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Image transforms of Image compression. Presenter: Cheng-Jin Kuo 郭政錦 Advisor: Jian-Jiun Ding, Ph. D. Professor 丁建均教授 Digital Image & Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, ROC. Outline. Introduction - PowerPoint PPT Presentation
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Image transforms of Image compression
Presenter: Cheng-Jin Kuo 郭政錦Advisor: Jian-Jiun Ding, Ph. D.
Professor 丁建均教授Digital Image & Signal Processing Lab
Graduate Institute of Communication EngineeringNational Taiwan University, Taipei, Taiwan, ROC
Outline
IntroductionImage compression schemeImage TransformOrthogonal TransformDCT transformSubband TransformHaar Wavelet transform
Introduction
• Image types:bi-level imagegrayscale imagecolor image : e.g. RGB, YCbCrcontinuous-tone image : -natural scene; -image noise; -clouds, mountains, surface of lakes;
Introduction
discrete-tone image(graphical image or synthetic image) :
-artificial image;
-sharp and well-defined edges;
-high contrasted from the background; cartoon-like image:
-uniform color;
Introduction
• The principle of Image compression:
removing the redundancy
-the neighboring pixels are highly correlated
-the correlation is called spatial redundancy
Image compression scheme Arithmetic coding,
Huffman coding,
1.Orthogonal transform(Walsh-Hamadard transform, RLE, …….
DCT, …)
2.Subband transform(wavelet transform, …)
quantization error
image transform quantizer encoder
Compressed image filedecoder
Inverse transform
image’
Image transform
• Two properties and main goals:
-to reduce image redundancy
-to isolate the various freq. of the image
(identify the important parts of the image)
Image transform
• Two main types:
-orthogonal transform:
e.g. Walsh-Hdamard transform, DCT
-subband transform:
e.g. Wavelet transform
Orthogonal transform
• Orthogonal matrix W
C=W . D
Reducing redundancyIsolating frequencies
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
w w w w
w w w w
w w w w
w w w w
1
2
3
4
c
c
c
c
1
2
3
4
d
d
d
d
Orthogonal transform
• One choice of W:
(Walsh-Hadamard transform)
C=W . D
• W should be Invertible (for inverse transform)
• Other properties?
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
Orthogonal transform
• Reducing redundancy (Energy weighted)
• example: d=[5 6 7 8]
after multiply by W/2 c=[13 -2 0 -1]
energy of d = energy of c= 174
• energy ratio of the first index:
d:25/174 =14%
c:169/174 =97%
Orthogonal transform
• Reducing redundancy (Energy weighted)
• d=[4 6 5 2] ; c=[8.5 1.5 -2.5 0.5] ; E=81
In general, we ignore several smallest elements in d’, and get c=[8.5 0 -2.5 0]
quantize it and get the inverse
c=[3 5.5 5.5 3]
E=81.75
• Property 1: should be large while others, small.1c
Orthogonal transform
• Isolating frequencies (freq. weighted)
• example:
d=[1 0 0 1]c=[2 0 2 0] W=
d=0.5[1 1 1 1]+0.5[1 -1 -1 1]
d=[0 0.33 -0.33 -1]c=[0 2.66 0 1.33]
d=0.66[1 1 -1 -1]+0.33[1 -1 1 -1]
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
Orthogonal transform
• Isolating frequencies (freq. weighted)
• Property 2: should correspond to zero freq. while other coefficients correspond to higher and higher freq.
W= , W=
(Walsh-Hadamard transform)
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 jc
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
1 1 1 1 1 1 1 1
Orthogonal transform
• So how do we choose W?
Invertible matrixCoefficients in the first row are all positiveEach row represents the different freq.
Orthogonal matrix
Orthogonal transform
L L L L L L L L L S S S S S S S L S S S S S S S
L L L L L L L L L S S S S S S S S s s s s s s s
L L L L L L L L L S S S S S S S S s s s s s s s
L L L L L L L L L S S S S S S S S s s s s s s s
L L L L L L L L L S S S S S S S S s s s s s s s
L L L L L L L L L S S S S S S S S s s s s s s s
L L L L L L L L L S S S S S S S S s s s s s s s
L L L L L L L L L S S S S S S S S s s s s s s s
Discrete Cosine Transform
• W matrix of DCT:
• W=
3 5 7 9 11 13 1516 16 16 16 16 16 16 16
cos 0 1 1 1 1 1 1 1 1
cos1 0.981 0.831 0.556 0.195 0.195 0.556 0.831 0.981
cos 2 0.924 0.383 0.383 0.924 0.924 0.383 0.383 0.924
cos3 0.831 0.195 0.981 0.556 0.556 0.981 0.195 0.831
cos 4
0.707 0.707 0.707 0.707 0.707 0.707 0.707 0.707
cos5 0.556 0.981 0.195 0.831 0.831 0.195 0.981 0.556
cos 6 0.383 0.924 0.924 0.383 0.383 0.924 0.924 0.383
cos 7 0.195 0.556 0.831 0.981 0.981 0.831 0.556 0.195
Discrete Cosine Transform
• 1D DCT: , for f=0~7
= , f=0
1 , f>0
• Inverse DCT(IDCT):
7
0
1 (2 1)cos[ ]
2 16f f tt
t fG C p
fC 12
7
0
1 (2 1)cos[ ], 0,1,......,7
2 16t j jj
t jP C G for t
Discrete Cosine Transform
• 2D DCT:
• Inverse DCT(IDCT):
1 1
0 0
1 (2 1) (2 1)cos[ ]cos[ ] ,for 0 , 1
2 22
n n
ij i j xyx y
y j x iG C C p i j n
n nn
7
0
1 (2 1)cos[ ], for t=0,1, ..., 7
2 16t j jj
t jP C G
Discrete Cosine Transform
Subband Transform
• Separate the high freq. and the low freq. by subband decomposition
Subband Transform
• Filter each row and downsample the filter output to obtain two N x M/2 images.
• Filter each column and downsample the filter output to obtain four N/2 x M/2 images
Haar wavelet transform
• Haar wavelet transform:
Average : resolutionDifference : detail
Example for one dimension
Haar wavelet transform
• Example: data=(5 7 6 5 3 4 6 9)
-average:(5+7)/2, (6+5)/2, (3+4)/2, (6+9)/2
-detail coefficients:
(5-7)/2, (6-5)/2, (3-4)/2, (6-9)/2
• n’= (6 5.5 3.5 7.5 | -1 0.5 -0.5 -1.5)
• n’’= (23/4 22/4 | 0.25 -2 -1 0.5 -0.5 -1.5)
• n’’’= (45/8 | 1/8 0.25 -2 -1 0.5 -0.5 -1.5)
Haar wavelet transform
Subband Transform
Subband Transform
• The standard image wavelet transform
• The Pyramid image wavelet transform
Subband Transform
HL
LH HH
LL
Subband Transform
Reference• David Salomon, Coding for Data and Computer
Communication, Springer, 2005.• A. Uhl, A. Pommer, Image and Video Encryption,
Springer, 2005• David Salomon, Data Compression - The Complete
Reference 3rd Edition, Springer, 2004.• Khalid Sayood, Introduction to Data Compression 2nd
Edition, Morgan Kaufmann, 2000.• J.Goswami, A.Chan, Fundamentals of Wavelets – Theory,
Algorithms, and Application, Wiley Interscience, 1999• C.S. Burrus, R. A. Gopinath, H. Guo, Introduction to
Wavelets and Wavelet Transforms – A Primer, Prentice-Hall, 1998