Mathematical models based on partial differential equations, the calculus of variations, and associated numerical techniques have had great success in image processing, especially in segmentation and denoising problems. This project will extend some of the most successful of these models, such as total variation based image denoising, to curve and surface denoising tasks that are of fundamental importance in computer vision and computer graphics applications. Accordingly, a central theme of this project is to find novel and natural ways to generalize models originally designed for processing images to processing curves and surfaces. This leads to curvature dependent functionals that need to be minimized over geometric objects. The project will draw on a variety of numerical techniques, such as the level set method, to develop algorithms for the solution of these challenging computational problems. It will also develop new variational image denoising and segmentation models that are more effective than current ones in multiscale decomposition of images.
The problems addressed by this project form a crucial step in diverse applications of image processing, computer vision, and computer graphics. In particular, surface denoising is a preliminary first step in many automatic detection and recognition tasks that involve three dimensional shapes, such as face recognition and target identification. It is also an essential component of algorithms that fit surfaces to volumetric data, which is needed in many medical imaging applications.
