Image Processing On Implicit Surfaces Based On PDEs

Image processing has been a hotspot research field in recent thirty years which contains many problems such as denoising, deblurring, edge detection, segmentation, inpainting, and so on. Processing techniques for planar images are well developed in the past decades and many efficient algorithms are proposed. Among them, PDE-based methods developed in recent years attract the attentions of researchers especially. However, there is little work on processing images on surfaces. In many applications (especially in computer graphics), one should consider not only geometric information but also color information and texture features on the surface when describing an object. Therefore color and texture processing is needed when the geometry information of an object is being processed. In this paper we discuss several problems about processing images on implicit surfaces.Classical methods for planar image processing are difficult to be applied to processing images on implicit surfaces. In this thesis, we generalize PDE-based methods for planar image processing to image processing on implicit surfaces via differential geometry and variational framework. The basic approach is that, we construct some fit energy functionals at first, then deduce the corresponding PDEs via variational methods. After image data defined on implicit surfaces are extrapolated to some narrow bands, these PDEs are solved numerically. Our work includes the following aspects:1. By intrinsic gradient operators of implicit surfaces, we present isotropic and anisotropic frameworks for processing images on implicit surfaces. For isotropic framework, a general energy functional is proposed at first, and then its variation and corresponding Euler-Lagrange equation are deduced. For anisotropic framework we directly construct a general PDE.2. According to the theory of vision phycology and Bayesian framework, we propose Weberized intrinsic TV model for denoising images on implicit surfaces.3. According to the theory of vision phycology and Bayesian framework, constraint energy functional is constructed and then an intrinsic TV model for inpainting images on implicit surfaces is proposed.4. Enlightened by the scale-space theory of planar images, we put forward micro-...