Partial differential equations in image analysis: Applications, implementations, and extensions (Mathematical morphology).

摘要

Partial Differential Equations (PDE's) have played important roles in image processing and image analysis for years. Using PDE's, one can model the images in a continuous domain, see existing methods in a different viewpoint, and combine multiple algorithms together. Furthermore, high accuracy and stability can be naturally obtained.; In this thesis, we propose some interesting applications, implementations and extensions of the PDE's in image analysis.; First, we extend the concept of level-lines to vector-valued images. Consistent with the scalar case, we define the level-lines of vector-valued images as the integral curves of the directions of minimal vectorial change. These directions and the magnitude of change are computed using classical Riemannian geometry. To exemplify the use of this new concept, we propose a novel approach to scalar representation of vector images using PDE's.; Second, we show the connections between vector median filters, morphological operators and geometric PDE's. We'll show how the concept of Median filter is related to morphological inf-sup operators and then extend this concept to vector-valued images. Considering a lexicographic order to define orders between vectors in R N, we also provide new definitions of the discrete and continuous versions of Vector Median Filters by means of a coupled system of geometric PDE's.; Third, we gives an example of the use of PDE-based segmentation techniques to detect the boundaries of skin lesions, which helps the dermatologist diagnose Malignant Melanoma, or skin cancers. A novel, morphological approach to remove the hairs covering the object of interest is also presented.; Fourth, we propose a very simple algorithm to skeletonize given gray-scale images using modified erosion/dilation equations without the need of pre-segmentation or addition of shock capturing schemes. With this scheme, the skeleton of multiple objects can be simultaneously computed.; Finally, we introduce our software, ImageRepair, which is an integration of many PDE-based image processing routines. This MS-Windows based user-friendly system is designed to assist with the analysis of images. This is a first of this kind of software that is made available to the research community. The fundamentals of some selected features of this software package are described with a few screen-shots.