Eurographics Symposium on Geometry Processing 2008Pierre Alliez and Szymon Rusinkiewicz(Guest Editors)

Volume 27(2008), Number 5

Maximum Entropy Coordinates for Arbitrary Polytopes

K. Hormann1 and N. Sukumar2

1Department of Informatics, Clausthal University of Technology, Germany2Department of Civil & Environmental Engineering, University of California, Davis, USA

AbstractBarycentric coordinates can be used to express any point inside a triangle as a unique convex combination of thetriangle’s vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices ofa triangle. In recent years, the ideas of barycentric coordinates and barycentric interpolation have been extendedto arbitrary polygons in the plane and general polytopes in higher dimensions, which in turn has led to novelsolutions in applications like mesh parameterization, image warping, and mesh deformation. In this paper weintroduce a new generalization of barycentric coordinates that stems from the maximum entropy principle. Thecoordinates are guaranteed to be positive inside any planar polygon, can be evaluated efficiently by solving aconvex optimization problem with Newton’s method, and experimental evidence indicates that they are smoothinside the domain. Moreover, the construction of these coordinates can be extended to arbitrary polyhedra andhigher-dimensional polytopes.

Categories and Subject Descriptors(according to ACM CCS): G.1.1 [Numerical Analysis]: Interpolation formulasG.1.6 [Numerical Analysis]: Constrained optimization I.3.5 [Computer Graphics]: Geometric algorithms

1. Introduction

Barycentric coordinates were first introduced by Möbius asa special kind of homogeneous coordinates with respectto the vertices of a simplex [Möb27]. While unique forsimplices, they can be generalized in several ways to ar-bitrary polygons [Wac75, Flo03, MLD05, FHK06], polyhe-dra [FKR05,JSW05,LBS06], higher dimensional polytopes[War96,JLW07], and even curves [Bel06,SJW07,DF08].

Let Ω⊂Rd be an arbitrary polytope (e.g., a polygon inR2

or a polyhedron inR3), with verticesv1, . . . ,vn. The func-tionsbi : Ω→ R, i = 1, . . . ,n are calledbarycentric coordi-nateswith respect toΩ if they form apartition of unity,

n

∑i=1

bi(v) = 1, (1)

allow to write any pointv ∈ Ω as anaffine combinationofthe vertices,

n

∑i=1

bi(v)vi = v, (2)

and satisfy theLagrange property

bi(v j ) = δi j . (3)

Due to the property in (3), barycentric coordinates can beused as basis functions forbarycentric interpolation: indeed,it is clear that the function

f (v) =n

∑i=1

bi(v) fi (4)

interpolates the datafi at the verticesvi for i = 1, . . . ,n,and properties (1) and (2) further guarantee the reproduc-tion of affine functions by barycentric interpolation. Obvi-ously, the interpolantf inherits the smoothness from thefunctionsbi and if thebi can be evaluated efficiently, thenso canf . Barycentric interpolation has many useful applica-tions, ranging from Gouraud and Phong shading, renderingof quadrilaterals [HT04], image warping [HF06,WSHD07],and mesh deformation [JSW05,LKCOL07,JMD∗07,LS08]to generalized Bézier surfaces [LD89, LS07] and finite ele-ment applications [AO06,SM06,WBG07,MP07,TS08].

Many of these applications require or at least benefit fromthe barycentric coordinates beingnon-negative,

bi(v)≥ 0, (5)

so that (2) and (4) becomeconvex combinationsand sof (v)is guaranteed to lie inside the convex hull of the datafi .

c© 2008 The Author(s)Journal compilationc© 2008 The Eurographics Association and Blackwell Publishing Ltd.Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and350 Main Street, Malden, MA 02148, USA.

K. Hormann & N. Sukumar / Maximum Entropy Coordinates for Arbitrary Polytopes

MVC PMVC HC MEC-1 MEC-2

Figure 1: Barycentric coordinates for the green (top row) and the red vertex (bottom row) of an L-shaped polygon. Note thatthe MVC for the green vertex is negative inside the grey region of the polygon and that the PMVC for that vertex is constantzero there. Moreover, PMVC are only C0 along the dashed lines and C1 along the dotted lines.

1.1. Related Work

If Ω is a simplex (e.g., a triangle inR2 or a tetrahedronin R3), then the barycentric coordinates are uniquely de-termined by conditions (1) and (2), and they automaticallysatisfy properties (3) and (5). For general polytopes, thisuniqueness breaks down and the non-negativity is not alwaysguaranteed by the several constructions that have been pro-posed in the past.

Wachspress [Wac75] was the first to come up witha generalization of barycentric coordinates for finite ele-ment applications. TheseWachspress coordinatesas well astheir higher-dimensional extensions [War96,WSHD07] sat-isfy (5) as long asΩ is convexand can further be evalu-ated efficiently [MLBD02]. For planar polygons, alternativegeneralizations are thediscrete harmonic coordinates[PP93,EDD∗95] and themetric coordinates[MLD05, SM06], butthey both can be negative even inside a convex polygon.

A major breakthrough came with the advent ofmeanvalue coordinates(MVC) that were discovered by Floater[Flo03] in the context of mesh parameterization and latergeneralized to polyhedra [FKR05, JSW05, LBS07]. In con-trast to Wachspress coordinates, they are well-defined evenif Ω is non-convex [HF06], but they can take on negativevalues then (see Figure1).

By modifying the transfinite description of mean value co-ordinates [JSW05, Bel06], Lipman et al. [LKCOL07] wereable to overcome this drawback, which significantly im-proves the results in applications like mesh deformation.Their positive mean value coordinates(PMVC) no longerhave a simple closed form, but they can still be evaluated ef-ficiently with the GPU. However, these coordinates are onlypiecewise smooth (see Figure1).

Up to now, the only known barycentric coordinates thatare smooth and non-negative for arbitrary polytopes are theharmonic coordinates(HC), which have first been men-tioned by Floater et al. [FHK06] and later realized by Joshiet al. [JMD∗07] for animating characters. The drawback ofharmonic coordinates is that they are rather costly to evalu-ate because they require to compute the solution of Laplace’sequation subject to suitable Dirichlet boundary conditions.

Another approach for constructing generalized barycen-tric coordinates that has been suggested independently bySukumar [Suk04] and Arroyo and Ortiz [AO06] is basedon Jaynes’s principle of maximum entropy [Jay57]. By con-struction, these coordinates are non-negative and always sat-isfy conditions (1) and (2), but the Lagrange property (3)holds only ifΩ is strictly convex(see Figure2).

1.2. Contribution

In this paper we show how to adapt the maximum entropyapproach in order to get non-negative barycentric coordi-nates for arbitrary polygons that also satisfy the Lagrangeproperty and can thus be used for barycentric interpolation.In contrast to PMVC, these new coordinates are smooth andunlike HC, they can be evaluated directly.

After a brief introduction to informational entropic mea-sures and the maximum entropy formalism (Section2), wereview how to derive barycentric coordinates fromprior es-timatesby maximizing the Shannon-Jaynes entropy with re-spect to linear constraints (Section3). We then present twochoices of appropriate prior functions (Section4) that yieldmaximum entropy coordinates(MEC) with all the desiredproperties for arbitrary polygons (see Figure1) and explainhow to extend the construction to higher dimensions. Wefurther describe how Newton’s method can be used to ef-ficiently evaluate these coordinates (Section5). Finally, wecompare MEC with previous constructions (MVC, PMVC,and HC) by studying some application examples (Section6)and discuss their limitations as well as interesting open ques-tions for future research (Section7).

2. Principle of Maximum Entropy

Shannon [Sha48] introduced the concept of entropy as ameasure of uncertainty in information theory, with an eyeon its applications in communication theory. TheShannonentropyof a discrete probability distribution is

H(p) = <− ln p> =−n

∑i=1

pi ln pi , (6)

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K. Hormann & N. Sukumar / Maximum Entropy Coordinates for Arbitrary Polytopes

MVC / PMVC HC MEC-1 MEC-2 MEC-constant MEC-Gaussian

Figure 2: Barycentric coordinates for the rightmost vertex of this convex polygon. For convex polygons, MVC and PMVC arethe same, and the maximum entropy coordinates based on constant [Suk04] and Gaussian priors [AO06] (with β = 5 in thisexample) satisfy the Lagrange property. Note that the coordinate derived from constant priors is very steep near the vertex andwould lose the Lagrange property if the polygon wasweakly convexand the vertex and its neighbours were collinear.

where< ·> is the expectation operator,pi = p(xi) is theprobability of the occurrence of the eventxi , andpi ln pi

.= 0if pi = 0. Note that the above form ofH satisfies the ax-iomatic requirements of an uncertainty measure [Khi57].

As a means for least-biased statistical inference in thepresence of testable (known) constraints, Jaynes used theShannon entropy to propose theprinciple of maximum en-tropy [Jay57]. While Jaynes’s initial emphasis was on ap-plications in statistical mechanics, the principle has broaderappeal and can be applied to any ill-posed problem that re-quires inductive inference [Jay03]. In a nutshell, maximizingentropy provides theleast-biasedstatistical inference solu-tion when insufficient information is available.

To illustrate this statement, consider a coin toss experi-ment and letpH and pT be the unknown probabilities ofheads and tails, respectively. Here, the only known constraintis pH + pT = 1, which involves two unknowns but only oneequation. But if we regularize the problem by maximizingthe Shannon entropy (6) subject to this constraint, then theunique maximum entropy solution ispH = pT = 1/2, whichis consistent with our expectations for an unbiased coin.

It was later recognized that forH to be invariant underinvertible mappings of the continuous random variablex, thegeneral form of the continuous entropy should be

H(p,m) =−Z

p(x) ln

(p(x)m(x)

)dx,

wherem is called ap-estimateor prior distribution [KL51,Jay63,SJ80]. In this paper we use the discrete version of theShannon-Jaynes entropy functional:

H(p,m) =−n

∑i=1

pi ln

(pi

mi

). (7)

In the literature, the quantityD(p‖m) =−H(p,m) is knownas the Kullback-Leibler (KL) distance. If the KL-distance isadopted as the objective functional, the variational princi-ple is known as the principle of minimum relative entropy[SJ80]. Obviously, maximizing the Shannon-Jaynes entropyfunctionalH(p,m) is equivalent to minimizing the relativeentropy functionalD(p‖m).

3. Maximum Entropy Coordinates

Historically, discrete probability measures have been seen asweights and hence their association with the construction ofbarycentric coordinates is natural. As in condition (1), dis-crete probability measures sum to one, and condition (2) isthe counterpart of the expectation value of the first moment(or mean) of a discrete probability distribution being known.

Sukumar [Suk04] adopted the Shannon entropy (6) toconstruct non-negative barycentric coordinates for strictlyconvex polygons, whereas Arroyo and Ortiz [AO06] useda modified entropy functional in the variational principle toderive basis functions for meshfree methods. The modifiedentropy chosen in [AO06] is a linear combination of Rajan’sfunctional [Raj94] and the Shannon entropy, and the solu-tion of the variational problem provides a smooth transitionfrom Delaunay interpolation as a limiting case at one end toglobal maximum entropy approximation at the other end ofthe spectrum.

Sukumar and Wright [SW07b] later realized that bothconstructions can be described in a unifying framework thatuses the Shannon-Jaynes entropy functional with a prior (7).The variational formulation for maximum entropy coordi-nates in general then is: findb = (b1, . . . ,bn) : Ω → Rn

+ asthe solution of the constrained optimization problem

maxb(v)∈Rn

+

H(b,m), H(b,m) =−n

∑i=1

bi(v) ln

(bi(v)mi(v)

)(8a)

subject to the linear precision conditions

n

∑i=1

bi(v) = 1, (8b)

n

∑i=1

bi(v)(vi −v) = 0 (8c)

for any v ∈ Ω. In (8a), Rn+ is the non-negative orthant and

mi : Ω → R+ is a prior estimate forbi . Note that if Ω isa simplex inRd, thenn = d + 1 and the linear constraintsyield a unique solution for thebi . Forn> d+1, which is thecase of interest in this paper, the linear constraints form anunder-determined system.

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K. Hormann & N. Sukumar / Maximum Entropy Coordinates for Arbitrary Polytopes

MVC PMVC HC MEC-1 MEC-1 prior

Figure 3: Barycentric coordinates for the lower left vertex of this non-convex polygon and prior estimate of MEC-1. In thisexample, all functions are between zero and one. Note that PMVC is constant zero over the region shaded in red and also attainsa local maximum just above the vertex in the centre.

In the present context, the prior functions can be seenas weight functions associated with each vertexvi , and thevariational principle provides acorrectionthat modifies theweight functionsmi in a minimal (i.e., least-biased) way toform coordinatesbi that satisfy the constant and linear pre-cision conditions. For non-negativemi , the objective func-tionalH(b,m) is strictly concave, and therefore the problemposed in (8) admits a unique non-negative solution forbi .Note that this construction of barycentric coordinates is alsovalid for pointsv inside the convex hullΩc = [v1, . . . ,vn] ofthe polytope as long as the priors are well-defined overΩc.

It now turns out that the coordinates of Sukumar [Suk04]are the solutions of (8) for constant priorsmi(v)≡ 1 and thatGaussian prior functionsmi(v) = exp(−β‖vi−v‖2) result inthe coordinates proposed by Arroyo and Ortiz [AO06] (seeFigure 2). We further note that one obtainsbi(v) = mi(v)for i = 1, . . . ,n if the priors mi a priori satisfy the lin-ear constraints. Moreover, if we omit the linear precisioncondition (8c) and use just the constant precision con-dition (8b) as a constraint, then the solution of (8) isbi(v) = mi(v)/∑n

j=1 mj (v), which we recognize as havingthe same form as the Shepard function [She68].

The key ingredient in the maximum entropy formulationis the selection of the prior functionsmi , and this flexibilitypresents the possibility of designing tailored barycentric co-ordinates. The next section describes the specific choices forprior functions that we make to construct non-negative bary-centric coordinates for arbitrary polygons and polyhedra.

4. Prior Functions

To obtain non-negative barycentric coordinates for an arbi-trary polygonΩ, we first construct non-negative priorsmithat possess the desired boundary properties. We then usethese within the entropy formulation to obtain functionsbithat are non-negative, interpolate on the boundary ofΩ, andare linearly precise.

To this end, letei be the edge of the polygon betweenviandvi+1 and consider theedge weight function

ρi(v) = ‖v−vi‖+‖v−vi+1‖−‖vi+1−vi‖ (9)

that vanishes alongei and is positive elsewhere, by virtueof the triangle inequality. Note thatρi is also used in theconstruction of metric coordinates [MLD05,SM06]. It thenfollows that the product

πi(v) = ∏j 6=i−1,i

ρ j (v)

is non-negative and vanishes on all edges that are not adja-cent to vertexvi . The same clearly holds for the prior func-tionsmi that we derive from theπi through normalization,

mi(v) =πi(v)

∑nj=1 π j (v)

.

Note that after dividing both the numerator and the denomi-nator by the product of allρ j , we get the equivalent form

mi(v) =πi(v)

∑nj=1 π j (v)

with πi(v) =1

ρi−1(v)ρi(v), (10)

which renders it amenable to stable numerical computations.

Now solving the optimization problem (8) with thesemias input yields functionsbi with all the desired properties.Due to the linear constraints (8b) and (8c), thebi clearly sat-isfy conditions (1) and (2) and as themi are non-negative,then so are thebi . It remains to be shown that thebi in-herit the Lagrange property from the prior functionsmi . Ifmi(v∗) = 0 for somei ∈ 1, . . . ,n andv∗ ∈ Ω, then solv-ing (8) givesbi(v∗) = 0 because

limv→v∗

bi(v) ln

(bi(v)mi(v)

)is zero ifbi(v∗) = 0 and diverges otherwise. Therefore, thefunctionsbi have the same zero sets as their prior estimatesmi , and the only function that does not vanish at vertexviis bi . The Lagrange property (3) now follows because of (1)and we similarly conclude that thebi are linear on the edgesof Ω. Hence, the functionsbi are non-negative barycentriccoordinates. We call them “MEC-1” and Figure3 shows anexample of such a functionbi and its prior estimatemi .

We would like to point out that this construction ofbarycentric coordinates is rather general and also works forany other choice of edge weight functions as long asρi is

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K. Hormann & N. Sukumar / Maximum Entropy Coordinates for Arbitrary Polytopes

source image MVC PMVC HC MEC-1 MEC-2

Figure 4: Image warping using different barycentric coordinates and backward mapping (cf. [HF06]). The bottom row showsa close-up of the region around the concave vertex of the target polygon. Note that the result using PMVC is only C0.

non-negative and vanishes alongei . For example, using thealternative edge weight functions

ρi(v) = ‖v−vi‖ · ‖v−vi+1‖+(v−vi) · (v−vi+1)

in (10) gives the barycentric coordinates that we refer to as“MEC-2” in our examples.

Moreover, the construction can be extended to higher di-mensions. For example, ifΩ is a polyhedron inR3 with con-vex planar faces, then we first define for each facefi a func-tion ρi that vanishes onfi and is positive elsewhere. Iffi hask verticesvi1, . . . ,vik , then the analogue of the edge weightfunction (9) is theface weight function

ρi(v) =k

∑j=1

A(vi j ,vi j+1,v)−A(vi1, . . . ,vik),

whereA(v1, . . . ,vm) denotes the area of the polygon withverticesv1, . . . ,vm. The prior functionmi for any vertexviof Ω is then computed as in (10) with the product in the de-nominator ofπi ranging over all faces adjacent tovi .

5. Numerical Algorithm

In order to use the proposed maximum entropy coordinatesfor practical applications, it is essential to be able to effi-ciently solve the constrained optimization problem in (8).We resort to the method of Lagrange multipliers to first con-vert (8) into an unconstrained problem and then use New-ton’s method to solve it.

Let λ0 ∈ R be the Lagrange multiplier for the con-straint (8b) andλ ∈Rd be the Lagrange multipliers for thedconstraints (8c). Then the Lagrangian for problem (8) is

L(b;λ0,λ) =n

∑i=1−bi(v) ln

(bi(v)mi(v)

)−λ0

( n

∑i=1

bi(v)−1

)−λ ·

( n

∑i=1

bi(v)vi

),

wherevi = vi −v. On setting the first variation ofL to zero,namelyδL(b;λ0,λ) = 0, we obtain[

−1− ln

(bi(v)mi(v)

)−λ0−λ · vi

]δbi(v) = 0

for i = 1, . . . ,n, and since the variationδbi(v) is arbitrary, theterm within the bracket must be identically equal to zero:

−1− ln

(bi(v)mi(v)

)−λ0−λ · vi = 0.

Therefore,

bi(v) =mi(v)exp(−λ · vi)

Z,

where the substitution lnZ = 1+ λ0 has been made (Z isknown as thepartition function in statistical mechanics).Now, on using condition (8b), we obtain [SW07b]

bi(v) =Zi(λ)Z(λ)

,

Zi(λ) = mi(v)exp(−λ · vi),

Z(λ) =n

∑j=1

Z j (λ).

(11)

Note thatλ, Zi(λ), andZ(λ) implicitly depend onv and thatonce the Lagrange multipliersλ = (λ1, . . . ,λd) are deter-mined, thenbi(v) can be obtained from (11). To computeλwe note that thebi(v) in (11) must satisfy (8c), which leadsto thed non-linear equations

1Z(λ)

n

∑i=1

Zi(λ)vi = 0 (12)

in thed unknownsλ1, . . . ,λd. The solution of (12) is equiv-alent to solving the dual unconstrained optimization prob-lem [BV04]

λ∗ = argminF(λ), F(λ) = lnZ(λ),

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K. Hormann & N. Sukumar / Maximum Entropy Coordinates for Arbitrary Polytopes

source mesh MVC PMVC HC MEC

Figure 5: Mesh deformation using different barycentric coordinates and a control polyhedron with triangular faces. The resultsfor MVC, PMVC, and HC are taken from [LKCOL07].

whereλ∗ is the optimal solution. SinceF is strictly convexin Ω, Newton’s method is the natural choice. The steps inthe Newton algorithm to compute the maximum entropy co-ordinatesbi(v) of anyv∈Ω are as follows:

1. For givenv, compute and store ˜vi = vi−v; also functionsto compute the prior functionsmi(v) are available;

2. Start with iteration counterk= 0, the initial guessλ0 = 0,and letε be the convergence tolerance. The convergencetolerance dictates to what accuracy the linear precisionconditions are satisfied. A value ofε in the range 10−3 to10−10 is suitable;

3. Computegk = ∇λF(λk) andHk = ∇λ∇λF(λk), whichare the gradient and Hessian ofF , respectively;

4. Determine Newton search direction∆λk =−(Hk)−1

gk;

5. Update:λk+1 = λk + α∆λk, whereα is the step size. ForNewton’s method (dampedor guarded), a line search al-gorithm [BF04] is used to determineα if the error isgreater than 10−4, otherwiseα is set to unity;

6. Check convergence: if‖gk+1‖ > ε, then increment theiteration counterk and goto 3, else continue;

7. Setλ∗ = λk+1 and computebi(v) from Equation (11).

Due to the quadratic convergence of Newton’s method, only3 to 7 iterations are needed to obtain an accuracy of 10−10.

6. Results and Discussion

The advantages of MEC over previous barycentric coordi-nates are threefold and we present several examples to illus-trate them.

Firstly, MEC are non-negative for arbitrary polytopes,which is important as it guarantees the convex hull propertyof the barycentric interpolation (4). PMVC and HC sharethis property with MEC, but MVC can be negative insidea non-convex domain (see Figure1). In some applications,this can lead to undesired results: for example, the inter-polation of colour values can yield values outside the validrange [HF06] and it can lead to severe artefacts in mesh de-formation in some extreme cases [LKCOL07]. However, inmany situations, the influence of negative weights is not no-ticeable (see Figure4).

Secondly, like MVC and HC, MEC are smooth insidethe domain, which in turn leads to well-behaved results in

MECm MVC PMVC HC

ε = 10−5 ε = 10−10

50 K 0.07 0.33 2.44 4.52 5.09100 K 0.15 0.67 6.75 8.99 10.5200 K 0.32 1.42 17.3 18.7 24.3400 K 0.59 2.61 195 34.6 39.7

Table 1: Timings (in sec.) for evaluating the barycentric in-terpolant at m points inside the L-shaped polygon with n= 6vertices from Figure1.

applications that build on barycentric interpolation. PMVC,however, are only piecewise smooth and have discontinuousderivatives along certain lines inside a non-convex polygon(see Figure1), and similarly inside non-convex polyhedra. Inparticular, this happens along the lines defined by a concavevertex and its two neighbours and can lead to visible “kinks”if used for image warping (for example, see Figure4).

Thirdly, MEC can be evaluatedlocally at any pointv∈Ω,though not as efficiently as MVC or PMVC, and the costscales reasonably with the desired accuracy. In contrast, theevaluation of HC requires the solution of aglobal approx-imation problem, and the computational cost substantiallyincreases if high accuracy is desired.

For planar polygons, we implemented all methods consid-ered in this paper in the following way. For MVC, we use thepseudo code suggested in [HF06], and for PMVC we first de-termine the segments ofΩ that are “visible” from some pointv ∈ Ω and then use the MVC formulas, restricted to thesesegments. In order to evaluate HC, we usetriangle [She02]to triangulateΩ with mvertices andTAUCS [Tol03] to solvethe linear system arising from the piecewise linear finite ele-ment discretization of the Laplacian. The details of our MECimplementation are described in Section5.

The timings in Table1 were measured on a 2 GHz IntelPentium M with 1 GB of RAM and confirm that the cost forcomputing MVC, PMVC, and MEC atm points grows lin-early with m and slightly worse for HC, because the latterrequires to solve a linear system with a sparse system matrixof sizem×m. The significantly larger constant of propor-tionality for HC in the last row is due to the fact that thelinear system solver required more memory than availableas RAM and thus started swapping data to the hard disk.

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K. Hormann & N. Sukumar / Maximum Entropy Coordinates for Arbitrary Polytopes

source mesh MEC

Figure 6: Mesh deformation using a control polyhedron withquadrilateral faces.

However, we point out that HC can be computed moreefficiently by using boundary element methods [Rus08],which requires solving a linear system with a densek× ksystem matrix (withn< km, wheren andmare the num-bers of polygon vertices and interior points, respectively).The timings also show that it is not very costly to increasethe accuracy of MEC due to the quadratic convergence rateof Newton’s method.

We also implemented MEC for polyhedra inR3 and Fig-ures5 and6 show the results of using them for mesh defor-mation. It took about 100 sec. to compute then = 51 MECof the m = 48485 vertices of the horse, and about 140 sec.for the n = 218 MEC of them = 15002 vertices of the ar-madillo, confirming again that the evaluation cost dependslinearly both onn andm. Note that MEC can also be usedif the control polyhedron consists of quadrilateral faces, asshown in Figure6. Moreover, the mesh to be deformed isnot required to be inside the control polyhedron but only in-side its convex hull (see Section3), which is the case in thisexample.

7. Conclusions

Maximum entropy coordinates offer a new way of general-izing barycentric coordinates to arbitrary polytopes. By con-struction, MEC are non-negative and have affine precision,and we have shown that they also satisfy the Lagrange prop-erty as long as the corresponding prior functions have thecorrect zero set. Due to this flexibility, there is a lot of roomfor improvement in designing “good” prior functions. Forexample, it might be possible to manipulate the derivativeof the barycentric coordinates in a prescribed way by mod-ifying the priors so as to allow for Hermite interpolation asin [LS08].

A good prior will probably need to account for the localgeometry of the polytope, and it should also have local sup-port. The construction in (10) seems to be a good recipe forconstructing priors and in essence, any function that mea-sures the distance to an edge (or a face inR3) in some way

can be used as a weight functionρi . Due to the close con-nection between distance fields and level set functions, webelieve that it is worth investigating the possible use of levelsets for the design of prior functions in future work.

Although confirmed by the many numerical results thatwe ran, it also remains to prove the smoothness of MEC. Forthe MEC, based on Gaussian prior functions, Arroyo andOrtiz [AO06] could prove smoothness, and it is generallyassumed that the MEC are as smooth as the prior functions.A first step towards a proof has been taken by Sukumar andWets [SW07a] who establishedC0-continuity of MEC forany set ofCk prior functions,k≥ 0.

We would finally like to mention again that MEC are onlydefined inside the convex hull of the polytopeΩ and noteverywhere inRd like MVC and HC (see [Rus08]), becausethe optimization problem (8) does not have a feasible non-negative solution at points outside the convex hull.

Acknowledgements

The authors would like to thank Johannes Kopf for kindlyproviding us with the MVC, PMVC, and HC data for theexample in Figure5. This work was partially supported bythe US NSF through contract grant CMMI-0626481.

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