Tuesday, 10.3.2015, 11:30
A discrete model of the Ricci flow for images is presented, using a purely combinatorial method for calculating the Ricci curvature, based on R. Forman's work on generalized Laplacians for cell complexes. The adaptation of Forman's curvature function for cell complexes to images is natural and straightforward, using geometric properties of the image as a surface embedded in Euclidean space. It is shown that the framework presented is far more applicable to images than other existing models. This work focuses on analyzing the proposed scheme and the different methods of its implementation. Several weighting methods for the calculation of the Ricci curvature are suggested and compared, and different implementations of the discrete Ricci flow are explored. The model is analyzed in terms of curvature validity, conformality, convergence and numerical stability. Simulation results show that the discrete Ricci flow has a powerful effect on the image and its curvature, and evolves the surface metric in a way that is consistent with theoretical expectations. Finally, some applications for image processing and computer vision are proposed, such as forward-and-backward Ricci flow, high dynamic range imaging, and change detection in aerial images.