Geometric and Multiscale Aspects of Image Denoising Models

基于偏微分方程、变分法和相关数值技术的数学模型在图像处理中取得了巨大的成功，特别是在分割和去噪问题上。本项目将把其中一些最成功的模型，如基于全变分的图像去噪，扩展到在计算机视觉和计算机图形学应用中至关重要的曲线和曲面去噪任务。因此，这个项目的一个中心主题是找到新颖和自然的方法来将最初为处理图像而设计的模型推广到处理曲线和曲面。这导致需要最小化几何对象上的曲率相关泛函。该项目将利用各种数值技术，如水平集方法，来开发算法来解决这些棘手的计算问题。它还将开发新的变分图像去噪和分割模型，这些模型在图像的多尺度分解中比现有的模型更有效。本项目所解决的问题是图像处理、计算机视觉和计算机图形学的各种应用的关键一步。特别是，表面去噪是许多涉及三维形状的自动检测和识别任务的第一步，如人脸识别和目标识别。它也是将曲面与体数据进行匹配的算法的重要组成部分，这在许多医学成像应用中都是必需的。

Mathematical models based on partial differential equations, the calculusof variations, and associated numerical techniques have had great successin image processing, especially in segmentation and denoising problems.This project will extend some of the most successful of these models, suchas total variation based image denoising, to curve and surface denoisingtasks that are of fundamental importance in computer vision and computergraphics applications. Accordingly, a central theme of this project is tofind novel and natural ways to generalize models originally designed forprocessing images to processing curves and surfaces. This leads tocurvature dependent functionals that need to be minimized over geometricobjects. The project will draw on a variety of numerical techniques, suchas the level set method, to develop algorithms for the solution of thesechallenging computational problems. It will also develop new variationalimage denoising and segmentation models that are more effective thancurrent ones in multiscale decomposition of images.The problems addressed by this project form a crucial step in diverseapplications of image processing, computer vision, and computer graphics.In particular, surface denoising is a preliminary first step in manyautomatic detection and recognition tasks that involve three dimensionalshapes, such as face recognition and target identification. It is also anessential component of algorithms that fit surfaces to volumetric data,which is needed in many medical imaging applications.