ENEE631 Digital Image Processing (Spring'04)
Unitary TransformsUnitary Transforms
Spring ’04 Instructor: Min Wu
ECE Department, Univ. of Maryland, College Park
www.ajconline.umd.edu (select ENEE631 S’04) [email protected]
Based on ENEE631 Based on ENEE631 Spring’04Spring’04Section 9Section 9
ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [3]
Image Transform: A RevisitImage Transform: A Revisit
With A Coding PerspectiveWith A Coding Perspective
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [4]
Why Do Transforms?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 as measurement– E.g., blurred images, radiology images (medical and astrophysics)– Often need inverse transform– May need to get assistance from other transforms
For efficient storage and transmission– Pick a few “representatives” (basis) – Just store/send the “contribution” from each basis
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [5]
Basic Process of Transform CodingBasic Process of Transform Coding
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Figure is from slides at Gonzalez/ Woods DIP book website (Chapter 8)
ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [6]
1-D DFT and Vector Form 1-D DFT and Vector Form
{ z(n) } { Z(k) }
n, k = 0, 1, …, N-1WN = exp{ - j2 / N }
~ complex conjugate of primitive Nth root of unity
Vector form and interpretation for inverse transform
z = k Z(k) ak
ak = [ 1 WN-k WN
-2k … WN-(N-1)k ]T / N
Basis vectors – ak
H = ak* T = [ 1 WN
k 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 )
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [7]
Basis Vectors and Basis ImagesBasis Vectors and Basis Images
A basis for a vector space ~ a set of vectors– 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
A vector space consists of a set of vectors, a field of scalars, a vector addition operation, and a scalar multiplication operation.
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [8]
1-D Unitary Transform1-D Unitary Transform
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-1 = A*T – Denote A*T as AH ~ “Hermitian”
– x = A-1 y = A*T y = ai*T y(i)
– Hermitian of row vectors of A form a set of orthonormal basis vectorsai
*T = [a*(i,0), …, a*(i,N-1)] T
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
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [9]
Exercise – Question for Today: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
cossin
sincos
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inv(A1) = [2, –3; -1, 2]
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [10]
Properties of 1-D Unitary Transform Properties of 1-D Unitary Transform y = A xy = A x
Energy Conservation
– || y ||2 = || x ||2
|| y ||2 = || Ax ||2= (Ax)*T (Ax)= x*T A*T A x = x*T x = || x ||2
Rotation
– 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
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [11]
Properties of 1-D Unitary Transform (cont’d)Properties of 1-D Unitary Transform (cont’d)
Energy Compaction– Many common unitary transforms tend to pack a large fraction of signal
energy into just a few transform coefficients
Decorrelation– 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 in a few lectures
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [12]
Review: 1-D Discrete Cosine Transform (DCT)Review: 1-D Discrete Cosine Transform (DCT)
Transform matrix C– c(k,n) = (0) for k=0 – c(k,n) = (k) cos[(2n+1)/2N] for k>0
C is real and orthogonal– rows of C form 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
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [13]
Periodicity Implied by DFT and DCTPeriodicity Implied by DFT and DCT
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Figure is from slides at Gonzalez/ Woods DIP book website (Chapter 8)
ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [14]
Example of 1-D DCT Example of 1-D DCT
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [15]
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Example of 1-D DCT (cont’d)Example of 1-D DCT (cont’d)
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Transform coeff.
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Reconstructions
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From Ken Lam’s DCT talk 2001 (HK Polytech)
ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [17]
2-D DCT2-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
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
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [18]
2-D Transform: General Case2-D Transform: General Case
A general 2-D linear transform {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!
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [20]
2-D Separable Unitary Transforms2-D Separable Unitary Transforms
Restrict 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 = AM
H Y AN*
Complexity ~ O(N3)– May further reduce complexity if unitary transf. has fast implementation
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [21]
Basis ImagesBasis Images
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)} & getting X
{ a*(k0 ,m)a*(l0 ,n) }m,n
in matrix form A*k,l = a*k al*T ~ a*k is kth column vector of AH
trasnf. coeff. y(k,l) is the inner product of A*k,l with the image
Exercise– Unitary transform or not?– Find basis images– Represent an image X with basis images (Jain’s e.g.5.1, pp137)
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [22]
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Common Unitary Transforms and Basis ImagesCommon Unitary Transforms and Basis Images
DFT DCT Haar transform K-L transform
See also: Jain’s Fig.5.2 pp136
ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [24]
2-D DFT2-D DFT
2-D DFT is Separable
– Y = F X F X = F* Y F*
– Basis images Bk,l = (ak ) (al
)T
where ak = [ 1 WN-k WN
-2k … WN-(N-1)k ]T / N
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [25]
Summary and Review (1)Summary and Review (1)
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 coeff.
to represent the original signal
2-D transform of a matrix– Rewrite the matrix into a vector and apply 1-D transform– Separable transform allows applying transform to rows then columns
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [26]
Summary and Review (1) cont’dSummary and Review (1) 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 orthonomal basis– Used for representing the space via their linear combinations– Many possible sets of basis and orthonomal 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
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [29]
Optimal TransformOptimal Transform
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [30]
Optimal TransformOptimal Transform
Recall: Why use transform in coding/compression?– Decorrelate the correlated data– Pack energy into a small number of coefficients
– Interested in unitary/orthogonal or approximate orthogonal transforms Energy preservation s.t. quantization effects can be better understood
and controlled
Unitary transforms we’ve dealt so far are data independent– Transform basis/filters are not depending on the signals we are processing
What unitary transform gives the best energy compaction and decorrelation?
– “Optimal” in a statistical sense to allow the codec works well with many images
Signal statistics would play an important role
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [31]
Review: Correlation After a Linear TransformReview: Correlation After a Linear Transform
Consider an Nx1 zero-mean random vector x
– Covariance (autocorrelation) matrix Rx = E[ x xH ] give ideas of correlation between elements Rx is a diagonal matrix for if all N r.v.’s are uncorrelated
Apply a linear transform to x: y = A x
What is the correlation matrix for y ?
Ry = E[ y yH ] = E[ (Ax) (Ax)H ] = E[ A x xH AH ]
= A E[ x xH ] AH = A Rx AH
Decorrelation: try to search for A that can produce a decorrelated y (equiv. a diagonal correlation matrix Ry )
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [32]
K-L Transform (Principal Component Analysis)K-L Transform (Principal Component Analysis)
Eigen decomposition of Rx: Rx uk = k uk
– Recall the properties of Rx
Hermitian (conjugate symmetric RH = R); Nonnegative definite (real non-negative eigen values)
Karhunen-Loeve Transform (KLT) y = UH x x = U y with U = [ u1, … uN ]
– KLT is a unitary transform with basis vectors in U being the orthonormalized eigenvectors of Rx
– UH Rx U = diag{1, 2, … , N} i.e. KLT performs decorrelation
– Often order {ui} so that 1 2 … N
– Also known as the Hotelling transform or the Principle Component Analysis (PCA)
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [33]
Properties of K-L TransformProperties of K-L Transform
Decorrelation
– E[ y yH ]= E[ (UH x) (UH x)H ]= UH E[ x xH ] U = diag{1, 2, … , N}
– Note: Other matrices (unitary or nonunitary) may also decorrelate the transformed sequence [Jain’s e.g.5.7 pp166]
Minimizing MSE under basis restriction
– If only allow to keep m coefficients for any 1 m N, what’s the best way to minimize reconstruction error?
Keep the coefficients w.r.t. the eigenvectors of the first m largest eigenvalues
Theorem 5.1 and Proof in Jain’s Book (pp166)
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [34]
KLT Basis RestrictionKLT Basis Restriction Basis restriction
– Keep only a subset of m transform coefficients and then perform inverse transform (1 m N)
– Basis restriction error: MSE between original & new sequences
Goal: to find the forward and backward transform matrices to minimize the restriction error for each and every m– The minimum is achieved by KLT arranged according to the
decreasing order of the eigenvalues of R
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [35]
K-L Transform for ImagesK-L Transform for Images
Work with 2-D autocorrelation function
– R(m,n; m’,n’)= E[ x(m, n) x(m’, n’) ] for all 0 m, m’, n, n’ N-1– K-L Basis images is the orthonormalized eigenfunctions of R
Rewrite images into vector form (N2x1)
– Need solve the eigen problem for N2xN2 matrix! ~ O(N 6)
Reduced computation for separable R
– R(m,n; m’,n’)= r1(m,m’) r2(n,n’)
– Only need solve the eigen problem for two NxN matrices ~ O(N3)– KLT can now be performed separably on rows and columns
Reducing the transform complexity from O(N4) to O(N3)
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [36]
Pros and Cons of K-L TransformPros and Cons of K-L Transform
Optimality
– Decorrelation and MMSE for the same# of partial coeff.
Data dependent
– Have to estimate the 2nd-order statistics to determine the transform– Can we get data-independent transform with similar performance?
DCT
Applications
– (non-universal) compression– pattern recognition: e.g., eigen faces– analyze the principal (“dominating”) components
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [37]
Energy Compaction of DCT vs. KLTEnergy Compaction of DCT vs. KLT
DCT has excellent energy compaction for highly correlated data
DCT is a good replacement for K-L– Close to optimal for highly correlated data– Not depend on specific data like K-L does– Fast algorithm available
[ref and statistics: Jain’s pp153, 168-175]
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [38]
Energy Compaction of DCT vs. KLT (cont’d)Energy Compaction of DCT vs. KLT (cont’d)
Preliminaries– The matrices R, R-1, and R-1 share the same eigen vectors
– DCT basis vectors are eigenvectors of a symmetric tri-diagonal matrix Qc
– Covariance matrix R of 1st-order stationary Markov sequence with has an inverse in the form of symmetric tri-diagonal matrix
DCT is close to KLT on 1st-order stationary Markov
– For highly correlated sequence, a scaled version of R-1 approx. Qc
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ENEE631 Digital Image Processing (Spring'04) Lec12 – Unitary Transform [41]
Summary and Review on Unitary TransformSummary and Review on Unitary Transform
Representation with orthonormal basis Unitary transform
– Preserve energy
Common unitary transforms
– DFT, DCT, Haar, KLT
Which transform to choose?
– Depend on need in particular task/application– DFT ~ reflect physical meaning of frequency or spatial frequency– KLT ~ optimal in energy compaction– DCT ~ real-to-real, and close to KLT’s energy compaction
=> A comparison table in Jain’s book Table 5.3
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