Salt And Pepper Noise Removal Based On Minimization Of A Functional With A Weighted Parameter

In our real world, images exist everywhere. Using some equipments, we can obtain images which can be stored in some forms, such as the digital image in computers. There is always noise in the obtained images. Before some image processing, we need to remove the noise in order to restore the original image as possible as we can. Therefore, the old topic, image denoising,has great attraction. A number of scholars work on the topic. The study of image denoising is of great importance.The principal sources of noise in digital images arise during image acquisitionand/or transmission. For example, salt and pepper noise is caused by malfunctioning pixels in camera sensors, faulty memory locations in hardware,or transmission in a noisy channel.In general, image contamination by noise can be seen as a degradation process, which can be described as the following model:where f(x, y) is the original input image, g(x, y) is the contaminated image, and n(x, y) is the additive noise term. In the equivalent frequency domain representation, the model can be rewritten as:where the terms in capital letters are the Fourier transforms of the correspondingterms in spatial domain representation. Reversely, image denoising is a restoration process. The denoising technique uses the reverse process, and try to reconstruct or restore the original image by using prior knowledge. The aim of denoising is to obtain the approximation to the original image. We hope that the result can approximate the true image as possible as we can. The salt and pepper noise will appear as a light dot or a dark dot in the image. Let x_i,j, for (i,j)∈A≡{1, 2,…, M}×{1, 2,…, N}, be the gray level of a true M-by-N image x at pixel location (i,j), and [s_min s_max] be the dynamic range of x. Denote by y the image contaminated by salt and pepper noise. Then, the observed gray level at pixel location (i, j) is givenbywhere r = p + q defines the noise level.There is a great deal of work on the study of restoration the image contaminatedby salt and pepper noise. These method all have their advantages and disadvantages. In this paper, we deal the denoising problem with two stages. In the first stage, AMF is used to detect the noise candidates, and the noise candidates set is defined aswhere y is the noisy image, and (?) is the result of AMF. In the second stage, minimizing the following functionalwhere V_ij is the set of our closet neighbors of the pixel at position (i,j)∈A, u∈R^|N|, andφ_αsatisfies some assumptions. Let x be the true image, u^* be the restored image, and u be the minimizer of F_α(u). Then, we will set The general conjugate gradient method applied to (?)f(x) has the followingform. Given x₀, letwhere g_k = (?)f(x_k), d_k is the k-th conjugate direction, andα_k is the step length which is determined by line-search. Since line search is costly, researchersproposed the conjugate gradient method without line search which substitute the line search with a fixed formula:where {Q_k} is a sequence of positive definite matrices which satisfy some requirement, andδis a parameter. When f(x) has some good properties, the method has a global convergence.In our numerical experiment, we present the result for 30%, 50%, 70% and 90% salt and pepper noise. The best weight parameterβand the correspondingPSNR, iteration steps and CPU time are provided. Some pictures are also presented.