Least squares conformal maps for automatic texture atlas generation

TLDR

A texture atlas efficiently represents color for 3D paint systems, but existing methods for triangulated surfaces are limited by many small charts with simple borders. This paper introduces a quasi‑conformal parameterization based on a least‑squares approximation of the Cauchy‑Riemann equations to eliminate chart discontinuities and enable large, complex‑border charts. The method decomposes the model into disc‑like charts, parameterizes each via the least‑squares quasi‑conformal approach, segments the model into natural‑shaped charts, and packs them efficiently in texture space. The resulting objective function uniquely minimizes angle deformation, avoids triangle flips, is numerically stable and efficiently minimized, and the approach successfully paints both scanned and modeled datasets.

Abstract

A Texture Atlas is an efficient color representation for 3D Paint Systems. The model to be textured is decomposed into charts homeomorphic to discs, each chart is parameterized, and the unfolded charts are packed in texture space. Existing texture atlas methods for triangulated surfaces suffer from several limitations, requiring them to generate a large number of small charts with simple borders. The discontinuities between the charts cause artifacts, and make it difficult to paint large areas with regular patterns.In this paper, our main contribution is a new quasi-conformal parameterization method, based on a least-squares approximation of the Cauchy-Riemann equations. The so-defined objective function minimizes angle deformations, and we prove the following properties: the minimum is unique, independent of a similarity in texture space, independent of the resolution of the mesh and cannot generate triangle flips. The function is numerically well behaved and can therefore be very efficiently minimized. Our approach is robust, and can parameterize large charts with complex borders.We also introduce segmentation methods to decompose the model into charts with natural shapes, and a new packing algorithm to gather them in texture space. We demonstrate our approach applied to paint both scanned and modeled data sets.