Feature-aligned harmonic volumetric mapping using MFS

doi:10.1016/j.cag.2010.03.004

Computers & Graphics

Volume 34, Issue 3, June 2010, Pages 242-251

https://doi.org/10.1016/j.cag.2010.03.004 Get rights and content

Abstract

We present an efficient adaptive method to compute the harmonic volumetric mapping, which establishes a smooth correspondence between two given solid objects of the same topology. We solve a sequence of charge systems based on the harmonic function theory and the method of fundamental solutions (MFS) for designing the map with boundary and feature constraints. Compared to the previous harmonic volumetric mapping computation using MFS, this new scheme is more efficient and accurate, and can support feature alignment and adaptive refinement. Our harmonic volumetric mapping paradigm is therefore more effective for practical shape modeling applications and can handle heterogeneous volumetric data. We demonstrate the efficacy of this new framework on handling volumetric data with heterogeneous structure and nontrivial topological types.

Introduction

The rapid advancement of 3D scanning techniques makes it easier to acquire massive 3D data nowadays. When datasets can be acquired in an explosive rate, computational techniques only evolve modestly. As a result, 3D data matching, analyzing, and searching become bottleneck for their efficient processing. Compared with 2D images, 3D shapes have many distinctions including larger sets of degrees of freedom and spatial variations in terms of geometry, topology, feature, and material. A viable approach for the effective shape matching and analyzing is to establish the correspondence between objects of interest, which can be computed by either solving a non-rigid bijective registration between given objects or composing two parameterizations from both objects onto one common domain. The key is to compute a mapping from one domain to another. When it is enough to purely consider boundary surfaces of the 3D data, one can focus on mapping 2d-manifolds (surfaces). Surface mapping seeks a bijection between two 2-manifolds with similar topology, aiming for least distortion (using length-, angle-, or area-preserving as the criterion) which dictates its effects in applications. Surface parameterization and inter-surface mapping have been extensively studied, playing important roles in computer graphics, and serving as ubiquitous tools for many valuable applications. For example, in computer graphics, it has been used for texture mapping, texture transfer, and morphing animation. In geometric modeling, it has been used for detail transfer, surface editing, mesh simplification. In CAGD, it has been used to construct the parametric domain for continuous representations such as splines. In visualization, complicated geometric structures may be better visualized and analyzed by mapping surfaces and their properties to a simpler domain. In vision and medical imaging, it has been used for surface matching, data completion, and so on. Surveys of surface mapping and their applications are given in [13], [42].

Solid volumetric data have richer contents than those of the boundary surface. When the data processing or analysis are related to material, intensity, or any other structural information defined over the whole 3D region of the object (instead of on just its boundary shell), we need to consider the shape as a 3-manifold and study the volumetric mapping. Therefore, volumetric mapping can also benefit aforementioned applications. Because of its importance, volumetric mapping and parameterization has gained greater interest in recent years, and a few related research work has been conducted towards various applications such as shape registration [47], [29], [30], volumetric deformation [21], [20], [32], [5], and trivariate spline construction [33], and so on. Although many valuable concepts and demos have been presented, all indicating the importance of this technique, its study has just started and is far from adequate. Several key limitations of existing algorithms prevent them from being applied into real applications with complex scenarios.

Generality: It is desirable that the mapping is general and can handle 3D shapes with variant topological types. Volumetric data from real scenarios usually have nontrivial topology, and most existing parameterization techniques [47], [33] focus on topological solid-sphere shapes. Refs. [29], [30] used the fundamental solution methods to compute harmonic volumetric mapping between 3D objects with general topology.

Efficiency: Solving the discretized vector field over a 3D voxelized domain or over a tetrahedral mesh usually is much slower than the surface mapping computation. The fundamental solution method of [29] is a boundary method. It reduces the volumetric mapping computation from the whole 3D domain to the degree of freedom with the boundary size, to be solved by a linear system of equations. However, it is still very time consuming to solve because the coefficient matrix is dense and ill-conditioned.

Heterogeneity: Most existing methods consider the volumetric mapping from homogeneous viewpoints and only compute the mapping purely based on geometry, without taking into account the interior structure and features. It is desirable to develop the capability of the mapping algorithm that can accommodate heterogeneous structures and integrate domain expertise in geometric modeling and processing.

In order to tackle these aforementioned limitations, this paper improves the algorithm of fundamental solution methods in mapping computation [30], and seeks a general and effective mapping computation algorithm with better efficiency, accuracy, and heterogeneity. We compute harmonic volumetric mapping by improving the fundamental solution methods of [30], and the side-by-side comparison shows that our new approach is more efficient and accurate. Furthermore, it supports feature alignment, which is important for many practical volumetric data processing tasks.

The main contributions of this work include:

•
We use multiple fundamental solution systems and an adaptive refinement scheme for the computation of harmonic volumetric mapping. Compared to [30], this computation efficiency is greatly improved, so that large and complex data can be parameterized in the new framework. In the mean time, with an adaptive sampling scheme, the new computation also converges to a better boundary fitting result in salient manners.
•
Our feature alignment scheme supports the computation of volumetric mapping composed by constrained harmonic functions that allow the alignment between various types of features including 0-manifolds (feature points), 1-manifolds (feature lines, such as skeletons), 2-manifolds (iso-surfaces).

The paper is organized as follows. Section 2 briefly reviews related literature. Then we introduce the theory and algorithms of our methods in Section 3, and address important implementation issues in Section 4. In Section 5, we demonstrate some experimental results, discuss and compare our algorithms with existing volumetric mapping methods, especially [30], and show the large efficiency/accuracy improvement over the current method. We also show a direct application on hex meshing. Finally, we conclude our work in Section 6.

Section snippets

Related work

Harmonic maps and surface parameterization: Surface mapping computes a one-to-one continuous map between a 2-manifold and a target domain with low distortions. It plays a critical role in various applications of graphics, CAGD, visualization, vision, medical imaging, and physical simulation. Having been extensively studied in the literature of surface parameterization, harmonic maps are usually addressed from the point of view of minimizing Dirichlet Energy. Its discrete version was first

Theory and algorithm

A volumetric map $\overset{\Rightarrow}{f}$ between two 3-manifolds embedding in $R^{3}$ is a bijective mapping $\overset{\Rightarrow}{f} : M_{1} \to M_{2}, M_{1} \subset R^{3}, M_{2} \subset R^{3}$ . The boundary constraint is a surface mapping $\overset{\Rightarrow}{f^{'}}$ from the boundary surface of the first solid object M₁, denoted as $\partial M_{1}$ , to the boundary surface of M₂, denoted as $\partial M_{2}$ . The mapping $\overset{\Rightarrow}{f} (p) = q (p \in M_{1}, q \in M_{2})$ is composed of three real functions in three axis directions, i.e., $\overset{\Rightarrow}{f} = (f^{1}, f^{2}, f^{3})$ . Each real function fⁱ (i=1,2,3) maps the point p to q(q₁, q₂, q₃)'s corresponding component q_i. This problem is

Source points and collocation points placement

In order to set up the coefficient matrix for boundary fitting, first we need to place source points and collocation points. The n_s source points $\overset{\Rightarrow}{Q} = {Q_{1}, Q_{2}, \dots, Q_{n_{s}}}$ are particles in the exterior of M₁ and n_c collocation points $\overset{\Rightarrow}{P} = {P_{1}, P_{2}, \dots, P_{n_{c}}}$ are evaluation points on the boundary $\partial M_{1}$ . We solve the weights (charge amount) distribution w_i, i=1, …, n_s on all source points $\overset{\Rightarrow}{Q}$ so that $\overset{\Rightarrow}{f} (P_{i})$ satisfies the boundary condition approximately.

The distribution of source and collocation points greatly

Experimental results and applications

We conduct a few volumetric mapping experiments over various volumetric data, with different sizes, topology and geometry complexities. We illustrate some of these mapping results in Fig. 4, Fig. 5. We use the color-encoded distance field to visualize the mapping result. When a map $\overset{\Rightarrow}{f} : M_{1} \to M_{2}$ is computed, the color-encoded (red indicates the maximum while blue indicates the minimum, see Fig. 5(h)) distance field defined on one region can be transferred to another region, by plotting the color of

Conclusion

We present a feature-aligned volumetric harmonic mapping computation algorithm using methods of fundamental solutions. The map $\overset{\Rightarrow}{f}$ is composed of a set of harmonic functions ${\overset{\Rightarrow}{f_{i}}}$ which can be efficiently solved. Also, our adaptive source/collocation points placement improves the numerical issue of MFS solving. Therefore, our algorithm largely improves the existing harmonic volumetric mapping computation algorithm using MFS [30]. The new algorithm has better efficiency and accuracy, and it

Acknowledgements

We would like to thank the anonymous reviewers for their helpful comments and suggestions. Discussions with Hong Qin and Warren Waggenspack inspired this work. The head of David model is from Stanford Michelangelo project, male and female models are from Cyberware. Other models are from AIM@SHAPE Shape Repository. This work is supported by Louisiana Board of Regents Research Competitiveness Subprogram (RCS) LEQSF(2009-12)-RD-A-06 and PFund: NSF(2009)-PFUND-133. Huanhuan Xu is supported in part

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Technical SectionFeature-aligned harmonic volumetric mapping using MFS

Abstract

Introduction

Section snippets

Related work

Theory and algorithm

Source points and collocation points placement

Experimental results and applications

Conclusion

Acknowledgements

Computer Methods in Applied Mechanics and Engineering

Computer Aided Geometric Design

Computers and Graphics

Graphical Models

Computers and Mathematics with Applications

Computer Aided Design

Skeleton extraction by mesh contraction

ACM Transactions on Graphics

The boundary element methods in engineering

Variational harmonic maps for space deformation

ACM Transactions on Graphics

Curve-skeleton properties, applications, and algorithms

IEEE Transactions on Visualization and Computer Graphics

Intrinsic parameterizations of surface meshes

Computer Graphics Forum

The method of fundamental solution for elliptic boundary value problems

Advances in Computational Mathematics

Meshless thin-shell simulation based on global conformal parameterization

IEEE Transactions on Visualization and Computer Graphics

Mean value coordinates for closed triangular meshes

SIGGRAPH

Three-dimensional geometric metamorphosis based on harmonic maps

The Visual Computing

Technical Section
Feature-aligned harmonic volumetric mapping using MFS