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# Digital Image Processing ECE 6258

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This 0 page Class Notes was uploaded by Cassidy Effertz on Monday November 2, 2015. The Class Notes belongs to ECE 6258 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/233897/ece-6258-georgia-institute-of-technology-main-campus in ELECTRICAL AND COMPUTER ENGINEERING at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

ECE6258 Lecture 20 Unitary Transforms msznus ECESZSE Rune M M eeee m I A General Coding Structure OVERALL ENCODER msznus ECESZSE Rune M M eeeeee m Unitary Trans forms I Sort samples of Xn1n2 in an MXN image or block of an image into a column vector of length MN I Compute transform coefficients yAx where A is a matrix of size MNXMN l The transform is unitary iff l IfA is realvalued ie AA the transform is orthogonal m znus ECESZSE mien Me Energy conservation with unitary transforms I For any unitary transform yAx we obtain JUIMWB 3133578 quotquot 7 a 1M I Interpretation every unitary transform is simply a rotation of the coordinate system I Vector lengths energies are conserved msznus ECESZSE Rune M Me Energy distribution for unitary transforms EigenmatriX of the autocorrelation matrix I Energy is conserved but often will be unevemy Definition eigenmatrix Dof autocorrelation matrix RXX distributed among coefficients D Dis unitaW D The columns of form an orthonormal set of eigenvectors of RXX Autocorrelatlon matrix Rxxt1gt sz 0 RW E yyH E AXXHAH ARXXAH A 0 A 0 AMN l Mean squared values of the coefficients 2 lie on the dlagonal Of Ryy39 u RXX is a normal matrix ie RXXHRXXRX XX so a unitary E 1121 Ryyii ARmAHL39J eigenmatrix exists a RXX is symmetric nonnegative definite hence 120 for all I 1062003 ECE 6258 Russell M Mersereau 5 1062003 ECE 6258 Russell M Mersereau 6 I Karhunen Loeve transform Basis images and eigenimages 39 unitaw transform With matrix I For any unitary transform the inverse transform A H x AHy can be interpreted in terms of the superposition of basis where the columns of Dare ordered according to decreasing images of Size MN eigenvalues Transform coefficients are pairwise uncorrelated If the transform is a KL transform the basis images which are R AR XAH deR d H A A the eigenvectors of the autocorrelation matrix Rxx are called yy x xx l i eigenimages I Energy concentration property No other unitary transform packs as much energy into the rst J coef cients for all J D I If the energy concentration property works well only a limited number of eigenimages is needed to approximate a set of Mean squared approximation error by choosing only rst J images with sma error coef cients is minimized D 1062003 ECE 6258 Russell M Mersereau 7 1062003 ECE 6258 Russell M Mersereau 8 I Eigenfaces I The first 16 eigenfaces from a training set of 20 faces Can be used for face recognition by nearest neighbor search in 16d face space Can be used to generate faces by adjusting 16 coefficients mszuus Emsst Russe M M eeeeee m 9 Finding eigeneyes eigennoses eigenrnouths m zuus ECESZSE Rm M M eeeeee m m Using eigeneyes eigennoses eigenmouths m s zuus I Results Source MIT Media Lab m zuus Computing eigenimages from a training set I How can we work with an MN x MN autocorrelation matrix D Use training set 11 F2 FL in de ne training set matrix S1 1 F2 FL in Calculate R liFFHilSSH l F Lgt1 z 7 L Problem 1 Training set size should be L MN If LltMN autocorrelation matrix is rankdeficient Problem 2 Finding eigenvectors of an MN x MN matrix I Can we find a small set of the most important eigenimages from a small training set LltltMN 1062003 ECE 6258 Russell M Mersereau 13 I Sirovich and Kirby method Instead of eigenvectors of SS consider the eigenvectors of SHS ie SHS v My Premultiply both sides by S SS Sv uSV By inspection we find that Sv are eigenvectors of 88 For LltltMN this saves computation by El Computing the LxL matrix SHS Computing L eigenvectors v of SHS Computing eigenimages corresponding to the LOS L largest eigenvalues using Sv 1062003 ECE 6258 Russell M Mersereau 14 Block Wise image processing Subdivide image into i small blocks Process each block independently from the 7 others 39 7 1 Typical block sizes 8x8 2 x quot 16x16 1062003 ECE 6258 Russell M Mersereau 15 I Separable blockwise transforms Image block written as a square matrix This can only be done it the transform is separable in n1 and n2 Inverse transform x A X AH 1062003 ECE 6258 Russell M Mersereau 16 I Haar transform Haar transform matrix for sizes N248 H111 11 o 2000 r 2171 11 072000 ii Eo 0200 Hr1117 o 07200 11 0 8 1710 0020 llli o 1710 00720 Hr441710 1710 0002 lilO E 17107 00072 Can be computed by taking sums and differences Fast algorithms by recursively applying Hr2 1062003 ECE 6258 Russell M Mersereau I Hadamard transform I Transform matrices can be resursively generated 7 i 1 1 1 HinHrzi lil Hd4 Hdg Hd2 Hdg Hd4 HdQ Hdg Hd2 Hd I Example 1 1 1 1 l HdB E Notethat Hadamard 1 coeffICIents 1 need reordering 1 to concentrate energy 1062003 ECE 6258 Russell M Mersereau Hadamard basis functions FIGURE 829 III1 llmlwnnunl I mi HinduHum N 4 Int rum ul ullvl l I llck I ll m mp lv Source Gonzalez and Woods SoumaGonzaiezandJNoods 1062003 ECE 6258 Russell M Mersereau 19 Discrete Fourier transform Only a minor change in the normalization to make the DFT unitary 27rn1k1 i 27T7 L2k2 1 klak2 chnlan2e j N1 6 N2 1062003 ECE 6258 Russell M Mersereau

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