Dynamic Mesh Optimization for Polygonized Implicit Surfaceswith Sharp Features

Yutaka Ohtake Alexander Belyaev Alexander PaskoComputer Graphics Group, Max-Planck-Institut fur Informatik

Stuhlsatzenhausweg 85, 66123 Saarbrucken, GermanyE-mails: ohtake,belyaev @mpi-sb.mpg.de

Shape Modeling Lab, University of Aizu, Aizu-Wakamatsu 965-8580, JapanE-mail: belyaev@u-aizu.ac.jp

Department of Digital Media, Faculty of Computer and Information Sciences, Hosei University3-7-2 Kajino-cho, Koganei-shi, Tokyo 184-8584, Japan

E-mail: pasko@k.hosei.ac.jp

Keywords: mesh evolution, implicit surface.

Abstract

The paper presents a novel approach for accurate polygo-nization of implicit surfaces with sharp features. The ap-proach is based on mesh evolution towards a given implicitsurface with simultaneous control of the mesh vertex posi-tions and mesh normals. Given an initial polygonization ofan implicit surface, a mesh evolution process initialized bythe polygonization is used. The evolving mesh converges toa limit mesh which delivers a high quality approximation ofthe implicit surface. For analyzing how close the evolvingmesh approaches the implicit surface we use two error met-rics. The metrics measure deviations of the mesh verticesfrom the implicit surface and deviations of mesh normalsfrom the normals of the implicit surface.

1 IntroductionImplicit surfaces are widely used in various applications,

including engineering [26], computer graphics [6], andmathematics [28]. Given a field function ,an implicit surface is defined as an isosurface (level set) ofthe field function, .

For visualization purposes, polygonization of an implicitsurface is often required. It includes sampling the functionat selected points, estimating the positions of the mesh ver-tices, and connecting them to form polygons. This usuallyresults in space aliasing (faceting) which is a common prob-lem arising whenever some continuous object is represented

by a set of discrete samples. Such faceting is most pervasivefor shapes with sharp features (edges, corners, spikes, etc.).

In this paper, we develop an approach for accurate poly-gonization of implicit surfaces with sharp features. Givenan implicit surface and its (usually rough) initial triangu-lation, we define a mesh evolution process initialized bythe triangulation and evolving the mesh towards the im-plicit surface. The process also fits the mesh normals toimplict surface normals and therefore the evolving mesh ap-proaches a high quality polygonization of the implicit sur-face. Fig. 1 shows an initial triangulation of an implicit sur-face with sharp edges (left) and a triangulated surface ob-

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The Visual Computer, 2002

tained after completeing the mesh evolution process (right).Sharp features, edges and corners, appear naturally if the

implicit surface is constructed using set operations (union,intersection, difference). The defining function for suchsurface can be obtained by applying min/max functions ormore general -functions [25, 26, 29, 23] to the definingfunctions of the arguments of set operations. If the binarytree of operations is available, then the sharp features can befound by a numerical method that analyzes the functions ofboth arguments for each operation [36]. However, if the re-sulting function is evaluated by a procedure making the treeof operations not available to the application, this method isnot applicable.

An elegant approach for fast feature-sensitive surface ex-traction from volume data described by directed distancefields was recently proposed in [15]. Unfortunately the mostimplicit surface modelers still work with the traditional onefunction representation.

One possible antializing approach consists of adaptive-resolution polygonization, in which the sampling rateadapts to the changing feature size (curvature scale) of a sur-face being polygonized [5, 12] (see also references therein).

Another remedy to reduce apparent faceting is smooth-ing. To date, the most powerful approaches for smoothingpolygonal meshes are geometric signal processing [32, 33]and mesh evolution by diffusion flows [28, 10, 9, 8, 3, 21].However, these smoothing techniques explore only localmesh properties and, therefore, are not best suited for iso-surface smoothing when the original field function

is available.Our basic idea is simple. Given a field function

and an initial polyginization (a triangle mesh) ofthe implicit surface , we smooth the mesh

with simultaneous minimization of

– the distances from the mesh vertices to the isosurface;

– disparities between the isosurface normals and meshnormals.

In this work, we improve our dynamic mesh approach [22]developed for optimizing polygonizations of implicit sur-faces with sharp features.

We act on the vertices of the evolving mesh by threeforces. Two forces optimize positions of the vertices ac-cording to the values of the function and its gradient at thevertices. The third force improves mesh regularity. Com-bination of these three forces allows us to achieve an accu-rate approximation of the implicit surface by a high qualitymesh.

The leftmost and rightmost images of Fig. 2 demonstratewireframe images of the initial and final triangle surfacesexposed in Fig. 1, respectively. The two middle images ofFig. 2 show intermediate stages of the evolution from theinitial mesh to the final mesh. Note that mesh vertices andedges align with sharp features of the implicit surface.

The idea to use meshes whose vertices act as dynamicparticle system was proposed in [11, 13] for 2D and in [17]for 3D and since then was extensively used for segmenta-tion and tracking in multidimensional images and volumedata (see, for example, [4, 16, 18] for recent achievements inshape modeling and analysis with deformable contours andsurfaces). Recently dynamic meshes were applied to sur-face extraction from distance volume datasets [35]. How-ever, in contrast to previous works, our method is developedto obtain an accurate mesh approximation of an implicit sur-face with sharp edges and corners. It is achieved by a simul-taneous adjustment of the mesh vertices and mesh normals.

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2 Mesh Evolution Toward Implicit SurfaceConsider a family of meshes evolving according to

the following equation

(1)

where is a mesh updating operator according to displace-ment vectors defined at the mesh vertices. In many situ-ations (1) can be considered as an explicit approximation ofan evolution of a family of smooth surfaces.

Consider an implicit surface andits polygonization . We want to construct a mesh updat-ing operator such that the mesh evolution process (1) with

taken as the initial condition converges to the surface,, as .

We implement the operator satisfying the above re-quirement as the superposition of three mesh updating op-erators (mesh flows) depending on and local mesh prop-erties:

These local operators are defined by the following vertexupdate procedures

-flow (2)-flow (3)-flow (4)

where denotes the set of neighbors of vertex in ,, , are vector functions (forces).In the subsequent subsections we define these vector

functions such that -flow corrects the mesh normals ac-cording to the implicit surface normals, -flow pushesmesh vertices toward the implicit surface ,and -flow equalizes the mesh sampling rate.

Fig. 4 demonstrates how these mesh updating operators(flows) act on a triangle mesh. The top row demonstratesfragments of the triangulated block model shown in the leftimage of Fig. 1. The bottom row shows the same fragmentsafter a number of repeated applications of the operators ,

, and .

2.1 Moving mesh vertices toward the implicitsurface

The simplest way to move mesh vertices toward a givenimplicit surface is to introduce an attractingforce defined via the gradient of . In our implementationwe define -force at a mesh vertex by

where denotes the gradient, is the sum of areas ofall mesh triangles incident with , and is a small posi-tive parameter. Since the implicit surface is

the minimal level set of the function , -force defined as above moves the mesh vertices towards thesurface.

A similar force was used in [34] in con-nection with adaptive-resolution polygonization of implicitsurfaces.

Fig. 3 demonstrates a 2D level set defininga square and the associated -force.

The value of is chosen to ensure numerical stability of(3). Based on an analogy with a stability analysis for thefirst-order linear partial differential equations (the Courant-Friedrichs-Levy stability criterion [24]) we select suchthat

where is a constant independent of the mesh and function. In our current implementation,the time step-size is de-

fined by

(5)

The gradient can be computed analytically only forsimple functions . So we use standardcentral-difference formulas to estimate the gradient.

2.2 Mesh optimization according to implicitsurface normals

A better approximation of the implicit surfaceis achieved if we modify positions of

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the mesh vertices such that the triangle normals are closeto corresponding implicit surface normals. Fig. 5 illustratesadvantages of such mesh optimization. The left imageshows approximating a smooth curve by a polygonal linewhose vertices are close to the curve. The right imagedemonstrates the same curve approximated by a polygonalline whose vertices and normals are close to curve pointsand normals.

An appropriate error function will be introduced and dis-cussed in Subsection 2.4

Consider a unit vector field

It is orthogonal to the implicit surface atthe implicit surface points. Fig. 6 presents a 2D example:a square defined as an implicit surface and itsassociated unit vector field .

The figure demonstrates also that smoothness of the nor-malized gradient vector field depends on functions cho-sen for basic Boolean operations. We use -functions[25, 26, 30] since they have better differential propertiesthan the commonly used functions.

We want to move mesh vertices such that for everymesh triangle the mesh normal becomes closer to

, where is the centroid of .It is achieved by the vertex update operator (2) with

where is the projection of

the vector on the direction, denotes thearea of , and the summations are taken over all -incidenttriangles, see Fig. 7.

A similar vertex update operator was used in [21] forcrease enhancement purposes.

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n(S)

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angle T

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P

2.3 Mesh relaxationTo improve mesh regularity we use the tangential com-

ponent of the Laplacian smoothing flow. The Laplacianflow, in its simplest form, moves repeatedly each mesh ver-tex by a displacement equal to a positive scale factor timesthe average of the neighboring vertices. Consider a meshvertex and its neighbors , , . The Laplacian flowis determined by the umbrella vector given by

See Fig. 8.

1

2

3...

n

PQ

Q

Q

QP

(P)U

The vertex update operator equalizing the mesh sam-pling rate is defined by (4) with

(6)

where is the mesh normal at vertex and is a positiveconstant. In our current implementation we use .

Fig. 9 demonstrates how the tangential component of theumbrella operator improves the mesh sampling quality.

However, the -flow (4) does not preserve sharp edges,as it is demonstrated in Fig. 10.

Thus dealing with an implicit surface with sharp featureswe have to switch off the mesh quality improving -flow

when the flow interferes with the mesh-to-surface conver-gence process.

Fig. 11 demonstrates how -flow (2) and -flow (3)transform a polygonized sphere (top-left image) toward acube (bottom-right image) defined as an implicit surface. Ifonly -flow is applied, the sphere transforms to a polygonalsurface shown in the top-right image. If the mesh is movedby -flow only, the sphere transforms to a polygonal sur-face shown in the bottom-left image. A perfect metamor-phosis of the sphere to the cube is achieved if at first we usethe composition of the three introduced flowsand then shift to the flow according to an analysis ofan error function defined and studied in the next subsectionsee Fig. 12.

2.4 Error analysisTo analyze how close the evolving mesh approaches

the implicit surface , as , we introduceerror estimator functions measuring the deviation of themesh vertices from and characterizing thedeviation of the mesh triangle normals from the normalizedgradient vector field .

The squared distance from a mesh vertex to the im-

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0 iterations 10 iterations 50 iterations 100 iterations 200 iterations

plicit surface can be estimated by the first-order approximation [31] . So we define

by

where the sum is taken over , all mesh trianglesincident with , and denotes the area of .

The deviation of the mesh triangle normals from

is estimated by

where the summations are taken over all the mesh tri-angles , is the unit normal of ,

, where is the centroid of .Note that mimics the sum of squared distances from

the mesh vertices to the implicit surface and gives aweighted sum (integral) of squared differences between themesh and surface normals since

Note that mimics the squared norm usedwidely in the theory of partial differential equations [1] andfor error analysis of finite element methods [7].

As we noted in the previous subsection, the mesh motionby the tangential component of the umbrella operator ( -flow) destroys sharp edges, see Fig. 10. We switch off themesh regularization by -flow after has stabilized.

Fig. 13 demonstrates the behaviors of and for theblock model mesh evolution (see Fig. 1 and Fig. 2) and thesphere-to-cube transformation (see Fig. 12).

3 Further Improvements3.1 Modified - and -flows

Alternatively, at the final stage of our mesh optimizationprocedure, a combination of modified - and -flows canbe used instead of -flow.

Consider a mesh vertex , a mesh triangle incidentwith , and two other vertices of the triangle, and .Let denote the triangle centroid. Letus measure the deviation of the triangle normal from

by

(7)

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and define a local energy associated with by

where the sum is taken over all mesh triangles incidentwith .

The -flow updates the position of such that the localenergy takes its minimal value. We minimizeby a gradient descent method. Let bethe antigradient of the energy with respect to

. The updated position of is approximated byfive consecutive iterations

started from the initial position of . Here is thedegree of vertex .

Computing the antigradient of with respect tois trivial if we know the antigradients of

for the triangles surrounding . Computing the antigra-dient of is simple. The antigradient of the area of thetriangle is given by , where is the unitvector defining the height direction from to the oppositeside , as seen in Fig. 14. Now let us note thatis proportional to the difference between the area ofand the area of the projection of onto a plane or-

thogonal to . Thus the antigradient of has cleargeometric meaning and can be easily found.

P

P

P

n(T)

T1

2

e

C

m(C)

The -flow is defined by -force given by

where the step-size parameter is given by (5). It can beconsidered as a simplified antigradient of the energy

where

Fig. 15 compares andmesh optimization procedures. (the computational timeswere measured on a Mobile Pentium III 500 MHz com-puter). The latter procedure requires less iterations, worksfaster, and produces better reconstruction of sharp edges.The visual comparison is also confirmed by the comparisonof graphs of the error functions and , as seen in Fig. 16.

3.2 Adaptive Mesh SubdivisionIf the initial polygonization is not dense enough, small

surface features cannot be well reconstructed. So we uselinear one-to-four subdivision of those mesh triangleswhere the mesh normals have large deviations fromthe implicit surface normals .

Consider a triangle and its imaginary one-to-foursubdivision into four triangles with centroids ,

, as seen in Fig. 17.It is convenient to measure the deviation of from

the implicit surface normals by a slight modification of (7)

where is the area of triangle . If is greaterthan a user-specified threshold, triangle is subdivided.

7

The adaptive subdivision procedure allows us to gener-ate high-quality polygonal approximations of complex im-plicit surfaces starting from a coarse initial mesh, as seen inFig. 18.

4 Conclusion and Future WorkWe have presented a novel approach for accurate poly-

gonization of implicit surfaces with sharp features. The ap-proach is based on mesh evolution towards a given implicitsurface with simultaneous control of the mesh vertex posi-tions and mesh normals.

In this paper, we haven’t considered the case when avoxelized data set is given instead of a field function

defined for every . We refer an interestedreader to our recent work [19]. Optimizing isosurfaces withboundaries was also studied in [19].

The presented implementation of the approach can beimproved in many directions. For example, if the samplingrate of the initial mesh is comparable with the size of afeature of a given implicit surface, then the feature can belost during the mesh-to-surface evolution process. Fig. 19demonstrates a Doraemon 1 model built with HyperFun [2]and then polygonized by the marching cubes method (top-left image) and the result after applying the mesh-to-surfaceevolution process developed in this paper (top-right image).Note that the mesh evolution removes the cat pupils (see thebottom images). One way around this problem consists ofdeveloping a better mesh subdivision procedure. A com-bination with a dynamic mesh connectivity approach [14]may be also useful.

The speed of convergence of a dynamic mesh towards animplicit surface with sharp features dependson differential properties of the function .The min/max functions used commonly [27] for the basicBoolean operations, union and intersection, have poor dif-ferential properties. For example, andare not differentiable along the line . Whereas cer-tain -functions [25, 26, 30] also define the union and in-tersection and, in addition, possess much better differentialproperties. Fig. 6 demonstrates advantages of -functionsover the min/max functions. -functions are implementedin the HyperFun geometric modeling language and support-ing software [2].

A variational approach to optimizing polygonized im-plicit surfaces constitutes another promising direction forfuture research. Our first steps in this direction [20] are veryencouraging.

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