Dual Laplacian morphing for triangular meshes - … · morphing approach for 3D triangular meshes ... present a solution to the interpolation problem of 2D polygon morphing, ... Our

COMPUTER ANIMATION AND VIRTUAL WORLDSComp. Anim. Virtual Worlds (in press)Published online in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/cav.182...........................................................................................Dual Laplacian morphing fortriangular meshes

By Jianwei Hu, Ligang Liu * and Guozhao Wang..........................................................................

Recently, animations with deforming objects have been frequently used in various computergraphics applications. Morphing of objects is one of the techniques which realize shapetransformation between two or more existing objects. In this paper, we present a novelmorphing approach for 3D triangular meshes with the same topology. The basic idea of ourmethod is to interpolate the mean curvature flow of the input meshes as the curvature flowLaplacian operator encodes the intrinsic local information of the mesh. The in-between meshesare recovered from the interpolated mean curvature flow in the dual mesh domain due to thesimplicity of the neighborhood structure of dual mesh vertices. Our approach can generatevisual pleasing and physical plausible morphing sequences and avoid the shrinkage and kinksappeared in the linear interpolation method. Experimental results are presented to showthe applicability and flexibility of our approach. Copyright © 2007 John Wiley & Sons, Ltd.

Received: 11 May 2007; Accepted: 14 May 2007

KEY WORDS: mesh morphing; Laplacian coordinates; vertex path problem; dual mesh

Introduction

Mesh morphing, or shape interpolation, as a processof smoothly transforming one 3D geometric object intoanother, has been widely used for enhancing visualeffects in computer animation. This technique hasalso been used for geometric modeling, advertising,medicine, and entertainment.1

Related Work

A variety of techniques have been developed for 3Dobjects morphing in the literature.2,3 Boundary-basedmesh morphing requires the solutions to two mainsub-problems: vertex correspondence problem, whichis to find a correspondence between vertices of thetwo shapes, and vertex path problem, which is to findpaths that the corresponding vertices traverse during themorphing process.2,3

Most of previous algorithms for mesh morphingare mainly concerned with the vertex correspondence

*Correspondence to: L. G. Liu, Department of Mathematics,Zhejiang University, Hangzhou 310027, China.E-mail: [email protected]

problem. The typical approaches first map the mesheson a common base domain and then compute theoverlapped triangulations on the based domain. Aplane is chosen as the base domain for a non-closedmesh patch4,5 and a sphere is chosen as the basedomain for a closed, genus zero mesh.6,7 A moregeneral approach is to parameterize the meshes over acommon intermediate simplicial mesh.8–11 The meshesare partitioned into matching patches with an identicalinter-patch connectivity using a set of consistent cuts.Then each patch is mapped onto the corresponding facein the based domain. Recently, the work of Zhang et al.12

directly maps the connectivity of one mesh onto anothermesh surface using the least squares meshes13 to createthe compatible meshes.

After building a one-to-one vertex correspondencebetween two meshes, many approaches simply adoptlinear interpolation to generate vertex paths. However,shrinkage and kinks may occur in the morph sequencewhen linear interpolation approach is used, as shownin Figure 1, because the large rotation can not correctlybe represented by linear interpolation. To addressthis problem, several researchers consider non-linearapproaches to interpolate the vertices. Sederberg et al.14

present a solution to the interpolation problem of 2Dpolygon morphing, in which the geometric information,

............................................................................................Copyright © 2007 John Wiley & Sons, Ltd.

J. HU, L. G. LIU AND G. WANG...........................................................................................

Figure 1. Comparisons among different morphing techniques. (a) shows the source horse mesh in different views; (e) shows thetarget horse mesh in different views. The first row shows the morphing sequence using linear interpolation method, the secondrow shows the morphing sequence using Laplacian morphing method,18 and the third row shows the morphing sequence using

our approach.

such as edge lengths and angles between edges, areinterpolated. A generalization for 3D meshes is given byLiu and Wang.15 The work of16,17 use dihedral angels andedge lengths to interpolate two polyhedra. Alexa18 andFu et al.19 apply Laplacian coordinates to mesh morphingand Xu et al.20 use the Poisson equation to interpolategradient fields of the meshes. Some researchers considerthe interior information of given shapes and proposemethods to control local volumes distortions.21–23

Our Approach

In this paper, we present a novel approach for interpolat-ing two 3D meshes given that the vertex correspondencesbetween them have been well established using existingmethods.9,10,12

Laplacian vector information provides an elegantlocal intrinsic feature descriptor for shape. It hasrecently been successfully applied in surface editingand deformation.24–26 Our key observation is that themean curvature flow Laplacian coordinates encode theintrinsic detail information of the mesh, which is inspiredby the work of Au et al.26 The coefficients of themean curvature flow Laplacian operator capture thelocal parameterization information and the magnitudeof the mean curvature flow Laplacian coordinatescaptures the local geometry information. And themesh geometry can be recovered from the meancurvature field and the coefficients of the curvatureflow Laplacian operator. By interpolating the intrinsicparameterization information and geometry informationof the two meshes, the intermediate meshes can then bereconstructed. Therefore, the morphed meshes preservethe intrinsic information of both meshes, avoiding

the shrinkage and kinks problems appeared in thelinear interpolation method. Furthermore, we proposeto interpolate the intrinsic information in the dualdomain of the input meshes. As the dual vertices ofa triangular mesh always have valence three, the localintrinsic parameterization and geometry information areuniquely defined, which makes our approach very stable.

Our approach has two main advantages over existinginterpolation approaches. First, we formulate thepath problem as mean curvature flow interpolation.Mean curvature flow Laplacian operator represents theintrinsic information of the shape. Second, we proposea novel shape interpolation approach based on the dualLaplacian interpolation. The in-between meshes arereconstructed from the interpolated mean curvatureflow which avoids shrinkage and local wrinkles.

Overview

The rest of the paper is organized as follows. In ‘Pre-liminaries’ section , some preliminaries are presented.The next section describes our dual Laplacian morphingapproach. Experimental results are presented in thefollowing section. Finally conclusions are presented.

Preliminaries

In this section, we briefly review Laplacian operator andthe curvature flow-based dual Laplacian system.

Laplacian Operator

LetV = (v1, v2, . . . , vn) be the vertex positions of the inputmesh, and Ni be the set of adjacent vertex positions of vi.

............................................................................................Copyright © 2007 John Wiley & Sons, Ltd. Comp. Anim. Virtual Worlds (in press)

DOI: 10.1002/cav

DUAL LAPLACIAN MORPHING...........................................................................................

The Laplacian coordinate of vertex vi is

li =∑j∈Ni

wij(vj − vi) (1)

where wij is the weight of edge (i, j) corresponding tovi. Many weighting schemes have been proposed.27,28

Since the Laplacian coordinate li is the weighted averagedifference vector of its adjacent vertices to vi, it describesthe local geometry at the vertex vi. Its matrix form isl = LV , where L is an 3n × 3n matrix with elementsderived from wij . We refer to L as the Laplace operator andits elements as the Laplacian coefficients.

Given some handle positions, Laplacian surfaceediting can deform the input mesh in the least squaressense.24 The deformed mesh can be obtained by solvinga sparse linear system AV ′ = b where A is the Laplacianoperator with handle positions information and b isa vector consisting of l and handle positions. As theLaplacian vector is rotation variant, the Laplacian surfaceediting may face suffer the transformation problem.Various editing systems based on Laplacian coordinateshave tried to adopt different approaches to solve thistransformation problem.24,25

Curvature Flow LaplacianOperator

Curvature flow Laplacian operator is proposed tosmooth surface by Desbrun et al.28 It smoothes the surfaceby moving along the surface normal with a speed equalto the mean curvature which alleviates the problem ofvertices drifting in the tangential planes.

The discrete version of the curvature flow Laplacianoperator is derived as in Equation (1) with

li = 4Aikini, wij = cot αij + cot βij

where ni and ki are respectively the unit normal and themean curvature at the vertex vi; αi and βi are the twoangles opposite the edge (i, j), and Ai is the sum of theareas of the triangles adjacent to the vertex vi.

This Laplacian coordinate can be viewed as anapproximation of the integrated mean curvature normalat the vertex vi. It consists of two types of information:the parameterization information which is capturedby the Laplacian coefficients (wij = cot αij + cot βij) andthe geometry information which is encoded by themagnitudes of the Laplacian coordinates (4Aikini).29 Inother words, the parameterization information describesthe shapes of the local features (adjacent triangles)around the vertex, while the geometry informationexpresses the sizes of the local features.

Based on the curvature flow Laplacian operator,Au et al.29 present an iterative framework to solvethe transformation problem of Laplacian-based meshediting. The deformed mesh is reconstructed by keepingthe coefficients and the magnitudes of the curvature flowLaplacian coordinates with the original one as much aspossible.

Dual Laplacian Coordinates

In general, the one-ring neighbors of a vertex arenot coplanar and the Laplacian coordinates may havetangential components since there is no common normaldirection for all planes formed by the one-ring neighbors.The curvature flow Laplacian coordinates will containtangential component in some of these planes andcause tangential drifts which might cause an instabilityproblem in the reconstruction process.

To address the instability problem, Au et al.26 proposea dual Laplacian editing framework by using thecurvature flow Laplacian operator in the dual domain30

of the input mesh. As the dual vertices of a triangularmesh always have valence three, there is a uniquedefinition of normal and tangent space at each vertex interms of its one-ring neighbors.

Let V = ( v1, v2, . . . , vn) be the vertices of the dualmesh. The curvature flow Laplacian operator of the dualmesh is written as

li = −hini =∑

j∈{1,2,3}wij (vij − vi) (2)

where the normal ni at the vertex vi is perpendicular tothe plane determined by the neighbor vertices vi1,vi2, andvi3; hi is the distance from vi to the plane, as shown inFigure 2.

Figure 2. Illustration of the one-ring structure of dualvertices.26


DOI: 10.1002/cav


Note that the parameterization information isrepresented by the Laplacian matrix L with elementsof weights {wij} and the geometry information isrepresented by h = {hi} for the dual mesh.

Our Morphing Approach

In this section, we describe our novel approach formesh morphing. The main idea is to interpolate the twointrinsic properties (the parameterization informationand the geometry information) of the given meshesduring the morphing process. We will show thatmorphing in the dual domain can eliminate theinstability in the reconstruction process.

We assume that the input meshes to be processedare triangular meshes and the compatible meshesare generated by a pre-processing step.9 We firstconvert them into their dual domains and obtaintwo corresponding compatible dual meshes. Thenwe interpolate the parameterization and geometryinformation of the dual meshes. The intermediate dualmesh can be reconstructed by the interpolated intrinsicinformation from an initial dual mesh iteratively.

Dual Laplacian MorphingFramework

Let M0 and M1 be the input source and targetmeshes respectively. Given time t (0 < t < 1), oursystem interpolates the parameterization and geometryinformation of the input meshes and generates anintermediate mesh Mt . The main steps are outlined asfollows:

1. Generate compatible meshes M0 and M1 for two inputmeshes using the previous method.9

2. Build the corresponding dual meshes M0 and M1 forcompatible meshes.

3. Compute the parameterization information pt andthe geometry information ht at the given time t byinterpolating the corresponding intrinsic informationof M0 and M1.

4. Reconstruct the dual mesh Mt by the intrinsicinformation pt and ht and obtain the intermediateinterpolated mesh Mt from Mt .

The details of the above steps will be described in thefollowing section.

Dual Laplacian Morphing

Intermediate Intrinsic Information. We obtain theintermediate intrinsic information at time t by linearlyinterpolating the corresponding intrinsic information asfollows:

Lt = (1 − t)L0 + tL1,

ht = (1 − t)h0 + th1

where h0, L0 and h1, L1 are respectively the parameteri-zation and geometry information of M0 and M1.

Note that the dual Laplacian coordinates lt maybe either inward or outward of the surface. And theelements of the vector ht can also be positive or negative.

Reconstruction of Intermediate Dual Mesh. Weuse a similar iterative process as in Reference [26]to reconstruct the dual mesh from the intermediateintrinsic information. An initial mesh M0

t is obtainedby the Laplacian morphing technique18 or by the linearinterpolation method.

Starting from the initial mesh M0t , we iteratively update

the intermediate dual mesh using the following two stepsby fixing one vertex position unchanged:

Step 1. Update the dual Laplacian coordinates: Weupdate the dual Laplacian coordinates by fixing eachlength to the interpolated geometry information ht ,and compute the new dual Laplacian coordinateslk+1t from the old one lkt . For each dual vertex vk+1

t,i , itscorresponding dual Laplacian coordinate is definedas

lk+1t,i = (

ht,i/hkt,i

) · lkt,i (3)

where ht,i is the ith element of ht , hkt,i is the signed

length of lkt,i.Step 2. Update the dual vertex positions: To interpo-

late the parameterization information, we thencompute the intermediate dual vertex positionsV k+1

t using the current dual Laplacian coordinateslk+1t ; that is, we solve the following sparse linear

system:

At · V k+1t = bk

t (4)

where bkt , At is derived from lkt , Lt , and the fixed vertex

position.


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Figure 3. Morphing sequence between fandisk and cube.

Figure 4. Morphing sequence between dinosaur and horse.

Normal Adjusting. We found that the intermediatedual mesh might collapse and the iteration might notbe convergent in the iteration when we use Equations 3and 4 to iteratively update the vertex positions of theintermediate dual mesh. Note that each intermediatedual mesh in a time step has different geometry. Thedirection of dual Laplacian coordinate of each dualvertex must be adjusted, in order to keep its directionperpendicular to the plane determined by the one-ringneighbors of this vertex. So we improve Equation 3 as

lk+1t,i = −ht,i · nk

t,i (5)

where nkt,j is the unit normal of ith vertex of the

the intermediate dual mesh at kth iteration step. We

terminate the iteration when the maximum ratio ofthe changes in vertex positions between two successiveiteration steps is less than a given threshold.

Experimental Results

A variety of 3D triangular models have been tested on acomputer with a 3.0 GHz CPU and 512 MB RAM. Severalexamples are demonstrated in Figures 3–6. Figure 3shows the morphing process between the fandisk anda cube in which the sharp features of fandisk arepreserved during the whole morphing process. Figure 4illustrates the transitions from a dinosaur to a horse.Note that the pleased interpolation of the two models

Figure 5. Morphing sequence between two human poses.

Figure 6. Morphing sequence between mug and torus.


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Morphing #Vertices #Edges #Faces Timemodels (sec)

Fandisk & cube 6583 19 743 13 162 0.937Dino & horse 10 189 30 561 20 374 1.547Man & woman 25 172 75 510 50 340 4.218Mug & torus 38 400 115 200 76 800 9.937

Table 1. Per-iteration running time for theexamples used in this paper

eliminates the shrinkage which may be caused by linearinterpolation. Figure 5 demonstrates the human posemorphing. The skeletons rotated and deformed naturallywithout specifying additional bone information. Ourmethod can also work well on high genus mesh objects,see Figure 6. Table 1 shows the computation time ofeach iteration step for the examples in the paper. Theaccompany live video shows the animation sequence ofthese morphing.

Conclusions

In this paper, we have proposed a novel morphingmethod based on dual Laplacian coordinates. The mainidea is to interpolate the parameterization and geometryinformation of meshes in dual domain. Our frameworkiteratively updates the vertex positions. The iterationsconverge quickly and return a visual pleasing result.We illustrate the superiority of our approach by manyexamples.

The most time-consuming part of our algorithm issolving the sparse linear system. And the results dependon the quality of the input compatible meshes in ourcurrent implementation. The proposed approach couldnot preserve the sharp features of the input models welldue to the use of dual meshes. We will work on this issuein the future.

ACKNOWLEDGMENTS

We would like to thank Dr Hongxin Zhang and Dr DongXu for providing the human models used in the paper. Thiswork is supported by National Natural Science Foundationof China (No. 60473130, 60503067), and Foundation of StateKey Basic Research 973 Development Programming Item (No.2004CB318000) of China.

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Authors’ biographies:

Jianwei Hu is a PhD student at the Departmentof Mathematics in Zhejiang University, from whichhe received the BSc degree in 2004. His researchinterests include digital geometry processing and imageprocessing.

Ligang Liu received the BS degree in mathematicsin 1996 from Zhejiang University, China. In 2001, hereceived the PhD degree in mathematics, also fromZhejiang University. From 2001 to 2004, he was anAssociate Researcher at the Internet Graphics Group,Microsoft Research Asia. Since 2004, he has been anAssociate Professor in the Department of Mathematicsat Zhejiang University. His current research interestsinclude geometric modeling and processing, interactivecomputer graphics, and image processing.

Guozhao Wang is a Professor at the department ofapplied mathematics at Zhejiang University, China. Heobtained a MS in applied mathematics from ZhejiangUniversity. His research interests include computeraided geometric design, computer graphics and medicalvisualization.


DOI: 10.1002/cav

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Dual Laplacian morphing for triangular meshes - … · morphing approach for 3D triangular meshes ... present a solution to the interpolation problem of 2D polygon morphing, ... Our