[PDF] Least Squares Affine Transitions for Global Parameterization



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Least Squares Affine Transitions for Global

Parameterization

Ana Maria Vintescu

LTCI - Télécom ParisTech -

Institut Mines-Telecom

75013 Paris, France

vintescu@telecom-paristech.frFlorent Dupont

Université de Lyon, CNRS

Université Lyon 1, LIRIS UMR 5205

69622 Villeurbanne, France

florent.dupont@liris.cnrs.frGuillaume Lavoué

Université de Lyon, CNRS

INSA-Lyon, LIRIS UMR 5205

69621 Villeurbanne, France

glavoue@liris.cnrs.fr

Pooran Memari

LIX UMR 7161, CNRS, École

Polytechnique, Université Paris Saclay

91128 Palaiseau Cedex - France

memari@lix.polytechnique.frJulien Tierny

Sorbonne Universités, UPMC Univ Paris

06, CNRS, LIP6 UMR 7606

75005 Paris, France

julien.tierny@lip6.fr

ABSTRACT

This paper presents an efficient algorithm for

a global parameterization of triangular surf acemeshes. In contrast

to previous techniques which achieve global parameterization through the optimization of non-linear systems of

equations, our algorithm is solely based on solving at most two linear equation systems, in the least square sense.

Therefore, in terms of running time the unfolding procedure is highly efficient. Our approach is direct - it solves

for the planar UV coordinates of each vertex directly - hence avoiding any numerically challenging planar recon-

struction in a post-process. This results in a robust unfolding algorithm. Curvature prescription for user-provided

cone singularities can either be specified manually , or suggested automatically by our approach. Experiments on a

variety of surface meshes demonstrate the runtime efficiency of our algorithm and the quality of its unfolding. To

demonstrate the utility and versatility of our approach, we apply it to seamless texturing.

The propos edalgorithm

is computationally efficient, robust and results in a parameterization with acceptable metric distortion.

0.1 Keywords

surface parameterization, geometry processing ,triangular mesh, mesh unfolding

1 INTRODUCTION

Surface parameterization represents a main topic in ge- ometry processing and computer graphics fields. It is defined as a one-to-one mapping between a surface and typically a 2D plane , where geometrical tasks can be carried out more efficiently. The most important ap- plication of surface parameterization are texture map- ping, texture synthesis, re-meshing, and morphing. In ordertounfoldasurfacetotheplane, itmusthaveadisk topology; for a closed surface this requirement implies cutting it into a s ingleor multiple disk topology charts.

Cutting

can result in visual artif actsdue to the discon- tinuities across the boundaries of the charts. To this

extent, methods forglobal parameterizationof triangu-Permission to make digital or hard copies of all or part of

this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or re- publish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.lated surfaces have been proposed.W ithinthis frame- work, the global parameterization of a surface with disk topology can be defined as a homeomorphism from the surface to a subset of the plane, such that the discrete Gaussian curvature, i.e. the difference between 2pand the incident triangles" sum of angles at a v ertex,is zero everywhere except for a few vertices calledcone singu- larities.These can be thought of distortion absorbers, being chosen as vertices of the mesh where large area distortion can be predicted prior to the actual parame- terization, [Kha05]. Several approaches based on metric scaling have been proposed in the past to address global parameterization [Jin08, Yan09]. However, these methods mostly rely on non-linear solvers and are hence computationally ex- pensive. Linearized approximations, although compu- tationally attractive, are imprecise (the target metric is only approximated and therefore is not guaranteed to be flat). More importantly, the planar coordinates of the surface vertices (the actual output) are not the variables that are optimized by this family of techniques . We will refer to those as indirect methods[Ben08].Indeed theyfocus on the surf acemetric, i.e. the edge lengths, and later reconstruct the planar coordinates in a post- process. However, thisreconstructionpost-processmay be computationally expensive and, more importantly, numerically challenging. This paper addresses these nique which is fast by employing linear solvers, which minimizes angular distortion through imposed confor- mality, and reduces the area distortion through the use of cone singularities. Our method issimpleanddirect. It directly solves for the 2D coordinates. Thus, it does not suffer from numerical instabilities due to angle-to- uv or scaling factors-to-uv conversions , as found with indirect approaches, [She06]. In contrast to more com- putationally expensive techniques based on non-linear solvers, thecomputationalspeedofourapproachmakes it a good candidate for interactive applications, such as user-driven parameterization improvement for instance, where the users could interacti velya djustthe number and locations of the cones.

Contributions

This paper makes the following new contributions:

1.A fast and robust global parameterization algo-

rithm:Our method is direct (hence robust), non- iterative and only relies on the solving of at most two linear systems.

2.Automatic curvature prescription:Given a list of

cone singularities, we present a fast algorithm to au- tomatically evaluate relevant curvature prescriptions at the cone singularities. The next section presents related work. Next, we in- troduce the method and its preliminaries. Sec. 4 and

5 present the proposed global parameterization algo-

rithm in detail. The modeling of the linear systems is described in Sec. 6, while experimental results are re- ported in Sec. 7. To demonstrate the utility and ver- satility of our technique, we present its application to seamless texturing in Sec. 8 and finally, Sec. 9 con- cludes the paper.

2 RELATED WORK

In the following, we will only focus on surface parame- terization techniques that are related to our work. We refer the reader to survey articles [Flo05, She06] for further reading. Most existing parameterization meth- ods focus onconformalparameterizations (where an- gle distortion is minimized). Several methods [Des02,

Lev02, Liu08, Ray03] focus on parameterizing sur-

faces of disk topology while reducing angular distor- tion. These methods employ linear solvers for the min- imization of energy functions (that are discrete analo- gous to Laplace and Cauchy-Riemann equations) de- fined in terms of the 2D coordinates of the vertices in(a)LSGP(b)MIQ

Figure 1: Comparison between our approach (a) and

Mixed-Integer Quadrangulation (MIQ) [Bom09] (b).

On this example, the MIQ approach generates many

boundary self-intersections, see the unfolded blue boundary. the mesh. These are thereforedirectmethods. They al- low a free boundary setting but pin two vertices to avoid a non-trivial solution (a more recent approach removes such necessity through a spectral embedding [Mul08]). Indirectmethods [She05, She00, Zay07] aim at mini- mizing the difference between the initial angles of thequotesdbs_dbs4.pdfusesText_8