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Computers & Graphics 34 (2010) 242–251

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Computers & Graphics

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Technical Section

Feature-aligned harmonic volumetric mapping using MFS

Xin Li a,b,�, Huanhuan Xu a,b, Shenghua Wan a,b, Zhao Yin c, Wuyi Yu c

a Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, LA 70803, USAb Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USAc Department of Automation, Xiamen University, Xiamen, Fujian 361005, China

a r t i c l e i n f o

Article history:

Received 8 December 2009

Received in revised form

15 February 2010

Accepted 7 March 2010

Keywords:

Geometric modeling

Volume and multidimensional modeling

Correspondence and registration

Harmonic volumetric mapping

93/$ - see front matter Published by Elsevier

016/j.cag.2010.03.004

esponding author at: Department of Electrica

a State University, Baton Rouge, LA 70803,

225 578 4831.

ail address: xinli@lsu.edu (X. Li).

a b s t r a c t

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.

Published by Elsevier Ltd.

1. Introduction

The rapid advancement of 3D scanning techniques makes iteasier to acquire massive 3D data nowadays. When datasets canbe acquired in an explosive rate, computational techniques onlyevolve modestly. As a result, 3D data matching, analyzing, andsearching become bottleneck for their efficient processing.Compared with 2D images, 3D shapes have many distinctionsincluding larger sets of degrees of freedom and spatial variationsin terms of geometry, topology, feature, and material. A viableapproach for the effective shape matching and analyzing is toestablish the correspondence between objects of interest, whichcan be computed by either solving a non-rigid bijective registra-tion between given objects or composing two parameterizationsfrom both objects onto one common domain. The key is tocompute a mapping from one domain to another. When it isenough to purely consider boundary surfaces of the 3D data, onecan focus on mapping 2d-manifolds (surfaces). Surface mappingseeks 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 applica-tions. Surface parameterization and inter-surface mapping havebeen extensively studied, playing important roles in computergraphics, and serving as ubiquitous tools for many valuableapplications. For example, in computer graphics, it has been usedfor texture mapping, texture transfer, and morphing animation. In

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geometric modeling, it has been used for detail transfer, surfaceediting, mesh simplification. In CAGD, it has been used toconstruct the parametric domain for continuous representationssuch as splines. In visualization, complicated geometric structuresmay be better visualized and analyzed by mapping surfaces andtheir properties to a simpler domain. In vision and medicalimaging, it has been used for surface matching, data completion,and so on. Surveys of surface mapping and their applications aregiven in [13,42].

Solid volumetric data have richer contents than those of theboundary surface. When the data processing or analysis arerelated to material, intensity, or any other structural informationdefined over the whole 3D region of the object (instead of on justits boundary shell), we need to consider the shape as a 3-manifoldand study the volumetric mapping. Therefore, volumetric map-ping can also benefit aforementioned applications. Because of itsimportance, volumetric mapping and parameterization hasgained greater interest in recent years, and a few related researchwork has been conducted towards various applications such asshape 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 pre-sented, all indicating the importance of this technique, its studyhas just started and is far from adequate. Several key limitationsof existing algorithms prevent them from being applied into realapplications with complex scenarios.

Generality: It is desirable that the mapping is general and canhandle 3D shapes with variant topological types. Volumetric datafrom real scenarios usually have nontrivial topology, and mostexisting parameterization techniques [47,33] focus on topologicalsolid-sphere shapes. Refs. [29,30] used the fundamental solution

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X. Li et al. / Computers & Graphics 34 (2010) 242–251 243

methods to compute harmonic volumetric mapping between 3Dobjects with general topology.

Efficiency: Solving the discretized vector field over a 3Dvoxelized domain or over a tetrahedral mesh usually is muchslower than the surface mapping computation. The fundamentalsolution method of [29] is a boundary method. It reduces thevolumetric mapping computation from the whole 3D domain to thedegree of freedom with the boundary size, to be solved by a linearsystem of equations. However, it is still very time consuming tosolve because the coefficient matrix is dense and ill-conditioned.

Heterogeneity: Most existing methods consider the volumetricmapping from homogeneous viewpoints and only compute themapping purely based on geometry, without taking into accountthe interior structure and features. It is desirable to develop thecapability of the mapping algorithm that can accommodateheterogeneous structures and integrate domain expertise ingeometric modeling and processing.

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

The main contributions of this work include:

�

We use multiple fundamental solution systems and anadaptive refinement scheme for the computation of harmonicvolumetric mapping. Compared to [30], this computationefficiency is greatly improved, so that large and complex datacan be parameterized in the new framework. In the mean time,with an adaptive sampling scheme, the new computation alsoconverges to a better boundary fitting result in salientmanners. � Our feature alignment scheme supports the computation of

volumetric mapping composed by constrained harmonicfunctions that allow the alignment between various types offeatures 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 reviewsrelated literature. Then we introduce the theory and algorithms ofour methods in Section 3, and address important implementationissues in Section 4. In Section 5, we demonstrate some experi-mental results, discuss and compare our algorithms with existingvolumetric mapping methods, especially [30], and show the largeefficiency/accuracy improvement over the current method. Wealso show a direct application on hex meshing. Finally, weconclude our work in Section 6.

2. Related work

Harmonic maps and surface parameterization: Surface mappingcomputes a one-to-one continuous map between a 2-manifoldand a target domain with low distortions. It plays a critical role invarious applications of graphics, CAGD, visualization, vision,medical imaging, and physical simulation. Having been exten-sively studied in the literature of surface parameterization,harmonic maps are usually addressed from the point of view ofminimizing Dirichlet Energy. Its discrete version was firstproposed by Pinkall and Polthier [38] and later introduced tocomputer graphics field in work of Eck et al. [10]. By discretizingthe energy defined in [38], Desbrun et al. [8] constructed free-

boundary harmonic maps. Harmonic maps are directly used forshape blending [22] and in later shape morphing applications[26,39,24,41]. A lot of effective surface manipulation techniquesand parameterization paradigms might be generalized onto 3-manifolds. A thorough survey on surface parameterizationtechniques is beyond the scope of this work, and we refer readersto nice survey reports of [13,42,18] for details.

Volumetric mapping: Solid geometry and volumetric data havericher contents than that of surfaces, and volumetric parameteriza-tion applies well to the aforementioned problems when volumetricdata are of most significance. Furthermore, in scientific computa-tion, physics-based simulation, fluid dynamics, and materialmodeling, we are primarily handling volumetric data, so volumetricparameterization is an enabling technology with great applicationpotentials. The previously mentioned problems will certainlybenefit from an effective volumetric shape representation/modelingparadigm. Because of its importance, volumetric parameterizationhas been gaining greater interest in recent years, a few relatedresearch work has been conducted towards various applicationssuch as shape registration [47,29], volume deformation [21,20], andtrivariate spline construction [33]. Wang et al. [47] parameterizedsolid shapes over solid spheres by a variational algorithm thatiteratively reduces the discrete harmonic energy defined overtetrahedral meshes, the harmonic energy is rigorously deduced butthe optimization is prone to getting stuck on local minima and itonly focuses on topological solid sphere shapes such as humanbrain datasets. Ju et al. generalized the mean value coordinates [12]from surfaces to volumes for a smooth volumetric interpolation,Joshi et al. [20] presented harmonic coordinates for volumetricinterpolation and deformation purposes, their method guaranteedthe non-negative weights and therefore led to a more pleasinginterpolation result in concave regions compared with that in [21].Li et al. [29,30] used Green’s functions and fundamental solutionmethods to map solid shape onto general target domains. Martinet al. [33] parameterized genus-zero tetrahedral meshes onto acylinder, and used it for trivariate spline construction. Lipman et al.[32] and Ben-Chen et al. [5] also used Green’s functions to generateharmonic functions to guide space deformation.

Boundary method and MFS: We construct the mapping througha meshless procedure by using a boundary method called method

of fundamental solution (MFS). Notable work among boundarymethods for solving elliptic partial differential equations (PDEs)includes the classical boundary integral equation and boundaryelement method (BIE/BEM), which has been widely used in manyengineering applications [3], and was introduced into computergraphics for the simulation of deformable objects in [19]. One ofthe major advantages of the BIE/BEM over the traditional finiteelement method (FEM) and finite difference method (FDM) is thatonly boundary discretization is required rather than the entiredomain discretization needed for solving the PDEs numerically.Compared with the BIE/BEM approach, the MFS uses only thefundamental solution in the construction of the solution of aproblem, without using any integrals over boundary elements.Furthermore, the MFS is a meshless method, since only boundarynodes are necessary for all the computation. ‘‘Meshless’’ has theadvantage of simplicity that neither domain nor mesh connectiv-ity is required in storage and computation; so it becomes veryattractive in scientific computing and modeling [4,15]. A com-prehensive review of the MFS and kernel functions for solvingmany elliptic PDE problems was documented in [11].

3. Theory and algorithm

A volumetric map ~f between two 3-manifolds embedding inR3 is a bijective mapping ~f : M1-M2;M1 �R3;M2 �R3. The

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boundary constraint is a surface mapping ~f 0 from the boundarysurface of the first solid object M1, denoted as @M1, to theboundary surface of M2, denoted as @M2. The mapping~f ðpÞ ¼ qðpAM1;qAM2Þ is composed of three real functions inthree axis directions, i.e., ~f ¼ ðf 1; f 2; f 3Þ. Each real function f i

(i¼1,2,3) maps the point p to q(q1, q2, q3)’s correspondingcomponent qi. This problem is then reduced to the computationof real functions f i (i¼1,2,3), with the given boundary surfacemapping constraints ~f 0 ¼ ðf 01; f 02; f 03Þ. We want the volumetricmapping to follow the boundary constraints and minimize aspecific metric distortion. In this work, our object is to minimizethe harmonic energy under the Dirichlet boundary conditiondiscussed above, defined by the boundary surface mapping.

3.1. Harmonic volumetric mapping

Harmonicity of a mapping characterizes the smoothness of thetransformation, which is a natural phenomenon that depicts theminimized physical energy that arises from the differencebetween two shapes. In the surface case, a harmonic map (withboundary loop mapping predetermined) finds the functions withthe vanishing Laplacian everywhere, and it minimizes the Dirichletenergy and leads to a minimal surface [38,10]. Intuitively speak-ing, finding a harmonic map between two surfaces with fixedboundary correspondence is like computing the physical deforma-tion of a rubber membrane. The membrane has the source surfaceas its relaxed shape configuration, and is arbitrarily wrapped ontothe target shape with the boundary constraint enforced; then welet go the membrane so that it freely flows over the target shape,its final status indicates a harmonic map.

Similarly, for harmonic volumetric mapping, if we fix theboundary map which is now a surface mapping between shells ofthe two given solid objects, we are computing the smooth mappingof the interior region by enforcing the vanishing 3D Laplacian. Thisis equivalent to computing the final stable configuration of a solidrubber subject to its boundary surface constraint.

In formulation, given two volumetric regions M1 �R3 andM2 �R3 and a one-to-one mapping ~f 0 between their boundarysurfaces @M1 and @M2: ~f 0 ðpÞ ¼ q;pA@M1;qA@M2, we seek amapping ~f : M1-M2 such that

D~f ðpÞ ¼ 0; pAM1,~f ðpÞ ¼ ~f 0 ðpÞ; pA@M1;

(

where D is the Laplace operator, defined for real function f in R3

as

Df ¼r � rf ¼@2f

@x2þ@2f

@y2þ@2f

@z2,

and D~f ¼ 0 for~f ¼ ðf 1; f 2; f 3Þ is equivalent to Df i ¼ 0 for all i¼1,2,3.Variational approaches have been proposed [47] for solving

the harmonic volumetric map over tetrahedral meshes. However,like the surface mapping, the quality of numerical solution in thistype of methods heavily depends on the mesh quality. In thesurface case, it is well known that the discrete harmonic map [10]could lead to non-bijective mapping locally when skinny trianglesexist and cotangent weights become negative. Similarly, harmonicweights [47] defined over edges of a tetrahedral mesh, which arederived via finite element analysis has this same problem. It hasbeen proven that if the mesh satisfies the Delaunay criterion, theneven it contains obtuse triangles, the parameterization obtainedusing the cotangent weights will be bijective. However, to ourbest knowledge, there is no similar result on the discrete weightover tetrahedral meshes. A mesh-free procedure is more attrac-tive for irregular geometries specifically with nontrivial topology/structures due to its flexibility and simplicity.

On the other hand, the linear nature of Laplacian equationsindicates that the boundary-based methods are most suitablesince the interior is now determined in an exact manner. In otherwords, according to the maximum principle of harmonic func-tions, the value of a harmonic function never reaches maximal orminimal values in the interior region of the domain, and values inthese interior regions are fully determined by the boundarycondition. The method of fundamental solution (MFS), based onGreen’s theory is a natural boundary mesh-free method to solvethis problem. MFS can be viewed as a modified Trefftz method,and the basic idea is to approximate the solution by a linearcombination of fundamental solutions with sources locatedoutside the problem domain. Refs. [29,30] applied MFS in thecomputation of harmonic volumetric mapping, where three linearsystems with one single coefficient matrix are solved to get theharmonic volumetric mapping between two 3D objects.

Compared to mesh-based variational methods such as [47], (1)the MFS method is a meshless boundary method, which is moreefficient than this conventional mesh based FEM method, andwith both time complexity and storage complexity greatlyreduced; (2) MFS is more general and can flexibly handlevolumetric datasets with complicated topologies, includingtopological noise; (3) the new MFS framework can also handleheterogeneous materials instead of just homogeneous shapes.

3.2. Method of fundamental solutions

We briefly review the idea of MFS in solving harmonicvolumetric maps and define the notations that are used in ouralgorithms.

MFS in harmonic volumetric mapping: We seek three harmonicfunctions ðf 1; f 2; f 3Þ : M1-M2, with Df i ¼ 0. Since D is a linear self-adjoint differential operator and M1 is a bounded domain in R3,we can compute its Green function. A fundamental solution of thisdifferential equation is a function Kðx;x0Þ such that

DKðx,x0Þ ¼ dðx,x0Þ, x,x0AR3,

where dðx;x0Þ is the Dirac delta function, the kernel K is definedeverywhere except the singularity point at x¼ x0.

Then we have

f iðxÞ ¼

ZKðx,x0Þgiðx0Þdx0:

Such a kernel function K is known as Green’s functionassociated with the 3D Laplacian operator D, and has the formula:Kðx;x0Þ ¼ ð1=4pÞ1=jx�x0j, where jx�x0j denotes the distancebetween the points x and x0.

Following this scheme, solving the aforementioned harmonicmapping ~f is like designing electric fields. For each harmonicfunction f i (for each axis direction), we compute a particle system.The outcome electric potential field of any particle system isalways a real harmonic function (guaranteed by the Kernelfunction), and we only need to find the particle system that fitsthe boundary condition, indicating the boundary surface map-ping, coupled with three axis components. This process of solvingthe best particle system simulates the computation of f i.

Suppose we have a particle system, and consider an electronicparticle Qs (called a singularity point, or a source point) outside thedomain M1, the corresponding fundamental solution for 3DLaplacian equation (i.e. its potential) on a point p can beformulated as

Kðp,Q sÞ ¼1

4p1

jp�Q sj, ð1Þ

where jp�Q sj denotes the distance between the point p and thisparticle Qs.

ARTICLE IN PRESS

Fig. 1. Heterogeneous volumetric mapping between head–skull–brain and

polycube-sphere. (a) The extracted and cleaned volumetric shape has three

salient iso-surfaces: head, skull, and brain. A target domain (d) is generated to test

the efficacy of our mapping with iso-surface constraints. (d) has a sphere, a

polycube skull, and a cube inside, corresponding to three iso-layers in (a). (b) and

X. Li et al. / Computers & Graphics 34 (2010) 242–251 245

Therefore, considering the entire particle system {Qs} with aset of source points, the MFS equation to evaluate f i on an interioror boundary point p is

f ið ~wi ,~Q ;pÞ ¼Xns

n ¼ 1

win � Kðp,Q nÞ, pAM1, ð2Þ

where suppose we have ns source points in the exterior of M1, ~Q isthe 3ns-dimensional vector concatenating positions of all ns 3Dsource points, and ~wi ¼ ðwi

1;wi2; . . . ;w

insÞT is the ns-dimensional

vector to be determined, which indicates charge amountdistribution on these source points.

Boundary fitting: When every source point Q nAR3;

n¼ 1; . . . ;ns is outside of M1, any charge distribution guaranteesthe vanishing Laplacian Df iðpÞ ¼ 0; 8pAM1, only that fi mightviolate the boundary conditions. Source points {Qn} should lieoutside of M1, namely, locate on the boundary surface @fM1 of aregion fM1 that contains M1 (i.e. M1 �

fM1 �R3). In [30], an offsetsurface @fM1 of M1 is created (by first computing implicit distancefield dð@M1Þ with respect to @M1 [25] and then generating thepolygonization [6] on the implicit surface dð@M1Þþd¼ 0), and aset of source points are uniformly sampled on this @fM1 . Then itsolves the charge amount on each particle such that the potentialfield approximates the boundary condition. The boundary condi-tion is the surface mapping, whose fitting is conducted over a setof evaluation points (also called the collocation points or constraint

points) fpiAM1g. Three particle systems with their chargedistribution solved in this fitting process compose the volumetricmap ~f ¼ ðf 1; f 2; f 3Þ.

Limitations of the aforementioned routine: The above algorithmof [30] has following two key limitations:

(c) show the 30% and 60% morphing from (a) to (d), generated by linear

interpolation.

� Computation efficiency and fitting accuracy: Three dense linearsystems need to be solved. Suppose we have nc collocation(evaluation) points and ns source points, we have a Awi

¼ bi

system where the dimension of the coefficient matrix A isnc�ns. A is dense since every source point contributes to everyconstraint point. Furthermore, A is ill-conditioned. As sug-gested in [40,29,30], singular value decomposition (SVD), due toits stableness against the ill-conditioned system, is chosen tosolve this system. However, SVD decomposition is slow forlarge matrices. For example, the solver of [30] needs more thanone day when both nc and ns exceed 20k vertices. Therefore,when handling complex volumetric data, we have to restrict nc

and ns. This causes the salient decrease on the boundary fittingaccuracy. Because now we either (1) lack enough particles fordesigning fine potential fields to well fit the boundarycondition (when ns is picked to be small), or (2) lack enoughevaluation points to sample the shape variance on theboundary (when nc is picked to be small).

� Feature alignment: Like other volumetric mapping methods

[47,21,30] focuses on homogeneous volumetric regions whereboundary surface map is the only constraint in the mappingcomputation. In real scenarios, volumetric data usually containdifferent materials and densities, or have salient structureinside its interior region (see Fig. 1(a) for example). Theseinformation or structures are usually meaningful and shouldbe considered. Therefore, a scheme that can properly handleheterogeneous structure is worthwhile, so that we will be ableto align or match similar material/intensity when necessary.

New computation scheme: To tackle these two limitations, weimprove [30] as follows.

�

Instead of one function ~f : M1-M2, we compute a set of

harmonic functions ~fi such that their sum approximates ~f . The

computation for each ~fi is more efficient and numerically more

stable. Each ~fi is harmonic, and therefore their sumP

i~fi is also

harmonic. Each subsequent ~fi aims to refine the existing mapPk ¼ i�1k ¼ 0

~fk towards the exact boundary condition ~f 0 (which

could need a too big system [29] to solve within only one shot).Now we use less constraint points and source points to

compute each ~fi , and therefore the solving is much faster, the

boundary condition for each ~fi is dfi ¼~f 0�

Pk ¼ i�1k ¼ 0

~fk . This

greatly improves the speed of the mapping computation, andmakes the MFS practical for large volumetric data.

� As we will demonstrate in our experiments, with only a few ~fi ,

the fitting accuracy can usually beat the algorithm of [29], andwe can keep refining it using more ~fi when necessary. Also,unlike the [30] that conducts uniform sampling over the offsetsurface @M1, we conduct the sampling adaptively following thegeometry of the given shapes. Together with the multi-MFSscheme, our scheme intuitively allows a flexible placement ofsource points. It is well known that in MFS, the location ofsource points constitutes a key issue and it has large impact onnumerical stability of the MFS computation. The locations ofsource points are either preassigned or determined along withthe coefficients of the linear combination. Most papers placesource points on a surface outside M1 uniformly and solve leastsquare linear systems, but this not always guarantees that thecomputed solution converges to the exact solution as thenumber of source points increases. The MFS with movingsource points has been considered by several authors (e.g.[34,11]). This leads to a slow nonlinear optimization, and still,it has been reported that the initial placement of the sources isusually very important in the convergence of these algorithms,as they converge to the first local minima encountered.

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In this paper,we also follow the preassign-approach that leadsto linear systems. Unlike the placement scheme in [30], weallow the removal and adaptive adding-in of new source pointsaccording to the result in the previous round, and this mimicsthe adjustment of the source points during the MFS solving.We will show that our new scheme also improves the accuracy

of the mapping computation using MFS.

� Feature alignment: We allow the setting of constraint points

during the mapping. In real applications, three types ofconstraints are very useful: 0-manifolds (feature points),1-manifolds (feature curves or skeletons), and 2-manifolds(iso-surfaces). These constraints are treated as a part ofboundary fitting. We apply an adaptive scheme to balancethe feature constraint and boundary constraint.

3.3. Algorithm pipeline

Our algorithm pipeline is as follows. The input is two givensolid objects M1, M2 and their boundary surface mapping

~f 0 ¼ ðf 01,f 02,f 03Þ : @M1-@M2:

The output is a harmonic volumetric mapping composed of a setof harmonic functions:

~f : M1-M2 ¼Xi ¼ nf

i ¼ 1

~fi ¼Xnf

i ¼ 1

ðf 1i ,f 2

i ,f 3i Þ,

such that on the boundary surface pA@M1, ~f ðpÞ ¼ ~f 0 ðpÞ and in theinterior region: r2~f ¼ 0.

Each harmonic real function f ij, i¼ 1,y,nf, j¼ 1,2,3 is solved by

one linear system Aijwi

j¼ bi

j. In the following algorithm, we omitthe indices i and j (e.g. using the notation A instead of Ai

j) forsimplicity, assuming this will not cause any ambiguity:

(1)

Place source points and collocation points (Section 4.1). (2) Compute the coefficient matrix A, whose (u,v)th element

Auv¼K(Pu, Qv) (Eq. (1)) for the collocation point Pu and sourcepoint Qv.

(3)

Decompose A using singular value decomposition A¼URV�.The decomposed results U;R;V� are used to solve the fittingsystem.

(4)

Set the boundary condition b at the right hand side of Auv w ¼b, and b¼{ bk}, where bk is the boundary constraint evaluatedon each collocation point.

Table 1

For each ~fi , this algorithm solves Aijwi

j¼ bi

j. When i¼1, theboundary condition is set to be a low-resolution surface mapping

from M1 to M2. For i41, we use a higher-resolution surfacemapping, and also apply the refined boundary fitting

d~f 0 ¼ ~f 0�Pi ¼ nf�1

i ¼ 1~f 0i .

Note that our algorithm takes the boundary surface mapping ~f 0

as an input. We briefly discuss how to obtain such a surfacemapping in Section 4.3.

Source and collocation points placement using geometry-adaptive (GA) sampling

and uniform (UN) sampling.

S-Pts C-Pts Boundary fitting error Collocation error

UN UN 0.0563971760 0.0499722402

GA UN 0.0546669675 0.0270761958

UN GA 0.0544260284 0.0469717138

GA GA 0.0484185840 0.0449218356

The experiment is conducted on mapping vase-lion model with 40k vertices to a

solid sphere, cRatio¼0.05, sRatio¼0.05, and the offset surface is 0.15 times object

size distant. The boundary fitting error indicates the boundary mapping quality.

Using geometry-adaptive sampling leads to less boundary error.

4. Implementation and discussion

4.1. 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 ns

source points ~Q ¼ fQ 1;Q 2; . . . ;Q nsg are particles in the exterior of

M1 and nc collocation points ~P ¼ fP1;P2; . . . ;Pnc g are evaluationpoints on the boundary @M1. We solve the weights (charge

amount) distribution wi, i¼1, y, ns on all source points ~Q so that~f ðPiÞ satisfies the boundary condition approximately.

The distribution of source and collocation points greatly affectsthe numerical stability and therefore the mapping efficiency andquality. The boundary error is sensitive to the collocation andsource points, so appropriately sampling ~Q and ~P is critical. In 2Dcases, theoretic studies have been conducted for analytical andsimply connected domains, for example, when M1 �R2 is a planardisk [34], uniformly sampling both collocation and source pointsis ideal and leads to exponential decreasing on boundary fittingerrors. When M1 �R2 is analytic, there is also discussion on theexistence of optimal placement [23], and one suggestion is to takea conformal mapping C from the unit disk D to M1, and place ~Qand ~P on C’s images of the evenly sampled points on theparametric circle. However, for more complicated domain shapesin 2D, or in our 3D case that M1 �R3, the optimal placement isstill unknown. Ref. [30] shows that placing source points on anearby offset surface produces more accurate mapping result. Wealso adopt the offset surfaces scheme but add in adaptivity,following both the geometry and sequential fitting errors.

4.1.1. Sampling collocation points

In 2D analytical boundary scenarios [11], uniformly sampled ~Pusually leads to a stable system and good approximation, thereforeis suggested as the strategy for preassigning collocation points. Ref.[30] follows this strategy, and uses the uniform sampling schemeof [36] to generate evenly distributed collocation points on @M1.The total number of collocation points (source points) is controlledby an aspect ratio cRatio¼ nc=nð@M1Þ ðsRatio¼ ns=nð@M1ÞÞ, wherenð@M1Þ is the total vertex number of @M1.

However, our multi-level MFS solving shows that for most 3Dpiecewise-linear domains, adaptively sampling these collocationpoints following local geometry could lead to better convergenceon boundary fitting errors. Intuitively, more evaluation pointsshall be placed on highly detailed regions for better sampling theboundary variance. Our geometry adaptive sampling algorithm isas follows:

(1)

Tessellate the boundary surface where we need to do thesampling: it is the domain boundary @M1 for collocationpoints, and offset surface @fM1 for source points.

(2)

Refine M1 (for collocation points sampling) or @fM1 (for sourcepoints sampling) by subdivision and get a dense mesh @M0.

(3)

Conduct surface simplification on @M0 using the quadric errormetric [14], which efficiently produces a good-qualityapproximated simplified mesh @M�. The vertex number of@M� is determined by our sampling budget.

(4)

Vertices of mesh @M� are used as sampling points.

Table 1 illustrates our experiments conducted on the sphericalmapping of a vase-lion model with 40k vertices, and the mapping

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X. Li et al. / Computers & Graphics 34 (2010) 242–251 247

is for cRatio¼0.05, sRatio¼0.05, and the offset distance is 0.15times object size. It clearly shows the advantage of geometry-adaptive sampling over uniform sampling in placement of bothcollocation points and source points.

4.1.2. Sampling source points

We place source points following three aspects.

�

Geometry-adaptive sampling: In the coarsest level (f0 computa-tion), we conduct geometry-adaptive sampling on sourcepoints to determine their locations. � Even partitioning: In finer levels, we partition sampled source

points {Qv} into several subsets evenly. Each time weonly use a subset of charge points for efficient boundaryfitting.

� Adaptive deletion/insertion: Meanwhile, in each step, we

remove redundant source points by analyzing the diagonalmatrix from the SVD decomposition (see Section 4.2 fordetails), and adaptively add in extra source points near theregions with large fitting errors by projecting badly fittedboundary points onto the offset surface.

4.2. Solving MFS linear systems by SVD

The boundary fitting is reduced to solving linear systemsA~w ¼~b. A can be dense and ill-conditioned [40], so regular linearsystem solvers such as Gaussian elimination, LU, and QRdecompositions usually fail to produce a stable solution. Assuggested in [40,30], we use singular value decomposition (SVD)to decompose A. There are three reasons.

(1)

It generates accurate and stable results when the coefficientmatrix is highly ill-conditioned.

(2)

It flexibly gets to the least-square solution for over con-strained boundary conditions (which always happen in ourmulti-round MFS solving).

(3)

Furthermore, in our approach, we also use the diagonal matrixR to adaptively remove redundant singularity points. Whenthe singular value is small (in all our experiment, we set thethreshold to 1e�5), the corresponding source point does notcontribute much to the potential field, and therefore weremove them in the source point set from subsequent linearsystem solving and MFS evaluations.

4.3. Surface mapping as boundary condition

The boundary condition of our harmonic volumetric mappingis a surface mapping between @M1 and @M2. Existing inter-surfacemapping techniques [22,26,35,39,24,41,48] can be used forcreating the boundary surface mapping. Although surfacemapping is not the focus on this paper, as discussed in [30], it isdesirable to have a low distorted surface mapping, and [30]illustrates an example that larger angular distortion oftentimesleads to worse volumetric mapping result. However, it is stillunknown that how exactly quality of boundary mapping andinterior mapping relate. Intuitively, area preserving is alsoimportant (salient shrinkage on boundary mapping shall lead tolarge volume distortion). Ref. [30] uses [27] for boundary surfacemapping computation which leads to least angular distortion, inour work, we use [28] to generate boundary surface mapping,which could (1) better balance area stretch and angle distortion,and (2) allow surface feature points and curves alignment whichbetter fits our current framework.

4.4. Feature alignment

Feature and structure constraints are important issues inprocessing many real volumetric data. Specifically, three types offeatures are commonly considered: 0-manifolds (feature points),1-manifolds (feature lines), and 2-manifolds (iso-surfaces).

Feature point alignment: Since we are using a meshlessparadigm, all the feature alignment shall naturally be handledin terms of points. We can simply add the feature constraintdefined on each point into the linear system as a new boundarycondition. So each new pair of feature points for alignmentcorresponds to an additional row in the coefficient matrix A andthe boundary condition vector b.

Feature line alignment: For feature lines (for instance, skele-tons) matching, we similarly sample and put in point-by-pointconstraints to A. Feature curves such as skeletons are usuallyrepresented as a piecewise graph. Fig. 3(a) shows an example, inwhich one wants the volumetric mapping follows skeletonstructure and is guided by the movement or deformation of theskeleton. Existing skeleton extraction algorithms [9,37,2,17]usually do not guarantee that the extracted skeletons fromdifferent objects are isomorphic even when these shapes are verysimilar. However, most skeletonization algorithms can preservethe homotopic structure of the shape and therefore their skeletalgraphs are topologically equivalent. Detail discussion on featurelines extraction, their topology, and their matching is far beyondthe scope of this work, and we refer readers to the survey paper[7]. In our experiment (Fig. 3), we use [9] to extract the skeletonand apply 1-manifold restamping algorithm of [1] to get densepoint-by-point correspondence between skeletons of the cyber-ware Male and Female models. Fig. 3(b) visualizes the volumetricmap result using the color-encoded distance field. The distance ofeach interior point to the boundary surface is computed andcolor-encoded, such a color is transferred to its correspondingpoint under the mapping. This visualizes the mapping behavior.(c) illustrates the skeleton fitting: in the interior of the targetFemale model, the green curve is its skeleton Sk2. Sample pointsfrom the skeleton of the source Male model should match Sk2, andthe red points shows their images under the mapping. The rootedmean square fitting error is 0.5%.

Feature surface alignment: With two feature surfaces to match,we need to compute inter-surface maps between them. The inter-surface parameterization methods, which we used to generateboundary surface correspondence, can be applied to get such amap. Then we simply include corresponding point pairs as theboundary conditions. Figs. 1 and 2 show an example of avolumetric mapping over the heterogeneous data head–skull–brain model, which has three salient iso-surfaces: the outerboundary is a genus-0 (head) surface, and the interior skull iso-surface is genus-2, within which there is a genus-0 brain surface.We generate a parametric domain (d) to test the efficacy of ourmapping on heterogeneous 3D data with iso-surface constraints.The outer head boundary surface is mapped onto a sphereboundary, the skull iso-surface is constrained on the polycubeskull, while the brain iso-surface is mapped to a small cube inside.(b,c) show the 30% and 60% morphing from (a) to (d), generatedby linear interpolation. (e,g) show two cross-sections on thepolycube-sphere domain, and (f,h) show their correspondingcross-sections on the head–skull–brain model. The point clouds in(e)–(i) show sample points on the iso-surface (e, g), and theirimages after the volumetric mapping (f–i). Locations of thesefeature points in (g–i) demonstrate that the iso-surfaceconstraints are precisely fitted, and the volumetric mappingalign the feature surface very well (Fig. 2(j)).

Feature alignment as soft constraints and weights: As discussedabove, features are aligned in a least square sense together with

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Fig. 2. Heterogeneous volumetric mapping between head–skull–brain and

polycube-sphere (cont.). (e,g) show cross-sections on the polycube-sphere

domain, (f,h,i) show corresponding cross-sections on the head-skull-brain model.

The point clouds in (e)–(i) are the sampled feature points on these iso-surfaces

(e,g) and their images (f,h,i) of the volumetric mapping. The color-encoding in (f,h)

visualizes the mapping via the transferred distance field of (e,g). In (j), we zoom in

the brain iso-surface (the grey transparent surface) and its fitting, green points are

images of sampled points on the interior cube in (e). The RMSE here is 0.57%.

Fig. 3. Volumetric mapping from the solid cyberware male model to female model. Thei

in (b) via one cross section: the distance field color-encoding on the female (right) mod

skeleton fitting: the green curve shows the target skeleton (i.e. the skeleton of the fema

skeleton RMSE fitting error is 0.54%.

X. Li et al. / Computers & Graphics 34 (2010) 242–251248

the boundary fitting process, so they are treated as softconstraints. Compared with the massive point number on theboundary, if feature points (or samplings on feature lines) areconsidered as ordinary collocation points, they might be over-shadowed by boundary collocation points during the fitting. Webalance this by assigning each sample feature point an extraweight w. This is equivalent to enforcing this feature w times. Inall our experiments, we take w¼20 for feature points. Thiseffectively leads to more precise feature alignment.

Unlike the traditional FEM-based methods that simply fixesfeature vertices to enforce the constraints, in this section wediscuss our method that blends several harmonic functions to getthe feature-aligned map. In each iterative refinement, we use aharmonic function, so the resultant map, i.e. the summation ofthese functions is still globally harmonic, and there is no obviousflip-over or discontinuity around the feature regions. In the meantime, while the feature constraints is precisely enforced, theboundary fitting accuracy could decrease a little bit (i.e. the RMSEincreases slightly on the boundary).

5. Experimental results and applications

We conduct a few volumetric mapping experiments overvarious volumetric data, with different sizes, topology andgeometry complexities. We illustrate some of these mappingresults in Figs. 4 and 5. We use the color-encoded distance field tovisualize the mapping result. When a map ~f : M1-M2 iscomputed, the color-encoded (red indicates the maximum whileblue indicates the minimum, see Fig. 5(h)) distance field definedon one region can be transferred to another region, by plotting thecolor of a point PAM1 on its corresponding image ~f ðPÞAM2 (orinversely, plotting the color of PAM2 on ~f ðPÞAM1). Thisvisualization shows the effect of the map. For example, whenwe transfer the distance field defined on the Max-Planck model(Fig. 4(c)) to the sphere, we can see a color-encoded head-shapedlevel-set in (d), while the original distance field of a sphere isconcentric as shown in (f).

We also conduct thorough comparison between our methodand the algorithm of [30]. Table 2 illustrates the side-by-sidestatistics. Using MFS [30], the cRatio (and sRatio), indicating theratio of the number of collocation points (and source points) overthe number of boundary points are listed in the CR (and SR)

r skeletons are illustrated in (a). The mapping with skeleton matched are illustrated

el is transferred to its corresponding point on the male (left) model. (c) shows the

le), images of sampled points on the male skeleton are visualized as red points. The

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Fig. 4. Spherical volumetric mapping. A solid sphere (a) is mapped to a solid Max-Planck model (b), the color-coded distance field on the Max-Planck model (c) is

transferred onto corresponding points on the sphere (d) for visualizing the map. An Omotondo model (e) is mapped onto the solid sphere (a), and we transfer the color-

encoded distance field of sphere (f) onto the Omotondo region (g). The color-coding scheme is illustrated in (h), red indicates maximum values while blue indicates

minimum values.

Fig. 5. Mapping between solid objects and polycubes. Polycubes (a,e) are mapped to two-torus (b) and kitten (f), respectively. Color-encoded distance field of (c,g) are

transferred under the mapping to (d,h).

Table 2Efficiency and accuracy comparison.

Models (vertex #) CR SR Time (s) RMSE CRn SRn nf Timen (s) RMSEn

Omotondo/Sphere (3k) 0.4 0.8 220.41 0.01649 0.4 0.8 6 42.41 0.001514

PCube/2-Torus (6.6k) 0.4 0.8 2393.10 0.00490 0.4 0.8 6 426.23 0.00475

Male/Female (6.3k) – – – – 0.2 2 10 1000.68 0.00485

PCube/Skull (29k) – – – – 0.03 0.03 7 14.42 0.01764

Vaselion/Sphere (40k) – – – – 0.1 0.2 6 1440.39 0.0230

PCube/Kitten (80k) – – – – 0.05 0.05 6 666.48 0.0139

PCube/Horse (100k) – – – – 0.01 0.02 6 23.82 0.01449

A lot of 3D solid models have been tested, from data with small vertex size to large size. Using MFS [30]: CR and SR (cRatio and sRatio of the mapping), computation time

(in second), and the RMSE (rooted mean square error of the boundary fitting) are listed; using the new computation framework, in the same row we list the statistics of the

corresponding CRn, SRn, nf (number of harmonic maps we solved), computation time, and RMSEn.

X. Li et al. / Computers & Graphics 34 (2010) 242–251 249

column; the computation times (in seconds) and rooted meansquare errors (object size normalized to a unit box) are also given.Using multiple MFS computation proposed in this work, we listthe number of harmonic functions we computed in the column ofnf, and show corresponding ratios, time, and errors in columns ofCR�, SR�, Time�, RMSE�, respectively. As we addressedpreviously, due to the efficiency issue, MFS [30] is not able tohandle large volumetric data or feature alignment, thereforestatistics on corresponding cells are blank. The statistics showthat our algorithm improves the computation of [30] in bothefficiency and accuracy.

Fig. 6 plots the reducing of boundary error during eachiterative step i of computing mapping functions ~fi .

5.1. Hex-remeshing

A direct application for volumetric mapping is hex-meshgeneration. Regular mesh structure is highly desirable for finiteelement analysis and physically based deformations/simulations,because regular meshes provide great efficiency for geometryprocessing and physically based computation [44]. Given a 3Dsolid data M, we first generate a solid polycube model P, then wecompute the surface mapping ~f 0 : @M-@P and volumetric map-ping~f : M-P. With~f we can transfer the regular structure on P toM. On a solid polycube, a regular hexahedral structure can be

easily generated, and since [43] introduced the concept of surfacepolycube map, several techniques [45,46,31,16] have beenproposed to (automatically) construct polycube, and the sur-face-polycube mapping. In all of our experiments, we constructour polycubes using the algorithm of [45]. Fig. 7 illustrates anexample of using a unit solid cube to remesh the solid David head.The original model M2 is shown in (a), and the hex mesh of theparametric cube M1 is shown in (b). We compute the volumetricmap~f : M1-M2 from the cube to David head. Then~f ðM1Þ is a solidwith the hex connectivity of M1 and the head shape of M2, and it isthe remeshed David head, as illustrate in (c) and (d). Fig. 8 showsa few more examples. A hex-remeshed two-hole torus is shown in(a). The hex-mesh structure of the polycube (b) is used to remeshthe kitten, shown in (c,d). Polycube (e) is used to remesh theChinese horse model (f–h). (f, g) visualize the result hex-mesh inits interior regions from two different cross-sections.

6. Conclusion

We present a feature-aligned volumetric harmonic mappingcomputation algorithm using methods of fundamental solutions.The map ~f is composed of a set of harmonic functions f~fi g whichcan be efficiently solved. Also, our adaptive source/collocationpoints placement improves the numerical issue of MFS solving.Therefore, our algorithm largely improves the existing harmonic

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X. Li et al. / Computers & Graphics 34 (2010) 242–251250

volumetric mapping computation algorithm using MFS [30]. Thenew algorithm has better efficiency and accuracy, and it supportsfeature points, curves, or surfaces alignment, which is importantfor integrating/matching heterogeneous volumetric data thathave intrinsic interior structure. We demonstrate that harmonic

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60.012

0.014

0.016

0.018

0.02

0.022

0.024

0.026

0.028

Itertation

Mea

n E

rror

Polycube/SkullOmotondo/SpherePolycube/HorsePolycube/KittenVaselion/Sphere

Fig. 6. Boundary rooted mean square error of volumetric mappings.

Fig. 7. Hex remeshing of the solid David head. (a) The original mesh structure of the D

(c) The remeshed David head and (d) a cross-section to show the interior structure.

Fig. 8. Hex remeshing. (a) illustrates a hex-remeshed solid two-torus using the polycub

(b). The remeshed kitten is illustrated in (c, d). (e)–(h) show the hex-remeshing for a s

volumetric mapping can be conducted on large data, hetero-geneous data, and data with feature to match, which can not behandled properly in [30].

6.1. Future work

We plan to explore the application of our volumetric mappingframework in realistic scenarios such as heterogeneous volu-metric medical data registration. And we would like to furtherexplore and seek to improve the MFS method in solving harmonicvolumetric mapping. Several directions might be interesting.

�

avid

e of

olid

Bijective volumetric mapping via domain decomposition: Thebijectiveness of harmonic volumetric parameterization hasbeen proved to exist in several special types of shape domains.We plan to explore effective mapping computation frame-works through domain decomposition methods.

� Free boundary volumetric map: Our current harmonic volu-

metric map depends on the boundary surface mapping. Asdiscussed in Section 4.3, the effect of volumetric map is closelyrelated to the boundary surface mapping. In the future work,we plan to explore free-boundary volumetric mapping, eitherallowing boundary vertices flowing over the boundary surfacein a variational manner or including them as unknownvariables in the MFS solving. A free-boundary volumetricmapping could have significantly less distortion.

� Source and collocation points placement: We have demonstrated

our adaptive placement of source and collocation points leads

head. (b) A simple cube domain that the hexahedral mesh is generated upon.

Fig. 5(a). The hex mesh on the polycube for remeshing solid kitten is shown in

Chinese horse model.

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X. Li et al. / Computers & Graphics 34 (2010) 242–251 251

to better convergence of boundary fitting errors in comparedwith generally suggested uniform sampling scheme (and alsothe algorithm in [30]). It is worthwhile to further explore thisproblem since it directly dictates the numerical effectivenessof the mapping computation. Ref. [23] discussed 2D Dirichletproblems over simply connect domains, and suggested ascheme based on planar conformal mapping: when we havea conformal map C from M1 to a regular exterior shape like acircle fM1

0

, an ideal placement of collocation points is on theimages of the inverse conformal map C�1

ðpÞ, where {p} areevenly sampled points on fM1

0

. We plan to explore this schemein the future in the R3 case via surface conformal mapping.

Acknowledgements

We would like to thank the anonymous reviewers for theirhelpful comments and suggestions. Discussions with Hong Qinand Warren Waggenspack inspired this work. The head of Davidmodel is from Stanford Michelangelo project, male and femalemodels are from Cyberware. Other models are from AIM@SHAPEShape Repository. This work is supported by Louisiana Board ofRegents Research Competitiveness Subprogram (RCS)LEQSF(2009-12)-RD-A-06 and PFund: NSF(2009)-PFUND-133.Huanhuan Xu is supported in part by the Mark and CarolynGuidry doctoral fellowship from Electrical and Computer En-gineering Department of LSU.

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