Abstract
A new Combinatorial Ricci curvature and Laplacian operators for grayscale images are introduced and tested on 2D medical images. These notions are based upon more general concepts developed by R. Forman. Further applications are also suggested.
Highlights
Curvature analysis plays a major role in Image Processing, Computer Graphics, Computer Vision and their related fields, for many applications, such as reconstruction, segmentation and recognition, to list only a few
Before commencing any experiments with the combinatorial Ricci curvature in the context of images, we had to choose a set of weights for the 2, 1- and 0dimensional cells of the picture, that is for squares, their common edges and the vertices of the tilling of the image by the pixels
The choice of weights was motivated by two factors: the context of Image Processing, where a natural choice for weights imposes itself and the desire to employ solely standard weights
Summary
Curvature analysis plays a major role in Image Processing, Computer Graphics, Computer Vision and their related fields, for many applications, such as reconstruction, segmentation and recognition, to list only a few (see, e.g. [12], [13]). It is important to note that in dimension n = 2, that is in the case that is the most relevant for classical Image Processing and its related fields, Ricci curvature reduces to sectional (and scalar!) curvature, i.e. to the classical Gauss curvature. Both in the more classical context, as well as in the new directions mentioned above, smooth surfaces and/or their polygonal approximations considered. While we succinctly present some of the more general facts residing in Forman’s work, in this paper we concentrate solely on the case of grayscale images with their very special combinatorics and weights, and defer the study of higher dimensional images and their curvatures and Laplacians for further study [11]
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