Geometric and Multiscale Aspects of Image Denoising Models

基于偏微分方程、变分演算和相关数值技术的数学模型在图像处理中取得了巨大成功，特别是在分割和去噪问题上。该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表面去噪任务。因此，该项目的中心主题是寻找新颖和自然的方法，将最初设计用于处理图像的模型推广到处理曲线和曲面。这导致需要在几何对象上最小化曲率相关函数。该项目将利用各种数值技术，如水平集方法，来开发解决这些具有挑战性的计算问题的算法。在多尺度图像分解方面，将开发出比现有模型更有效的变分图像去噪和分割模型。该项目解决的问题构成了图像处理、计算机视觉和计算机图形学等多种应用的关键一步。特别是，在许多涉及三维形状的自动检测和识别任务中，表面去噪是初步的第一步，例如人脸识别和目标识别。它也是将曲面拟合到体积数据的算法的重要组成部分，这在许多医学成像应用中是必需的。

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.