1. 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

2. Outline Introduction Image compression scheme Image Transform
Orthogonal Transform
DCT transform
Subband Transform
Haar Wavelet transform

3. Introduction Image types: bi-level image grayscale image
color image : e.g. RGB, YCbCr
continuous-tone image :
-natural scene;
-image noise;
-clouds, mountains, surface of lakes;

4. 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;

5. Introduction The principle of Image compression:
removing the redundancy
-the neighboring pixels are highly correlated
-the correlation is called spatial redundancy

6. 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 file
image’
Inverse transform
decoder

7. 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)

8. Image transform Two main types: -orthogonal transform:
e.g. Walsh-Hdamard transform, DCT
-subband transform:
e.g. Wavelet transform

9. Orthogonal transform Orthogonal matrix W  C=W．D Reducing redundancy
Isolating frequencies

10. Orthogonal transform One choice of W: (Walsh-Hadamard transform) C=W．D
W should be Invertible (for inverse transform)
Other properties?

11. Orthogonal transform Reducing redundancy (Energy weighted)
example: d=[ ]
after multiply by W/2 c=[ ]
energy of d = energy of c= 174
energy ratio of the first index:
d:25/174 =14%
c:169/174 =97%

12. Orthogonal transform Reducing redundancy (Energy weighted)
d=[ ] ; c=[ ] ; E=81
In general, we ignore several smallest elements in d’, and get c=[ ]
quantize it and get the inverse
c=[ ]
E=81.75
Property 1: should be large while others, small.

13. Orthogonal transform Isolating frequencies (freq. weighted) example:
d=[ ]c=[ ] W=
d=0.5[ ]+0.5[ ]
d=[ ]c=[ ]
d=0.66[ ]+0.33[ ]

14. 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)

15. Orthogonal transform So how do we choose W? Invertible matrix
Coefficients in the first row are all positive
Each row represents the different freq.
Orthogonal matrix

16. Orthogonal transform

17. Discrete Cosine Transform
W matrix of DCT: W=

18. Discrete Cosine Transform
1D DCT: , for f=0~7
= , f=0
1 , f>0
Inverse DCT(IDCT):

19. Discrete Cosine Transform
2D DCT:
Inverse DCT(IDCT):

20. Discrete Cosine Transform

21. Subband Transform
Separate the high freq. and the low freq. by subband decomposition

22. 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

23. Haar wavelet transform
Average : resolution
Difference : detail
Example for one dimension

24. Haar wavelet transform
Example: data=( )
-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’= ( | )
n’’= (23/4 22/4 | )
n’’’= (45/8 | 1/ )

25. Haar wavelet transform

26. Subband Transform

27. Subband Transform The standard image wavelet transform
The Pyramid image wavelet transform

28. Subband Transform

29. Subband Transform

30. 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