| In this thesis we study some applications of geometric partial differential equations and differential invariants for problems in computer vision and image processing.; A pixel matching approach to the stereo-disparity problem is presented based on total variation. Two curvature driven evolution equations can be simultaneously solved to get two disparity functions one for each left and right image. The advantage is that the disparity functions are not excessively smooth and therefore preserve depth information in the images much better.; We also present a new knowledge-based approach for segmentation. A priori knowledge about the number of objects present in the image, e.g., for SAR images target, shadow, and background terrain, is introduced via Bayes' rule. Posterior probabilities obtained in this way are then anisotropically smoothed, and the image segmentation is obtained via MAP classifications of the smoothed data. When segmenting sequences of images, these smoothed posterior probabilities are used as prior probabilities in the succeeding frame.; We next give an explicit method for mapping any simply connected surface diffeomorphic to the sphere onto the sphere in a manner which preserves angles. This technique relies on certain conformal mappings from differential geometry. Our method provides a new way to automatically assign texture coordinates to complex undulating surfaces. We demonstrate a finite element method that can be used to apply our mapping technique to a triangulated geometric description of a surface. Such a technique has applications to surface warping in medical imaging, and in particular virtual colonoscopy, and brain flattening for functional magnetic resonance imaging. Extensions to surfaces with other topologies are given, as well as methods for improving these flattening maps.; Finally, we discuss some results from the classical theory of differential invariants for invariant object recognition. This involves the computation of signature curves and more generally signature manifolds which contain the minimal number of invariants to determine the object up to given group transformation. Numerical schemes for the efficient computation of some of these invariants are also indicated. |
