A Class Of Forward-backward Singular Diffusion Equations For Image Processing

As a kind of intuitive, convenient media to reflect the real world, image is widely applied to multimedia, digital medical treatment, artificial intelligence. In many application fields, there is a huge demand for clear and high quality images. Therefore, it has very important significance to restore images by denoising and improving quality. In this paper, we deal with image denoising and image enhancement by using forward and backward diffusion equations. We analyze our models by partial differential equation theories,and design numerical algorithms of partial differential equations. We implement forward–backward diffusion equations based models for image denoising and enhancement, and compare them with other existing algorithms. The main contents are as following:To deal with additive noise, we propose a new variational framework consisting of a non convex functional and linear growth convex functional, and then propose a forward–backward diffusion equation model for noise removal. We consider the discontinuous points and introduce the Young measure theory to prove the existence of weak solution of the evolution equation. Under some conditions, we prove the uniqueness, stability,extremum and comparison principle of Young measure solution to obtain the asymptotic property. In numerical aspects, we propose two numerical discrete schemes, the PM scheme(PMS) and the AOS scheme. In addition, we propose the similarity of edges to measure the restoration performance, which is closer to the real human eye feelings.Experimental results illustrate the proposed model avoid the stair case effect of TV model and spot effect of PM model, and perform well in edge enhancement.Then we focus on the problem of multiplicative noise removal. We first propose a gray level indicator based on the properties of the noise to obtain an adaptive total variation regularizer, and then introduce a new fidelity term which is global convex to derive an adaptive total variation model for multiplicative noise removal. In theoretical aspect, we prove the existence, uniqueness, and comparison principle of the minimizer for the variational problem. Furthermore, Because of the nonlinearity and singularity, we introduce a new weak solution, regularize the initial data and equation twice and approximate the total variation by using p-Laplace equation and establish the existence, uniqueness, and long-time behavior of the associated evolution equation. In numerical aspects, we propose two type of numerical scheme for the evolution equation. One is similar to the numerical scheme of total variation equation and the other one is approximating the numerical scheme of total variation equation by using the numerical scheme of p-Laplace equation.We also utilize the dynamic balance parameter calculation formula to reduce the number of parameters to promote efficiency. Our algorithm performs among the best and even outperforms other classical algorithms most of the time both visually and quantitatively.As traditional coherence-enhancing diffusion filtering(CED) completes the interrupted lines and gaps but at the cost of reducing the image contrast, we introduce a source term into CED filtering to restore the initial image and the contrast lost by pure diffusion filters. Meanwhile, the new model is also suitable for dealing with white noise.We assessed our method in terms of the theoretical and numerical properties changed by the source term. We prove the existence and uniqueness by using Schauder fixed point theorem and introduce a parameter to control the source term. In our numerical assessment, we implement our approach using an explicit scheme, which was accelerated by fast explicit diffusion. Compared with CED filtering, our proposed approach performance better on restoring fingerprint and significantly enhance the texture. At last, using the similar method as before, we propose the image segmentation model based on the forward-backward diffusion equation with source terms.