Transcript Digital Image Processing

Digital Image Processing
Unitary Transforms
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Duong Anh Duc - Digital Image Processing
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Unitary Transforms
 Sort samples f(x,y) in an MxN image (or a rectangular
block in the image) into colunm vector of length MN
 Compute transform coefficients


c  Af
where A is a matrix of size MNxMN
1
 The transform A is unitary, iff A 
A

A

*T
H
Hermitian conjugate
 If A is real-valued, i.e., A-1=A*, transform is
„orthonormal“
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Energy conservation with unitary
transforms


 For any unitary transform c  Af we obtain
2
 2 H   H H 
c  c c  f A Af  f
 Interpretation: every unitary transform is simply
a rotation of the coordinate system.
 Vector lengths („energies“) are conserved.
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Energy distribution for unitary
transforms
 Energy is conserved, but often will be unevenly
distributed among coefficients.
 Autocorrelation matrix
  

 H H
H
H
Rcc  E c c  E Af  f A  AR ff A
 Mean squared values („average energies“) of
the coefficients ci are on the diagonal of Rcc
 

E c  Rcc i ,i  AR ff A
2
i
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H

i ,i
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Eigenmatrix of the autocorrelation
matrix
Definition: eigenmatrix F of autocorrelation matrix Rff
 F is unitary
 The columns of F form an orthonormalized set of eigenvectors
of Rff, i.e.,
Rff F  FL
0 
 0


1


is a diagonal matrix of eigenvalues.
L




0


MN 1 

 Rff is symmetric nonnegative definite, hence i  0 for all i
 Rff is normal matrix, i.e., R Hff R ff  R ff R Hff , hence unitary
eigenmatrix exists
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Karhunen-Loeve transform
 Unitary transform with matrix
A = FH
where the columns of F are ordered according to
decreasing eigenvalues.
 Transform coefficients are pairwise uncorrelated
Rcc = ARffAH = FHRffF = FHFL = L
 Energy concentration property:
No other unitary transform packs as much energy into the
first J coefficients, where J is arbitrary
 Mean squared approximation error by choosing only first J
coefficients is minimized.

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Optimum energy concentration by KL
transform
 To show optimum energy concentration property, consider the truncated

coefficient vector

b  IJc
where IJ contain ones on the first J diagonal positions, else zeros.
 Energy in first J coefficients for arbitrary transform A


J 1
E  Tr Rbb   Tr I J Rcc I J   Tr I J AR ff A I J   akT R ff ak*
H
k 0
where akT is the k - th row of A.
 Lagrangian cost function to enforce unit-length basis vectors
J 1


J 1
J 1

L  E   k 1  a a   a R ff a   k 1  akT ak*
k 0
T
k
*
k
k 0
T
k
*
k

k 0
 Differentiating L with respect to aj yields neccessary condition
R ff a*j   j a*j for all j  J
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Basis images and eigenimages
 For any unitary transform,
the inverse transform


f A c
H
can be interpreted in terms of the superposition of
„basis images“ (columns of AH) of size MN.
 If the transform is a KL transform, the basis images,
which are the eigenvectors of the autocorrelation
matrix Rff , are called „eigenimages.“
 If energy concentration works well, only a limited
number of eigenimages is needed to approximate a
set of images with small error. These eigenimages
form an optimal linear subspace of dimensionality J.
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Computing eigenimages from a
training set
 How to measure MNxMN
autocorrelation matrix?




Use training set 1 , 2 ,, L
 

Define training set matrix S  1 , 2 ,, L 
and calculate
1 L 
1
R 


L
l 1
H
l l

L
SS H
 Problem 1: Training set size should be L >> MN
If L < MN, autocorrelation matrix Rff is rank - deficient
 Problem 2: Finding eigenvectors of an MNxMN matrix.
 Can we find a small set of the most important eigenimages
from a small training set L << MN
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Sirovich and Kirby method
 Instead of eigenvectors of SSH, consider the
eigenvectors of SHS, i.e.,

H 
S Svi  i vi


 Premultiply both sides by S: SS H Svi  i Svi

 By inspection, we find that Svi are eigenvectors of SSH
 For this gives rise to great computational savings, by
Computing the LxL matrix SHS

 Computing L eigenvectors Svi of SHS
 Computing eigenimages corresponding to the L0  L largest

eigenvalues according as Svi

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Example: eigenfaces
 The first 8 eigenfaces obtained from a training set of
500 frontal views of human faces.
 Can be used for face recognition by nearest neighbor
search in 8-d „face space.“
 Can be used to generate faces by adjusting 8
coefficients.
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Gender recognition using eigenfaces
 Task: Male or female?
 Eigenimages from a data base of 20 and 20
female training images
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Gender recognition using eigenfaces
(cont.)
 Recognition accuracy using 8 eigenimages
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Block-wise image processing
 Subdivide image into
small blocks
 Process each block
independently from the
others
 Typical blocksizes: 8x8,
16x16
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Separable blockwise transforms
 Image block written as a square matrix f
(NxN coefficients)
T
c = A .f.A
 This can only be done, if transform is
separable in x and y, i.e.,
(N xN )
AT = AA
 Inverse transform
f = A*.c.AH
2
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Haar transform
 Haar transform matrix for sizes N=2,4,8
1 1 1 


Hr2 
2 1  1
1 1
2

1 1 1  2
Hr4 

4 1  1
0
1  1
0

1

1

1

0 
1 1

0  Hr8  8 1


2 
1

 2 
1
1

1
2
1
2
1  2
1  2
1
0
1
0
1
0
1
0
0
0
0
0
2
2
 2
 2
2
0
0
0 

2 0
0
0 

0
2
0
0 
0 2 0
0 

0
0
2
0 
0
0 2 0 

0
0
0
2 
0
0
0  2 
 Can be computed by taking sums and differences
 Fast algorithms by recursively applying Hr2
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Haar transform example
Original Cameraman
256x256
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256x256 Haar transform
of Cameraman
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Haar transform example
Original Einstein
256x256
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256x256 Haar transform
of Einstein
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Haar transform example
Original Lena
512x512
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512x512 Haar transform
of Lena
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Hadamard transform
 Transform matrices can be recursively generated
1 1 1 


2 1  1
Hd 4  Hd 2  Hd 2
Hd 2  Hr2 
Hd 8  Hd 4  Hd 2  Hd 2  Hd 2  Hd 2


1
Example

1
1

1 1
Hd 8 

8 1
1

1
1

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1
1
1
1
1
1
1
1
1
1
1
1
1 1 1
1 1 1
1
1
1 1
1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1
1
1
1
1
1

 1
 1

1
 1
1

1
 1
Note that Hadamard
Coefficients need
reordering to concentrate
energy
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Discrete Fourier transform
 j 2 

 For DFT of order N, define WN  exp 

 N 
 Transform
WN0 WN0
WN0
 0
matrix
WN2
WN WN1
1  0
2
DTFN 
W
W
N
N
N
 


 0
N 1
WN2( N 1)
WN WN







N 1
WN

WN2( N 1) 



WN( N 1)( N 1) 
WN0
 Definition for general N, fast algorithms for N=2m
 DFT coefficients are complex (even if input image is
real)
 Inverse transform DFTN-1 = DTFH = DTF*
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Discrete cosine transform
 Transform matrix


1

0  n  N 1  k  0





N
DCTN  ak ,n ak ,n  


2

2n  1k 



cos
 0  n  N  1  1  k  N  1

2N




 N
 Can be interpreted as DFT
of a mirror-extended image block
(shown here for 2-d DCT)
 Transform is used in many
coding standards (JPEG, MPEG)
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Basis images of an 8x8 DCT
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Blockwise DCT example
Original Lena
256x256
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Blockwise 8x8 DCT
of Lena
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Comparison of block transforms
Comparison of 1-D
basis functions for
block size N=8
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Comparison of block transforms (cont.)
DCT
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Hadamard
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Haar
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Transform Coding
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Transform Coding (cont.)
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DCT coding artifacts
DCT coding with increasingly coarse quantization, block size 8x8
quantizer stepsize
for AC coefficients: 25
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quantizer stepsize
for AC coefficients: 100
Duong Anh Duc - Digital Image Processing
quantizer stepsize
for AC coefficients: 200
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