Recent advances in the analysis of high-dimensional signal data have triggered an
increasing interest in geometry-based methods for nonlinear dimensionality reduction …

A new Combinatorial Ricci curvature and Laplacian operators for grayscale images are
introduced and tested on 2D synthetic, natural and medical images. The effectiveness of the …

Abstract Representation of one-dimensional (1D) signals as surfaces and higher-
dimensional manifolds reveals geometric structures that can enhance assessment of signal …

Motivated by the methods and results of manifold sampling based on Ricci curvature, we
propose a similar approach for networks. To this end, we make an appeal to three types of …

We introduce a novel method of image sampling based on viewing grayscale images as
manifolds with density, and sampling them according to the generalized Ricci curvature …

Clustering and sampling are key methods for the study of relational data. Learning efficient
representations of such data relies on the identification of major geometric and topological …

We consider the problem of better approximating surfaces by triangular meshes. The
approximating triangulations are regarded as finite metric spaces and the approximated …

This book introduces differential geometry and cutting-edge findings from the discipline by
incorporating both classical approaches and modern discrete differential geometry across …

Relationships that exist between the classical, Shannon-type, and geometric-based
approaches to sampling are investigated. Some aspects of coding and communication …

A discrete version of the Ricci flow, applicable to images, is introduced and applied in image
denoising and in single-image-based enhancement and super-resolution. This flow is …