The manipulation of surface homeomorphisms is an important aspect in 3D modeling and surface processing. Every homeomorphic surface map can be considered as a quasiconformal map, with its local non-conformal distortion given by its Beltrami differential. As a generalization of conformal maps, quasiconformal maps are of great interest in mathematical study and real applications. Efficient and ac...

The Beltrami image flow is an effective nonlinear filter, often used in color image processing. It was shown to be closely related to the median, total variation, and bilateral filters. It treats the image as a two-dimensional manifold embedded in a hybrid spatial-feature space. Minimization of the image surface area yields the Beltrami flow. The corresponding diffusion operator is anisotropic ...

Beltrami equations L¯t(g)=μ(·,t)Lt(g) on S3 (where Lt, |t|<1, are the Rossi operators i.e., Lt spans globally nonembeddable CR structure H(t) discovered by H. Rossi) derived such that to describe quasiconformal mappings f:S3→N⊂C2 from sphere S3,H(t). Using Greiner–Kohn–Stein solution Lewy equation and Bargmann representations of Heisenberg group, we solve for Sobolev-type solutions gt gt−v∈W...

Mesh segmentation offers a desirable divide-and-conquer strategy for many graphics applications. In this paper, we present a novel, efficient, and intrinsic method to segment meshes following the minima rule. The eigenfunctions of the Laplace-Beltrami operator define locality and volume-shape preserving functions over a surface. Inspired on Manifold learning theory, we use these functions as th...

The convergence property of the discrete Laplace-Beltrami operator is the foundation of convergence analysis of the numerical simulation process of some geometric partial differential equations which involve the operator. The aim of this paper is to review several already used discrete Laplace-Beltrami operators over triangulated surface and study numerically as well as theoretically their conv...

In this work, we propose a fast and simple approach to obtain a spherical parameterization of a certain class of closed surfaces without holes. Our approach relies on empirical findings that can be mathematically investigated, to a certain extent, by using Laplace-Beltrami Operator and associated geometrical tools. The mapping proposed here is defined by considering only the three first non-tri...

Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed f...

Abstract We study the removable singularities for solutions to the Beltrami equation ∂f = μ∂f , where μ is a bounded function, ‖μ‖∞ ≤ K−1 K+1 < 1, and such that μ ∈ W 1,p for some p ≤ 2. Our results are based on an extended version of the well known Weyl’s lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly s...