Shape Interpolation with Multiresolution Analysis

Junho Kim�, Sungyul Choe†, and Seungyong Lee‡

Department of Computer Science and Engineering

Pohang University of Science and Technology (POSTECH)

Pohang, 790-784, Korea

Abstract

This paper presents a novel approach for shape interpo-lation in 3D morphing with irregular meshes. We assumethat the metameshes of the source and target meshes aregiven. The basic idea is that the multiresolution analysisof metameshes with detail encoding helps to generate nat-ural in-between meshes. We construct mesh hierarchies bysimplifying the given metameshes to the base meshes withthe error metric reflecting both the source and target geom-etry. The geometric differences between successive levels inthe hierarchies are encoded as details represented in localframes. An in-between mesh is synthesized by adding upthe blended details to the interpolated base mesh. Exper-imental results demonstrate that the generated in-betweenmeshes have reasonable blended shapes of the source andtarget meshes.

Keywords: Mesh morphing, Shape interpolation, Multires-olution mesh analysis.

1. Introduction

Morphing between two input models generates an an-imation in which the source model smoothly changes tothe target. Morphing generally consists of two problems,the correspondence problem and the interpolation problem.The correspondence problem concerns finding a one-to-onemapping between the components of two given models. Theinterpolation problem is involved with generating naturalin-between models by interpolating pairs of correspondingcomponents.

In this paper, we present a novel solution for the interpo-lation problem of 3D mesh morphing. We assume that themetameshes of the source and target meshes have been ob-tained by using a previous mesh morphing technique. The

�victor@postech.ac.kr, http://home.postech.ac.kr/�victor†ggalssam@postech.ac.kr, http://home.postech.ac.kr/�ggalssam‡leesy@postech.ac.kr, http://www.postech.ac.kr/�leesy

basic idea of our shape interpolation technique is that nat-ural in-between meshes can be generated by blending mul-tiresolution details of the source and target metameshes in acoarse-to-fine manner. To obtain the multiresolution details,we use the approach of Guskov et al. [6] proposed for meshsignal processing. In this paper, to adapt the approach forshape interpolation, we introduce a new error metric for anedge collapse transformation and a new encoding schemefor the details.

Given the source and target metameshes, we obtain mul-tiresolution hierarchies of the meshes by applying the samesequence of edge collapses to the meshes. The order ofedge collapses is determined by the error metric that re-flects both the source and target geometry. For each edgecollapse transformation, the geometric change of a meshinduced by the transformation is encoded as a detail rep-resented in a local frame. To generate an in-between mesh,we first interpolate the source and target base meshes to ob-tain an in-between base mesh. Then, an in-between mesh issynthesized by progressively adding up the blended detailsto the in-between base mesh.

In Section 2, we briefly review the related work. Sec-tion 3 explains the simplification process for multiresolu-tion analysis of metameshes. The encoding scheme to rep-resent the details in the analysis is described in Section 4.Section 5 presents the in-between mesh generation process.We show experimental results in Section 6, discuss the lim-itations of the current technique in Section 7, and concludein Section 8.

2. Related Work

In this section, we review the related work that consid-ered the interpolation problem in morphing. For a survey ofmorphing techniques, including the solutions for the corre-spondence problem, refer to [19, 12].

Hughes proposed a volume morphing technique wheredifferent interpolation schedules are applied to the fre-quency subbands of input volumes [9]. The idea of sep-arating an input model into frequency subbands is similar

to the approach of this paper. However, the frequency sub-bands are used only for scheduling the global and detailedshape changes in the transformation sequence. In contrast,in this paper, the frequency subbands are used to provide ablending hierarchy that supports a complicated transforma-tion.

Sederberg et al. proposed a solution for the interpolationproblem in 2D polygon morphing which interpolates intrin-sic parameters such as lengths and angles instead of vertexpositions [14]. Sun et al. extended the solution to handleinterpolation in 3D polyheral morphing [15]. Goldstein andGotsman proposed a 2D polygon morphing technique thatincludes multiresolution interpolation based on curve evo-lution [5]. Surazhsky and Gotsman proposed an interpola-tion technique for two compatible planar triangulations [16]and applied the technique to the interpolation of two stickfigures [17]. Alexa et al. introduced the concept of rigidityinto shape interpolation [1]. Two input objects are decom-posed into isomorphic sets of simplexes, and each pair ofcorresponding simplexes is interpolated so that the overallshape is transformed as rigidly as possible. Although theseprevious approaches are excellent for 2D polygon morph-ing, their 3D extensions for mesh morphing are rather com-plicated. In contrast, the approach proposed in this paper issimple and effectively handles the 3D case.

3. Multiresolution Hierarchy

Let MS and MT be the given source and targetmetameshes, which have been obtained by merging the ver-tices and edges of two input meshes for morphing. Thereexist a one-to-one correspondence between the vertex setsof MS and MT . The edge sets of MS and MT are isomorphicto each other, where the corresponding vertices of MS andMT have the same connectivity. Only the vertex positionsare different among MS and MT . We assume that MS andMT are 2-manifold triangular meshes.

To perform multiresolution detail encoding of MS andMT , we first obtain multiresolution hierarchies of MS andMT . Similar to the progressive mesh construction [7], weapply an edge collapse sequence to MS and MT , which sim-plifies MS and MT to base meshes M

0S and M

0T , respectively.

To obtain the same multiresolution hierarchy, we apply thesame sequence of edge collapses to both MS and MT .

Compared to the simplification of a single mesh, si-multaneous simplification of MS and MT is rather compli-cated, because MS and MT generally have different geome-try. The error metric used for determining the edge collapsesequence must reflect the geometry of both MS and MT sothat the simplification errors for MS and MT are comparableto each other. Moreover, we can consider the geometry ofMT as an attribute field (e.g., texture) defined on the surfaceof MS, where the vertex positions of MT give the sample val-

ues of the field at the corresponding vertices of MS. Then, ifthe simplification process of MS much distorts the field, themultiresolution details of MT will also be much distortedwhen they are sampled on the surfaces of meshes in the hi-erarchy of MS. We can consider the distortions of the detailsof MS with respect to MT in the same way. In this case, largedistortions of the details will have bad effects when the de-tails are blended and added up to generate an in-betweenmesh. Hence, the distortions must be reflected on the errormetric for simplification of MS and MT .

In this paper, we regard the vertex positions of MT andMS as attributes of the corresponding vertices of MS and MT ,respectively. Several elegant techniques were developed formesh simplification considering attributes on meshes [2, 4,8, 13]. We design our error metric for simplification withthe approach proposed by Hoppe [8], which uses the sumof quadric errors from the geometry and attributes.

Before the simplification process starts, the vertex posi-tions in MS and MT are normalized into a unit cube. Thequadric errors for MS are obtained from the vertex positionsin MS, which provide the geometry information, and thevertex positions in MT , which are considered as attributeson the corresponding vertices in MS. The quadric errors forMT are derived in a similar way. The error metric for sim-plification is defined as the sum of the quadric errors fromMS and MT .

In this paper, we use half-edge collapses for mesh simpli-fication, because the multiresolution analysis in Section 4 isbased on a half-edge collapse. In the simplification process,the order of half-edge collapses is determined by comparingthe error metrics for possible half-edge collapses. Since thenew vertex position is fixed for a half-edge collapse, we caneasily evaluate the error metric for each half-edge collapse.

A simple alternative to the proposed approach is to usethe sum of quadric error metrics [3] obtained by indepen-dently considering the geometry of MS and MT . With thesimple approach, we can obtain comparable simplificationerrors for MS and MT . However, the approach does not con-sider the detail distortions of MS and MT measured with re-spect to the other mesh. The approach proposed in this pa-per reduces such distortions by including the attribute errorsin the error metric. As the result, the triangle shapes in thecorresponding simplified meshes from MS and MT becomesimilar in the simplification process. Note that the detaildistortions between MS and MT are minimized when the lo-cal parameterizations of MS and MT are the same. Figure1 compares the simplified meshes obtained by the proposedand the simple approaches.

4. Multiresolution Analysis

With an edge collapse sequence to simplify the givenmetameshes MS and MT , we obtain multiresolution hierar-

M0 M10 M25 M200 M1000

(a) proposed error metric

(b) sum of geometric quadric errors

Figure 1. Simplification process for metameshes

chies of MS and MT . The multiresolution details of MS andMT can be derived by analyzing the differences among themeshes in the hierarchies of MS and MT , respectively. LetMl and Ml�1 be two meshes at adjacent levels in the hier-archy of MS, where M

l is the coarser mesh. In this section,we propose an encoding scheme for the detail of M l�1 fromMl . The same scheme can be applied to the hierarchy ofMT .

The detail of Ml�1 from Ml can be considered as a fre-quency subband information when we apply a frequencyanalysis to MS. Recently, geometric algorithms were intro-duced for the frequency analysis of 2-manifold meshes [18,11, 6, 10]. The algorithms are based on the smoothing fil-ters that approximate the Laplacian operator. The frequencysubbands of a given mesh are obtained by evaluating thedifferences between the meshes before and after smooth-ing. Among the algorithms, we adopt the signal process-ing framework proposed by Guskov et al. [6] because it isproper to handle the multiresolution hierarchy from an edgecollapse sequence, used in this paper.

In the framework proposed by Guskov et al. [6], the de-tail of Ml�1 from Ml is obtained as follows. Let v be thevertex of Ml�1 which has been removed by the half-edgecollapse that reduces Ml�1 to Ml . Let M̃l�1 be the smoothedversion of Ml�1 determined by the geometry of M l�1. Toobtain M̃l�1, we first subdivide Ml to create the vertex vand compute the position of v by using a smoothing opera-

tor that reflects the geometry of Ml�1. Then, we apply thesmoothing operator to the 1-ring neighbor vertices of v, andthe resulting mesh is M̃l�1. The detail of Ml�1 from Ml isdefined as the difference vectors between M̃l�1 and Ml�1 atthe vertex v and its 1-ring neighbor vertices. In [6], the dif-ference vectors are represented in local frames that dependon Ml .

When we apply multiresolution shape interpolation toMS and MT , the details of MS and MT are interpolated andincrementally added to the in-between base mesh. The lo-cal frames used to represent the details play an importantrole in detail interpolation because the interpolation resultsdepend on the local frames. In the multiresolution shape in-terpolation, the geometry of an in-between mesh is locallyrotated and scaled against MS and MT . Hence, the localframes should reflect the local orientation and scale of thegeometry. Also, the local frames should be consistent overthe entire mesh; that is, local frames for the details at ad-jacent vertices should not have much different orientationsand scales.

In this paper, we propose a new encoding scheme for thedetail of Ml�1 from Ml . Let d be the difference vector of avertex v for which the position difference between M̃l�1 andM̃l�1 is computed. We represent d as a linear combinationof the normal vector n of v and unit direction vectors u iassociated with the 1-ring neighbor edges ei of v (see Figure2). That is,

d � w0�α0n��k

∑i�1

wi�αiui�� (1)

where k is the degree of v. The vectors n and u i are derivedfrom the smoothed mesh M̃l�1. Values αi, i � 1� � � � �k, arethe lengths of the 1-ring neighbor edges e i, and α0 is theaverage length, ∑ki�1 wi�k. The weights wi are determinedto satisfy Eq. (1). Among possibly many solutions for wi,we choose the one which minimizes ∑ki�0 w2i . We can obtainthe least-square solution by using pseudoinverse.

n

d

u1

u2

u3

u4

u5

u6

Figure 2. Detail encoding with a local frame

The weights wi obtained by Eq. (1) provide an encodingof the detail at a vertex v which reflects the orientation andscale of the local geometry around v. In Eq. (1), the localorientations around v are represented by the unit vectors nand ui. With the values αi, the local scales around v arecounted in determining wi.

5. Multiresolution Interpolation

Once the base meshes and the multiresolution detailsof the given metameshes MS and MT have been obtained,the construction of an in-between mesh consists of twosteps; base mesh interpolation and incremental add-up ofthe blended details. We first obtain an in-between base meshby interpolating the base meshes of MS and MT . Then, fromthe coarsest to the finest levels of multiresolution hierar-chies of MS and MT , we blend the details of MS and MTat each level and add the blended details to the intermediatemesh constructed up to the level.

For the base mesh interpolation, we use the linear inter-polation of the vertex positions in the base meshes M 0S andM0T . To minimize distortion from the linear interpolation,we use a rigid transformation that aligns M0S with M

0T . The

rigid transformation is composed of a rotation and a transla-tion, and we compute the transformation by minimizing thesum of the distances between corresponding vertices afterthe transformation. Although the as-rigid-as-possible inter-polation [1] may be an excellent solution for the interpola-tion, we use linear interpolation in this paper for simplicity.Since the base meshes are simple, linear interpolation usu-ally generates reasonable in-between base meshes.

Now we explain how to add the blended details to anintermediate in-between mesh in the multiresolution shapeinterpolation. Let MlI and M

l�1I be the intermediate in-

between meshes before and after adding the details, respec-tively.

As in the mesh signal processing framework ofGuskov et al. [6], we first derive a smooth version M̃l�1I ofMl�1I by using subdivision and a smoothing operator. How-ever, in this case, the smoothing operator is determined bythe geometry of Ml�1I , which we want to construct. To re-solve this problem, we derive the smoothing operator forMl�1I by interpolating those operators for M

l�1S and M

l�1T .

The interpolated operator can be obtained by using thelengths and areas of the interpolated triangles to computethe coefficients of the smoothing operator. See [6] for de-tails. Since an in-between mesh has a blended shape amongMS and MT , we can expect that the interpolated smoothingoperator works well for the generation of M̃l�1I .

After M̃l�1I has been obtained, we blend the details atlevel l in the multiresolution hierarchy of MS and MT . Asdescribed in Section 4, each of the details of MS and MT ata vertex v is represented by the weights wi. To obtain thedetail of M̃l�1I at the vertex v, we interpolate the weightswi between MS and MT . The final difference vector for thedetail at v is determined by linearly combining the normalvector n of v and unit direction vectors u i associated withthe 1-ring neighbor edges of v in M̃l�1I , as described in Eq.(1). The position of the vertex v in M l�1I is obtained byadding the difference vector to the position of v in M̃l�1I .

By repeating this process for all vertices in M̃l�1I which havedetails, we derived the mesh Ml�1I .

Figure 3 illustrates the process of the incremental gen-eration of an in-between mesh. As shown in Figure 3, themultiresolution hierarchy of the in-between mesh is reason-able, although we construct it by blending the details ob-tained from the source and target meshes.

M30 M50 M100 M400 M1000

Figure 3. Multiresolution shape interpolation pro-cess: Top, middle, and bottom rows are the mul-tiresolution hierarchies of the source, in-betweenat time 0.5, and target meshes, respectively.

6. Results

Figure 4 compares in-between meshes obtained by ourmultiresolution shape interpolation with linear interpola-tion. In the example, the target mesh was constructed byapplying a bending deformation to the source mesh. Thebase meshes used for the multiresolution shape interpola-tion contain 27 faces. Figure 5 shows another example. Thetarget mesh is also a bent version of the source, while thebase meshes consist of 6 triangles. As shown in Figures 4and 5, multiresolution shape interpolation generates morenatural in-between meshes than linear interpolation. In Fig-ure 6, a statue model is twisted to construct the target mesh.In this example, we use the base meshes with 9 triangles.Figure 6 shows that the head of statue does not shrink in themultiresolution shape interpolation.

Figure 7 shows the result of the multiresolution shapeinterpolation between mannequin and Venus models. In thisexample, we used the base meshes that contain 34 triangles.

Figure 4. Shape interpolation between a cow andits bent version (top: linear interpolation, bottom:multiresolution shape interpolation)

Figure 5. Shape interpolation for a stretched andcrooked horse leg (top: linear interpolation, bot-tom: multiresolution shape interpolation)

Figure 6. Shape interpolation for a twisted statue(top: linear interpolation, bottom: multiresolutionshape interpolation)

Figure 7. Multiresolution shape interpolation be-tween mannequin and Venus models

7. Discussion

As demonstrated in Section 6, the technique proposedin this paper can produce natural shape interpolation formeshes with different geometry. However, for robust han-dling of drastic geometric differences between meshes, theproposed technique may need some improvements.

A more consistent approach will be helpful for analysisand blending of the geometric signals on the given meshesMS and MT . In this paper, we simplify MS and MT bysimultaneously considering the geometry of both MS andMT . However, the multiresolution analysis of the geomet-ric signals are independently performed on MS and MT , andthe independent details are blended later to construct an in-between mesh. A more plausible approach would be that thegeometric signals on MS and MT are analyzed on a commondomain and an in-between mesh is synthesized on the com-mon domain by blending the correlated details of MS andMT .

In the proposed technique, the resolution of the basemeshes of MS and MT should be properly selected to ob-tain a natural in-between mesh. When the resolution is toohigh, the generated in-between mesh may not much differfrom the result of linear interpolation. On the other hand,if the resolution is too low, the base meshes may not reflectthe geometry of MS and MT , and the resulting in-betweenmeshes may not naturally interpolate the shapes of MS andMT . For the examples in this paper, we tried several basemesh resolutions and chose the one with the best result. Tohandle the problem of the base mesh resolution, we are in-vestigating a shape description metric, which measures thedifference of a simplified mesh from the original. With sucha metric, a semi-automatic approach could be developed todetermine the base mesh resolution that produces naturalshape interpolation.

In the multiresolution shape interpolation, the shape ofan in-between mesh is much influenced by the base meshinterpolation technique. In this paper, we used simple lin-ear interpolation for the base meshes and obtained success-ful results on the examples shown in Section 6. However, anadvanced approach considering the rigidity in the base meshinterpolation will be helpful to generate natural in-betweenmeshes, especially when the given meshes have much dif-

ferent geometry.

8. Conclusion

In this paper, we proposed a shape interpolation tech-nique based on a multiresolution signal processing frame-work for meshes. We presented an effective approach toconstruct multiresolution hierarchies of the source and tar-get meshes, which considers the geometry information ofone mesh as the attributes of the other mesh in the sim-plification process. We also introduced a novel encodingscheme for the details extracted from the multiresolutionhierarchies. For future work, we will extend our multires-olution shape interpolation framework to skin deformationof an animation character.

Acknowledgments

This work was supported by the Korea Ministry of Ed-ucation through the Brain Korea 21 program and the KIPAGame Animation Research Center.

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