Discrete Complex Structure on Surfel Surfaces

Abstract

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.

The boundary value problem for discrete analytic functions

“…A quadrilateral lattice is a finite graph Q ⊂ C with rectilinear edges such that each bounded face is a quadrilateral (not necessarily convex). Depending on the shape of faces, one speaks about square, rhombic, or orthogonal lattices (the latter is quadrilateral lattices such that the diagonals of each face are orthogonal); see A complex-valued function f on the vertices of Q is called discrete analytic [25], if the difference quotients along the two diagonals of each face are equal, i. e.,…”

Complex Barycentric Coordinates with Applications to Planar Shape Deformation

“…One theoretical issue which we haven't addressed at all, is the connection between complex barycentric coordinates and the so‐called “primal/dual ratio”. As Mercat [Mer08] pointed out, complex primal/dual ratios will arise when the primal and dual edges are not orthogonal. We believe more insight into complex barycentric coordinates can be gained by studying more these concepts.…”

Smooth 2D Coordinate Systems on Discrete Surfaces

“…Indeed, these faces are planar squares and all the ρ coefficients equal 1. Therefore a more meaningful discrete conformal structure has to be defined, using extrinsic or non local data such as a given normal vector [14]: we compute a normal vector of each face using for instance the method described in [8], or coming from the scanned data. It allows us to determine the tangent plane of the surface in each surfel.…”