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4/8/2017 ENEE631 Spring’09 Lecture 10 (2/25/2009) Unitary Transform xxx Lecture: xxx. – xx slides, about xx min Spring ’09 Instructor: Min Wu Electrical and Computer Engineering Department, University of Maryland, College Park bb.eng.umd.edu (select ENEE631 S’09)

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**UMCP ENEE631 Slides (created by M.Wu © 2004)**

4/8/2017 Overview Last Time: MMSE Quantizer for non-uniform and uniform source Companding: Quantizer with pre- and post- nonlinear transformation Quantizer in predictive coding Today: Vector vs. Scalar Quantizer Revisit image transform from a coding and basis perspective => Unitary transform DCT transform Logistics: (1) mid-term exam (2) Assign#3 to be posted UMCP ENEE631 Slides (created by M.Wu © 2004) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Recap: Scalar Quantizer**

4/8/2017 … Recap: Scalar Quantizer Quantize one sample at a time Quantizer … Input/Output response Input x Output Q(x) … quantization error M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Recap: MMSE Quantizer Example – Gaussian Source**

4/8/2017 Recap: MMSE Quantizer Example – Gaussian Source Truncated Gaussian N(0,1) with L=16 quantizer Start with uniform quantizer Use iterative algorithm. optimum thresholds (red) and reconstruction values (blue) From B. Liu PU EE488 F’06 M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Vector Quantization Encode a set of values together**

4/8/2017 Vector Quantization vector quantization of 2 elements Encode a set of values together Find the representative combinations Encode the indices of combinations Scalar vs. Vector quantization SQ is simpler in implementation VQ allows flexible partition of coding cells VQ could naturally explore the correlation between elements Stages to build vector quantizer Codebook design Encoder Decoder UMCP ENEE631 Slides (created by M.Wu © 2001) scalar quantization of 2 elements Signal Sample-1 Signal Sample-2 From Bovik’s Handbook Sec.5.3 M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Outline of Core Parts in VQ**

4/8/2017 Outline of Core Parts in VQ Design codebook Optimization formulation is similar to MMSE scalar quantizer Given a set of representative points “Nearest neighbor” rule to determine partition boundaries Given a set of partition boundaries “Probability centroid” rule to determine representative points that minimizes mean distortion in each cell Search for codeword at encoder Tedious exhaustive search Design codebook with special structures to speed up encoding E.g., tree-structured VQ Reference: A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, Kluwer Publisher. R. M. Gray, ``Vector Quantization,'' IEEE ASSP Magazine, pp , April 1984. UMCP ENEE631 Slides (created by M.Wu © 2001/2004) vector quantization of 2 elements Wang’s book Section 8.6 M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Recap: List of Compression Tools**

4/8/2017 Recap: List of Compression Tools Lossless encoding tools Entropy coding: Huffman, Arithmetic coding, Lemple-Ziv, … Run-length coding Lossy tools for reducing bit rate Quantization: scalar quantizer vs. vector quantizer Truncations: discard unimportant parts of data Facilitating compression via Prediction Convert the full signal to prediction residue with smaller dynamic range Encode prediction parameters and residues with less bits Be careful: use quantized version available to decoder when designing encoder Facilitating compression via Transforms Transform into a domain with improved energy compaction UMCP ENEE631 Slides (created by M.Wu © 2004) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Image Transform: A Revisit With A Coding Perspective**

4/8/2017 Image Transform: A Revisit With A Coding Perspective UMCP ENEE631 Slides (created by M.Wu © 2004) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**UMCP ENEE631 Slides (created by M.Wu © 2001)**

4/8/2017 Why Do Transforms? Fast computation E.g., convolution vs. multiplication for filter with wide support Conceptual insights for various image processing E.g., spatial frequency info. (smooth, moderate change, fast change, etc.) Obtain transformed data from measurement E.g., blurred images, radiology images (medical and astrophysics) Often need to perform an inverse transform to obtain the actual data For efficient storage and transmission Pick a few “representatives” (basis) Just store/send the major “contribution” from some basis image/vector => Examine a segment of signal samples together UMCP ENEE631 Slides (created by M.Wu © 2001) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Basic Process of Transform Coding**

4/8/2017 Basic Process of Transform Coding UMCP ENEE631 Slides (created by M.Wu © 2004) Figure is from slides at Gonzalez/ Woods DIP book website (Chapter 8) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Basis Vectors and Basis Images**

4/8/2017 A vector space consists of a set of vectors, a field of scalars, a vector addition operation, and a scalar multiplication operation. Basis Vectors and Basis Images A basis for a vector space ~ a set of vectors that is Linearly independent ~ ai vi = 0 if and only if all ai=0 Uniquely represent every vector in the space by their linear combination ~ bi vi ( “spanning set” {vi} ) Orthonormal basis Orthogonality ~ inner product <x, y> = y*T x= 0 Normalized length ~ || x ||2 = <x, x> = x*T x= 1 Inner product for 2-D arrays <F, G> = m n f(m,n) g*(m,n) = G1*T F1 (rewrite matrix into vector) !! Don’t do FG ~ may not even be a valid operation for MxN matrices! 2D Basis Matrices (Basis Images) Represent any images of the same size as a linear combination of basis images UMCP ENEE631 Slides (created by M.Wu © 2001) “basis” vectors can be viewed as “building blocks” to construct all vectors in a vector space M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Standard / “Trivial” Basis**

4/8/2017 Standard / “Trivial” Basis Standard basis vectors Standard basis images M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Matrix/Vector Form of 1-D DFT**

4/8/2017 Matrix/Vector Form of 1-D DFT { z(n) } { Z(k) } n, k = 0, 1, …, N-1, WN = exp{ - j2 / N } ~ complex conjugate of primitive Nth root of unity UMCP ENEE631 Slides (created by M.Wu © 2001/2004) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Matrix/Vector Form of 1-D DFT (cont’d)**

4/8/2017 Matrix/Vector Form of 1-D DFT (cont’d) { z(n) } { Z(k) } n, k = 0, 1, …, N-1, WN = exp{ - j2 / N } “transform kernels” are complex exponentials Vector form and interpretation for inverse transform z = k Z(k) ak where ak = [ 1, WN-k , WN-2k , … WN-(N-1)k ]T / N Basis vectors akH = ak* T = [ 1, WNk , WN2k , … WN(N-1)k ] / N Use akH as row vectors to construct a matrix F Z = F z z = F*T Z = F* Z F is symmetric (FT=F) and unitary (F-1 = FH where FH = F*T ) UMCP ENEE631 Slides (created by M.Wu © 2001/2004) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**UMCP ENEE631 Slides (created by M.Wu © 2001)**

4/8/2017 1-D Unitary Transform Consider linear invertible transform 1-D sequence { x(0), x(1), …, x(N-1) } as a vector y = A x and A is invertible Unitary matrix: A is unitary if A-1 = A*T = AH Denote A*T as AH ~ “Hermitian” x = A-1 y = A*T y = ai*T y(i) Hermitian of row vectors of unitary matrix A form a set of orthonormal basis vectors {ai*T } Think: how about column vectors of A? Orthogonal matrix ~ A-1 = AT Real-valued unitary matrix is also an orthogonal matrix Row vectors of real orthogonal matrix A form orthonormal basis vectors UMCP ENEE631 Slides (created by M.Wu © 2001) If A is unitary, its inverse A^(-1) which equals to A^H is also unitary: because by applying definition of unitary, (A^-1)^H=(A^H)^H=A=(A^-1)^-1 As A^H’s row vector Hermitian is A’s column vector, we have A’s column vectors also form a set of orthonormal basis and this set of basis is used to represent any vector y in the transform domain (in addition to its row vector Hermitian forming orthonormal basis as stated in the slides) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**UMCP ENEE631 Slides (created by M.Wu © 2001/2004)**

4/8/2017 Exercise: Is each matrix here unitary or orthogonal? If yes, what are the basis vectors? UMCP ENEE631 Slides (created by M.Wu © 2001/2004) For real matrix: check if A x A^H = I ? For A3: A^-1 not equal to A’ 1: n; 2: y. inv(A3) = [2, –3; -1, 2]; Check A A’ = I ? M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Exercise – Question for Today:**

4/8/2017 Exercise – Question for Today: Which is unitary or orthogonal? 2-D DFT Write the forward and inverse transform Express in terms of matrix-vector form Find basis images UMCP ENEE631 Slides (created by M.Wu © 2001/2004) For real matrix: check if A x A’ = I ? inv(A1) = [2, –3; -1, 2] Check A A’ = I ? M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Properties of 1-D Unitary Transform y = A x**

4/8/2017 Properties of 1-D Unitary Transform y = A x Energy Conservation: || y ||2 = || x ||2 Proof: || y ||2 = || Ax ||2= (Ax)*T (Ax)= x*T A*T A x = x*T x = || x ||2 Interpretation: The angles between vectors are preserved A unitary transformation is a rotation of a vector in an N-dimension space, i.e., a rotation of basis coordinates UMCP ENEE631 Slides (created by M.Wu © 2001/2004) Determinant and all eigenvalues of unitary matrix have unit magnitude. Here N-dimensional space may be complex-valued (i.e. with complex field of scalar) such as DFT. M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Properties of 1-D Unitary Transform (cont’d)**

4/8/2017 Properties of 1-D Unitary Transform (cont’d) Energy Compaction and Decorrelation Many commonly used unitary transforms tend to pack a large fraction of signal energy into just a few transform coefficients Highly correlated input elements quite uncorrelated output coefficients Covariance matrix E[ ( y – E(y) ) ( y – E(y) )*T ] small correlation implies small off-diagonal terms Example: recall the effect of DFT Question: What unitary transform gives the best compaction and decorrelation? => Will revisit this issue UMCP ENEE631 Slides (created by M.Wu © 2001/2004) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**1-D Discrete Cosine Transform (DCT)**

4/8/2017 1-D Discrete Cosine Transform (DCT) Linear transform matrix C c(k,n) = (0) for k=0 c(k,n) = (k) cos[(2n+1)/2N] for k>0 => Compare “transform kernels” of DCT vs. DFT C is real and orthogonal rows of C form an orthonormal basis C is not symmetric! DCT is not the real part of unitary DFT! See Assignment#3 related to DFT of a symmetrically extended signal UMCP ENEE631 Slides (created by M.Wu © 2001/2004) “kernel function” used in the transform M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Periodicity Implied by DFT and DCT**

4/8/2017 Periodicity Implied by DFT and DCT UMCP ENEE631 Slides (created by M.Wu © 2004) Gives better energy compaction – eliminating the discontinuity on the boundaries. Figure is from slides at Gonzalez/ Woods DIP book website (Chapter 8) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**UMCP ENEE631 Slides (created by M.Wu © 2001)**

4/8/2017 From Ken Lam’s DCT talk 2001 (HK Polytech) Example of 1-D DCT 100 50 -50 -100 n z(n) 100 50 -50 -100 k Z(k) UMCP ENEE631 Slides (created by M.Wu © 2001) DCT M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Example of 1-D DCT (cont’d): N = 8**

4/8/2017 From Ken Lam’s DCT talk 2001 (HK Polytech) Example of 1-D DCT (cont’d): N = 8 1.0 0.0 -1.0 Basis vectors 100 -100 u=0 u=0 to 1 u=0 to 4 u=0 to 5 u=0 to 2 u=0 to 3 u=0 to 6 u=0 to 7 Reconstructions n z(n) Original signal UMCP ENEE631 Slides (created by M.Wu © 2001) Spatial freq of DCT basis are reflected in the # of zero crossings k Z(k) Transform coeff. M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**UMCP ENEE631 Slides (created by M.Wu © 2001)**

4/8/2017 Fast DCT via FFT Define new sequence reorder odd and even elements Split DCT sum into odd and even terms Other real-value fast algorithms UMCP ENEE631 Slides (created by M.Wu © 2001) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Relation between DCT and DFT – see assign#3**

M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**UMCP ENEE631 Slides (created by M.Wu © 2001)**

4/8/2017 2-D DCT Separable orthogonal transform Apply 1-D DCT to each row, then to each column Y = C X CT X = CT Y C = mn y(m,n) Bm,n Set y(m,n)=1 and rest as zeros to obtain basis image Bm,n ~ outer product of C’s mth & nth rows DCT basis images: Equivalent to represent an NxN image with a set of orthonormal NxN “basis images” Each DCT coefficient indicates the contribution from (or similarity to) the corresponding basis image UMCP ENEE631 Slides (created by M.Wu © 2001) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**UMCP ENEE631 Slides (created by M.Wu © 2001)**

4/8/2017 2-D DCT Separable orthogonal transform Apply 1-D DCT to each row, then to each column Y = C X CT X = CT Y C = mn y(m,n) Bm,n Set y(m,n)=1 and rest as zeros to obtain basis image Bm,n ~ outer product of C’s mth & nth rows DCT basis images: Equivalent to represent an NxN image with a set of orthonormal NxN “basis images” Each DCT coefficient indicates the contribution from (or similarity to) the corresponding basis image UMCP ENEE631 Slides (created by M.Wu © 2001) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**2-D Transform: General Case**

4/8/2017 2-D Transform: General Case A general 2-D linear transform with kernel {ak,l(m,n)} y(k,l) is a transform coefficient for Image {x(m,n)} {y(k,l)} is “Transformed Image” Equiv to rewriting all from 2-D to 1-D and applying 1-D transform Computational complexity N2 values to compute N2 terms in summation per output coefficient O(N4) for transforming an NxN image! UMCP ENEE631 Slides (created by M.Wu © 2001/2004) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**UMCP ENEE631 Slides (created by M.Wu © 2001)**

4/8/2017 2-D Linear Transforms Image transform kernel {ak,l(m,n)} y(k,l) is a transform coefficient for Image {x(m,n)} {y(k,l)} is “Transformed Image” Equiv to rewriting all from 2-D to 1-D and applying 1-D transform May generalize the transform (series expansion) from NxN to NxM Orthonomality condition Assure any truncated expansion of the form will minimize sum of squared errors when y(k,l) take values as above UMCP ENEE631 Slides (created by M.Wu © 2001) Completeness condition assure zero error when taking full basis a1 a2 a3 orth. proj. gives min. distance M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**2-D Separable Unitary Transforms**

4/8/2017 2-D Separable Unitary Transforms Focus our attention to separable transform ak,l(m,n) = ak(m) bl(n) , denote this as a(k,m) b(l,n) Use 1-D unitary transform as building block {ak(m)}k and {bl(n)}l are 1-D complete orthonormal sets of basis vectors use as row vectors to obtain unitary matrices A={a(k,m)} & B={b(l,n)} Apply to columns and rows Y = A X BT often choose same unitary matrix as A and B (e.g., 2-D DFT) For square NxN image A: Y = A X AT X = AH Y A* For rectangular MxN image A: Y = AM X AN T X = AMH Y AN* Complexity ~ O(N3) May further reduce complexity if unitary transf. has fast implementation UMCP ENEE631 Slides (created by M.Wu © 2001) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Basis Images for Separable Transform**

4/8/2017 Basis Images for Separable Transform X = AH Y A* => x(m,n) = k l a*(k,m)a*(l,n) y(k,l) Represent X with NxN basis images weighted by coeff. Y Obtain basis image by setting Y={(k-k0, l-l0)} and getting X { a*(k0 ,m)a*(l0 ,n) }m,n Basis image in matrix form A*k,l = a*k al*T ~ a*k is kth column vector of AH transf. coeff. y(k,l) is the inner product of the basis A*k,l with image X UMCP ENEE631 Slides (created by M.Wu © 2001) For non-squared image, e.g. MxN, the size of row and col transform basis would be different and the basis images would be rectangular. M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Exercise on Basis Images**

4/8/2017 Exercise on Basis Images For 2-D separable unitary transform: Y = A X AT => X = AH Y A* Represent X with NxN basis images weighted by coeff. Y Obtain basis image by setting Y={(k-k0, l-l0)} & getting X In matrix form A*k,l = a*k al*T ~ a*k is kth column vector of AH Exercise: A is Unitary transform or not? If so, find basis images Represent an image X with basis images UMCP ENEE631 Slides (created by M.Wu © 2001) (Jain’s e.g.5.1, pp137: A’ [5, –1; – 2, 0] A; outer product of columns of AH : [1,1]’[1 1]/2, …) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Review and Exercise on Basis Images**

4/8/2017 Review and Exercise on Basis Images Exercise: A is Unitary transform or not? Find basis images Represent an image X with basis images UMCP ENEE631 Slides (created by M.Wu © 2001) (Jain’s e.g.5.1, pp137: A’ [5, –1; – 2, 0] A; outer product of columns of AH : [1,1]’[1 1]/2, …) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**UMCP ENEE631 Slides (created by M.Wu © 2001)**

4/8/2017 (1.n; 2. y.) Exercise: Unitary or not? And find basis for the unitary matrix Find basis images and represent image X with basis images X = AH Y A* (separable) => x(m,n) = k l a*(k,m)a*(l,n) y(k,l) Represent X with NxN basis images weighted by coeff. Y Obtain basis image { a*(k0 ,m)a*(l0 ,n) }m,n by setting Y={(k-k0, l-l0)} & getting X In matrix form A*k,l = a*k al*T ~ a*k is kth column vector of A*T (akT is kth row vector of A) Transform coeff. y(k,l) is the inner product of A*k,l with the image UMCP ENEE631 Slides (created by M.Wu © 2001) Jain’s e.g.5.1, pp137 A’ [5 –1;-2 0] A : [1,1]’[1 1]/2, … M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**UMCP ENEE631 Slides (created by M.Wu © 2001/2004)**

4/8/2017 Review: 2-D DFT Recall: 2-D DFT is Separable Y = F X F X = F* Y F* Basis images Bk,l = (ak ) (al )T ~ outer product of two vectors where ak = [ 1 WN-k WN-2k … WN-(N-1)k ]T / N UMCP ENEE631 Slides (created by M.Wu © 2001/2004) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Visualizing Fourier Basis Images**

4/8/2017 Visualizing Fourier Basis Images Fourier basis uses complex exponentials Their real and imaginary parts give smoothly varying sinusoidal patterns in different frequencies and orientations exp[ j 2 (ux + vy) ] = cos[2 (ux + vy)] + j sin[2 (ux + vy)] v Real (cos) part (u, v) (1, 0) (1, 1) (0, 5) Imaginary (sin) part Figures from Mani Thomas U.Del CISC489/689 2D FT M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**8x8 DFT Basis Images Figures from John Woods’ book.**

4/8/2017 8x8 DFT Basis Images Note at multiple of 4, the basis is real valued. Figures from John Woods’ book. M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**UMCP ENEE631 Slides (created by M.Wu © 2001)**

4/8/2017 Summary and Review 1-D transform of a vector Represent an N-sample sequence as a vector in N-dimension vector space Transform Different representation of this vector in the space via different basis e.g., 1-D DFT from time domain to frequency domain Forward transform In the form of inner product Project a vector onto a new set of basis to obtain N “coefficients” Inverse transform Use linear combination of basis vectors weighted by transform coefficents to represent the original signal 2-D transform of a matrix Generally can rewrite the matrix into a long vector & apply 1-D transform Separable transform allows applying transform to rows then columns UMCP ENEE631 Slides (created by M.Wu © 2001) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Summary and Review (cont’d)**

4/8/2017 Summary and Review (cont’d) Vector/matrix representation of 1-D & 2-D sampled signal Representing an image as a matrix or sometimes as a long vector Basis functions/vectors and orthonormal basis Used for representing the space via their linear combinations Many possible sets of basis and orthonormal basis Unitary transform on input x ~ A-1 = A*T y = A x x = A-1 y = A*T y = ai*T y(i) ~ represented by basis vectors {ai*T} Rows (and columns) of a unitary matrix form an orthonormal basis General 2-D transform and separable unitary 2-D transform 2-D transform involves O(N4) computation Separable: Y = A X AT = (A X) AT ~ O(N3) computation Apply 1-D transform to all columns, then apply 1-D transform to rows For non-square image of size MxN: Y = AMxM X ATNxN ; basis images MxN UMCP ENEE631 Slides (created by M.Wu © 2001) If A is unitary, then A’ is also unitary: (A’) (A’)^H = (A’) (A^H)’ = (A^H A)’ = I M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Summary of Today’s Lecture**

4/8/2017 Summary of Today’s Lecture Unitary transform and properties Basis vectors and Basis images DCT transform Next time: putting together … Transform coding and JPEG compression standard Readings Gonzalez’s 3/e book 8.1; 8.2.1, , (till before motion); ; ; Jain’s book (on unitary transform) To read more: Woods’ book 4.2, 4.3, 4.5 UMCP ENEE631 Slides (created by M.Wu © 2004) Wang’s book 9.1 Gonzalez’s 2/e book 8.5.2, M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Clarifications “Dimension” Eigenvalues of unitary transform**

4/8/2017 Clarifications “Dimension” Dimension of a signal ~ # of index variables audio and speech is 1-D signal, image is 2-D, video is 3-D Dimension of a vector space ~ # of basis vectors in it Eigenvalues of unitary transform All eigenvalues have unit magnitude (could be complex valued) By definition of eigenvalues ~ A x = x By energy preservation of unitary ~ || A x || = ||x|| Eigenvalues here are different from the eigenvalues in K-L transform K-L concerns the eigen of covariance matrix of random vector Eigenvectors ~ we generally consider the orthonormalized ones M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Summary of Today’s Lecture**

4/8/2017 Summary of Today’s Lecture Optimal bit allocation Optimal transform - KLT Next Lecture: Video coding Readings on Wavelet Results on rate-distortion theory: Jain’s book Section 2.13 Further exploration – Tutorial on rate-distortion by Ortega-Ramchandran in IEEE Sig. Proc. Magazine, Nov. 98 KLT: Jain’s book 2.9, 5.6, 5.11, 5.14, (further exploration – 5.12, 5.13) UMCP ENEE631 Slides (created by M.Wu © 2004) M. Wu: ENEE631 Digital Image Processing (Spring'09)

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**Common Unitary Transforms and Basis Images**

4/8/2017 Common Unitary Transforms and Basis Images DFT DCT Haar transform K-L transform UMCP ENEE631 Slides (created by M.Wu © 2001) See also: Jain’s Fig.5.2 pp136 M. Wu: ENEE631 Digital Image Processing (Spring'09)

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4/8/2017 Hadamard Transform From Mani Thomas U.Del CISC489/689 2D FT M. Wu: ENEE631 Digital Image Processing (Spring'09)

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