PDE-based Image Denoising And Image Segmentation

Since the1960s, with the increasing popularity and advancement of electronics and computer technology, in particular multimedia technology and information technology, image processing has become a widespread concern, and more and more research into image processing technologies has been conducted in computer vision and simulation, medical diagnosis, remote sensing and astronomical observations among other fields.At present, image processing methods are divided into the following three categories: Fourier and Wavelet transform methods based, probability and statistical methods based and variational partial differential equations (PDEs) based. Because of dealing directly with geometry such as image gradient and curvature, since1990s, PDE-based mathemat-ical imaging models and their numerical solution methods have gained much success and fast development in the last two decades. Research areas in image processing include: image segmentation, image denoising and deblurring (inverse de-convolution), image de-composition. image inpainting, image reconstruction and image texture classification and so on. Our works in this PhD project focus on two PDE-based applications:image de-noising and segmentation. After discussing some basic and useful background material in Chapters1-2, the rest of the thesis contains these main contributions:In Chapter3, an iterative multiplier method is proposed to accurately solve KKT system for the classical constrained ROF image denoising model. After constructing an Lagrange energy functional, the convexity of the functional and existence and uniqueness of the solution for minimization problem are analyzed and proved when the multiplier in energy functional is fixed, the monotonicity of the constrained functional with the multi-pliers also is given. A new multipliers updating algorithm is proposed to choose a better value in some feasible multipliers interval. Then a multigrid with some Krylov accelerat-ed subspace method is employed to solve the Euler-Lagrange equation to determine next feasible multipliers interval, alternately implementing these two processes and ultimately obtaining the solution of the KKT system. Experimental results show that the method yields better denoising results, in comparing with the Landi augmented multiplier method for choosing the smoothing parameter.In Chapter4. an improved multigrid method is proposed to solve the3D Chan-Vese non-convex model. The main goal is to deal with low efficiency of the traditional algorithm and a multigrid method with standard smoothers for the3D Chan-Vese image segmen-tation problem. Three adaptive smoothers are proposed to improve the performance of some smoothers generalized from2D work. A local Fourier analysis is used to choose the adaptive parameters. Tests show that the new smoothers can eliminate effectively the effect of high frequency oscillation of error. They also show the new multigrid has faster computation speed and better performance than the time marching method and the widely used AOS method.In Chapter5, a selective segmentation approach based on local image features is proposed to overcome the global statistics of the similar features of the Chan-Vese model and the weakness of the Badshah-Chen segmentation model. Our approach captures local image features surrounding the present zero level set to determine the evolution of the level set so the features far away from the zero level set do not control the motion of the level set. Numerical experiments show that this improved method can extract the desired objects with complex structures with excellent robustness.In Chapter6, we generalize the above2D selection segmentation approach based on local image features to the3D case. Such a directly generalized3D model may not work or may be inefficient with a general initial level set. As our first step, an efficient and yet. robust initial guess strategy is proposed. The strategy uses only a few markers on two or three2D slices to construct a polyhedral surface approximating the object shape. In addition, a new marching tube method restricting the evolving computation in a narrow region, combined with a multigrid re-initializing method to force the level set function to be a signed distance function, is employed to reduce the computational cost. Numerical experiments show the proposed algorithm has high efficiency and robust performance for3D medical extraction of various realistic organs (3D shapes).