a"> </span>framework.<span class="_"> </span>Howe<span class="_ _2"></span>ver<span class="_ _3"></span>,<span class="_ _a"> </span>these<span class="_ _b"> </span>latter<span class="_ _a"> </span>works<span class="_ _a"> </span>were<span class="_ _b"> </span>not<span class="_ _a"> </span>con-</div><div class="t m0 xa h6 y50 ff1 fs4 fc0 sc0 ls0 ws0">cerned<span class="_ _1"> </span>with<span class="_"> </span>quantization<span class="_ _1"> </span>and<span class="_ _1"> </span>bitrates:<span class="_"> </span>compression<span class="_ _1"> </span>results<span class="_ _1"> </span>from</div><div class="t m0 xa h6 y51 ff1 fs4 fc0 sc0 ls0 ws0">a<span class="_ _1"> </span>reduced<span class="_ _1"> </span>number<span class="_ _1"> </span>of<span class="_"> </span>nonzero<span class="_ _10"> </span>wavelet<span class="_ _10"> </span>coeff<span class="_ _2"></span>icients,<span class="_ _1"> </span>and<span class="_ _1"> </span>not<span class="_ _1"> </span>from</div><div class="t m0 xa h6 y52 ff1 fs4 fc0 sc0 ls0 ws0">an<span class="_ _a"> </span>explicit<span class="_ _a"> </span>design<span class="_ _a"> </span>of<span class="_ _b"> </span>a<span class="_ _a"> </span>coder<span class="_ _3"></span>.</div><div class="t m0 xb h6 y53 ff1 fs4 fc0 sc0 ls0 ws0">The<span class="_ _b"> </span>intuition<span class="_ _b"> </span>behind<span class="_ _b"> </span>using<span class="_ _b"> </span>lossy<span class="_ _b"> </span>compression<span class="_ _b"> </span>for<span class="_ _b"> </span>denoising</div><div class="t m0 xa h6 y54 ff1 fs4 fc0 sc0 ls0 ws0">may<span class="_ _b"> </span>be<span class="_ _b"> </span>explained<span class="_ _b"> </span>as<span class="_ _b"> </span>follows.<span class="_ _b"> </span>A<span class="_ _b"> </span>signal<span class="_ _b"> </span>typically<span class="_ _b"> </span>has<span class="_ _b"> </span>structural</div><div class="t m0 xa h6 y55 ff1 fs4 fc0 sc0 ls0 ws0">correlations<span class="_ _1"> </span>that<span class="_ _1"> </span>a<span class="_"> </span>good<span class="_ _10"> </span>coder<span class="_"> </span>can<span class="_ _1"> </span>exploit<span class="_ _10"> </span>to<span class="_"> </span>yield<span class="_ _1"> </span>a<span class="_ _1"> </span>concise<span class="_ _1"> </span>rep-</div><div class="t m0 xa h6 y56 ff1 fs4 fc0 sc0 ls0 ws0">resentation.<span class="_ _a"> </span>White<span class="_ _b"> </span>noise,<span class="_ _a"> </span>howe<span class="_ _3"></span>ver<span class="_ _2"></span>,<span class="_ _a"> </span>does<span class="_ _b"> </span>not<span class="_ _a"> </span>have<span class="_"> </span>structural<span class="_ _b"> </span>re-</div><div class="t m0 xa h6 y57 ff1 fs4 fc0 sc0 ls0 ws0">dundancies<span class="_"> </span>and<span class="_"> </span>thus<span class="_ _a"> </span>is<span class="_ _a"> </span>not<span class="_"> </span>easily<span class="_ _a"> </span>compressable.<span class="_ _a"> </span>Hence,<span class="_"> </span>a<span class="_ _a"> </span>good</div><div class="t m0 xa h6 y58 ff1 fs4 fc0 sc0 ls0 ws0">compression<span class="_ _9"> </span>method<span class="_ _8"> </span>can<span class="_ _8"> </span>provide<span class="_ _9"> </span>a<span class="_ _8"> </span>suitable<span class="_ _8"> </span>model<span class="_ _8"> </span>for<span class="_ _8"> </span>distin-</div><div class="t m0 xa h6 y59 ff1 fs4 fc0 sc0 ls0 ws0">guishing<span class="_ _b"> </span>between<span class="_ _b"> </span>signal<span class="_ _b"> </span>and<span class="_ _9"> </span>noise.<span class="_ _b"> </span>The<span class="_ _b"> </span>discussion<span class="_ _9"> </span>will<span class="_ _b"> </span>be<span class="_ _b"> </span>re-</div><div class="t m0 xa h6 y5a ff1 fs4 fc0 sc0 ls0 ws0">stricted<span class="_ _b"> </span>to<span class="_ _9"> </span>wa<span class="_ _2"></span>velet-based<span class="_ _b"> </span>coders,<span class="_ _9"> </span>though<span class="_ _b"> </span>these<span class="_ _9"> </span>insights<span class="_ _9"> </span>can<span class="_ _b"> </span>be</div><div class="t m0 xa h6 y5b ff1 fs4 fc0 sc0 ls0 ws0">extended<span class="_"> </span>to<span class="_"> </span>other<span class="_"> </span>transform-domain<span class="_"> </span>coders<span class="_"> </span>as<span class="_"> </span>well.<span class="_"> </span>A<span class="_"> </span>concrete</div><div class="t m0 xa h6 y5c ff1 fs4 fc0 sc0 ls0 ws0">connection<span class="_ _10"> </span>between<span class="_ _1"> </span>lossy<span class="_ _10"> </span>compression<span class="_ _1"> </span>and<span class="_ _10"> </span>denoising<span class="_ _1"> </span>can<span class="_ _1"> </span>easily</div><div class="t m0 xa h6 y5d ff1 fs4 fc0 sc0 ls0 ws0">be<span class="_ _1"> </span>seen<span class="_"> </span>when<span class="_ _10"> </span>one<span class="_"> </span>examines<span class="_ _10"> </span>the<span class="_"> </span>similarity<span class="_ _1"> </span>between<span class="_ _1"> </span>thresholding</div><div class="t m0 xa h6 y5e ff1 fs4 fc0 sc0 ls0 ws0">and<span class="_ _10"> </span>quantization,<span class="_ _10"> </span>the<span class="_ _10"> </span>latter<span class="_ _1"> </span>of<span class="_ _10"> </span>which<span class="_ _10"> </span>is<span class="_ _10"> </span>a<span class="_ _1"> </span>necessary<span class="_ _10"> </span>step<span class="_ _10"> </span>in<span class="_ _10"> </span>a<span class="_ _10"> </span>prac-</div><div class="t m0 xa h6 y5f ff1 fs4 fc0 sc0 ls0 ws0">tical<span class="_ _10"> </span>lossy<span class="_ _10"> </span>coder<span class="_ _3"></span>.<span class="_ _10"> </span>That<span class="_ _1"> </span>is,<span class="_ _10"> </span>the<span class="_ _10"> </span>quantization<span class="_ _10"> </span>of<span class="_ _10"> </span>wav<span class="_ _2"></span>elet<span class="_ _10"> </span>coeff<span class="_ _3"></span>icients</div><div class="t m0 xa h6 y60 ff2 fs4 fc0 sc0 ls0 ws0">with<span class="_ _a"> </span>a<span class="_ _b"> </span>zer<span class="_ _3"></span>o-zone<span class="_ _a"> </span><span class="ff1">is<span class="_ _b"> </span>an<span class="_ _a"> </span>approximation<span class="_ _b"> </span>to<span class="_"> </span>the<span class="_ _b"> </span>thresholding<span class="_ _a"> </span>func-</span></div><div class="t m0 xa h6 y61 ff1 fs4 fc0 sc0 ls0 ws0">tion<span class="_ _b"> </span>(see<span class="_ _9"> </span>Fig.<span class="_ _9"> </span>1).<span class="_ _b"> </span>Thus,<span class="_ _9"> </span>provided<span class="_ _b"> </span>that<span class="_ _9"> </span>the<span class="_ _9"> </span>quantization<span class="_ _b"> </span>outside</div><div class="t m0 xa h6 y62 ff1 fs4 fc0 sc0 ls0 ws0">of<span class="_"> </span>the<span class="_ _1"> </span>zero-zone<span class="_ _1"> </span>does<span class="_"> </span>not<span class="_ _1"> </span>introduce<span class="_"> </span>significant<span class="_ _1"> </span>distortion,<span class="_"> </span>it<span class="_ _1"> </span>fol-</div><div class="t m0 xa h6 y63 ff1 fs4 fc0 sc0 ls0 ws0">lows<span class="_"> </span>that<span class="_"> </span>w<span class="_ _2"></span>av<span class="_ _2"></span>elet-based<span class="_"> </span>lossy<span class="_"> </span>compression<span class="_"> </span>achie<span class="_ _2"></span>ves<span class="_"> </span>denoising.</div><div class="t m0 xa h6 y64 ff1 fs4 fc0 sc0 ls0 ws0">W<span class="_ _3"></span>ith<span class="_ _1"> </span>this<span class="_ _1"> </span>connection<span class="_ _10"> </span>in<span class="_ _1"> </span>mind,<span class="_ _1"> </span>this<span class="_ _1"> </span>paper<span class="_ _10"> </span>is<span class="_ _1"> </span>about<span class="_ _1"> </span>wa<span class="_ _2"></span>velet<span class="_ _10"> </span>thresh-</div><div class="t m0 xa h6 y65 ff1 fs4 fc0 sc0 ls0 ws0">olding<span class="_ _1"> </span>for<span class="_"> </span>image<span class="_ _1"> </span>denoising<span class="_"> </span>and<span class="_ _1"> </span>also<span class="_"> </span>for<span class="_ _1"> </span>lossy<span class="_ _1"> </span>compression.<span class="_"> </span>The</div><div class="t m0 xa h6 y66 ff1 fs4 fc0 sc0 ls0 ws0">threshold<span class="_ _b"> </span>choice<span class="_ _9"> </span>aids<span class="_ _9"> </span>the<span class="_ _9"> </span>lossy<span class="_ _9"> </span>coder<span class="_ _9"> </span>to<span class="_ _9"> </span>choose<span class="_ _9"> </span>its<span class="_ _b"> </span>zero-zone,</div><div class="t m0 xa h6 y67 ff1 fs4 fc0 sc0 ls0 ws0">and<span class="_ _f"> </span>the<span class="_ _8"> </span>resulting<span class="_ _f"> </span>coder<span class="_ _f"> </span>achieves<span class="_ _8"> </span>simultaneous<span class="_ _f"> </span>denoising<span class="_ _f"> </span>and</div><div class="t m0 xa h6 y68 ff1 fs4 fc0 sc0 ls0 ws0">compression<span class="_"> </span>if<span class="_ _a"> </span>such<span class="_ _b"> </span>property<span class="_"> </span>is<span class="_ _a"> </span>desired.</div><div class="t m0 xc h8 y69 ff1 fs5 fc0 sc0 ls0 ws0">1057&#8211;7149/00$10.00<span class="_ _12"> </span>&#169;<span class="_ _12"> </span>2000<span class="_ _12"> </span>IEEE</div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div> </body> </html>
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