Maximum Likelihood Coordinates

Abstract

Any point inside a d‐dimensional simplex can be expressed in a unique way as a convex combination of the simplex's vertices, and the coefficients of this combination are called the barycentric coordinates of the point. The idea of barycentric coordinates extends to general polytopes with n vertices, but they are no longer unique if n > d+1. Several constructions of such generalized barycentric coordinates have been proposed, in particular for polygons and polyhedra, but most approaches cannot guarantee the non‐negativity of the coordinates, which is important for applications like image warping and mesh deformation. We present a novel construction of non‐negative and smooth generalized barycentric coordinates for arbitrary simple polygons, which extends to higher dimensions and can include isolated interior points. Our approach is inspired by maximum entropy coordinates, as it also uses a statistical model to define coordinates for convex polygons, but our generalization to non‐convex shapes is different and based instead on the project‐and‐smooth idea of iterative coordinates. We show that our coordinates and their gradients can be evaluated efficiently and provide several examples that illustrate their advantages over previous constructions.

A Survey on Cage‐based Deformation of 3D Models

“…Coordinates that are at least C 1 and non-negative for arbitrary polygons include harmonic [JMD*07], maximum entropy [HS08], maximum likelihood [CDH23], positive and smooth Gordon-Wixom [WLMD19], iterative [DCH20], and local barycentric coordinates [ZDL*14; TDZ19], but they do not have a closed form and must be approximated by some numerical procedure. Some of the 2D constructions can be extended to the 3D setting, which is required for cage-based deformation, and we review these constructions in detail in Sections 5 to 8.…”

“…. , λN C ) of the origin with respect to the vertices of a spherical cage Ĉ, and CHANG et al [CDH23] propose to find the latter by maximizing the function…”

“…( 21). The resulting MLC have all desired properties, and CHANG et al [CDH23] further explain how the global minimizer η of F can be used to compute the derivatives of MLC at x. However, if x is close to a cage face, then it may happen that the spherical cage Ĉ does not contain the origin, and then the procedure above does not give proper GBC of x with respect to C.…”