1. 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)

2. 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)

3. 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)

4. 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)

5. 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)

6. 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)

7. 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)

8. 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)

9. 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)

10. 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)

11. 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)

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

13. 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)

14. 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)

15. 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)

16. 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)

17. 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)

18. 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)

19. 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)

20. 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)

21. 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)

22. 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)

23. 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)

24. 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)

25. Relation between DCT and DFT – see assign#3
M. Wu: ENEE631 Digital Image Processing (Spring'09)

26. 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 = mn 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)

27. 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 = mn 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)

28. 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)

29. 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)

30. 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)

31. 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)

32. 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)

33. 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)

34. 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)

35. 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)

36. 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)

37. 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)

38. 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)

39. 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)

40. 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)

41. 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)

42. 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)

43. 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)

44. 4/8/2017
Hadamard Transform
From Mani Thomas U.Del CISC489/689 2D FT
M. Wu: ENEE631 Digital Image Processing (Spring'09)