Elsevier

Computers & Graphics

Volume 35, Issue 3, June 2011, Pages 639-649
Computers & Graphics

SMI 2011: Full Paper
A topology-preserving optimization algorithm for polycube mapping

https://doi.org/10.1016/j.cag.2011.03.018Get rights and content

Abstract

We present an effective optimization framework to compute polycube mapping. Composed of a set of small cubes, a polycube well approximates the geometry of the free-form model yet possesses great regularity; therefore, it can serve as a nice parametric domain for free-form shape modeling and analysis. Generally, the more cubes are used to construct the polycube, the better the shape can be approximated and parameterized with less distortion. However, corner points of a polycube domain are singularities of this parametric representation, so a polycube domain having too many corners is undesirable. We develop an iterative algorithm to seek for the optimal polycube domain and mapping, with the constraint on using a restricted number of cubes (therefore restricted number of corner points). We also use our polycube mapping framework to compute an optimal common polycube domain for multiple objects simultaneously for lowly distorted consistent parameterization.

Graphical abstract

An optimal common polycube domain for multiple objects for consistent arameterization.

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Highlights

► An effective optimization framework to compute polycube mapping. ► An iterative algorithm to optimize polycube domain and mapping. ► Optimal common polycube for multiple objects for consistent parameterization.

Introduction

Computing the parameterization of 3D shapes (surfaces/solids) on specific canonical domains is an important problem in shape modeling, and it can facilitate many computer graphics and geometric processing tasks. Polycube mapping was first introduced by [33]. It parameterizes a closed surface onto a polycube domain, which is composed of a set of small cubes. A polycube has the same topology of the given surface, and it is usually constructed to approximate the geometry of the surface. Therefore, the surface parameterization on a polycube domain often has much smaller distortion than that on a planar domain. Meanwhile, the polycube domain still possesses great regularity; each subpatch is a rectangle; transitions between adjacent patches are simple rotation and translation except on corner points. Due to many of these advantages, the polycube mapping has been used in many graphics and shape modeling applications such as texture mapping [33] and synthesis [21], shape morphing [11], spline construction [34], [35], and volumetric matching [23], [39], [40].

Intuitively, the more cubes one uses to construct the polycube, the better the domain can approximate the original model, which brings parameterization very small area and angle distortion. However, corner points are singularity points of the parameterization. They are undesirable in many tasks such as spline construction [34], [35], physics-based simulations [15], etc. On the other hand, if one uses fewer cubes to construct a simpler domain with fewer corner points, the parameterization will possess larger distortion due to the dissimilarity of geometric structures between the model and the domain shape. Therefore, when a fundamental question is asked: What is the optimal polycube domain? A reasonable answer can be an optimized balancing between the singularity number and mapping distortion. More specifically, we try to solve the following problem: given a surface S and a budget n of the singularity point number, what is the optimal shape of the polycube domain P so that the parameterization f:SP has the least distortion and P has no more than n corners?

Depending on applications, different metrics (angle distortion, area distortion, isometry distortion, etc.) have been studied and used to measure the mapping quality. Harmonic functions are most widely used in constructing lowly distorted mapping. With a fixed boundary condition, a function ϕ(x,y) is harmonic if it is a solution of Laplace's equation. When a boundary condition is given, ϕ is a minimizer of the Dirichlet energy [28], [10] and it possesses great smoothness. For example, conformal parameterization can be constructed by two conjugate harmonic functions [14], [31]. In this paper, we use harmonic functions to construct polycube mapping, minimizing a metric energy composed of shape-preserving and area-preserving terms. The framework is general and can be used for other metrics. A similar idea, proposed by Pietroni et al. [27], considered the trade off between the mapping distortion and the simplicity of the domain, solves the surface parameterization over abstract domains by locally optimizing the mapping on subregions then globally smoothing it.

Now the optimal polycube maps can be formulated as solving argminE(P,f) for a given shape S, where energy function E is defined on any mapping f:SP and P is a polycube with n corners. Since the domain P is part of the optimization, it is extremely difficult. We restrict our optimization to a subspace of this problem, which we call a topology-preserving polycube mapping. Specifically, given an initial polycube domain P={Pi}, the topology of the polycube P is defined by its dual graph (see Fig. 1) DM={DV,DE}. DV={dv1,,dvn} are nodes corresponding to rectangle subpatches {Pi}. DE is a set of edges: an edge [dvi,dvj] is in DE, if Pi and Pj are adjacent to each other. We say two polycubes P={P1,,Pn} and Q={Q1,,Qm} are topologically equivalent, if their dual graphs DP and DQ are isomorphic. Therefore, given an initial polycube P, our goal is to find the optimal polycube P and the mapping f that minimizes distortion E(P,f), in the same topological equivalence class (without changing the structure of its dual graph).

This paper has three main contributions.

  • We formulate the above optimal polycube mapping problem, and present a polycube mapping computation framework based on the given restricted complexity of polycube domain.

  • We develop efficient optimization solvers to seek the topology-preserving optimal polycube domain and mapping iteratively.

  • We extend the polycube optimization algorithm to multiple objects, for the construction of the common optimal domain for multiple models.

Section snippets

Related work

Surface parameterization using harmonic functions: Theories and technologies in surface parameterization have been widely studied and they have been playing a critical role in many geometric processing tasks in graphics, CAGD, visualization, vision, medical imaging, physical simulation, etc. Many effective techniques have been developed to solve the parameterization under different distortion metrics with different boundary conditions. A thorough review is beyond the scope of this paper, and we

Algorithms overview

A polycube domain P is composed of a set of rectangular patches Pi. A polycube map is therefore composed of a set of rectangular maps. We use the harmonicity and area distortion to measure the mapping quality and optimize the domain shape as well as the mapping.

Ideally, given a metric, we shall simultaneously optimize the polycube domain P as well as the mapping f:SP to minimize the distortion E(f). We can formulate this as minimizing E(x,y)=E(x1,x2,,x3n,y1,y2,,y3n), with the constraints

Constructing initial polycube and mapping

The initial polycube can be constructed manually [33], [34], or automatically [24], [18]. We also use a simple voxelization algorithm (Section 4.1) to generate the polycube. Since this initial polycube and maps (Section 4.2) will be optimized to minimize the distortion, a simple, efficient, and adaptive (to different corner budgets) scheme such as this voxelization algorithm is sometimes enough. The following optimization framework is general, and can optimize an initial polycube mapping

Optimizing polycube domain

Given a polycube mapping f:SP={fi:SiPi} defined on a set of topological rectangle patches on S. We want to find the optimal re-scaled Pi so that mapping distortion is minimized. We use a distortion energy E composed of the harmonic energies Ht(f),t=0,1,2 and an area-stretching term A(f):Ht=PkHkt=Pkei,jPk12wij(ft(Xi)ft(Xj))2,A=PkFi,j,hPk(Δ(Ui,Uj,Uh))2Δ(Xi,Xj,Xh),E=H0+H1+H2+αA,where Δ(Xi,Xj,Xh) and Δ(Ui,Uj,Uh) denote the original area of triangle (vi,vj,vh) and the area of its image

Optimizing polycube mapping

In Section 5, we fix the corner point mapping f(VCS)VCP to optimize the shape of polycube domain. We further reduce the mapping distortion by moving vertices VCS (without ambiguity, we also call them corner points) over S. Any 2D manifold S can be parameterized to an atlas Ω={Ωi}, and locally any point on S: (x1,x2,x3)S can be represented as a 2D coordinate (r1,r2) on a local planar chart. We construct local parameterization gi:SiΩi by mapping the C-ring neighboring regions (in our

Polycube mapping for multiple objects

We also demonstrate an application of our polycube mapping framework in multiple objects mapping. Polycube can be used as a canonical base domain for multiple objects (preferably, these objects have the same topology and similar geometry). Our framework can be used to generate such a common regular domain, and multiple objects are parameterized onto this single polycube with low distortion. Multiple shapes can be analyzed, processed, and integrated over this single domain. Supposing we have a

Experimental results

We compare the property of our polycube mapping framework with existing methods and list them in Table 1. Our method generates the optimal polycube within the same topological class, and the complexity of the polycube is flexibly bounded by the given number of singularities. We test our optimization framework on a few 3D shapes. Fig. 9 shows the optimization on Bimba and Max-planck. The texture-mapped rectangular grids become closer to squares, indicating the reducing of angle distortion.

Fig. 10

Conclusion

We present an interactive optimization framework to solve the optimal polycube mapping problem. Because directly solving optimal polycube domain and mapping together is too expensive, we iteratively optimize polycube domain shape and polycube mapping separatively, to make full use of the available partial derivative information of the objective function. We develop an efficient nonlinear optimization algorithm with linear bound constraints for the first subproblem. For the second subproblem, we

Acknowledgments

This work is supported by Louisiana Board of Regents Research Competitiveness Subprogram (RCS) LEQSF(2009-12)-RD-A-06, PFund: NSF(2009)-PFund-133, LSU Faculty Research Grant 2010, and NSF DMS-1016204. We also thank AIM@Shape and Stanford Shape Repository for their datasets.

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