Dual Laplacian Morphing for Triangular Meshes Jianwei Hu

∗

Ligang Liu

Zhejiang University Hangzhou, P.R.China

Zhejiang University Hangzhou, P.R.China

[email protected]

[email protected]

[email protected]

ABSTRACT Recently, animations with deforming objects have been frequently used in various computer graphics applications. Morphing of objects is one of the techniques which realize shape transformation between two or more existing objects. In this paper, we present a novel morphing approach for 3D triangular meshes with the same topology. The basic idea of our method is to interpolate the mean curvature flow of the input meshes as the curvature flow Laplacian operator encodes the intrinsic local information of the mesh. The inbetween meshes are recovered from the interpolated mean curvature flow in the dual mesh domain due to the simplicity of the neighborhood structure of dual mesh vertices. Our approach can generate visual pleasing and physical plausible morphing sequences and avoid the shrinkage and kinks appeared in the linear interpolation method. Experimental results are presented to show the applicability and flexibility of our approach.

Keywords mesh morphing, Laplacian coordinates, vertex path problem, dual mesh

1.

Guozhao Wang

Zhejiang University Hangzhou, P.R.China

INTRODUCTION

Mesh morphing, or shape interpolation, as a process of smoothly transforming one 3D geometric object into another, has been widely used for enhancing visual effects in computer animation. This technique has also been used for geometric modeling, advertising, medicine, and entertainment[11].

1.1 Related work A variety of techniques have been developed for 3D objects morphing in the literature [8, 2]. Boundary based mesh morphing requires the solutions to two main sub-problems: vertex correspondence problem, which is to find a correspondence between vertices of the two shapes, and vertex ∗Corresponding author.

path problem, which is to find paths that the corresponding vertices traverse during the morphing process[8, 2]. Most of previous algorithms for mesh morphing are mainly concerned with the vertex correspondence problem. The typical approaches first map the meshes on a common base domain and then compute the overlapped triangulations on the based domain. A plane is chosen as the base domain for a non-closed mesh patch[12, 30] and a sphere is chosen as the base domain for a closed, genus zero mesh[13, 1]. A more general approach is to parameterize the meshes over a common intermediate simplicial mesh[15, 17, 18, 14]. The meshes are partitioned into matching patches with an identical inter-patch connectivity using a set of consistent cuts. Then each patch is mapped onto the corresponding face in the based domain. Recently, the work of [29] directly maps the connectivity of one mesh onto another mesh surface using the least squares meshes [22] to create the compatible meshes. After building a one-to-one vertex correspondence between two meshes, many approaches simply adopt linear interpolation to generate vertex paths. However, shrinkage and kinks may occur in the morph sequence when linear interpolation approach is used, as shown in Figure 1, because the large rotation can not correctly be represented by linear interpolation. To address this problem, several researchers consider non-linear approaches to interpolate the vertices. Sederberg et al. [19] presents a solution to the interpolation problem of 2D polygon morphing, in which the geometric information, such as edge lengths and angles between edges, are interpolated. A generalization for 3D meshes is given by [16]. The work of [24, 21] use dihedral angels and edge lengths to interpolate two polyhedra. Alexa [3] and Fu et al. [9] apply Laplacian coordinates to mesh morphing and Xu et al. [27] use the Poisson equation to interpolate gradient fields of the meshes. Some researchers consider the interior information of given shapes and propose methods to control local volumes distortions[20, 4, 25].

1.2 Our approach In this paper, we present a novel approach for interpolating two 3D meshes given that the vertex correspondences between them have been well established using existing methods[17, 18, 29]. Laplacian vector information provides an elegant local intrinsic feature descriptor for shape. It has recently been

Figure 1: Comparisons among different morphing techniques. (a) shows the source horse mesh in different views; (e) shows the target horse mesh in different views. The first row shows the morphing sequence using linear interpolation method, the second row shows the morphing sequence using Laplacian morphing method[3], and the third row shows the morphing sequence using our approach. successfully applied in surface editing and deformation[23, 28, 6]. Our key observation is that the mean curvature flow Laplacian coordinates encode the intrinsic detail information of the mesh, which is inspired by the work of Au et al. [6]. The coefficients of the mean curvature flow Laplacian operator capture the local parameterization information and the magnitude of the mean curvature flow Laplacian coordinates captures the local geometry information. And the mesh geometry can be recovered from the mean curvature field and the coefficients of the curvature flow Laplacian operator. By interpolating the intrinsic parameterization information and geometry information of the two meshes, the intermediate meshes can then be reconstructed. Therefore, the morphed meshes preserve the intrinsic information of both meshes, avoiding the shrinkage and kinks problems appeared in the linear interpolation method. Furthermore, we propose to interpolate the intrinsic information in the dual domain of the input meshes. As the dual vertices of a triangular mesh always have valence three, the local intrinsic parameterization and geometry information are uniquely defined, which makes our approach very stable. Our approach has two main advantages over existing interpolation approaches. First, we formulate the path problem as mean curvature flow interpolation. Mean curvature flow Laplacian operator represents the intrinsic information of the shape. Second, we propose a novel shape interpolation approach based on the dual Laplacian interpolation. The in-between meshes are reconstructed from the interpolated mean curvature flow which avoids shrinkage and local wrinkles.

1.3 Overview The rest of the paper is organized as follows. In Section 2, some preliminaries are presented. Section 3 describes our dual Laplacian morphing approach. Experimental results are presented in Section 4, followed by conclusions in Section 5.

2.

PRELIMINARIES

In this section, we briefly review Laplacian operator and the curvature flow based dual Laplacian system.

2.1 Laplacian operator Let V = (v1 , v2 , · · · , vn ) be the vertex positions of the input mesh, and Ni be the set of adjacent vertex positions of vi . The Laplacian coordinate of vertex vi is X li = wij (vj − vi ), (1) j∈Ni

where wij is the weight of edge (i, j) corresponding to vi . Many weighting schemes have been proposed [26, 7]. Since the Laplacian coordinate li is the weighted average difference vector of its adjacent vertices to vi , it describes the local geometry at the vertex vi . Its matrix form is l = LV , where L is an 3n × 3n matrix with elements derived from wij . We refer to L as the Laplace operator and its elements as the Laplacian coefficients. Given some handle positions, Laplacian surface editing can deform the input mesh in the least squares sense[23]. The deformed mesh can be obtained by solving a sparse linear system AV ′ = b where A is the Laplacian operator with handle positions information and b is a vector consisting of l and handle positions. As the Laplacian vector is rotation variant, the Laplacian surface editing may suffer the transformation problem. Various editing systems based on Laplacian coordinates have tried to adopt different approaches to solve this transformation problem[23, 28].

2.2 Curvature flow Laplacian operator Curvature flow Laplacian operator is proposed to smooth surface by Desbrun et al. [7]. It smoothes the surface by moving along the surface normal with a speed equal to the mean curvature which alleviates the problem of vertices drifting in the tangential planes. The discrete version of the curvature flow Laplacian operator is derived as in Eq. 1 with li = 4Ai ki ni , wij = cot αij + cot βij , where ni and ki are respectively the unit normal and the mean curvature at the vertex vi ; αi and βi are the two angles opposite the edge (i, j), and Ai is the sum of the areas of the triangles adjacent to the vertex vi .

Note that the parameterization information is represented by the Laplacian matrix L with elements of weights {wij } and the geometry information is represented by e h = {e hi } for the dual mesh.

3. OUR MORPHING APPROACH In this section, we describe our novel approach for mesh morphing. The main idea is to interpolate the two intrinsic properties (the parameterization information and the geometry information) of the given meshes during the morphing process. We will show that morphing in the dual domain can eliminate the instability in the reconstruction process.

Figure 2: Illustration of the one-ring structure of dual vertices[6]. This Laplacian coordinate can be viewed as an approximation of the integrated mean curvature normal at the vertex vi . It consists of two types of information: the parameterization information which is captured by the Laplacian coefficients (wij = cot αij + cot βij ) and the geometry information which is encoded by the magnitudes of the Laplacian coordinates (4Ai ki ni )[5]. In other words, the parameterization information describes the shapes of the local features (adjacent triangles) around the vertex, while the geometry information expresses the sizes of the local features. Based on the curvature flow Laplacian operator, Au et al. [5] present an iterative framework to solve the transformation problem of Laplacian-based mesh editing. The deformed mesh is reconstructed by keeping the coefficients and the magnitudes of the curvature flow Laplacian coordinates with the original one as much as possible.

2.3 Dual Laplacian coordinates In general, the one-ring neighbors of a vertex are not coplanar and the Laplacian coordinates may have tangential components since there is no common normal direction for all planes formed by the one-ring neighbors. The curvature flow Laplacian coordinates will contain tangential component in some of these planes and cause tangential drifts which might cause an instability problem in the reconstruction process. To address the instability problem, Au et al. [6] propose a dual Laplacian editing framework by using the curvature flow Laplacian operator in the dual domain [10] of the input mesh. As the dual vertices of a triangular mesh always have valence three, there is a unique definition of normal and tangent space at each vertex in terms of its one-ring neighbors. Let V = (e v1 , ve2 , · · · , ven ) be the vertices of the dual mesh. The curvature flow Laplacian operator of the dual mesh is written as X e li = −e hi n ei = w eij (e vij − vei ), (2) j∈{1,2,3}

where the normal n ei at the vertex vei is perpendicular to the plane determined by the neighbor vertices vei1 ,e vi2 , and vei3 ; e hi is the distance from e vi to the plane, as shown in Fig. 2.

We assume that the input meshes to be processed are triangular meshes and the compatible meshes are generated by a pre-processing step[17]. We first convert them into their dual domains and obtain two corresponding compatible dual meshes. Then we interpolate the parameterization and geometry information of the dual meshes. The intermediate dual mesh can be reconstructed by the interpolated intrinsic information from an initial dual mesh iteratively.

3.1 Dual Laplacian morphing framework Let M0 and M1 be the input source and target meshes respectively. Given time t (0 < t < 1), our system interpolates the parameterization and geometry information of the input meshes and generates an intermediate mesh Mt . The main steps are outlined as follows: c0 and M c1 for two input 1. Generate compatible meshes M meshes using the previous method[17]. f0 and M f1 for 2. Build the corresponding dual meshes M compatible meshes. 3. Compute the parameterization information pet and the geometry information e ht at the given time t by interf0 polating the corresponding intrinsic information of M f1 . and M

ft by the intrinsic infor4. Reconstruct the dual mesh M mation pet and e ht and obtain the intermediate interpoft . lated mesh Mt from M The details of the above steps will be described in the following section.

3.2 Dual Laplacian morphing 3.2.1 Intermediate intrinsic information We obtain the intermediate intrinsic information at time t by linearly interpolating the corresponding intrinsic information as follows: e t = (1 − t)L e 0 + tL e1 , L e ht = (1 − t)e h0 + te h1 , e e e e where h0 , L0 and h1 , L1 are respectively the parameterizaf0 and M f1 . tion and geometry information of M Note that the dual Laplacian coordinates e lt may be either inward or outward of the surface. And the elements of the vector e ht can also be positive or negative.

Table 1: Per-iteration running time for the examples used in this paper. Morphing Models #Vertices #Edges #Faces Time(sec) fandisk & cube 6583 19743 13162 0.937 dino & horse 10189 30561 20374 1.547 man & woman 25172 75510 50340 4.218 mug & torus 38400 115200 76800 9.937

3.2.2 Reconstruction of intermediate dual mesh We use a similar iterative process as in [6] to reconstruct the dual mesh from the intermediate intrinsic information. ft0 is obtained by the Laplacian morphing An initial mesh M technique [3] or by the linear interpolation method. ft0 , we iteratively update Starting from the initial mesh M the intermediate dual mesh using the following two steps by fixing one vertex position unchanged: Step 1. Update the dual Laplacian coordinates. We update the dual Laplacian coordinates by fixing each length to the interpolated geometry information e ht , and compute the new dual Laplacian coordinates e ltk+1 from the old one e ltk . k+1 For each dual vertex vet,i , its corresponding dual Laplacian coordinate is defined as k+1 k e lt,i = (e ht,i /e hkt,i ) · e lt,i

(3)

where e ht,i is the ith element of e ht , e hkt,i is the signed length k e of lt,i . Step 2. Update the dual vertex positions. To interpolate the parameterization information, we then compute the intermediate dual vertex positions Vetk+1 using the current dual Laplacian coordinates e ltk+1 ; that is, we solve the following sparse linear system: ft · Vetk+1 = e A bkt

(4)

e t , and the fixed vertex et is derived from e ltk , L where e bkt , A position.

3.2.3 Normal adjusting We found that the intermediate dual mesh might collapse and the iteration might not be convergent in the iteration when we use Equations 3 and 4 to iteratively update the vertex positions of the intermediate dual mesh. Note that each intermediate dual mesh in a time step has different geometry. The direction of dual Laplacian coordinate of each dual vertex must be adjusted, in order to keep its direction perpendicular to the plane determined by the one-ring neighbors of this vertex. So we improve Equation 3 as k+1 e lt,i = −e ht,i · n ekt,i ,

(5)

where n ekt,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 of the changes in vertex positions between two successive iteration steps is less than a given threshold.

4.

EXPERIMENTAL RESULTS

A variety of 3D triangular models have been tested on a computer with a 3.0GHz CPU and 512MB RAM. Table 1

provides the computation time of each iteration step for the models used in this paper. Several examples are demonstrated in Figure 3-6. Figure 3 shows the morphing process between the fandisk and a cube in which the sharp features of fandisk are preserved during the whole morphing process. Figure 4 illustrates the transitions from a dinosaur to a horse. Note that the pleased interpolation of the two models eliminates the shrinkage which may be caused by linear interpolation. Figure 5 demonstrates the human pose morphing. The skeletons rotated and deformed naturally without specifying additional bone information. Our method can also work well on high genus mesh objects, see Figure 6. The accompany live video shows the animation sequence of these morphing.

5. CONCLUSIONS In this paper, we have proposed a novel morphing method based on dual Laplacian coordinates. The main idea is to interpolate the parameterization and geometry information of meshes in dual domain. Our framework iteratively updates the vertex positions. The iterations converge quickly and return a visual pleasing result. We illustrate the superiority of our approach by many examples. The most time-consuming part of our algorithm is solving the sparse linear system. And the results depend on the quality of the input compatible meshes in our current implementation. In the future work we would like to work on eliminating local self-intersection during morphing.

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Figure 3: Morphing sequence between fandisk and cube.

Figure 4: Morphing sequence between dinosaur and horse.

Figure 5: Morphing sequence between two human poses.

Figure 6: Morphing sequence between mug and torus.

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