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Computers & Graphics 34 (2010) 187–197

Contents lists available at ScienceDirect

Computers & Graphics

0097-84

doi:10.1

� Corr

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journal homepage: www.elsevier.com/locate/cag

Technical Section

Volumetric parameterization of complex objects by respectingmultiple materials

Tobias Martin �, Elaine Cohen

School of Computing, University of Utah, Salt Lake City, Utah 84112, USA

a r t i c l e i n f o

Keywords:

Trivariate b-spline modeling and

generation

Volumetric parameterization

Model acquisition for simulation

93/$ - see front matter & 2010 Elsevier Ltd. A

016/j.cag.2010.03.011

esponding author.

ail address: martin@cs.utah.edu (T. Martin).

a b s t r a c t

In this paper we present a methodology to create higher order parametric trivariate representations

such as B-splines or T-splines, from closed triangle meshes with higher genus or bifurcations. The input

can consist of multiple interior boundaries which represent inner object material attributes.

Fundamental to our approach is the use of a midsurface in combination with harmonic functions to

decompose the object into a small number of trivariate tensor-product patches that respect material

attributes. The methodology is applicable to thin solid models which we extend using the flexibility of

harmonic functions and demonstrate our technique, among other objects, on a genus-1 pelvis data set

containing an interior triangle mesh separating the cortical part of the bone from the trabecular part.

Finally, a B-spline representation is generated from the parameterization.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Generating volumetric models suitable for simulation is anincreasingly important task in geometric modeling. In practice aclosed triangle mesh in three space is given which represents theboundary of a physical domain. Commonly, interior trianglemeshes are given that separate different materials within thephysical domain. For simulation, the goal is to create volumeelements in the region enclosed by the multiple boundary trianglemeshes. This process is referred to as model completion. Inpractice, two choices are usually given: Filling up the volumeusing a unstructured grid representation based on tetrahedral orhexahedral elements or using a structured grid representation.

Unstructured tetrahedra meshes are often used in practicebecause they are fast, almost fully automatic and work on anygeometric topology. However, these representations have draw-backs. For instance, multi-resolution algorithms such as refine-ment on an unstructured representation is more difficult than ona structured grid. Furthermore, if higher-order elements are usedin simulation, cross element continuity greater than C0 is difficultto achieve. Also, some simulation methods such as linearelasticity give better results on hexahedral meshes than ontetrahedral meshes with the same degree of complexity.

Structured grids do not share these problems, i.e. refinementcan be easily applied. Furthermore, the user can specify higherorders of continuity across elements more easily, making iteasier to achieve higher continuity on a structured representation

ll rights reserved.

compared to an unstructured representation. Therefore,structured representations are desired for many types of finiteelement simulations such as isogeometric analysis [19]. However,like unstructured grids, structured grids have limitations, inthat they require more user interaction to create them andalso depend more strongly on topology. This paper presents amethodology to create structured representations for a class ofobjects as discussed below.

This paper makes the following contributions:

(1)

A method to establish a volumetric parameterization forobjects represented with triangle meshes with higher genusand bifurcations based on a midsurface, suitable for bothB-spline and T-spline fitting. The volumetric parameterizationmethod is a hybrid technique inheriting positive aspects ofboth polar- and polycube-style parameterizations.

(2)

An isoparametric methodology to parameterize the volumethat respects interior boundaries of material attributes.

We assume that interior material boundaries are of similargeometric complexity as the exterior triangle mesh boundary andthat the interior material boundaries are nested within eachother. A demonstration of the approach on diverse objects such asa genus-1 pelvis and a genus-1 propeller is given. The pelvis dataset consists of an interior triangle mesh to separate the interiorsoft (cortical) material from the hard (trabecular) boundary layermaterial. The parameterization is used to fit a trivariate B-splineto the bone for a smooth transition from the cortical to thetrabecular part of the bone.

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Fig. 1. Parameterization of a genus-3 torus.

Fig. 2. Parameterization of a disc. Top left: polar-like with one degenerate point

(magenta); top right: polycube-like with four corner points (blue); bottom: blend

of the two parameterizations. (For interpretation of the references to colour in this

figure legend, the reader is referred to the web version of this article.)

T. Martin, E. Cohen / Computers & Graphics 34 (2010) 187–197188

Martin et al. [21] proposed a methodology to create trivariaterepresentations of generalized cylinders which also allowsrepresentation of interior materials attributes. However, theapproach works only on genus-0 objects and is suitable only forobjects with smaller shape overhangs such as local bifurcations.The main reason for these limitations is that only one trivariateB-spline is used to create the trivariate representation to avoid thenecessity of gluing patches. Similarly, Aigner et al. [1] applies thisconcept to a class of engineering objects, sweeping a parameter-ized planar surface along multiple guiding curves. The sweep inthese approaches correspond to a single skeletal line. Given atopologically more complex object, a single skeletal line isunsuitable. First, it ignores the genus of the object and it ignoresbranches that can result in parametric distortions.

In the following discussion the reader is referred to Fig. 2.There are two well understood ways to parameterize a disc.The first is a polar parameterization with high-quality elementsclose to the boundary, but with a degeneracy at the center.This type of parameterization is used in generalized cylinders [7]and was further generalized in [21]. The second way toparameterize a disc is to choose four corners at the boundaryof the disc, and decompose the boundary into four iso-para-metric segments. The interior elements have high quality, i.e.they are close to having right angles at the vertices, but theelements closer to the corners are more skewed. This may affectthe quality of the physical simulation in those corner regions.Polycubes [31,33,34] are generalizations of this kind of para-meterization.

In this paper, we propose a blend of these two parameteriza-tion types, i.e. near to the boundary the parameterization ispolar-like but in the interior, it has the advantages of polycubemaps. In this way, there are no parametric or geometricdegeneracies, i.e. each element is a geometric quadrilateral. Also,no corners are specified at the boundary and hence thequadrilaterals closer to the boundary have nearly right angles.

Our method reduces the dimensionality of the parameteriza-tion problem by parameterizing a simplified manifold basesurface lying in the interior of the object using 2D parameteriza-tion techniques and using that to parameterize the volume. Thesize of the base surface, derived from a midsurface, determineswhich of the above discussed parameterization methods arefavoured and can be controlled by the user according torequirements in simulation.

Midsurfaces, decomposing an object into two pieces, arecommon occurrences in modeling for simulation. Considerableefforts have been invested to find them on CAD models consistingmostly of flat faces. Finding a midsurface for a more general objectis an unsolved problem. In this paper we make an effort to findmidsurfaces suitable for volumetric parameterization.

The choice of the midsurface and its subsequent parameter-ization affects the volumetric parameterization of the inputobject. While this reduces the modeling time significantly, theclass of objects which can be parameterized such as a hand, apelvis or a genus-3 torus as shown in Fig. 1, is extended. Whilethese objects are not general solid models, we also show that ourmethod can be applied to more general models such as a genus-1propeller as shown in Fig. 16.

In Section 2, previous work is discussed and different choicesof possible parameterizations are presented. An overview ofour proposed framework is given in Section 3. Section 4 isconcerned with the construction and parameterization of themidsurface used in Section 5 to decompose the object ofinterest. Section 6 discusses the construction of the volu-metric parameterization of the object of interest. The paper isconcluded with results, discussion of limitations and extensions(Sections 7 and 8).

2. Previous work

Model completion is a hard problem for surfaces and evenmore difficult for volumes. If the input is a closed curve, theboundary of the physical domain, model completion fills theinterior area enclosed by the curve using an algorithm such as[22]. Accordingly, for volumes, a completion algorithm fills thevolume enclosed by the input surface (2-manifold) representingthe closed physical domain. Fig. 3 illustrates the input for a pelvisdata set with inner and outer boundary, represented by trianglemeshes. The input can have additional interior boundaries thatseparate different materials within the domain. In the following,we focus on the volume case and review relevant volumeparameterization techniques. For a detailed overview on surfaceparameterization techniques, often considered as a starting pointfor volumetric parameterization, the reader is referred to thesurveys [29,17]. Furthermore, since our approach is based on amidsurface we review medial axis-based meshing techniques.

The medial axis plays an important role in mesh generation.Armstrong et al. [3] shows that the medial axis has variousapplications in modeling for simulation. For instance, the medial

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Fig. 3. 3D input: exterior boundary and interior boundary of pelvis separating

trabecular from cortical bone material.

T. Martin, E. Cohen / Computers & Graphics 34 (2010) 187–197 189

axis is used to automatically identify features that are significantin mesh generation, dimensional reduction and detail removal. In[26] a medial axis-based mesh generator is described. After theconstruction of a simplified medial axis, quad dominant mesheson the medial axis are generated and extruded to the boundary byadvancing front schemes. In this case, advancing front schemescan cause inconsistency cases and for a more complex model, atleast in part because it is difficult to control the front, e.g. thefront may intersect itself resulting in degenerate elements.Furthermore, in addition to hexahedra, the resulting meshescontain prisms, pyramids and tetrahedrons.

In a similar approach Sheffer et al. [30] propose a method forautomatic hexahedral meshing based on the embedded Voronoigraph, containing the full symbolic information of the Voronoidiagram and the medial axis of the object. It is used to decomposethe object into sweepable subvolumes. Their approach is testedon CAD models with relatively simple medial axes. A moregeneral model with a much more complex medial axis containingmany small features (e.g. Fig. 3) might result in many subvolumesthereby reducing the structured volume behaviour of theresulting mesh. In this paper, we combine a base surface withharmonic functions to consistently parameterize subvolumes, inmany cases with low distortion.

A midsurface is used here because medial axis features such assmall local fins or local branches often are not necessary for lowdistortion parameterization and can complicate the methodology.A basic approach to generate a medial axis-based parameteriza-tion is to create paths from points on the surface to the simplifiedmedial axis. Two problems must be solved with such an approach.In the first problem, how to lay out the points on the surface tocreate a tensor-product like parameterization. The second oneconcerns dealing with local concavities that can result in theintersection of these paths. Han et al. [15] propose an inter-polation approach that decomposes the volume enclosed by asurface representation into a interior region and crest region andapplies it to a genus-0 kidney model.

Polycube-Maps were originally proposed by Tarini et al. in [31]to remove seams in texture mapping by making the texturetopology of the mesh compatible with the texture domain.Polycubes have been successfully used to construct manifoldand polycube splines [13,33] where the main challenge is toconstruct the polycube surface mesh for a given object. Wanget al. [34] propose a user-controllable framework where the userdirectly selects the corner points of the polycubes on the original3D surface and based on that choice create the polycube mapsapplying discrete ricci flow. Polycubes have been suggested forvolumetric parameterization [16] but have not been presented fora variety of volume models including those that can contain

interior material boundaries to which data fitting has to beapplied to construct a representation for simulation.

Harmonic functions are holomorphic 1-forms as defined inArbarello et al. [2] and were first introduced by Gu et al. [14] foruse in surface parameterization. In this paper, we use harmonicfunctions in combination with a midsurface to attain a para-meterization that is consistent with respect to inner materialboundaries in the parameterization. Harmonic functions havebeen shown useful in [21], among other approaches, where aharmonic vector field is used to trace paths from the exterior tothe skeleton to parameterize the volume of cylinder-like objects.The properties of harmonic functions allows a consistent extrac-tion of these paths (i.e. no self-intersection between adjacentpaths) even when the boundary triangle mesh has overhangs(local concavities). Furthermore, our approach is guaranteed tocreate hexahedral-only meshes suitable for B-spline [9] fitting.We also demonstrate with examples that our methodology can beused to create a representation suitable for T-spline andT-NURCCs refinement [27]. In our method, user input is requiredonly for the generation and parameterization of the 2D midsur-face, and so reduce the dimensionality of the problem.

In addition to [21], harmonic functions are widely used inmodeling and the meshing community. Dong et al. [12] uses themto generate quad meshes on triangle meshes with arbitrarytopology where the harmonic functions are generated from userspecified critical points on the triangle mesh. In [11], criticalpoints are automatically determined by using Laplacian eigen-functions defined on the surface. Tong et al. [32] automates thesurface parameterization problem with discrete differentialforms.

Having a midsurface for a specific object is crucial to thispaper. In general, a medial axis is difficult to compute because ofthe underlying algebraic complexity. The reader is referred toAttali et al. [4] for an extensive survey on the topic. In Section 4,we describe how we generate a medial related midsurface,suitable for our modeling technique.

2.1. Discrete harmonic functions

In this paper harmonic functions are used to establish avolumetric parameterization over a domain O respecting innermaterial attributes. In general, a harmonic function is a functionuAC2ðOÞ,u : O-R, with boundary @O, satisfying Laplace’s equa-tion, that is

r2u¼ 0, ð1Þ

where OARd and r2¼Pd

i @2=@x2

i is the Laplace operator. In ourcase d¼2,3, i.e. when d¼2 then r2

¼ @2=@x2þ@2=@y2 where @O isa poly line. When d¼3, then r2

¼ @2=@x2þ@2=@y2þ@2=@z2 where@O is a triangle mesh. u satisfies the maximum principle, i.e. itdoes not exhibit any local minima and maxima, so harmonicfunctions are suitable for tensor-product like parameterization asshown in [12,32,21].

In this paper, the finite element method (FEM) [18] is usedto discretize Eq. (1). The domain O is exactly represented with atetrahedral mesh ðH,T ,V,CÞ, where H is the set of tetrahedra,T the set of faces of the tetrahedra in H, and V �R3 the set ofvertices. C specifies the connectivity of the mesh.V can be decomposed into the sets VB and VI , where VB is the

set of boundary vertices that lie on the Dirichlet boundaries whichcan correspond to the exterior triangle boundary or any interiortriangular boundary. V I is the set of vertices for which the solutionis sought. The solution is of the form:

uðx,y,zÞ ¼X

vk AV I

ukfkðx,y,zÞþX

vk AVB

ukfkðx,y,zÞ, ð2Þ

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Fig. 4. 2D input consisting of an exterior domain boundary and an interior

material boundary. Boundaries are decomposed to divide the domain into

subregions.

T. Martin, E. Cohen / Computers & Graphics 34 (2010) 187–197190

where fiðx,y,zÞ are the linear hat functions associated with vertexi evaluating to 1 at vi and to 0 for the other vertices defining therespective tetrahedron. The weak Galerkin’s formulation is usedto form the linear system S~u ¼~f , where S is the stiffness matrixand ~f is a right-hand-side function. S is positive definite [18]allowing the linear system to be solved efficiently using thepreconditioned conjugate gradient method [5].

The value of u at a point p¼(x,y,z) inside a tetrahedron is thelinear combination uðpÞ ¼

P4j ¼ 1 ujfjðpÞ, where uj is associated

with vertex vj of the respective tetrahedron. Note, that j is a localindex. The gradient field ru over O is piecewise constant, i.e. ru

defined over a tetrahedron is the linear combination ruðpÞ ¼P4i ¼ 1 ujrfjðpÞ, where rfjðpÞ is constant.

2.2. Volumetric parameterization

Given domain OAR3, represented with the tetrahedral meshH, a parameterization is a function

F : Y-O, ð3Þ

where F is a bijection and YAR3 is the reference domain. In thediscrete case, two tetrahedral meshes HY and H are isomorphic ifthere is a correspondence between their vertices, edges, trianglesand tetrahedra such that corresponding edges join correspondingvertices, corresponding triangles join corresponding vertices andedges and corresponding tetrahedra join corresponding vertices,edges and triangles.

As an example, the reference domain Y of a trivariate B-splineis a single rectangular parallelepiped, whereas the domain of apolycube map [31] consisting of many rectangular parallelepi-peds, is more similar to its corresponding physical domain O.Given such a parameterization, if O has to be represented with ahigher-order trivariate B-spline, e.g. for purposes of applyingphysical analysis to O its domain has to be decomposed into a setof cubes where each cube maps to some piece of O. Since thecollection of cubes is irregular, there can be extraordinary pointson the surface so the T-spline generalization T-NURCCs is morenatural in this context, as it allows more complex referencedomains.

In general, parameterization strategies look for ways todecompose O into sub domains, where each can be parameterizedindependently. This makes the strategy more flexible in order torepresent more complex geometries with reduced parametricdistortion, however, establishing matching parameterizations andcontinuity across the boundary surfaces of the sub domains isoften more difficult to achieve. For instance, in the surface case,Ray et al. [25] propose a global parameterization method frominput vector fields where the cuts of O (i.e. the topology of thebase complex) emerge simultaneously from the global numericaloptimization process. In this paper, harmonic functions asdiscussed above are used to decompose O into subvolumes andestablish a parameterization of each subvolume so that adjacentsubvolumes have a matching parameterization where theyconnect while respecting interior material attributes in theparameterization.

2.3. Parameterization strategies

This paper describes an approach based on a midsurface,decomposing the object of interest into two regions which is usedto create a volumetric parameterization from an exteriorboundary that respects interior material boundaries in theparameterization. For better illustration of the proposedtechnique, this Section shows two parameterization strategieson a 2D domain that has neither holes nor bifurcations.

Fig. 4 illustrates the approach. The input consists of two closedpoly lines (piecewise linear line segments). The outer one definesthe physical domain and the inner one represents a materialboundary. The inner boundary has similar geometric complexityrelated to the outer boundary. The domain is discretized withtriangles and the inner boundary is embedded in thetriangulation. The goal is to parameterize this domain whilerespecting the interior boundary. A midcurve that lies inside theinnermost area, i.e. the area enclosed by the inner boundary, isconstructed based on the outer poly line. Then, paths are tracedfrom the corners of the midcurve to the outer poly line. Thisconfiguration decomposes the space into different regions(see Fig. 4). A region is enclosed in the general case by fourboundaries. Harmonic functions are used to parameterize eachregion so that the parameterization of a boundary between twoadjacent regions match.

In the following, ai, i¼1,2,3 refer to the three regions as labeledin Fig. 5a, and bi for i¼1,y,5 refer to the five regions as labeled inFig. 5 b. Note, that O¼ a1 [ a2 [ a3 ¼ b1 [ . . . [ b5, where O refersto the whole domain. Furthermore, si (Fig. 4) are open poly linesrepresented with vertices from an augmented triangle meshrepresenting O. The notation

fsigni ’u¼ x

means that the scalar xAR is assigned to the parametric directionu of the vertices of the collection of segments s1,s2,y,sn as part ofa Dirichlet boundary condition.

Strategy 1: Laplace’s Eq. (1) is solved over O for the para-metric v-direction with the following Dirichlet boundary:fs1,s2,s3,s4g’v¼ 1 where s1 [ s2 [ s3 [ s4 is the outer boun-dary; fs5,s6,s7,s8g’v¼ 0:5 where s5 [ s6 [ s7 [ s8 is the innerboundary. Finally, fs9g’v¼ 0, where s9 is a midcurve of theouter boundary. For the parametric u-direction, Eq. (1) is solvedfor the regions a1, a2 and a3 independently. Boundary conditionsfor a1 are: fs10,s14g’u¼ 0 and fs13,s17g’u¼ 1. Boundary condi-tions are set up for a2 analogously. Eq. (1) is solved for a3

by assigning u¼0 to s10 [ s14 [ s13 [ s17, and u¼1 tos11 [ s15 [ s16 [ s12.

Strategy 2: The parametric v-direction is constructed in twosteps. Eq. (1) is solved over the region b1 [ b2 [ b3 [ b4 with theboundary condition fs1,s2,s3,s4g’v¼ 1 and fs5,s6,s7,s8g’v¼ 0:5.Then, Eq. (1) is solved over region b5 with boundary conditionfs5g’v¼ 0 and fs7g’v¼ 1. The u-direction is constructedanalogously: for region b1, assign fs10g’u¼ 0 and fs13g’u¼ 1.

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Fig. 5. Parameterization strategies on 2D input.

T. Martin, E. Cohen / Computers & Graphics 34 (2010) 187–197 191

For region b2, fs10g’u¼ 0 and fs11g’u¼ 1. Boundary conditionsare assigned accordingly to region b3 and b4. For the inner regionb5, fs8g’u¼ 0 and fs6g’u¼ 1.

Both strategies are based on the same decomposition of O,where both have advantages and disadvantages. In the firststrategy, since a1 and a3 are enclosed by only three segmentseach, the resulting parameterization has a degeneracy at thecorner c9 and c10 of the medial axis. The second strategy avoidsthis degeneracy, since all regions have four boundary segments.However, the resulting parameterization contains four extra-ordinary points in the interior (Fig. 5b), i.e. there is no tensor-product behaviour in this region. Establishing a smooth functionis therefore difficult to accomplish. In 3D, next to extraordinarypoints there are also extraordinary edges having other than foursubvolumes attached to it. A smooth representation around theseedges is difficult to achieve. T-NURCCs [27], a generalization ofT-splines have been proposed for surfaces for dealing with thisproblem.

3. Framework overview

This section gives an overview of our methodology toparameterize domains in R3. Section 2.3 discussed two para-meterization strategies in 2D. Based on a 2D input domainboundary, the enclosed volume is decomposed into regions byintroducing additional segments si using a midsurface. While in2D, these segments are open poly lines, in 3D these segmentscorrespond to triangulated surfaces called Si, the faces.

Our framework takes n closed input triangle meshes T i fori¼1,y,n, where T iþ1 is nested within T i. T 1 represents theboundary of the physical domain O and interior triangle meshesT i for i¼2,y,n separate materials. The following framework stepsdescribe the generation of a set of trivariate tensor-productswhich represent O and respect the T i.

Step 1:

Fig. 6. Simplified medial axis’ of Olivier hand data set (left: tight cocone [10],

right: our approach).

Construct a base surfaceM with respect to T 1, so thatMlies within T n. Then, create a tetrahedral meshH from T 1

which embeds all the interior T i and M as tetrahedralfaces.

Step 2:

Create a harmonic scalar field w by solving Laplace’s Eq.(1) so that interior material boundaries are respected, i.e.rw is orthogonal to T i.

Step 3:

Given rw, sweep segments from the boundary of M toform surfaces that decompose the volume enclosed by T 1

into subvolumes.

Step 4: Establish u- and v-harmonic scalar fields on subvolumes

to parameterize H. The w scalar field from Step 2 is usedas the third parametric direction in the volumetricparameterization strategy.

Step 5:

Depending on the parameterization strategy, fit trivariateB-splines, T-splines or T-NURCCs, respectively, to theparameterized subvolumes.

The trivariate representation is constructed so that theboundaries closely approximate the input data. Due to thetensor-product nature of the representation, the resolution ofthe trivariate grids can be high. Therefore, as a post-processingstep, data reduction [20] is applied to the final trivariaterepresentation.

4. Midsurface construction

Section 2 discusses work that has been done to compute themedial axis of an object in 3D. In general, for a given trianglemesh, these approaches construct an approximate medial axis.For volumetric modeling however, often these medial axes are notsuitable as they contain features like local fins or other non-manifold topology. The reader is referred to Fig. 6 which shows amedial axis generated from the Olivier hand data set using thetight cocone software [10]. Often, a time-consuming post-processing step involving manual removal and change in medialaxis topology is necessary to clean up the medial axis for it to besuitable for volumetric parameterization.

Often in modeling, a simplified medial axis is sufficient when itat least captures the topology of the object. For instance, theapproach proposed in [21] is based on a skeleton line of theobject. Harmonic functions were used as an aid to consistently fillup the interior, especially useful if the object has local overhangsfar away from the medial axis.

In this paper we construct a manifold midsurface M0 (i.e. asingle sheet) that has its boundary on T 1 and decomposes each T i

into two volumes.M0 is constructed to have no fins or other non-manifold geometry. A base surface M is computed by trimmingM0. We call it a base surface, because its size and parameteriza-tion affects the resulting volumetric parameterization. With thisapproach to use a base surface for parameterization, the approachgiven in [21] is generalized.

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ConstructingM has four steps: (1) The midsurface sheetM0 isextracted, thus decomposing each Ti into two regions (Section4.1). (2) A trimmed M0, M0trim, is computed using a scalar fielddefined on M0 and parameterized (Section 4.2). (3) The medialaxis of the parametric domain ofM0trim is formed (Section 4.3). (4)Points of the 2D medial axis of the parameter domain ofM0trim areinserted into M0trim in order to construct the base surface M(Section 4.4).

4.1. Construction of M0

Given T 1 with genus g, the midsurface M0 is constructed bydefining a set S¼ fd1, . . . ,dgþ1g each of whose elements is aclosed path on T 1. Together they decompose T i into two regions.Path dj on T 1 is a piecewise linear function defined by a sequenceof vertices from T 1 and corresponds to a boundary of T 1’smidsurfaceM0. We present the construction ofM0 by referring toFig. 7 for illustration. �1 is assigned to the first region (green) and+1 is assigned to the second region (red). This Dirichlet boundarycondition is used to solve Eq. (1) over O. Let f be thecorresponding solution. Then, the isosurface M0 at isovaluef(x,y,z)¼0 is extracted using marching tetrahedra [8]. Given theproperties of Laplace’s Eq. (1), M0 is guaranteed to lie within T 1.Several approaches can be used to construct the paths definingthe boundary of M0, including, for example, (1) extraction ofsalient ridge lines from T 1, (2) connection of critical points usingMorse analysis. Both approaches require user assistance and arediscussed in the following.

The set of ridge lines for a given object define the complexity ofits medial axis. However, only a few salient ridge linescharacterize the global structure and topology of the object andmedial axis. Ridge extraction techniques such as [24] can be usedto extract ridges from T 1. However, it is difficult to extract closedridge lines on discrete data, since ridges require higher-orderinformation. A user process is often necessary to close them inorder to decompose T 1 into two regions. The paths to decomposeT 1 can refer to closed ridges or crests requiring T 1 to be doublycurved. For instance, if T 1 is an ellipsoid,M0 is a solid ellipse withboundary along the major equator of T 1. In this case, theboundary of M0 is equal to one of the ridge lines of T 1.

In our framework, we follow the method by Ni et al. [23] toextract the topological structure of T 1, that is, the user specifiescritical points, i.e. minimum points and maximum points on T 1 toconstruct a harmonic scalar field on T 1. Discrete Morse analysis is

Fig. 7. Olivier hand model (n¼1): midsurfaceM0 (yellow and blue area) and base

surface (blue area). Since n¼1,M0 ¼M0trim . (For interpretation of the references to

colour in this figure legend, the reader is referred to the web version of this

article.)

used to find saddle points on T 1. For every saddle, the gradientfield of the harmonic field is used to create paths connecting thecritical points, i.e. connecting saddles with saddles, saddles withminima and saddles with maxima. Paths reaching the sameminima or maxima are removed. The bifurcation emanating atevery saddle has a path which ends at either a maximum or aminimum or at a different saddle. Given such a path, a point onthe opposite side (in case there is no path defined on that sidealready) is chosen automatically to create a new path which endsat a minimum or a maximum. The remaining paths can be joinedat the critical points to create a single closed path to decomposeT 1 into two regions.

However, this approach sometimes fails to determine a set ofdesirable closed loops since determining the paths in S todecompose T 1 into two pieces suitable for volumetric parameter-ization is a difficult and unsolved problem in general. Suitable inthis context means that the final volumetric parameterizationcontains as little parametric distortion as possible. An alternativeis to have the user just draw boundary curves onto T 1 using atechnique such as [6], as was for instance done to parameterize apropeller from a triangulated CAD representation (Fig. 16).

Given a set of valid paths S, the resulting scalar field f definedon H, as computed above, can be used to formulate a qualitymeasure on the choice of S in the following way. For every vertexvk on T 1, trace a path hk through H using þrf or �rf dependingon which side of T 1 vk lies. Let H be the set containing thesepaths. A good choice of S results in H with a small variance of itspaths. A higher variance means that there are both shorter andlonger paths directly affecting the parametric distortion of the endresult.

Given a choice of paths S, f and the resulting H can becomputed relatively efficiently due to the linear basis used tocompute f and the paths in H which change piecewise constantlyoverH. This is a useful tool for the user to make a judgment on thechoice of S. It is clear that there are objects which cannot bemodelled with a single midsurface, i.e. the approach works onlyon some classes of models.

4.2. Parameterization of trimmed M0

In this work we assume that T iþ1 is contained within T i fori¼1,y,n�1. Furthermore, we require thatM is contained withinthe innermost triangle mesh boundary T n. Therefore, we computeM0trim ¼M0 \ T n, i.e. the boundaries of M0trim lie on T n. Note ifn¼1, M0trim ¼M0. M0trim is parameterized by adopting the 2Danalogue version of polycubes given in [31] to a flattened versionof M0 using a flattening method such as presented in [29].Similarly as in [34], the user picks corners on the boundaries ofM0 to decompose them into isoparametric sides acting asDirichlet boundary conditions to solve the 2D version of Eq. (1)in u and v. Then, the boundary of M0trim is decomposed into a setof segments sj where the two boundary vertices of sj are 2Dpolycube corners. Each segment is isoparametric in u or v. As inthe 3D case [34], the 2D polycube representation for M0trim isparameterized in u and v. An example of the parameterization ofM0trim is given in Fig. 8.

4.3. Construction of 2D medial axis on M0trim

Given the parameterized sheetM0trim the medial axis of its 2Dparameter domain boundary is computed. Since the domain hasaxis aligned boundaries this is quite straightforward. Those edgesof the medial axis that extend to the boundary of the parametricdomain are removed leaving only the medial sheet curves.

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Fig. 8. Polycube like parameterization of midsurface M0 on hand model;

(a) parameterization and (b) corresponding parameteric domain.

Fig. 9. A trim value m0 closer to zero (right) introduces parametric distortion.

Fig. 10. Parameterization of midsurface M0 of kitten model: (a) partial view of

parameterized midsurfaceM0; (b) corresponding parameter domain and (c) scalar

field f is used to trim M0 .

T. Martin, E. Cohen / Computers & Graphics 34 (2010) 187–197 193

4.4. Construction of M

Then, the image of the medial axis computed in Section 4.3 inM0trim are inserted into the mesh and used to construct a harmonicscalar field b evaluating to 1 at medial axis image points onM0trim

and 0 at the boundary ofM0trim. The user is given the flexibility tospecify an isovalue m0 where the parts onM0trim with bðx,y,zÞom0

are removed to create the final M. Fig. 9 illustrates differentchoices of m0 affecting the w-scalar field computed in Section 5and hence the element shapes. Finally, rb is used to move thecorners defined on the boundary of M0trim to the boundary of M.Fig. 10 illustrates these steps.

Fig. 11. Construction of Pþ ¼ fpþ1 , . . . ,pþn g and corresponding P�.

5. Decomposition of volume

This Section presents the method for decomposing the volumeenclosed by T 1 into subvolumes, one subvolume on the positiveside of M, one on the negative side of M, and a set of crestvolumes around the g boundaries of M in O with genus g. O is

decomposed using a surface S consisting of g disjoint surfacepieces. S is the surface shared by these subvolumes. Theconstruction of S and hence the decomposition of O involvesthe construction of interior faces Sþ and S�, where S ¼ Sþ [ S�.Sþ and S� are computed by following a harmonic gradient fieldfor each of the g boundaries of M until T 1 is reached, passingthrough the intermediate triangle meshes T 2, . . . ,T n along theway. Fig. 4 shows the 2D equivalent of this decomposition.

Since the base surface M respects only the topologicalstructure of O, unlike the true medial axis, the generation ofthese faces is challenging because in some regions, depending onthe distortion, a face has to move more than in regions whereMis closer to T 1. In the following we assume that T i andM have afeature aware and regular triangulation. Then, Sþ and S� aremerged to form S and then decomposed into faces that relate tothe corners ofM. The construction of Sþ and S� and the resultingdecomposition of S is discussed in Sections 5.1 and 5.2.

5.1. Construction of faces Sþ and S�

Sþ and S� are robustly constructed by solving Laplace’s Eq. (1)over H by constructing a harmonic scalar field w. H contains Mand all the T i as sub meshes with the Dirichlet boundaryconditions that w¼0 is assigned to T1 and w¼1 is assigned toM. Furthermore, w¼(i�1)/n is assigned to the interior boundaryT i. Given a point p on M and its normal ~n, rw can be used totrace two paths emanating from M and ending on T 1. The startpositions of these paths are pþe~n and p�e~n, respectively, where eis a small number and ~n is the normal of p. The harmonic nature ofw guarantees that these paths do not contain loops or end at localminima within O.

Let P¼ fp1,p2, . . . ,png be the vertices of a piecewise linearboundary of M where pi are the boundary points. From P,two additional poly lines Pþ ¼ fpþ1 ,pþ2 , . . . ,pþn g and P� ¼

fp�1 ,p�2 , . . . ,p�n g are constructed as discussed next. rwðpiÞ ¼~bi

where piAP and ~ni is the normal at pi.~t i is the boundary tangenton M at pi and ~bi is the corresponding binormal (see Fig. 11).

Given pi, we find pi+ and pi

� so that the angle between rwðpþi Þ

and ~bi is � y and between rwðp�i Þ and ~bi is ��y. The user choiceof y affects the crest region, i.e. how close the final Sþ and S� areto each other. For many model y¼ p=4 is a good choice.

As shown in Fig. 11, P ¼ fp1,p2, . . . ,png, where pi ¼ pi�e~bi forall piAP. A curve gi : ½0,1�-R3 is defined by

giðtÞ ¼

ð1�3tÞqþ

i þ3tqþi , 0rto1

3

McðtÞþpi,1

3rto

2

3

ð3�3tÞq�i þð3t�2Þq�

i ,2

3rto1,

8>>>>>><>>>>>>:

ð4Þ

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T. Martin, E. Cohen / Computers & Graphics 34 (2010) 187–197194

where cðtÞ ¼ h ðsinðð3t�3ÞpÞ,0,cosðð3t�3ÞpÞÞ and M is the 3�3rotation matrix defined by ~ni,

~bi and~t i. Furthermore, qþ

i ¼ piþe~ni,qþi ¼ piþe~ni and q

�

i ¼ pi�e~ni, q�i ¼ pi�e~ni.The start position pþi ¼ giðtÞ, where t A ½0,1=2�, is determined

such that the angle between giðtÞ and rwðgiðtÞÞ is close to y. pi� is

determined analogously on the interval [1/2,1].Surface Sþ is constructed incrementally: A surface mesh is

generated where the poly lines Pþ0 :¼ P and Pþ1 :¼ Pþ definequadrilateral elements having one edge at the boundary of M.Row Pþk is determined by computing, for every point pþ APþk�1, aline segment using rw starting at p+ and ending at a face of thetetrahedron in which p+ lies (pathological cases are robustlyhandled as in [21]). The shortest line segment with length d

determines the amount Pþk travels through H, i.e. jpþ�pþ j ¼ d

where pþ

is the new point in Pþk corresponding to p+ and lyingon the line segment corresponding to p+. (See Fig. 12.)

Once a new row has been computed it is triangulated with theprevious row. Furthermore, if the distance between two adjacentpoints in Pþk gets twice as large as the corresponding points inPþ0 , then a new column is inserted between them emanating atrow k. Accordingly, columns are removed when the distance ofthese points is less than half the distance of the correspondingpoints in Pþ0 . S� is analogously constructed using P and P�.

5.2. Construction of faces Sj

Once Sþ (red surface in Fig. 13) and S� (green surface inFig. 13) have been computed they are merged into the single faceS for each loop on the boundary ofM. After this process has been

Fig. 12. Construction of Sþ and S�.

Fig. 13. Decomposition of Olivier hand data set.

applied to all boundaries of M, the surfaces connecting theboundaries ofM with T 1 decompose the volume enclosed by T 1

into three subvolumes: (1) the volume H1 enclosed by all Sþ ,Mand T 1, (2) the volume H2 enclosed by all S�,M and T 1, and (3)the crest volume H3 from the volume which remains, i.e.H3 ¼H\ðH1 [H2Þ (A is the closure of A). Note that H3 consistsof g+1 disjoint pieces (g is the genus of T i).

The crest volumes are further divided: For every corner vertexcj defined on the boundary ofM (see Section 4.2), a corner surfaceCj (yellow surfaces in Fig. 13) is computed. Cj connects cj to T 1 bypassing through all the interior T i. One boundary curve of Cj lieson Sþ , the other on S�. Cj is computed analogously to Sþ and S�,by using scalar field w to propagate a poly line through thevolume starting at cj till T 1 is reached. Given the corners surfacesCj, S can be decomposed into faces Si, where the face Si

corresponds to the segment i of M as classified in Section 4.2.Furthermore, Si and the two corner surfaces at its boundaryenclose a surface on T 1 (Fig. 13). These surfaces on T 1 and thefaces Si are used to establish a parameterization over O asdiscussed in the following section.

The automatic approach proposed above to construct Sþ andS� is consistent, i.e. rw flows from M to T 1 without any localsinks as guaranteed by the choice of boundary conditions used tocompute w, and it also guarantees that Sþ and S� reach theexterior T 1 orthogonally, passing orthogonally through theinterior boundaries T i (i¼2,y,n). This means that the surfacesshared by two adjacent subvolumes are orthogonal to theboundaries T i, resulting in more orthogonal parameterizationsin these regions. This approach makes establishing continuitybetween two adjacent subvolumes easier, as discussed below.

6. Volumetric parameterization

In this Section the method to volumetrically parameterize thedomain O represented with H is presented. The 2D parameter-ization strategies from Section 2.3 are extended to 3D. While in2D, the boundaries were defined by poly lines referred to assegments, in 3D, boundaries are represented with faces Si asdiscussed above. Furthermore, as in 2D, the interior materialboundaries T i, i¼2,y,n are respected in the parameterization.Both strategies make use of the w-scalar field used to extract thefaces S.

Strategy 1: This strategy uses the parametric direction w

computed in the previous Section and requires only computingu- and v-scalar fields over H. As discussed above, every boundarysegment si of M (see Fig. 13) corresponds to a face Si and isconstant value of either u or v. T 1, Si and the two corner facesadjacent to Si together form the boundary of either a crestsubvolume Hucrest

i or Hvcresti . Let

Hucrest ¼[m

i ¼ 1

Hucresti , ð5Þ

where m is the number of faces Si. Hvcrest is defined analogously.Next, the scalar fields for u and v are constructed. They are

constructed analogously, so we present only the construction foru. For all edge segments si ofM with constant u-value ui, assign ui

to its respective Si embedding si. Solve Eq. (1) on Hu, whereHu ¼H\Hucrest .

Now, the subvolume H\ðHucrest [HvcrestÞ, i.e. H without thecrest subvolumes, has a parameterization. Hucrest has a parametricv-scalar field, and Hvcrest has a corresponding u-scalar field.Similar to the 2D case, the remaining scalar field is constructed byassigning a constant parameter value to Sþ and a constantparameter value to S� followed by solving Laplace’s Equation asabove. This step can be seen as consistently sweeping Sþ to S�

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Fig. 14. Second volumetric parameterization strategy for Olivier hand data set.

(Parametric cut in u, showing a part of the interior material boundary.)

Fig. 15. Different choices of corners (red), resulting in two different scalar fields on

a midsurface. (For interpretation of the references to colour in this figure legend,

the reader is referred to the web version of this article.)

Fig. 16. Parameterization of a CAD propeller of genus-1 with a single midsurface

(midsurface boundary in yellow). (For interpretation of the references to colour in

this figure legend, the reader is referred to the web version of this article.)

T. Martin, E. Cohen / Computers & Graphics 34 (2010) 187–197 195

through the crest region ofH, and since the first row of Sþ and S�are the same, i.e. the first row lies on the boundary of M, theresulting parameterization has a degeneracy along M.

Strategy 2: This strategy results in a parameterization with nodegeneracies, but with several internal extraordinary points. Avalue w0 is picked on the w-scalar field that is less than the w-value of T n and the respective isosurface, called T nþ1, at w0 isextracted. The parameterization for the volume enclosed by T 1

and T nþ1 follows strategy 1 while the parameterization for thevolume enclosed by T nþ1 is done differently. The choice of w0

often depends on simulation requirements, i.e. its choice affectshow much the resulting volumetric parameterization is polar-likeversus polycube-like. As in the 2D case (Fig. 5 b), in this strategy,Sþ and S� emanate from T nþ1 rather than fromM, by trimmingthose parts of Sþ and S� lying within the innermost trianglemesh boundary T nþ1 inH away. This implies thatM is not part ofthe resulting representation.

As discussed above, every isoparametric segment sj of Mcorresponds to a face Sj and two corner surfaces corresponding toits endpoints. Lets call this configuration S j. Let T j,nþ1 be the partof T nþ1 inside the crest region S j. Fig. 13 shows T j,nþ1 in color forthe case j¼1. The union of all T j,nþ1 refers to the crest regionT crest

nþ1 of T nþ1. Furthermore, let fT þnþ1,T �nþ1g ¼ T nþ1\T crestnþ1 where

fT þnþ1g refers to the sub triangle mesh of T nþ1 on the side whereSþ passes through T nþ1. T �nþ1 is the sub triangle mesh of T nþ1

on the side where the collection of S� pass through T nþ1. The w-scalar field over T nþ1 is computed by assigning w0 to T �nþ1 andw1 to T þnþ1 where w0ow1 followed by solving Eq. (1). Then,depending whether segment sj ofM is isoparametric in u or in v,the respective isovalue is assigned to the vertices defining T j,nþ1.The u- and v-scalar field can be computed for the volume enclosedby T nþ1 by solving Eq. (1) with these boundary conditions. Fig. 14shows a part of the volumetric parameterization for the handmodel using this parameterization strategy.

7. Modeling examples

In this Section we demonstrate our proposed methodology byestablishing a volumetric parameterization on a genus-1 pelvisdata set consisting of an exterior triangle mesh boundary and aninterior material boundary as illustrated in Fig. 3 using the firstparameterization strategy as discussed in the previous Section.

Parameterized subvolumes adjacent to each other are glued withC(0) continuity.

Given M, several parameterization choices are possible.Specific corner selections may result in a volumetric parameter-ization of H, such that the geometry of H is followed morenaturally. The reader is referred to Fig. 15 showing two differentu-scalar fields on a schematic genus-1 midsurface similar toM ofthe pelvis. A parameterization ofM on the right is chosen so thatthe handle region of M is similar to a sweep kind ofparameterization, where the bottom is similar to a polycubekind of parameterization. The resulting scalar field follows thegeometry of the midsurface more naturally compared to thescalar field on the left of the Figure which is polycube-like.

Midsurfaces have been applied mainly to thin solids. Wedemonstrate our technique on a triangulated CAD data set of apropeller (Fig. 16) where the user defined the boundary of Mmanually. A usual midsurface would be more complex, having asheet for the hub and fins for each of the blades. Instead, thechosen midsurface slices the propeller across the hub resulting in

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Fig. 17. Parameterized pelvis.

T. Martin, E. Cohen / Computers & Graphics 34 (2010) 187–197196

a small midsurface. With a minimal user interaction to choose thesize of M and corners on it, the object could be parameterizedefficiently. This example shows that the proposed midsurface-based parameterization technique can model more than thinsolids (Fig. 17).

8. Discussion and conclusion

In this paper we proposed a methodology to create volumetricparameterizations of triangle meshes with interior materialboundaries. The algorithm presents a generalization of themethod proposed in [21] and two parameterization strategiessuitable to fit trivariate tensor-product B-splines or T-splines tothe respective volumetric parameterization. Note that a B-splinerepresentation can be converted into a T-spline representation.Once converted, local refinement on the T-spline can be used toincrease the accuracy of the fit.

The volumetric parameterization is based on a midsurface,constructed as part of the algorithm and does not require timeconsuming clean-up procedures often required when simplifyinga medial axis. The use of harmonic functions allows the use of arelatively simple midsurface for more complex geometry such asthe pelvis model or the propeller in Fig. 16 to guarantee aconsistent parameterization resulting in a relative few number ofvolumetric sub patches. The harmonic nature of the parameter-ization guarantees that adjacent subvolumes are orthogonal tothe scalar field respecting interior material boundaries. While thealgorithm requires initial user input to specify corners on themidsurface, the rest of the algorithm proceeds automatically.

This has advantages, for instance, in that the user has controlover where corner vertices should be placed, which is oftenimportant in simulation where the critical regions on the domainof interest should be free of corners and degeneracies. Also, a goodcorner selection can result in a more appropriate parameteriza-tion where its gradient field follows the geometry more naturally,as was shown on the pelvis data set. However, placing corners canbe more challenging for the user on more complex input models.While in the current approach the user gets aid for the cornerselection, we are investigating how this initial step could be

further automated through a more in-depth analysis of thegeometry.

A further generalization should not require interior materialboundaries be contained within each other and also the casewhere the interior material boundaries are unrelated andgeometrically more complex than the exterior surface. Lastly,for a more general approach, to avoid distortions in theparameterization, the definition of the midsurface has to befurther generalized to include of multiple sheets.

Acknowledgments

This work was supported in part by NSF (CCF0541402). Allopinions, findings, conclusions or recommendations expressed inthis document are those of the authors and do not necessarilyreflect the views of the sponsoring agencies. The Olivier handmodel and the kitten model were acquired from the AIM@SHAPEShape Repository.

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