Study On Methods Of PDEs About Image Processing

In the past few decades, In the field of computer visualization and image analysis, the based-model of partial differential equations played an important role in the area of image processing. The idea of image processing using partial differential equations can be traced back to Gabor and Jain, but this method was really built up by means of the beginning study of Witkin and Koenderind. The concept of Scale Space(Scale Space)was introduced by them.A group of images can be stated by some scales in Scale Space. To a large extent, their contribution constituted bases of the theory of image processing of partial differential equations. In their studies, the multi-scale of image was obtained through the Gaussian smoothing, which is equivalent to the use of classic heat conduction equations which evolved figures to get an isotropic flow. In the late 80's, Hummel pointed out that the heat conduction equation is not the only parabolic equations which brought Scale Space, and gave the principle of scale space constituted: a Scale Space can be defined as long as meeting the greatest principles of the evolution equation. Perona and Malik proposed anisotropic diffusion equations(P-M model)that were the most influential in this area and could solve blurring the image of the characteristics of the edge when denoising figures. We know that the Gaussian image smoothing filter is the most common method,and the main steps of the method made convolution between the initial figure u₀ and Gaussian kernel,that is to say, set the initial gray-scale images for u₀=u(x, y, 0), using Gaussian kernelto made convolution with u₀, then obtained the smoothing pictures at time Koenderind pointed out, the smooth image by means of (2) is equivalent to the solution of isotropic heat conduction equation(3)satisfied the minimum-maximum principleFrom the mathematical point of view and image denoising, It's no doubt that Gaussian filtering was very simple and easily achievable, but its low-pass filtering characteristics of image reduced the high-frequency part of the spectrum consistently, made the results of treatment of the characteristics of apparently blurred, and result in seriously the image visibility.However, P-M model:where the diffusion coefficient was:Here div is divergence operator, (?) is the gradient operator, |(?)u| is the image of the gradient mode, the diffusion coefficient c(s) which is used to control the proliferation of speed is a non-negative function of | (?)u|. The ideal diffusion coefficient should make fast changes of anisotropic diffusion in flat gray area, rapid changes in the location of gray (that is, image features) slow and stop the spread. P-M models have the spread of selective smoothing, good balance between noise de-noising and edge protection. But P-M models have the inadequacy of the model itself that P-M is sick in mathematics. Therefore, by modifying the P-M equation to eliminate the pathological nature. Literature [30] pointed out that,in (6),(6)is known as the regularization P-M equation, also known as CLMC model. It has been proved that the type (6) is a full-posed problem in mathematics. Chapter II of this paper, the second-order partial differential equations of image denoising algorithms (ie, AOS algorithm) is based on regularization of the P-M model to start from, Simulation results show that with the increase in the number of iteration, image denoising have good results, but the characteristics of images has become increasingly marginal,appearing a "massive", which is a second-order partial differential equations of the inherent characteristics of image denoising, the algorithm of the AOS is no exception.As the second-order partial differential equations in image denoising, the image characteristics of the edge time become increasingly blurred over the evolution of time. In that case, can Image Denoising better , at the same time protect the characteristics of the edge of this image? The answer is yes. Yu-Li You and M. Kaveh proposed fourth-order partial differential equations used for image denoising, set image enhancement based function u , t is time variable. Consider the energy functionalshereΩis the image support, f(·)≥Ois an increasing function associated with the diffusion coeffcient assecond-order evolution equations associated with(7) rewrite(7), that is to say,fourth-order evolution equations associated with(10)Well, according to (11), we carry out simulation experiments using the image of u₀, the simulation results show that fourth-order partial differential equations denoising image While protecting good characteristic of the edge of the image, but after some time, the image will appear "spot".Now we have to ask, can reduce the order of partial differential equations,and the effect of simulation goes well in both denoising image and protecting the image of characteristics of the edge? Answer is yes. In recent years, the fractional order partial differential equations in computer vision etc. has some applications, although it is less prevalent than P-M models, T-V models in image processing. In fact, the fractional order differential appeared while the integer-order differential not at the same time, but fractional order differential equations encountered a number of obstacles in the practical applications, which is one of the reasons lagging behind in the use of integer-order partial differential equations.In this article raised and (7) similar to the energy functional:hereΩis image support and f(.)≥Ois an increasing functionassociated with the diffusion coeffcient asfractional-order evolution equations associated with(12)Takeα∈[1,2], using methods similar to the finite element solution of the development equation (14)). The so-called similar finite element method to select the basis function isΦ_ij(x, y), both sides of the development of equation (14) made inner product with based functionsΦ_ij(x,y), where i, j depends on image size M x N. According to the simulation methods, experimental results show that Compared to second-order and fourth-order partial differential equations, fractional order partial differential equations have more than the second-order and fourth-order partial differential equations Partial differential equations to obtain a better signal to noise ratio and visual effects in image denoising and edge protection.