Image Denoising And Compression Based On The Elliptic And Parabolic Systems

Image processing techniques have made a breakthrough over the last thirty years,thanks to the fast development of computer technology and the rapidly growing demand in the fields such as industry,medical diagnosis,aeronautics and astronautics.A typical example is the image processing method based on partial differential equations(PDEs),which has been widely used in image denoising,image segmentation,image fusion,et al.In this dissertation,we utilize elliptic system and parabolic equation(system)to model the problems of image denoising and color image compression,establish the wellposedness of solutions to the proposed PDE systems,and solve them by the finite difference scheme to verify the validity in image processing.First,this dissertation investigates an elliptic system coupled in source terms and its application in restoring images contaminated by Gaussian noise.The proposed system consists of the p(x)-Laplacian equation and the Poisson equation,where the p(x)-Laplacian equation is used for removing noise in images and the Poisson equation is used to update the source term of the p(x)-Laplacian equation.Under the assumptions that the variable exponent p(x)is allowed to be discontinuous and the noisy image f(x)belongs to L1(?),we prove the existence and uniqueness of weak solutions for the homogeneous Neumann boundary problem in a non-standard variable exponent Sobolev space when p-> max{1,n/3},where p-:= infx ??p(x)and n is the dimension of ?.Notice that the system is coupled in source terms,we utilize the "divide and conquer" strategy,which splits the system into two independent subproblems and utilizes the Schauder's fixedpoint theorem for connecting the subproblems and the system.The proof also provides an iterative scheme for solving the system numerically.Experimental results illustrate that the piecewise constant variable exponent constructed by the local K-means algorithm performs better than the commonly used smooth variable exponent in image denoising.Second,this dissertation proposes a color image compression model based on a linear degenerate reaction-diffusion system.Under the assumption that we have the luminance component and a few representative pixels extracted from the original color image,by explicitly introducing the relation between the luminance component and the original color image into diffusion system,we propose a linear reaction-diffusion system with Perona-Malik type diffusion coefficient to reconstruct the color image.The diffusion coefficient is a function of the luminance component,so that the restored color image and the luminance component have similar geometrical structure.It also leads to interior degeneration of the reaction-diffusion system.By the approximation method,we prove the existence and uniqueness of weak solutions for the proposed system with a specific class of diffusion coefficients in a weighted Sobolev space.The image compression model proposed in this chapter is based on the image colorization idea demonstrated as above.That is to say,in compression,we store only the compressed luminance component and a few representative pixels extracted from the original color image.In decompression,we reconstruct the color image by the proposed reaction-diffusion system.A local-optimal algorithm is also proposed for selecting representative pixels.It splits the original color image into a series of different size subimages and searches the optimal representative pixel in each subimage.Comparisons with recent colorization-based image compression methods,as well as transform-based JPEG and JPEG 2000 standards are performed to show the potential for successful compression applications of the proposed method.At last,this dissertation proposes a diffusion equation in non-divergence form for the removal of impulse noise in images.According to the property of impulse noise,it randomly corrupts only a portion of pixels in images.Existing PDE-based methods can not remain values of uncorrupted pixels unchanged in the process of removing noise.We introduce an impulse noise indicator into the diffusion equation to fix this problem.In the proposed equation,the impulse noise indicator keeps values of uncorrupted pixels unchanged,and the regularized Perona-Malik equation has the ability of removing impulse noise efficiently.The asymptotic behavior of the proposed equation observed in numerical experiments solves the problem of choosing the stop time in the finite difference scheme.It also enables us to extend the proposed equation for removing impulse noise and mixed Gaussian noise easily.Numerical results shows significant improvements in denoising when compared to recent median-type filters and PDE-based/variational methods.