1. Borrowed from UMD ENEE631 Spring’04
Unitary Transforms

2. Image Transform: A Revisit With A Coding Perspective
UMCP ENEE631 Slides (created by M.Wu © 2004)

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

4. 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)

5. UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
1-D DFT and Vector Form
{ z(n) }  { Z(k) }
n, k = 0, 1, …, N-1
WN = 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 
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)

6. Basis Vectors and Basis Images
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
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)

7. UMCP ENEE631 Slides (created by M.Wu © 2001)
1-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 vectors ai*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
UMCP ENEE631 Slides (created by M.Wu © 2001)

8. Properties of 1-D Unitary Transform y = 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
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
Determinant and all eigenvalues have unit magnitude

9. 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
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)

10. Review: 1-D Discrete Cosine Transform (DCT)
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
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

11. Periodicity Implied by DFT and DCT
UMCP ENEE631 Slides (created by M.Wu © 2004)
Figure is from slides at Gonzalez/ Woods DIP book website (Chapter 8)

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

13. Example of 1-D DCT (cont’d)
From Ken Lam’s DCT talk 2001 (HK Polytech)
Example of 1-D DCT (cont’d)
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) k
Z(k)
Transform coeff.

14. UMCP ENEE631 Slides (created by M.Wu © 2001)
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 = mn 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
UMCP ENEE631 Slides (created by M.Wu © 2001)

15. 2-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!
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)

16. 2-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 = AMH Y AN*
Complexity ~ O(N3)
May further reduce complexity if unitary transf. has fast implementation
UMCP ENEE631 Slides (created by M.Wu © 2001)

17. UMCP ENEE631 Slides (created by M.Wu © 2001)
Basis 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
UMCP ENEE631 Slides (created by M.Wu © 2001)
(Jain’s e.g.5.1, pp137)

18. 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

19. UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
2-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
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)

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

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

22. UMCP ENEE631 Slides (created by M.Wu © 2004)
Optimal Transform
UMCP ENEE631 Slides (created by M.Wu © 2004)

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

24. Review: 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 )
UMCP ENEE631 Slides (created by M.Wu © 2004)

25. 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)
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)

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

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

28. K-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)
UMCP ENEE631 Slides (created by M.Wu © 2001)

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

30. Energy 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, ]
UMCP ENEE631 Slides (created by M.Wu © 2004)

31. 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
UMCP ENEE631 Slides (created by M.Wu © 2004)
Markov sequence (AR model) x(n+1) = rho x(n) + z(n); correlation func r(n) = rho^|n| and |rho|<1 for stability/stationary

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