英语翻译To overcome this problem,the Wiener filter has been exte

英语翻译
To overcome this problem,the Wiener filter has been extended to multiple-bases representations for noise removal.Mihcak and Kozintsev^([1]) approached the signal estimation problem from the perspective of designing the Wiener filter in the wavelet domain.The technology indirectly yields an estimate of the signal subspace that is leveraged into the design of the filter.This paper studies the problem of nonlinear Wiener filtering in reproducing kernel Hilbert spaces via least square support vector regression,The method reflected new perspectives within the framework of kernel methods for denoising problem.Experimental results confirm a significant improvement in image denoising.
Least support squares vector regression is a new universal learning machine proposed by Suykens etal.^([2]) Let x∈R^d,y∈R,R^d represent input space,d is the dimension.By some nonlinear mapping ∅,x is mapping into some a prior chosen Hilbert space spanned by the linear combination of a set of functions.
with ∅(x):R^d→R.
Such that the following regularized risk function J is minimized:
The parameter γ is a positive regularization constant.After elimination of w,e one obtains the solution:
Where Y=[y_1⋯y_N],ρ_1=[1⋯1],α=[α_1⋯α_N] and Ω=K+γ^(-1)I .The resulting least support squares vector regression model for function estimation becomes:
where K(x,x_i)=∅(x) ∅(x_i)(i=1,⋯,N) is the kernel function and must satisfy the Mercer condition,^([3]) α are Lagrange multipliers and b almost equals the mean of y.
Consider a 2D image consisting of a matrix of M=N×N pixels,the observation image can be regarded as a function in pixel areas y=f(i.j); R^2→R^1,where input (i,j) is 2D vector equals to the row and column indices of that pixel,where output y is the approximated intensity value.^([4]) The Lagrange multipliers α_(i,j) of the observed image pixel y(i,j) can be easily calculated using Eq.(3).
where A=Ω^(-1),B=(I^T Ω^(-1))/(I^T Ω^(-1) I) and O_α is a N×N matrix defined by A(I-IB).Notice that,
the Lagrange multipliers α_(i,j) of the observed image pixel y(i,j) is determined by the multiplication of the matrix O_α and the observed image Y.That is,the Lagrange multipliers are influenced by the clean image S and random noise N.As in Eq.(4),the observed image can be reconstructed by a linear combination of kernels with weights equal to the values of Lagrange multipliers and an appropriate support vector regression can concentrate the signal energy into a number of support vectors(SVs) that α_(i,j) is nonzero.
The localization of SVs is particularly appropriate for imaging applications,where it is crucial to preserve fine details like edges and textures.Pixels with positive Lagrange values try to raise the grey levels of themselves and their neighbors,while those with negative Lagrange multipliers will try to reduce the grey levels and they appear darker.
Therefore,the Lagrange multipliers effectively weigh the kernel functions to estimate intensity value of image.Furthermore,random noise can be considered as forces that try to make Lagrange multipliers to oscillate above and below the standard value.The noise can be reduced by smoothing the value of Lagrange multipliers,whereas sharp edges may be preserved within certain ranges which rely on a suitable kernel function possessing the capability of nonlinear representation.
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