The Study On Image Restoration Algorithms

Image restoration is one of the important and difficulty research topics in digital image processing fields. The study on image restoration has important application value and becomes one of the hot spots in the research fields of image processing. Image restoration becomes necessary when we wish a crisp deblurred image for viewing or further processing. The degradation procedure is often modeled as the result of a convolution with a low-pass filter f(n1,n2)=(u*h)(n1,n2)+γ(n1,n2)(1) where f, u are the original image and the degraded image, respectively.γ is the noise introduced in the procedure of image acquisition, and it is generally assumed to be additive white Gaussian noise(AWGN) with variance a2, h denotes the point spread function(PSF).’*" denotes convolution. The process of degradation is known to be an ill-posed problem. Thus, to obtain a reasonable image estimate, a method of reducing controlling noise needs to be utilized.For the restoration of depredated image, The contributions of this paper mainly includes the following several aspects:I. Present an image restoration algorithm based on gradient. Image restoration from degraded image is known to be ill-posed. To stabilize the recovery, total variation(TV) regularization is often utilized for its beneficial in preserving the image edge property. We propose the image restoration algorithm based on gradient. The algorithm could obtain higher quality image compared with the traditional restoration method based on TV. The proposed algorithm mainly includes the following two steps:1) Difference images restoration with TV method.We utilize the TV model to process the difference images and could obtain most of the edge information of image while suppress the noise. a) Given blurred observation/with size N x M, the Fourier observation F f could be obtained through fast Fourier transform, then the discrete Fourier transform of horizontal difference image fx and vertical difference image fy are obtained as following:(F fx)k＝(1-e-2πik1/N)(Ff)k (F fx)k＝(1-e-2πik1/N)(Ff)k where Fdenotes the discrete Fourier transform.b) Recover the difference image by TV based image restoration. The horizontal and vertical degraded images have the following form: fx=h*ux+γx (2) fy=h*uy+γy (3) where (γx=F-1((1-e-2πik1/N)(Fη))(4)(γx=F-1((1-e-2πik1/N)(Fη))(5) The estimations ux, uy of ux and uy could be obtained by solving the above two equations 2) Original Image Restoration by the Simple Least Square Optimization. After obtain the ux and uy, the estimation u of u could be restored by the difference images ux, uy and degraded image F. The estimation u is calculated by solving the following least square optimization problem: u=argmin {||ux-ux||2+||uy-uy||2+β(||ux||2+||uy||2)+λ||u*h-f||2}(6)The solution of Eq.(6) could be written directly in Fourier domain: where A1=1-e-i2k1π/N, A2=1-e-i2k1π/M, U, Ux, Uy, U, H, F are the fast Fourier transform of u, ux, uy, u, h and F, respectively. A1, A2, H are the complex conjugate of A1, A2and H, respectively. The addition, multiplication, and division are all component-wise operators. Compared to minimizing Eq.(6) directly in the image space, which involves very large-matrix inversion, computation in the Fourier domain is much faster due to the simple component-wise division.Ⅱ. Present and edge-preserving image restoration algorithm based on guide filter.The guided filter could be expressed as a formula: q=guidfilter(I,p,ω,ε), where ω is the size of choosing kernel, ε>0is the regularization parameter,I is the guidance image, and p is the image for filtering, u is the filtered image. Both I and p are given beforehand according to the application, and they could be equal. The efficient iterative algorithm with the decouple of deblurring and denoising steps in the restoration process is proposed. 1) In the deblurring step:We propose two cost functions: where ue is a pre-estimated image, and λ>0is the regularization parameter. After the deblurring step, we obtain two images uI and up, where up contains the more leaked noise and more details than uI.2) In the denoising step:To suppress the amplified noise and artifacts introduced by the deblurring step, the guided filter is applied to smooth the estimated image up, and uI is used as the guidance image. That is to say, u=guidfilter(uI,up, ω,ε). In order to obtain a better recovery result, the deblurring step and denoising step are operated alternating iterative in the proposed algorithm.3) The Effective method to Compute the Parameters Automatically. In the deblurring step, the deblurred images depend greatly on the degree of regularization which is determined by the regularization parameter A. We present a simple but effective method to compute the parameter automatically. The proper parameter A is chosen by where the p is computed as follow: where μ(f) denotes the mean of f.Ⅲ. Present an image restoration algorithm based on separation.In the proposed algorithm, the image is divided into two parts:cartoon part and texture part. For the two parts of image, we first present the image restoration algorithm based on L0gradient minimization for the smooth part restoration and propose the wave atom-based wiener shrinkage filter to extract details for the texture part restoration. The approach we take for texture-preserving image deconvolution starts with the optimization frame work making use of the proposed L0gradient minimization based image restoration, which can globally control how many non-zero gradients are resulted in to approximate prominent structure in a sparsity-control manner. We obtain an approximation of main edges of original image with minimal loss of image detail components by the first step. This method could smooth image well, while maintain main edges and contour structure. This paper confine the discrete number of intensity change among neighboring pixels, which links mathematically to the L0norm for information sparsity pursuit. This idea also leads to an unconventional global optimization procedure involving a discrete metric, whose solution enables diversified edge manipulation according to saliency. The qualitative effect of our method is to thin salient edges, which makes them easier to be detected and more visually distinct. The proposed method mainly contains the following three steps.1) Image restoration of main edges and contour structure. We define the gradient▽up=((?)xup,(?)yup)T for each pixel p is calculated as pixel difference between neighboring pixels along the x and y directions, where p=(n1,n2). Our L0gradient measure is expressed as: C(u)=#{p||(?)xup|+|(?)yup|≠0}(12) It counts p whose magnitude|(?)xup|+|(?)yup|is not zero. With this definition, the initial estimation of smooth part of original image u is calculated by solving:The solver u1is an approximation of smooth parts of original image. 2) Extract details by wave atom-based wiener shrinkage filter.a.l) Fourier-Tikhonov shrinkage for residual image. After recovering contour image u1, the lose detail image could be written as residual image△n:△u(n1,n2)=u(n1,n2)-u1(n1,n2)(14)The FoRD estimation△uα of residual image Au has the following formation in the Fourier domain: where△Uα, THα are the2-D discrete Fourier transforms (DFTs) of residual estimate image△uα and leaking colored noise.a.2) Wave Atom-based Wiener Shrinkage Filter. The leaked noise γα in△uα becomes less after Fourier regularization, we apply the following Wiener shrinkage in the wave atom domain to reduce noise remaining in△uα: where ω△α,μe are the wave atom coefficients from another denoised estimate obtained by hard shrinkage to ω△uα,μ are the noise’s variance at wave atom subscript μ, Ψμ are the DFTs of φμ, β is regularization parameters. Computing the inverse wave atom-transform with ω△uα,μ, we can get the wave atom-based Wiener estimate△uαω.3) Reduce the ringing by the improved non-local means filter.Finally, we apply the improved non-local means filter in spatial domain to reduce the ringing, and this filter could suppress the leaked colored noise while preserving image details. The improved non-local means filter could surmount the problems of boundary effects, and be effective in regularizing the approximate restoration process. We obtain the restoration image by adding smooth image u1and residual image u(n1,n2)=u1(n1,n2)+△uαω(n1,n2)(17) We propose an improved non-local means filter by incorporating the spatial and intensity weights. Then the final restored result ufinal using improved non-local means filter can be calculated as following: where ΩB(i,j) denotes the set of points in the (2B+1) x (2B+1) window centered at (i,j), u(Ni,j) is the image patch centered at pixel location (i,j), and the size of this patch is (2L+1) x (2L+1), Ga is a Gaussian kernel, where a is the standard deviation.||u(nI,J)-u(Nt,l)||2,α2is the weighted Euclidean distance of the two pixels’ neighborhoods Ni,j and NI,j with equal size.