pca-sift.rar_pca-sift_sift-pca_图形图像处理

<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/6254db6047503a0a93cceb35/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/6254db6047503a0a93cceb35/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Color image retrieval technique with local features based</div><div class="t m0 x1 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">on orthogonal polynomials model and SIFT</div><div class="t m0 x1 h3 y3 ff1 fs1 fc0 sc0 ls0 ws0">J. Kalpana & R. Krishnamoorthi</div><div class="t m0 x1 h4 y4 ff3 fs2 fc0 sc0 ls0 ws0">Received: 27 December 2013 / Revised: 31 August 2014 /Accepted: 3 September 2014 /</div><div class="t m0 x1 h4 y5 ff3 fs2 fc0 sc0 ls0 ws0">Published online: 2 December 2014</div><div class="t m0 x1 h5 y6 ff4 fs2 fc0 sc0 ls0 ws0">#</div><div class="t m0 x2 h4 y7 ff3 fs2 fc0 sc0 ls0 ws0">Springer Science+Business Media New York 2014</div><div class="t m0 x1 h6 y8 ff1 fs1 fc0 sc0 ls0 ws0">Abstract In this paper, a new color image retrieval technique is proposed with local features</div><div class="t m0 x1 h6 y9 ff3 fs1 fc0 sc0 ls0 ws0">given by Scale Invariant Feature Transform (SIFT) key points that are described in the integer-</div><div class="t m0 x1 h6 ya ff3 fs1 fc0 sc0 ls0 ws0">natured, computationally light Orthogonal Polynomials Transform (OPT) domain. The trans-</div><div class="t m0 x1 h6 yb ff3 fs1 fc0 sc0 ls0 ws0">form’s point spread operators are derived from a generating function, modification to which</div><div class="t m0 x1 h6 yc ff3 fs1 fc0 sc0 ls0 ws0">has been proposed for alleviating computational complexity further. The expressive power of</div><div class="t m0 x1 h6 yd ff3 fs1 fc0 sc0 ls0 ws0">the transform coefficients has been exploited for forming the descriptors of SIFT key points of</div><div class="t m0 x1 h6 ye ff3 fs1 fc0 sc0 ls0 ws0">a given image. The key point descriptors, OPT-SIFT so formed have good expressive power,</div><div class="t m0 x1 h6 yf ff3 fs1 fc0 sc0 ls0 ws0">despite being shorter in length and having a reduced computational complexity. A retrieval</div><div class="t m0 x1 h6 y10 ff3 fs1 fc0 sc0 ls0 ws0">technique has been proposed based on OPT-SIFT features. The proposed retrieval technique</div><div class="t m0 x1 h6 y11 ff3 fs1 fc0 sc0 ls0 ws0">has been experimented with images from standard databases such COIL-100 and Corel and the</div><div class="t m0 x1 h6 y12 ff3 fs1 fc0 sc0 ls0 ws0">results demonstrate the superiority of the proposed descriptors when compared to other</div><div class="t m0 x1 h6 y13 ff3 fs1 fc0 sc0 ls0 ws0">descriptors.</div><div class="t m0 x1 h6 y14 ff1 fs1 fc0 sc0 ls0 ws0">Keywords Content based image retrieval (CBIR)</div><div class="t m0 x3 h6 y15 ff3 fs1 fc0 sc0 ls0 ws0">.</div><div class="t m0 x4 h6 y14 ff3 fs1 fc0 sc0 ls0 ws0">Dimensionality reduction</div><div class="t m0 x5 h6 y15 ff3 fs1 fc0 sc0 ls0 ws0">.</div><div class="t m0 x6 h6 y14 ff3 fs1 fc0 sc0 ls0 ws0">Orthogonal</div><div class="t m0 x1 h6 y16 ff3 fs1 fc0 sc0 ls0 ws0">polynomials</div><div class="t m0 x7 h6 y17 ff3 fs1 fc0 sc0 ls0 ws0">.</div><div class="t m0 x8 h6 y16 ff3 fs1 fc0 sc0 ls0 ws0">SIFT keypoint descriptors</div><div class="t m0 x1 h3 y18 ff1 fs1 fc0 sc0 ls0 ws0">1 Introduction</div><div class="t m0 x1 h6 y19 ff3 fs1 fc0 sc0 ls2 ws0">Several Content Based Image Retrieval (CBIR) techniques have been proposed based on various</div><div class="t m0 x1 h6 y1a ff3 fs1 fc0 sc0 ls4 ws0">global and local low level features. Ritendra [29] report that the trends of CBIR research have</div><div class="t m0 x1 h6 y1b ff3 fs1 fc0 sc0 ls7 ws0">switched from global features to local features, which is proved as a practical way to handle</div><div class="t m0 x1 h7 y1c ff5 fs1 fc0 sc0 ls0 ws0">“semantic gap”. Local features are more reliable for several reasons: − they isolate occlusions,</div><div class="t m0 x1 h6 y1d ff3 fs1 fc0 sc0 ls0 ws0">isolate clutter and the background and also accommodate partial appearance variation.</div><div class="t m0 x1 h6 y1e ff3 fs1 fc0 sc0 lsa ws0">Mikolajczyk et al. in [26] report that the method used for local feature identification does not</div><div class="t m0 x1 h6 y1f ff3 fs1 fc0 sc0 lsb ws0">affect the method employed to describe the local features. They also conclude that the description</div><div class="t m0 x1 h6 y20 ff3 fs1 fc0 sc0 lsd ws0">of the local features depends much on the representation used to model image patches around</div><div class="t m0 x1 h6 y21 ff3 fs1 fc0 sc0 lse ws0">interest points. This report gives scope for two independent processes - identification of local</div><div class="t m0 x1 h6 y22 ff3 fs1 fc0 sc0 ls4 ws0">features and description of the identified local features.</div><div class="t m0 x1 h4 y23 ff3 fs2 fc0 sc0 ls0 ws0">Multimed Tools Appl (2016) 75:49–69</div><div class="t m0 x1 h4 y24 ff3 fs2 fc0 sc0 ls0 ws0">DOI 10.1007/s11042-014-2262-1</div><div class="t m0 x1 h8 y25 ff3 fs2 fc0 sc0 ls11 ws0">J. Kalpana (*)</div><div class="t m0 x9 h9 y26 ff8 fs3 fc0 sc0 ls0 ws0">:</div><div class="t m0 xa h4 y25 ff3 fs2 fc0 sc0 ls0 ws0">R. Krishnamoorthi</div><div class="t m0 x1 h4 y27 ff3 fs2 fc0 sc0 ls0 ws0">Image Vision Laboratory, Dept. of CSE, Anna University of Technology, Tiruchirapalli 620 024, India</div><div class="t m0 x1 h4 y28 ff3 fs2 fc0 sc0 ls0 ws0">e-mail: kalpanalak@gmail.com</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[2.037103,0.000000,0.000000,2.037103,0.000000,0.000000]}'></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/6254db6047503a0a93cceb35/bg2.jpg"><div class="t m0 xb h6 y29 ff3 fs1 fc0 sc0 ls12 ws0">Among the techniques based on local features, SIFT [23] has been accepted as the state-of-art</div><div class="t m0 x1 h6 y2a ff3 fs1 fc0 sc0 ls0 ws0">technique. However, it is plagued by two drawbacks with respect to its descriptors - the</div><div class="t m0 x1 h6 y2b ff3 fs1 fc0 sc0 ls14 ws0">computational complexity and length of its descriptor. Many refinements to the original SIFT</div><div class="t m0 x1 h6 y2c ff3 fs1 fc0 sc0 ls15 ws0">descriptors have been proposed in order to overcome its drawbacks. For instance, a reduced-</div><div class="t m0 x1 h6 y2d ff3 fs1 fc0 sc0 ls16 ws0">length SIFT feature descriptor is possible as there is a scope for removal of redundancy within the</div><div class="t m0 x1 h6 y2e ff3 fs1 fc0 sc0 ls17 ws0">SIFT key point descriptors. Several techniques that aim at reduction in the length of the descriptor</div><div class="t m0 x1 h6 y2f ff3 fs1 fc0 sc0 lse ws0">have been attempted and one such attempt has been with the dimensionality reduction algorithm,</div><div class="t m0 x1 h6 y30 ff3 fs1 fc0 sc0 ls2 ws0">PCA, where the inherent drawback is its computational complexity [6]. Eventually, compara-</div><div class="t m0 x1 h6 y31 ff3 fs1 fc0 sc0 ls7 ws0">tively less computationally intensive transforms such as DST, DCT that closely approximate PCA</div><div class="t m0 x1 h6 y32 ff3 fs1 fc0 sc0 ls18 ws0">in removing the redundancy in the captured information. However, the computation of these</div><div class="t m0 x1 h6 y33 ff3 fs1 fc0 sc0 lse ws0">transformation matrices is still not that light. In this direction, Orthogonal Polynomials Transform</div><div class="t m0 x1 h6 y34 ff3 fs1 fc0 sc0 ls7 ws0">(OPT), with integer- natured coefficients and computational light-weightedness as its other merits</div><div class="t m0 x1 h6 y35 ff3 fs1 fc0 sc0 ls2 ws0">is considered for forming the descriptors of SIFT key points. Like in any other transform, OPT’s</div><div class="t m0 x1 h6 y36 ff3 fs1 fc0 sc0 ls19 ws0">lowest frequency transform coefficients take the greatest part of energy and corresponding</div><div class="t m0 x1 h6 y37 ff3 fs1 fc0 sc0 ls1b ws0">coefficients have particular information regarding low level features such as color , texture and</div><div class="t m0 x1 h6 y38 ff3 fs1 fc0 sc0 ls1c ws0">edge [18].</div><div class="t m0 xb h6 y39 ff3 fs1 fc0 sc0 ls0 ws0">In the proposed work, OPT-SIFT, localization of key points and description of the key</div><div class="t m0 x1 h6 y3a ff3 fs1 fc0 sc0 ls0 ws0">points are handled as two independent issues where localizing of key points is done as in</div><div class="t m0 x1 h6 y3b ff3 fs1 fc0 sc0 ls0 ws0">Lowe’s[23] implementation. The key points so identified are not described in the pixel domain</div><div class="t m0 x1 h6 y3c ff3 fs1 fc0 sc0 ls0 ws0">as in the original implementation, instead they are described in the orthogonal polynomials</div><div class="t m0 x1 h6 y3d ff3 fs1 fc0 sc0 ls0 ws0">transform domain using the transform’s low frequency coefficients. In this work, the paradigm</div><div class="t m0 x1 h6 y3e ff3 fs1 fc0 sc0 ls0 ws0">shift from the pixel- based descriptors to the transform coefficient-based descriptors is</div><div class="t m0 x1 h6 y3f ff3 fs1 fc0 sc0 ls1f ws0">attempted with a motive of reducing the length of the descriptor, taking advantage of the</div><div class="t m0 x1 h6 y40 ff3 fs1 fc0 sc0 ls0 ws0">inherent redundancy among adjoining pixels. Hence, in forming these descriptors, the basic</div><div class="t m0 x1 h6 y41 ff3 fs1 fc0 sc0 ls0 ws0">unit of processing is block instead of pixels as against the original proposal. The advantage of</div><div class="t m0 x1 h6 y42 ff3 fs1 fc0 sc0 ls0 ws0">descriptors formed in this way is that it provides a reduction in time complexity for calculating</div><div class="t m0 x1 h6 y43 ff3 fs1 fc0 sc0 ls0 ws0">the descriptors. Based on the identification of SIFT features and describing them in the OPT</div><div class="t m0 x1 h6 y44 ff3 fs1 fc0 sc0 ls0 ws0">domain, an image retrieval technique has been proposed.</div><div class="t m0 xb h6 y45 ff3 fs1 fc0 sc0 ls0 ws0">The main contributions of the work in this paper are: 1) Proposing a modification to the</div><div class="t m0 x1 h6 y46 ff3 fs1 fc0 sc0 ls0 ws0">earlier reported generating function of the orthogonal polynomials transform 2) Describing the</div><div class="t m0 x1 h6 y47 ff3 fs1 fc0 sc0 ls0 ws0">SIFT key points with block-based low level information obtained from the transform and 3)</div><div class="t m0 x1 h6 y48 ff3 fs1 fc0 sc0 ls0 ws0">Proposing an image retrieval technique based on the local features given by SIFT, described in</div><div class="t m0 x1 h6 y49 ff3 fs1 fc0 sc0 ls0 ws0">the orthogonal polynomials domain. The organization of the paper is as follows: In section 2,a</div><div class="t m0 x1 h6 y4a ff3 fs1 fc0 sc0 ls0 ws0">review of the literature is given and in section 3, the framework for orthogonal polynomials is</div><div class="t m0 x1 h6 y4b ff3 fs1 fc0 sc0 ls0 ws0">outlined and a modification suggested to the model is proposed. Section 4 describes the</div><div class="t m0 x1 h6 y4c ff3 fs1 fc0 sc0 ls0 ws0">proposed SIFT key point descriptor, OPT-SIFT and the image retrieval technique based on</div><div class="t m0 x1 h6 y4d ff3 fs1 fc0 sc0 ls0 ws0">it. In section 5, a cohesive summary of the experiments conducted is discussed along with their</div><div class="t m0 x1 h6 y4e ff3 fs1 fc0 sc0 ls0 ws0">results followed by concluding remarks in section 6.</div><div class="t m0 x1 h3 y4f ff1 fs1 fc0 sc0 ls0 ws0">2 Related works</div><div class="t m0 x1 h6 y50 ff3 fs1 fc0 sc0 ls0 ws0">SIFT has been adopted in image retrieval applications because of its capability to describe the</div><div class="t m0 x1 h6 y51 ff3 fs1 fc0 sc0 ls0 ws0">local image structures in a way that is robust against changes in viewpoint, illumination and</div><div class="t m0 x1 h6 y52 ff3 fs1 fc0 sc0 ls0 ws0">even some forms of affine distortion. It consists of two independent processes - localization of</div><div class="t m0 x1 h6 y53 ff3 fs1 fc0 sc0 ls0 ws0">key points and the description of them. Mikolajczyk et al. [26] have reported that the local</div><div class="t m0 x1 h6 y54 ff3 fs1 fc0 sc0 ls0 ws0">features given by SIFT [23] are the most robust and distinctive. The original SIFT approach</div><div class="t m0 x1 h6 y55 ff3 fs1 fc0 sc0 ls0 ws0">only uses grey scale intensities and therefore colour differences produce ambiguities. To</div><div class="t m0 x1 h6 y56 ff3 fs1 fc0 sc0 ls0 ws0">eliminate mismatches between objects of similar texture and geometry but differing colour,</div><div class="t m0 x1 h4 y23 ff3 fs2 fc0 sc0 ls21 ws0">50 Multimed T ools Appl (2016) 75:49–69</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[2.037103,0.000000,0.000000,2.037103,0.000000,0.000000]}'></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/6254db6047503a0a93cceb35/bg3.jpg"><div class="t m0 x1 h6 y29 ff3 fs1 fc0 sc0 ls0 ws0">colour information also needs to be augmented to the descriptor. Many proposals have been</div><div class="t m0 x1 h6 y2a ff3 fs1 fc0 sc0 ls0 ws0">made in this direction such as Colour SIFT [10], SIFT-CCH [8] but a majority of these</div><div class="t m0 x1 h6 y2b ff3 fs1 fc0 sc0 ls0 ws0">methods add heavily to the computational complexity of SIFT.</div><div class="t m0 xb h6 y2c ff3 fs1 fc0 sc0 lsf ws0">In addition, the computational complexity and the length of the descriptors of SIFT key points</div><div class="t m0 x1 h6 y2d ff3 fs1 fc0 sc0 ls1a ws0">prevent its application in computationally light environments such as mobile OS. T owar ds</div><div class="t m0 x1 h6 y2e ff3 fs1 fc0 sc0 ls0 ws0">alleviating the deficiencies, Bay et al. proposed Speeded-Up Robust Features (SURF) [4]</div><div class="t m0 x1 h6 y2f ff3 fs1 fc0 sc0 ls3 ws0">descriptor , the length of which is 64 dimensions but its discriminative power is not supreme as</div><div class="t m0 x1 h6 y30 ff3 fs1 fc0 sc0 ls15 ws0">reviewed by Mikolajczyk et al. in [26]. Gradient Location and Orientation Histogram (GLOH)</div><div class="t m0 x1 h6 y31 ff3 fs1 fc0 sc0 ls0 ws0">[26] has a good discriminative power but is computationally expensive. In yet another move</div><div class="t m0 x1 h6 y32 ff3 fs1 fc0 sc0 lse ws0">towards reducing the querying time, Gao et al. [9] added all the SIFT features of the image and</div><div class="t m0 x1 h6 y33 ff3 fs1 fc0 sc0 ls22 ws0">used the resultant 128 dimensional feature to represent the key point. Jone et al. [13] process only</div><div class="t m0 x1 h6 y34 ff3 fs1 fc0 sc0 ls9 ws0">partial information and determine two orientations along with only one diagonal edge. However,</div><div class="t m0 x1 h6 y35 ff3 fs1 fc0 sc0 lsa ws0">it has been observed that with both these methods, the discriminative property is less. In the event</div><div class="t m0 x1 h6 y36 ff3 fs1 fc0 sc0 lsf ws0">of failure of such approximations of the descriptors, several methods for dimensionality reduction</div><div class="t m0 x1 h6 y37 ff3 fs1 fc0 sc0 ls22 ws0">are reported based mainly on Principal Component Analysis (PCA) and Independent Component</div><div class="t m0 x1 h6 y38 ff3 fs1 fc0 sc0 ls9 ws0">Analysis (ICA). However, these techniques are time consuming [15]. Ke et al. in their work,</div><div class="t m0 x1 h6 y39 ff3 fs1 fc0 sc0 ls7 ws0">PCA-SIFT [14], PCA has been used for dimensionality reduction and the descriptor’slengthis</div><div class="t m0 x1 h6 y3a ff3 fs1 fc0 sc0 ls3 ws0">reduced from 128 to 36. However , computing eigenvectors is computationally intensive and if</div><div class="t m0 x1 h6 y3b ff3 fs1 fc0 sc0 ls4 ws0">done on-line can make the system relatively slow, in addition to being less distinctive.</div><div class="t m0 xb h6 y3c ff3 fs1 fc0 sc0 ls0 ws0">Considering the computational light-weightedness of the transforms when compared to</div><div class="t m0 x1 h6 y3d ff3 fs1 fc0 sc0 ls0 ws0">PCA, employing them is good choice from the perspective of dimensionality reduction. The</div><div class="t m0 x1 h6 y3e ff3 fs1 fc0 sc0 ls0 ws0">popular transforms include Discrete Fourier Transform (DFT) [2], Discrete Cosine Transform</div><div class="t m0 x1 h6 y3f ff3 fs1 fc0 sc0 ls0 ws0">(DCT) [1], Discrete Sine Transform (DST) [25], Wavelets Transform [24], Walsh-Hadamard</div><div class="t m0 x1 h6 y40 ff3 fs1 fc0 sc0 ls27 ws0">Transform [28] and Haar Transform [7]. Knowledge about low level features have been</div><div class="t m0 x1 h6 y41 ff3 fs1 fc0 sc0 ls0 ws0">derived from a subset of the transform coefficients. For instance, Chang et al. in their work</div><div class="t m0 x1 h6 y42 ff3 fs1 fc0 sc0 ls0 ws0">[5] have attempted to interpret the block edge (BE) patterns directly from the DCT coefficients.</div><div class="t m0 x1 h6 y43 ff3 fs1 fc0 sc0 ls0 ws0">Kim and Lee [16] have reported that the edge location is well captured by the polarities of the</div><div class="t m0 x1 h6 y44 ff3 fs1 fc0 sc0 ls0 ws0">projection of DCT coefficients. Shen and Sethi have examined the DCT coefficient patterns</div><div class="t m0 x1 h6 y45 ff3 fs1 fc0 sc0 ls0 ws0">induced from an ideal edge model in their work [30] and showed how the relative values or</div><div class="t m0 x1 h6 y46 ff3 fs1 fc0 sc0 ls0 ws0">signs of different coefficients of a block can be used to estimate variational edge parameters</div><div class="t m0 x1 h6 y47 ff3 fs1 fc0 sc0 ls0 ws0">such as orientation, strength, and offset. Their scheme has been extended and refined after-</div><div class="t m0 x1 h6 y48 ff3 fs1 fc0 sc0 ls0 ws0">wards by Lee et al. and Li et al. in their respective works [21] and [22]. Jiang et al. [12]</div><div class="t m0 x1 h6 y49 ff3 fs1 fc0 sc0 ls0 ws0">construct a run length histogram and classify only five edge patterns from the DCT coeffi-</div><div class="t m0 x1 h6 y4a ff3 fs1 fc0 sc0 ls0 ws0">cients. Pour et al. [27] create a histogram from a selective set of DCT coefficients and use it for</div><div class="t m0 x1 h6 y4b ff3 fs1 fc0 sc0 ls0 ws0">indexing the images. Similarly, in [3, 20, 31], partial DCT coefficients are selected to extract</div><div class="t m0 x1 h6 y4c ff3 fs1 fc0 sc0 ls0 ws0">the feature vectors. In [11] the discrete wavelet transform (DWT) structure to rearrange the</div><div class="t m0 x1 h6 y4d ff3 fs1 fc0 sc0 ls0 ws0">DCT coefficients because of the advantage of the DWT. In [33], multiple DCT coefficients are</div><div class="t m0 x1 h6 y4e ff3 fs1 fc0 sc0 ls0 ws0">extracted, corresponding coefficients are summed up and the resultant feature vector is formed.</div><div class="t m0 xb h6 y57 ff3 fs1 fc0 sc0 ls0 ws0">Capitalizing on the information that can be derived from the transform’s coefficients and the</div><div class="t m0 x1 h6 y58 ff3 fs1 fc0 sc0 ls0 ws0">inherent dimensionality reduction that can achieved with the usage of transforms, the authors</div><div class="t m0 x1 h6 y59 ff3 fs1 fc0 sc0 ls0 ws0">see a potential in utilizing the transform coefficients to form the descriptors for SIFT key</div><div class="t m0 x1 h6 y5a ff3 fs1 fc0 sc0 ls0 ws0">points. In the search for a suitable transform, the polynomials transform has been chosen,</div><div class="t m0 x1 h6 y5b ff3 fs1 fc0 sc0 ls0 ws0">considering its computational light-weightedness and integer-natured coefficients as its merits.</div><div class="t m0 x1 h6 y5c ff3 fs1 fc0 sc0 ls0 ws0">With this, a paradigm shift from pixel processing to block processing for the purpose of</div><div class="t m0 x1 h6 y5d ff3 fs1 fc0 sc0 ls0 ws0">forming the descriptors of SIFT key points has been attempted in this paper. The role of the</div><div class="t m0 x1 h6 y5e ff3 fs1 fc0 sc0 ls0 ws0">transform in this paper is two-fold: 1) address the curse of dimensionality and offer dimen-</div><div class="t m0 x1 h6 y5f ff3 fs1 fc0 sc0 ls0 ws0">sionality reduction through transform and other statistical learning parameters and 2) provide</div><div class="t m0 x1 h6 y60 ff3 fs1 fc0 sc0 ls0 ws0">coefficients that reflect the low level features of the image so that they can be used to form a</div><div class="t m0 x1 h6 y61 ff3 fs1 fc0 sc0 ls0 ws0">representation of the image.</div><div class="t m0 x1 h4 y23 ff3 fs2 fc0 sc0 ls0 ws0">Multimed Tools Appl (2016) 75:49–69 51</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[2.037103,0.000000,0.000000,2.037103,0.000000,0.000000]}'></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/6254db6047503a0a93cceb35/bg4.jpg"><div class="t m0 x1 h3 y29 ff1 fs1 fc0 sc0 ls0 ws0">3 Orthogonal polynomials model for colour images</div><div class="t m0 x1 h6 y2b ff3 fs1 fc0 sc0 ls29 ws0">The set of features of fered by different un itary transforms like Karhunen-Loeve Transform</div><div class="t m0 x1 h6 y2c ff3 fs1 fc0 sc0 ls2a ws0">(KLT), Discrete Fourier Transf orm (DFT), Discrete Cosine T ransform (DCT) provides features</div><div class="t m0 x1 h6 y2d ff3 fs1 fc0 sc0 ls8 ws0">that are efficient for classification purposes more than that of the original image. The Orthogonal</div><div class="t m0 x1 h6 y2e ff3 fs1 fc0 sc0 ls3 ws0">Polynomials Transform belongs to the family of orthogonal transforms and is preferred due to its</div><div class="t m0 x1 h6 y2f ff3 fs1 fc0 sc0 ls0 ws0">less computation complexity when compared to DFT, KLT and DCT. At first, the image</div><div class="t m0 x1 h6 y30 ff3 fs1 fc0 sc0 ls2 ws0">formation system is analyzed for the design of a suitable transform for color images.</div><div class="t m0 x1 h6 y32 ff3 fs1 fc0 sc0 ls0 ws0">3.1 Point-spread operators based on a class of orthogonal polynomials</div><div class="t m0 x1 h6 y34 ff3 fs1 fc0 sc0 ls0 ws0">In order to analyze a colour image for retrieval purposes, the colour image formation system is</div><div class="t m0 x1 h6 y35 ff3 fs1 fc0 sc0 ls0 ws0">considered. As per the classical definition of a colour image, let x and y be the two spatial</div><div class="t m0 x1 h6 y36 ff3 fs1 fc0 sc0 ls2c ws0">coordinates and z indicates the colour space, then the colour image formation I (x, y,z) can be</div><div class="t m0 x1 h6 y37 ff3 fs1 fc0 sc0 ls0 ws0">described as</div><div class="t m0 xc h7 y62 ff9 fs1 fc0 sc0 ls2e ws0">Ix; y; zðÞ¼∭f ξ; η; γðÞdμξðÞdμηðÞdμγðÞ ð1Þ</div><div class="t m0 x1 ha y63 ff3 fs1 fc0 sc0 ls0 ws0">where ξ, η, and γ are coordinates in the 3-D space, the object function f (ξ,η,γ) is integrable</div><div class="t m0 x1 ha y64 ff3 fs1 fc0 sc0 ls0 ws0">on a measure space and μ is a σ finite measure with an infinite number of points of increase.</div><div class="t m0 x1 h6 y65 ff3 fs1 fc0 sc0 ls35 ws0">The image I can be considered to be a signed measure on the ring of all measurable sets. It can be</div><div class="t m0 x1 ha y66 ff3 fs1 fc0 sc0 ls37 ws0">easily shown that if I is non-negative and finite valued on the σ ring ℜ of measurable sets then f</div><div class="t m0 x1 h6 y67 ff3 fs1 fc0 sc0 ls3b ws0">can be defined as a kind of derivative of I relative to a signed measure supported by a null set.</div><div class="t m0 x1 h6 y68 ff3 fs1 fc0 sc0 ls3c ws0">The three dimensional point-spread function M (x, y, z) is considered to be real valued function</div><div class="t m0 x1 h7 y69 ff3 fs1 fc0 sc0 ls38 ws0">defined for (x, y, z) ∈ X × Y × Z wh</div><div class="t m0 xd h6 y6a ff3 fs1 fc0 sc0 ls0 ws0">ere X, Yand Z are ordered subsets of real values. In the case of a</div><div class="t m0 x1 h6 y6b ff3 fs1 fc0 sc0 ls2b ws0">colour image of size (n × n × n)whereX (rows) consists of a finite set, which for convenience can</div><div class="t m0 x1 h6 y6c ff3 fs1 fc0 sc0 ls40 ws0">be labeled as {0, 1, …, n-1}, the function M(x,y,z) reduces to a sequence of functions</div><div class="t m0 xe hb y6d ff9 fs1 fc0 sc0 ls41 ws0">Mi; xðÞ¼u</div><div class="t m0 xf hc y6e ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x10 h7 y6f ff9 fs1 fc0 sc0 ls0 ws0">xðÞ; i ¼ 0; 1;::n−1 ð2Þ</div><div class="t m0 x1 h6 y70 ff3 fs1 fc0 sc0 ls45 ws0">where u</div><div class="t m2 x11 hd y71 ff9 fs5 fc0 sc0 ls0 ws0">i</div><div class="t m0 x12 h7 y72 ff3 fs1 fc0 sc0 ls0 ws0">(x) is a set of orthogonal polynomials function of degree 0,1,2.. (n−1) As shown in</div><div class="t m0 x1 h6 y73 ff3 fs1 fc0 sc0 ls38 ws0">equation (2), the process of colour image analysis can be viewed as the linear transformation</div><div class="t m0 x1 h6 y74 ff3 fs1 fc0 sc0 ls3c ws0">defined by the point-spread operator M (x, y)(M (i, t)=u</div><div class="t m2 x13 hd y75 ff9 fs5 fc0 sc0 ls0 ws0">i</div><div class="t m0 x14 h6 y76 ff3 fs1 fc0 sc0 ls0 ws0">(t))</div><div class="t m0 x15 ha y77 ffb fs1 fc0 sc0 ls0 ws0">β</div><div class="t m0 x9 he y78 ffa fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x16 hf y79 ff9 fs1 fc0 sc0 ls0 ws0">s; ς; η</div><div class="t m0 x17 hb y7a ffa fs1 fc0 sc0 ls49 ws0">ðÞ</div><div class="t m0 x18 hb y79 ffa fs1 fc0 sc0 ls0 ws0">¼</div><div class="t m3 x19 h10 y7b ffd fs7 fc0 sc0 ls0 ws0">Z</div><div class="t m0 xc h11 y7c ff9 fs4 fc0 sc0 ls0 ws0">x∈X</div><div class="t m3 x1a h10 y7b ffd fs7 fc0 sc0 ls0 ws0">Z</div><div class="t m0 x1b h11 y7c ff9 fs4 fc0 sc0 ls0 ws0">y∈Y</div><div class="t m3 x1c h10 y7b ffd fs7 fc0 sc0 ls0 ws0">Z</div><div class="t m0 x1d h11 y7c ff9 fs4 fc0 sc0 ls0 ws0">z∈Z</div><div class="t m0 x1e hf y79 ff9 fs1 fc0 sc0 ls0 ws0">M ς; x</div><div class="t m0 x1f hb y7a ffa fs1 fc0 sc0 ls4a ws0">ðÞ</div><div class="t m0 x20 hf y79 ff9 fs1 fc0 sc0 ls0 ws0">M η; y</div><div class="t m0 x21 hb y7a ffa fs1 fc0 sc0 ls4b ws0">ðÞ</div><div class="t m0 x22 hf y79 ff9 fs1 fc0 sc0 ls4c ws0">Ms; z</div><div class="t m0 x23 hb y7a ffa fs1 fc0 sc0 ls4d ws0">ðÞ</div><div class="t m0 x24 hf y79 ff9 fs1 fc0 sc0 ls2e ws0">Is; z</div><div class="t m0 x25 hb y7a ffa fs1 fc0 sc0 ls4d ws0">ðÞ</div><div class="t m0 x26 hf y79 ff9 fs1 fc0 sc0 ls2e ws0">Ix; y; z</div><div class="t m0 x27 hb y7a ffa fs1 fc0 sc0 ls4e ws0">ðÞ</div><div class="t m0 x28 hb y79 ff9 fs1 fc0 sc0 ls13 ws0">dxdydz ð3Þ</div><div class="t m0 x1 h6 y7d ff3 fs1 fc0 sc0 lsb ws0">where i, j and k are coordinates in the 3-D transformed space. Considering each of X, Y and Z to</div><div class="t m0 x1 h6 y7e ff3 fs1 fc0 sc0 ls22 ws0">be finite set of values {0, 1, …, n-1} equation (2) can be written in matrix notation as follows:</div><div class="t m0 x29 ha y7f ffb fs1 fc0 sc0 ls0 ws0">β</div><div class="t m0 x2a he y80 ffa fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x2a hc y81 ff9 fs4 fc0 sc0 ls50 ws0">ijk</div><div class="t m0 x2b h12 y82 ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x2b h12 y83 ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x2b h12 y84 ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x2c h12 y82 ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x2c h12 y83 ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x2c h12 y84 ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x2d h11 y85 ff9 fs4 fc0 sc0 ls0 ws0">n−1</div><div class="t m0 x2d h13 y86 ff9 fs4 fc0 sc0 ls0 ws0">i; j;k¼0</div><div class="t m0 x2e hb y87 ffa fs1 fc0 sc0 ls0 ws0">¼ M</div><div class="t m0 x20 hb y88 ffa fs1 fc0 sc0 ls51 ws0">jj</div><div class="t m0 x2f h7 y87 ff6 fs1 fc0 sc0 ls0 ws0">⊗ M</div><div class="t m0 x30 hb y88 ffa fs1 fc0 sc0 ls51 ws0">jj</div><div class="t m0 x31 h7 y87 ff6 fs1 fc0 sc0 ls0 ws0">⊗ M</div><div class="t m0 x32 hb y88 ffa fs1 fc0 sc0 ls51 ws0">jj</div><div class="t m0 x10 hb y87 ffa fs1 fc0 sc0 ls52 ws0">ðÞ</div><div class="t m0 x33 hc y89 ff9 fs4 fc0 sc0 ls0 ws0">t</div><div class="t m0 x34 h14 y87 ff9 fs1 fc0 sc0 ls0 ws0">I</div><div class="t m0 x35 hb y88 ffa fs1 fc0 sc0 ls53 ws0">jj</div><div class="t m0 x36 hb y87 ffa fs1 fc0 sc0 ls0 ws0">ð4Þ</div><div class="t m0 x1 h6 y8a ff3 fs1 fc0 sc0 ls22 ws0">where the point spread operator |M|is</div><div class="t m0 x37 h6 y8b ff3 fs1 fc0 sc0 ls0 ws0">M</div><div class="t m0 x38 hb y8c ffa fs1 fc0 sc0 ls51 ws0">jj</div><div class="t m0 x39 hb y8b ffa fs1 fc0 sc0 ls0 ws0">¼</div><div class="t m0 x3a h14 y8d ff9 fs1 fc0 sc0 ls0 ws0">u</div><div class="t m0 x3b h15 y8e ff3 fs4 fc0 sc0 ls0 ws0">0</div><div class="t m0 x3c h14 y8f ff9 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x3d h15 y8e ff3 fs4 fc0 sc0 ls0 ws0">0</div><div class="t m0 x3e hb y90 ffa fs1 fc0 sc0 ls55 ws0">ðÞ</div><div class="t m0 x3f h14 y91 ff9 fs1 fc0 sc0 ls0 ws0">u</div><div class="t m0 x40 h15 y8e ff3 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 x41 h14 y8f ff9 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x42 h15 y8e ff3 fs4 fc0 sc0 ls0 ws0">0</div><div class="t m0 x43 hb y90 ffa fs1 fc0 sc0 ls56 ws0">ðÞ</div><div class="t m0 x44 h7 y91 ff6 fs1 fc0 sc0 ls0 ws0">⋯ u</div><div class="t m0 x45 h11 y8e ff9 fs4 fc0 sc0 ls0 ws0">n−1</div><div class="t m0 x46 h14 y8f ff9 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x47 h15 y8e ff3 fs4 fc0 sc0 ls0 ws0">0</div><div class="t m0 x48 hb y90 ffa fs1 fc0 sc0 ls57 ws0">ðÞ</div><div class="t m0 x3a h14 y92 ff9 fs1 fc0 sc0 ls0 ws0">u</div><div class="t m0 x3b h15 y93 ff3 fs4 fc0 sc0 ls0 ws0">0</div><div class="t m0 x3c h14 y94 ff9 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x3d h15 y93 ff3 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 x3e hb y94 ffa fs1 fc0 sc0 ls55 ws0">ðÞu</div><div class="t m0 x40 h15 y93 ff3 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 x41 h14 y94 ff9 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x42 h15 y93 ff3 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 x43 h7 y94 ffa fs1 fc0 sc0 ls56 ws0">ðÞ ⋯ u</div><div class="t m0 x45 h11 y93 ff9 fs4 fc0 sc0 ls0 ws0">n−1</div><div class="t m0 x46 h14 y94 ff9 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x47 h15 y93 ff3 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 x48 hb y94 ffa fs1 fc0 sc0 ls57 ws0">ðÞ</div><div class="t m0 x49 h7 y95 ff6 fs1 fc0 sc0 ls0 ws0">⋮</div><div class="t m0 x4a h14 y96 ff9 fs1 fc0 sc0 ls0 ws0">u</div><div class="t m0 x4b h15 y97 ff3 fs4 fc0 sc0 ls0 ws0">0</div><div class="t m0 x4c h14 y98 ff9 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x1f h11 y97 ff9 fs4 fc0 sc0 ls0 ws0">n−1</div><div class="t m0 x4d hb y99 ffa fs1 fc0 sc0 ls58 ws0">ðÞ</div><div class="t m0 x40 h14 y9a ff9 fs1 fc0 sc0 ls0 ws0">u</div><div class="t m0 x4e h15 y97 ff3 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 x4f h14 y98 ff9 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x30 h11 y97 ff9 fs4 fc0 sc0 ls0 ws0">n−1</div><div class="t m0 x50 hb y99 ffa fs1 fc0 sc0 ls58 ws0">ðÞ</div><div class="t m0 x51 h7 y9a ff6 fs1 fc0 sc0 ls0 ws0">⋯ u</div><div class="t m0 x52 h11 y97 ff9 fs4 fc0 sc0 ls0 ws0">n−1</div><div class="t m0 x53 h14 y98 ff9 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x54 h11 y97 ff9 fs4 fc0 sc0 ls0 ws0">n−1</div><div class="t m0 x55 hb y99 ffa fs1 fc0 sc0 ls59 ws0">ðÞ</div><div class="t m0 x56 h12 y9b ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x56 h12 y9c ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x56 h12 y9d ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x56 h12 y9e ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x56 h12 y9f ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x56 h12 ya0 ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x56 h12 ya1 ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x56 h12 ya2 ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x57 h12 y9b ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x57 h12 y9c ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x57 h12 y9d ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x57 h12 y9e ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x57 h12 y9f ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x57 h12 ya0 ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x57 h12 ya1 ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x57 h12 ya2 ffd fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x36 hb ya3 ffa fs1 fc0 sc0 ls0 ws0">ð5Þ</div><div class="t m0 xb h7 ya4 ff6 fs1 fc0 sc0 ls0 ws0">⊗ is the outer product and |β</div><div class="t m2 x4d hd ya5 ff9 fs5 fc0 sc0 ls5b ws0">ijk</div><div class="t m2 x4d h16 ya6 ff5 fs5 fc0 sc0 ls0 ws0">′</div><div class="t m0 x58 h6 ya7 ff3 fs1 fc0 sc0 ls5c ws0">|bethen</div><div class="t m2 x59 h17 ya8 ff3 fs5 fc0 sc0 ls0 ws0">3</div><div class="t m0 x5a h6 ya7 ff3 fs1 fc0 sc0 ls5d ws0">matrices arranged in the dictionary</div><div class="t m0 x1 h6 ya9 ff3 fs1 fc0 sc0 ls5e ws0">sequence that takes the effect of individual R, G and B planes as well as interactions</div><div class="t m0 x1 h4 y23 ff3 fs2 fc0 sc0 ls21 ws0">52 Multimed T ools Appl (2016) 75:49–69</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[2.037103,0.000000,0.000000,2.037103,0.000000,0.000000]}'></div></div>

<div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6254db6047503a0a93cceb35/bg5.jpg"><div class="t m0 x1 ha y29 ff3 fs1 fc0 sc0 ls5f ws0">among the colour planes. |I| is the image and |β</div><div class="t m2 x5b hd yaa ff9 fs5 fc0 sc0 ls61 ws0">ijk</div><div class="t m2 x5b h16 yab ff5 fs5 fc0 sc0 ls0 ws0">′</div><div class="t m0 x51 h6 y29 ff3 fs1 fc0 sc0 ls62 ws0">| be the coefficients of transforma-</div><div class="t m0 x1 h6 y2a ff3 fs1 fc0 sc0 ls63 ws0">tion. We consider a set of orthogo nal polynomials u</div><div class="t m2 x5c h17 yac ff3 fs5 fc0 sc0 ls0 ws0">0</div><div class="t m0 x5d h6 yad ff3 fs1 fc0 sc0 ls0 ws0">(x), u</div><div class="t m2 x5e h17 yac ff3 fs5 fc0 sc0 ls0 ws0">1</div><div class="t m0 x5f h6 yad ff3 fs1 fc0 sc0 ls0 ws0">(x), …, u</div><div class="t m2 x60 h18 yac ff9 fs5 fc0 sc0 ls0 ws0">n−1</div><div class="t m0 x61 h6 yad ff3 fs1 fc0 sc0 ls0 ws0">(x)ofdegrees</div><div class="t m0 x1 h6 yae ff3 fs1 fc0 sc0 ls67 ws0">0, 1, 2, …, n-1, respectively. The generating formula for the polynomials is as</div><div class="t m0 x1 h6 yaf ff3 fs1 fc0 sc0 ls69 ws0">follows:</div><div class="t m0 x62 h14 yb0 ff9 fs1 fc0 sc0 ls0 ws0">u</div><div class="t m0 x19 h13 yb1 ff9 fs4 fc0 sc0 ls0 ws0">iþ1</div><div class="t m0 x63 h7 yb2 ff9 fs1 fc0 sc0 ls0 ws0">xðÞ¼ x−μðÞu</div><div class="t m0 x64 hc yb1 ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x3f h7 yb2 ff9 fs1 fc0 sc0 ls0 ws0">xðÞ−b</div><div class="t m0 x50 hc yb1 ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x3 hb yb2 ff9 fs1 fc0 sc0 ls0 ws0">nðÞu</div><div class="t m0 x5b h11 yb1 ff9 fs4 fc0 sc0 ls0 ws0">i−1</div><div class="t m0 x14 h7 yb2 ff9 fs1 fc0 sc0 ls0 ws0">xðÞ for i≥1 ð6Þ</div><div class="t m0 xb h6 yb3 ff3 fs1 fc0 sc0 ls0 ws0">Here u</div><div class="t m2 x65 h17 yb4 ff3 fs5 fc0 sc0 ls0 ws0">1</div><div class="t m0 x66 h7 yb5 ff3 fs1 fc0 sc0 ls0 ws0">(x)=x−μ,andu</div><div class="t m2 x67 h17 yb4 ff3 fs5 fc0 sc0 ls0 ws0">0</div><div class="t m0 x68 h6 yb5 ff3 fs1 fc0 sc0 ls0 ws0">(x)=1</div><div class="t m0 xb h6 yb6 ff3 fs1 fc0 sc0 ls0 ws0">Where:</div><div class="t m0 x69 h14 yb7 ff9 fs1 fc0 sc0 ls0 ws0">b</div><div class="t m0 x1b hc yb8 ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x68 hb yb9 ff9 fs1 fc0 sc0 ls0 ws0">nðÞ¼</div><div class="t m0 x6a h14 yba ff9 fs1 fc0 sc0 ls0 ws0">u</div><div class="t m0 x6b hc ybb ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x64 hf yba ff8 fs1 fc0 sc0 ls0 ws0">; u</div><div class="t m0 x6c hc ybb ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x3c hb ybc ffa fs1 fc0 sc0 ls70 ws0">hi</div><div class="t m0 x3e h14 ybd ff9 fs1 fc0 sc0 ls0 ws0">u</div><div class="t m0 x58 h11 ybe ff9 fs4 fc0 sc0 ls0 ws0">i−1</div><div class="t m0 x64 hf ybf ff8 fs1 fc0 sc0 ls0 ws0">; u</div><div class="t m0 x6c h11 ybe ff9 fs4 fc0 sc0 ls0 ws0">i−1</div><div class="t m0 x1e hb yc0 ffa fs1 fc0 sc0 ls71 ws0">hi</div><div class="t m0 x6d hb yc1 ffa fs1 fc0 sc0 ls0 ws0">¼</div><div class="t m0 x6e h12 yc2 ffd fs1 fc0 sc0 ls0 ws0">X</div><div class="t m0 x13 hc yc3 ff9 fs4 fc0 sc0 ls0 ws0">n</div><div class="t m0 x13 h13 yc4 ff9 fs4 fc0 sc0 ls0 ws0">x¼1</div><div class="t m0 x6f h14 yc5 ff9 fs1 fc0 sc0 ls0 ws0">u</div><div class="t m0 x70 h15 yc6 ff3 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x70 hc yc7 ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x71 hb yc5 ff9 fs1 fc0 sc0 ls0 ws0">xðÞ</div><div class="t m0 x59 h12 yc8 ffd fs1 fc0 sc0 ls0 ws0">X</div><div class="t m0 x23 hc yc9 ff9 fs4 fc0 sc0 ls0 ws0">n</div><div class="t m0 x23 h13 yca ff9 fs4 fc0 sc0 ls0 ws0">x¼1</div><div class="t m0 x72 h14 ycb ff9 fs1 fc0 sc0 ls0 ws0">u</div><div class="t m0 x73 h15 ycc ff3 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x73 h11 ycd ff9 fs4 fc0 sc0 ls0 ws0">i−1</div><div class="t m0 x48 hb ycb ff9 fs1 fc0 sc0 ls0 ws0">xðÞ</div><div class="t m0 x36 hb yce ffa fs1 fc0 sc0 ls0 ws0">ð7Þ</div><div class="t m0 x74 hb ycf ffb fs1 fc0 sc0 ls0 ws0">μ ¼</div><div class="t m0 x4e h6 yd0 ff3 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x4e h14 yd1 ff9 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x75 h12 yd2 ffd fs1 fc0 sc0 ls0 ws0">X</div><div class="t m0 x41 h13 yd3 ff9 fs4 fc0 sc0 ls0 ws0">t¼1</div><div class="t m0 x42 hc yd4 ff9 fs4 fc0 sc0 ls0 ws0">n</div><div class="t m0 x49 hb yd5 ff9 fs1 fc0 sc0 ls0 ws0">x ð8Þ</div><div class="t m0 xb h6 yd6 ff3 fs1 fc0 sc0 ls0 ws0">Considering the range of values of t to x=i, i=1,2,3,…,n,weget</div><div class="t m0 x19 h14 yd7 ff9 fs1 fc0 sc0 ls0 ws0">b</div><div class="t m0 x76 hc yd8 ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x77 hb yd9 ff9 fs1 fc0 sc0 ls0 ws0">nðÞ¼</div><div class="t m0 x2c h14 yda ff9 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x78 h15 ydb ff3 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x79 h14 yda ff9 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x3e h15 ydb ff3 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x3c h7 yda ff6 fs1 fc0 sc0 ls0 ws0">−i</div><div class="t m0 x2e h15 ydb ff3 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x3b hb yda ffa fs1 fc0 sc0 ls4e ws0">ðÞ</div><div class="t m0 x2c h6 ydc ff3 fs1 fc0 sc0 ls74 ws0">44i</div><div class="t m0 x4c h15 ydd ff3 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x58 h7 yde ff6 fs1 fc0 sc0 ls0 ws0">−1ðÞ</div><div class="t m0 x7a hb ydf ff8 fs1 fc0 sc0 ls0 ws0">; μ ¼</div><div class="t m0 x59 h6 yda ff3 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x59 h14 ye0 ff9 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x7b h12 ye1 ffd fs1 fc0 sc0 ls0 ws0">X</div><div class="t m0 x44 h13 ye2 ff9 fs4 fc0 sc0 ls0 ws0">x¼1</div><div class="t m0 x51 hc ye3 ff9 fs4 fc0 sc0 ls0 ws0">n</div><div class="t m0 x7c h14 yd9 ff9 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x70 hb ye4 ffa fs1 fc0 sc0 ls0 ws0">¼</div><div class="t m0 x48 hb ye5 ff9 fs1 fc0 sc0 ls0 ws0">n þ 1</div><div class="t m0 x7d h6 ye6 ff3 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x36 hb ye7 ffa fs1 fc0 sc0 ls0 ws0">ð9Þ</div><div class="t m0 xb h6 ye8 ff3 fs1 fc0 sc0 ls0 ws0">Point-spread operators |M| of different sizes can be constructed from the above orthogonal</div><div class="t m0 x1 h7 ye9 ff3 fs1 fc0 sc0 ls0 ws0">polynomials for n≥2andx</div><div class="t m2 x76 hd yea ff9 fs5 fc0 sc0 ls0 ws0">i</div><div class="t m0 x7e h6 yeb ff3 fs1 fc0 sc0 ls0 ws0">=i+1 using equation (5). Here, for the convenience of point-spread</div><div class="t m0 x1 h6 yec ff3 fs1 fc0 sc0 ls0 ws0">operations, the elements of |M| are scaled to make them integers. Having presented the</div><div class="t m0 x1 h6 yed ff3 fs1 fc0 sc0 ls0 ws0">framework for colour image analysis in terms of orthogonal polynomials model, an imple-</div><div class="t m0 x1 h6 yee ff3 fs1 fc0 sc0 ls0 ws0">mentation of the same that takes less time called Fast OPT is proposed and presented in the</div><div class="t m0 x1 h6 yef ff3 fs1 fc0 sc0 ls0 ws0">next subsection.</div><div class="t m0 x1 h6 yf0 ff3 fs1 fc0 sc0 ls0 ws0">3.2 Fast OPT</div><div class="t m0 x1 h6 yf1 ff3 fs1 fc0 sc0 ls77 ws0">Modifications to the existing transform have come into existence following the</div><div class="t m0 x1 h6 yf2 ff3 fs1 fc0 sc0 ls79 ws0">original proposals, with a clear objective, for instance, the Fast Fourier Transform</div><div class="t m0 x1 h6 yf3 ff3 fs1 fc0 sc0 ls7a ws0">(FFT) (Cool67) produces the same result as that of Discrete Fourier Transform (DFT)</div><div class="t m0 x1 h6 yf4 ff3 fs1 fc0 sc0 ls7a ws0">and the only difference is that FFT is much faster. Motivated by this predecessor of a</div><div class="t m0 x1 h6 yf5 ff3 fs1 fc0 sc0 ls7a ws0">fast implementation of a transform, in this subsection, an efficient and fast implemen-</div><div class="t m0 x1 h6 yf6 ff3 fs1 fc0 sc0 ls62 ws0">tation for calculating the point spread operator |M| for an arbitrary block of size ( B×</div><div class="t m0 x1 h6 yf7 ff9 fs1 fc0 sc0 ls0 ws0">B) devised by taking advantage of symmetry the orthog onal polynomials exhibit is</div><div class="t m0 x1 h6 yf8 ff3 fs1 fc0 sc0 ls7a ws0">presented. Among the B</div><div class="t m2 x7f h17 yf9 ff3 fs5 fc0 sc0 ls0 ws0">2</div><div class="t m0 x80 h6 yfa ff3 fs1 fc0 sc0 ls0 ws0">val ues of |M|, calculation of (B</div><div class="t m2 x81 h17 yf9 ff3 fs5 fc0 sc0 ls0 ws0">2</div><div class="t m0 x82 h7 yfa ff6 fs1 fc0 sc0 ls0 ws0">−B) values is sufficient</div><div class="t m0 x1 h6 yfb ff3 fs1 fc0 sc0 ls7c ws0">since u</div><div class="t m2 x83 h17 yfc ff3 fs5 fc0 sc0 ls0 ws0">0</div><div class="t m0 x84 h6 yfd ff3 fs1 fc0 sc0 ls0 ws0">(x)=1. On further inspection of the orthogonal polynomials operator | M|, the</div><div class="t m0 x1 h7 yfe ff3 fs1 fc0 sc0 ls7e ws0">same point spread operator ignoring the sign appears in rows j and (B−j+1). This</div><div class="t m0 x1 h6 yff ff3 fs1 fc0 sc0 ls7e ws0">implies that there are o nly (B</div><div class="t m2 x1d h17 y100 ff3 fs5 fc0 sc0 ls0 ws0">2</div><div class="t m0 x78 h7 y101 ff6 fs1 fc0 sc0 ls0 ws0">−B)/2 distinct elements even though |M| has (B</div><div class="t m2 x85 h17 y100 ff3 fs5 fc0 sc0 ls0 ws0">2</div><div class="t m0 x86 h7 y101 ff6 fs1 fc0 sc0 ls0 ws0">−B)</div><div class="t m0 x1 h6 y102 ff3 fs1 fc0 sc0 ls81 ws0">distinct entries. A sample of the point spread operator for a (8×8) block is shown in</div><div class="t m0 x1 h6 y103 ff3 fs1 fc0 sc0 ls81 ws0">equation (10), where C</div><div class="t m2 x87 hd y104 ff9 fs5 fc0 sc0 ls0 ws0">b</div><div class="t m0 x76 h7 y105 ff3 fs1 fc0 sc0 ls84 ws0">,0≤b≤27 denotes the orthogonal polynomials.</div><div class="t m0 x1 h4 y23 ff3 fs2 fc0 sc0 ls0 ws0">Multimed Tools Appl (2016) 75:49–69 53</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[2.037103,0.000000,0.000000,2.037103,0.000000,0.000000]}'></div></div>