1. Wavelet domain image denoising via support vector regression
Source: Electronics Letters,Volume:40,Issue: 2004 PP :
Authors: H. Cheng, J.W. Tian, J. Liu and Q.Z. Yu
Presented by: C.Y.Gun
Date: 4/7 2005

2. Outline Abstract Introduction( Mother Wavelet, Father Wavelet)
Support vector regression
Proposed theory and algorithm
Experimental results and discussion

3. Introduction Families of Wavelets：
Father wavelet ϕ(t) – generates scaling functions (low-pass filter)
Mother wavelet ψ(t) –generates wavelet functions (high-pass filter)
All other members of the wavelet family are scaling and translations of either the mother or the father wavelet

4. Introduction
Father wavelets (low-pass filters)

5. Introduction
Mother wavelets (high-pass filters)

6. Introduction
2-D DWT for Image

7. Introduction
2-D DWT for Image

8. Introduction
2-D DWT for Image

9. Support vector regression
Standard linear regression equation
The linear case is a special case of the nonlinear regression equation

10. Support vector regression
Idea : we define a « tube » of radius εaround the regression(ε≥ 0)
No error if y lays inside the « tube » or« band »

11. Support vector regression
We therefore define an ε-insensitive loss function L1 L2

12. Support vector regression
Graphical representation

13. Support vector regression
Slack variables ei are defined for each observation: e e e e

14. Support vector regression
Kernel methods：

15. Support vector regression
Basic kernels for vectorial data:
– Linear kernel:
(feature space is Q-dimensional if Q is the dim of ; Map is identity!)
– RBF-kernel:
(feature space is infinite dimensional)
– Polynomial kernel of degree two:
(feature space is d(d+1)/2 －dimensional if d is the dim of )

16. LS－SVM Regression
We define the following optimization problem:
Or:

17. LS－SVM Regression
From Least squares support vector machine classifiers
….(1)

18. LS－SVM Regression The Result LS-SVM model for function estimation is
….(2)

19. LS－SVM Regression

20. LS－SVM Regression
(1)
….(3)

21. Proposed theory and algorithm
Block matrix decompositions
The main formula we need concerns the inverse of a block matrix
=??

22. Proposed theory and algorithm= =
where

23. Proposed theory and algorithm
(3)

24. Proposed theory and algorithm

25. Proposed theory and algorithm3
f(-1,-1)
f(-1,0)
f(-1,1)
f(0,-1)
f(0,0)
f(0,1)
f(1,-1)
f(-1,-1)
f(-1,0)
f(-1,1)
f(0,-1)
f(0,0)
f(0,1)
f(1,-1) 3
SVR
DWT V
Original Image

26. Proposed theory and algorithm
where fm is the modified wavelet coefficient, p=0.3×max( f ) .
Max( f ) is the maximal value of the wavelet coefficient in that detail subband.

27. Experimental results and discussion

28. Experimental results and discussion

29. Reference
1 Mallat, S.G.: ‘A theory for multiresolution signal decomposition: the wavelet representation’, IEEE Trans. Pattern Anal. Mach. Intell., 1989,11, (7), pp. 674–693
2 Donoho, D.L., and Johnstone, I.M.: ‘Ideal spatial adaptation via waveletshrinkage’, Biometrica, 1994, 81, pp. 425–455
3 Chang, S.G., Yu, B., and Vetterli, M.: ‘Adaptive wavelet thresholding forimage denoising and compression’, IEEE Trans. Image Process., 2000, 9,pp. 1532–1546
4 Vapnik, V.: ‘The nature of statistical learning theory’ (Springer-Verlag,New York, 1995)
5 Suykens, J.A.K., and Vandewalle, J.: ‘Least squares support vectormachine classifiers’, Neural Process. Lett., 1999, 9, (3), pp. 293–300