Discrete Complex Structure on Surfel Surfaces

TL;DR

This paper develops a conformal parametrization theory for surfel-based digital surfaces, generalizing classical methods to handle complex ratios and extending Laplacian concepts to surfel surfaces.

Contribution

It introduces a novel conformal parametrization framework for surfel surfaces, extending existing theories for polyhedral surfaces to accommodate complex ratios.

Findings

Defines a conformal theory for surfel surfaces with normal vectors.

Generalizes the Laplacian to surfel surfaces with complex ratios.

Provides a mathematical foundation for surfel surface analysis.

Abstract

This paper defines a theory of conformal parametrization of digital surfaces made of surfels equipped with a normal vector. The main idea is to locally project each surfel to the tangent plane, therefore deforming its aspect-ratio. It is a generalization of the theory known for polyhedral surfaces. The main difference is that the conformal ratios that appear are no longer real in general. It yields a generalization of the standard Laplacian on weighted graphs.