Wavelet domain image denoising via support vector regression Source: Electronics Letters,Volume:40,Issue: 2004 PP :1479-1481 Authors: H. Cheng, J.W. Tian, J. Liu and Q.Z. Yu Presented by: C.Y.Gun Date: 4/7 2005

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

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

Introduction Father wavelets (low-pass filters)

Introduction Mother wavelets (high-pass filters)

Introduction 2-D DWT for Image

Introduction 2-D DWT for Image

Introduction 2-D DWT for Image

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

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 »

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

Support vector regression Graphical representation

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

Support vector regression Kernel methods：

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 )

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

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

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

LS－SVM Regression

LS－SVM Regression (1) ….(3)

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

Proposed theory and algorithm = = where

Proposed theory and algorithm (3)

Proposed theory and algorithm

Proposed theory and algorithm 3 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

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.

Experimental results and discussion

Experimental results and discussion

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