Some Applications Of Riemannian Geometry In Digital Image Processing

It is a significant and challenging subject to apply the theory of Riemann Geometry into the fields of digital image processing. In this thesis, we take the above mentioned subject to spread our work. The main work includes:1. Intrinsic image segmentation. We propose an image segmentation model based on the view of Riemannian Geometry. Endowing with a Riemannian metric that derived from the image gray scale information, the image space became a Riemannian manifold. Then we evolved a curve by the mean curvature flow using level set methods on the Riemannian manifold. Because the gray scale information had been embedded into the Riemannian manifold, the evolution was intrinsic. Calculations show that the method is an extension of existing models, furthermore, finer controls for the evolution can be achieved. Numerical experiments show the effectivity and some good features of the proposed method.2. The ROF model on Riemannian manifolds. We generalize an image processing model based on minimizing total variation introduced by L. Rudin, et al. to Riemannian manifolds, obtaining a model framework. Then we consider several special cases. The framework unifies many classical schemes and algorithms, and results in new and efficient schemes.3. Image objects detection on surfaces. We introduce the model and algorithm for segmentation method based on curve evolution constrained on surfaces. The surfaces are implicitly embedded into a volume data in R~3 as its zero level sets, and image information is also extended smoothly into the volume. The we evolve the volume data using level sets method, and the image objects are then detected. Owing the level sets method, the proposed model and algorithm properly deal with the problems caused by the topologies of the surfaces and also the problem sin the curve evolution.