Dynamic Surface Deformation and Modeling Using Rubber Sweepers

Transcription

1 Dynamic Surface Deformation and Modeling Using Rubber Sweepers Chengjun Li, Wenbing Ge, and Guoping Wang Peking University, Beijing, China, Homepage: Abstract. We introduce a new surface deformation and modeling method in this paper. Referring to the swept volume generation, the surface is pulled or pushed along a trajectory curve. The key point is the sweeping function. Surface points are moved to where they should be during sweeping operations according to the global parameter, which is determined by topological distance. An index factor controls how much the surface deforms around the handle point. The proposed method is easy to extend to fit different applications such as various constraints, local deformation and animations. 1 Introduction Surface modeling is one of the most popular topics in computer engineering and digital entertainment in recent years. Efficiently modeling and intuitively manipulating complex shapes are paramount to engineers and designers in geometric design, manufacturing, visualization, animation, and many other fields. Most of the real objects are too complicated to represent in simple expressions. The hard work in modeling is to start from simple objects and refine them towards the desired shapes. Powerful editing tools are required. Barr [1] first discussed global and local deformation of solid primitives. Because the transformation matrix in his method is a variable of spatial position, the deformation is independent to the representation of target objects. The subsequent approaches made by other researchers mostly inherit the same concept. Sederberg and Parry [2] introduced freeform deformations (FFD) which allow continuous space deformations with multiple points transformed. They define the lattice of control points for a Bézier volume. The embedded space is then smoothly deformed by interpolating the control point coordinates. Later Hsu et al. [3] proposed a way of doing direct manipulation of a single point or multiple points in space with FFD. Many other deformation methods are extended from FFD [4]. A major restriction of FFD is the regularity of the grid. The limited This research is supported by the National Grand Fundamental Research 973 Program of China (Grant No. 2004CB719403), the National High Technology Development 863 Program of China (Grant No. 2004AA115120), and the National Natural Science Foundation of China (Grant No ).

2 2 control points may not provide enough freedom for the deformation. Axial deformation [5] uses another parameterization method like FFD to embed objects, but it acts well mainly in scaling and twisting. In some different way, subdivision techniques provide a convenient surface representation of hierarchy. Zorin et al. [6] introduced a framework of multiresolution mesh editing based on subdivision. Analysis and Synthesis procedures build up the relation of different levels, which indicate editing extent, and editing results are transferred in between. Boier-Martin et al. [7] introduced another detail-preserving variational surface design method. The deformation constraints may be applied to position, tangent or normal. Subdivision hierarchy is also needed. Multi-resolution designs are free to control the region of interest, but still need some powerful deformation tools. In this paper, we introduce a new deformation method referring to solid sweeping. Our inputs are a surface mesh with topology structure and a couple of spline curves, and the output is a deformed surface indicating the dragging effects along these curves like a rubber band. Fig. 1 is an overlook of our goal. Our contributions include an intuitional modeling method and some extended techniques for special applications. This paper is organized as follows. We first review some related works in Sect. 2 and explain some terms. And then, the algorithm is discussed in detail in Sect. 3, including parameterizations, deformation constraints and region of interest. Sect. 4 gives some related topics of extension. Sect. 5 is the results and discussion and we conclude in Sect. 6. Fig. 1. Rubber sweeper deformation. Left: a bunny model and three guiding paths, and the long-eared and long-tailed bunny. Mesh segmentation and boundary constraint are applied to keep other parts unchanged. Right: bent fingers. The deformation is global with proper interest indices (both α = 4) 2 Related Works Sweeping is one of the most important modeling methods in the field of computeraided design (CAD), computer graphics and animation. It has two components:

3 3 an initial moving object and its trajectory curve (also called as a spine). The basic sweeping operation can be expressed as follows: S(t) = P (t) + F (t) C (1) where S(t) indicates the sweeping surface at t, P (t) is the spine, a smooth sweeping curve, and F (t) is the local moving frame indicating the directions of the moving object at t, and C is the moving object represented in local coordinates, such as a circle curve. If C is a curve, whether closed or open, it produces a surface during sweeping; if C is a solid object, that is, it has the difference between inner and outer sides, the silhouettes of C during sweeping compose the sweeping surface, and the swept result is called swept volume (Fig. 2). The objective of solid sweeping is to find the envelope surface of the initial moving object as the modeling result [8, 9]. Fig. 2. Swept volume (by Weinert et al. [9]). Up: discrete view. Down: smooth surface For most of the applications, P (t) is differentiable, otherwise the surface may not be smooth. There are several methods to compute F (t) according to P (t), such as Frenet frame method, Generalization Translation Frame method [10], projection normal method, rotation minimizing method, etc. Choi et al. [11] and Piegl et al. [12] make thorough discussions about these methods. It is not the main part of this paper, and we just use the Frenet frame method, although other methods are also applicable. In some extensions, F (t) may be a 4 4 matrix made up of translation, rotation and scaling. In this paper, we assume F (t) is a rigid rotation matrix. Angelidis et al. [13] present sweepers, a new class of space deformations suitable for interactive virtual sculpture. The user describes a basic deformation as a path through which a tool is moved. The tool causes a deformation of the working shape along the path. Nevertheless, as a sculpture tool, it needs the power of pulling. Our method deforms the surface directly with a virtual force along the path. The only difference between pull and push is the direction of force on the surface. It is easy to be integrated with mouse or other input devices in real-time applications.

4 4 3 Deformation Using Rubber Sweepers 3.1 Overview Given a surface mesh M, first we just consider one single spine curve P (t), 0 t 1, used as deformation sketch. Multiple curve deformation is discussed later. Usually we locate P (0) on one of the mesh vertices h, called handle point, where the deformation starts. If P (0) is not on the surface, the nearest point to P (0) of mesh M is the handle point instead. For each vertex v of M, the algorithm first computes its local coordinates L(v) and global parameterization G(v). According to Equation 1, we reposition each vertex to where it should be during a sweeping operation (Fig. 3). This can be expressed in Equation 2: T (v) = P (H(v)) + F (H(v)) L(v) (2) where H(v) is the sweeping parameter, usually a function of G(v), and T (v) is the new position of v. Fig. 3. Discrete sweeping deformation. Top left: initial mesh and trajectory. Top right: discrete sweeping. Bottom right: corresponding parts at different parameters. Bottom left: deformed mesh At the same time, the topology of the original mesh is retained to construct a deformed copy of the old one. Point or boundary constraints are also considered for some applications. Because the mesh is somewhat stretched, we can use an adaptive interpolating subdivision method to smooth and refine it. 3.2 Local Frame Parameterization Note that in sweeping operation, in Equation 1, C is represented in local coordinates. It is transformed by P (t) and F (t), or rather, translated and rotated by P (t) and F (t) respectively. For each vertex of the input mesh, we construct a local frame using P (0) as the origin and F (0) as the three axes. We get the vertex v s local coordinates L(v) using Equation 3: L(v) = F 1 (0) (v P (0)) (3)

5 5 3.3 Global Parameterization Before we talk about global parameterization, we first introduce the term - topological distance. We use the so-called topological distance D (v 1, v 2 ) in our method, which is the minimum distance along edges on the surface mesh between two vertices v 1 and v 2. It is different from geodesic distance in that it does not cross mesh polygons. It s well known that measuring geodesic distance is rather difficult and inefficient along complicated surface. The topological distance is somewhat a good approximation of geodesic distance. For each vertex v of the mesh M, the topological distance to vertex v, D (v, v ), can be computed using an FIFO queue structure (Algorithm I, Fig. 4). At the beginning all the distance values are set to invalid except v to be zero. The procedure loops ring by ring according to topology, taking v as the center. // Algorithm I: d[v] = D(v, v*) // init d[v] for each v of M if (v=v*) d[v]=0 else d[v]= 1 // init the FIFO queue Q Q.push(v*) // loop while size(q)>0 qc=size(q) for each v i in Q for each v j, direct neighbor of v i nd=d[v i]+ v j v i if (d[v j]<0) d[v j]=nd Q.push(v j) // push else if (nd<d[v j]) d[v j]=nd UpdateDistance(v j) Q.pop(qc) // pop the first qc elements Fig. 4. Topological distance algorithm: distance to a certain point In Algorithm I, the UpdateDistance(v j ) procedure updates the distances topologically from v j and ensures all the already defined are minimal, that is, they are the correct values up to the present. The pseudocode is presented in Appendix. Assume n is the number of vertices of M and the maximum valence of each vertex is a limited constant, then the cost is O(n) except for UpdateDistance(v j ). However, the topology-based spreading and uniformity of mesh vertices will reduce the updating cost remarkably.

6 6 Once the topological distance is available, we can define the global parameter of each vertex in respect to v as follows: G (v, v ) = D (v, v ) /max{d (v i, v ) v i M} (4) For some meshes with bad connectivity, the topological distance is not valid, so the direct Euclidean distance is feasible instead. But sometimes this will cause unexpected changes at far away in the sense of geodesic distance, such as the middle finger in Fig Sweeping Parameter and Region of Interest Index It is obvious that the farther a vertex to the handle point, the shorter distance it moves along the trajectory. Moreover, the farthest vertex seems to stay still. Thus we can define the sweeping parameter in Equation 2 as: H (v) = (1 G(v)) α (5) The index factor α is introduced to affect how much the handle point pushes or pulls the surface. The bigger the α, the more concentrated and sharper the surface near the handle point. Fig. 5 gives some examples of deformed spheres with different index factors. Fig. 5. Deformed spheres. Left: sphere and trajectory. Right four: α = 0.7, 1, 3, 7 4 Extended Applications The basic deformation algorithm introduced in the previous section can be easily extended to fit other applications. 4.1 Multiple Curve Deformation If there are more than one forces exerted to the surface, that is, more than one trajectory curves and corresponding handle points, the total effect need to be

7 7 the sum of them. The final deformation is a weighted average of each one in respect to mesh vertices, and the weight is the sweeping parameter: T (v) = H i (v)t i (v) / Hi (v) (6) For example, two-curve deformation is useful for symmetrical geometry, because two separate operations cannot achieve the same result. Since the index factor is crucial to the shape (Fig. 5), it values mush for the conjunct effect of multiple curves (Fig. 6). Note that the handle points will not be moved to the ends of the path curves because of averaging. Fig. 6. Two-curve deformation. Left: initial sphere and two curves. Middle left: α 1 = α 2 = 3. Middle right: α 1 = 9, α 2 = 3. Right: α 1 = α 2 = Boundary Constraint Sometimes it s necessary to keep the boundary of an open surface fixed. The deformation is only executed on inner surface from handle point to boundary. Therefore the vertices near the boundary move less than inner ones. The topological distance to the boundary B(v) of v is defined as follows: B(v) = min{d (v, v i ) v i is on the boundary} (7) Besides the definition in Equation 7, there is another way to get all the B(v)s in a similar method like the distance algorithm to a certain point. The only difference is that the spreading process starts from each boundary vertex rather than the handle point. Under the boundary constraint, the global parameter of each vertex v in respect to v is defined by G (v, v ) = D (v, v ) /(D (v, v ) + B(v)) (8) 4.3 Point Constraint Sometimes we must keep several vertices of M untouched and locked. The vertices between the handle point and those locked ones had to spread out regularly. For a certain vertex v, multiple locked vertices will affect it simultaneously.

8 8 Let {l i }(i = 1 k) be the locked, we can define v s global parameter using a harmonic equation as follows: G i (v, v ) = D (v, v ) /(D (v, v ) + D (v, l i )), i = 1 k (9) G (v, v ) = k / Gi (v, v ) 1 (10) 4.4 Adaptive Subdivision When the mesh is deformed, especially when stretched, its faces become some slender. We subdivide it repeatedly according to edge length. Dyn et al. [14] propose an interpolating triangular subdivision scheme called Butterfly scheme. Zorin et al. [15] later refine it to the Modified Butterfly scheme, which is more complicated but delicate. The edges of the input mesh are subdivided using Zorin s method if their corresponding ones of the deformed mesh are longer than some given threshold ε e. This continues until no edge is bigger than ε e. However, G (v, v ) depends totally on mesh topology, but the adaptive subdivision will change the topology unfortunately. There are two ways to handle this problem. We can just discard the difference of topological distance before and after subdivision, that is, after subdivision G (v, v ) is recalculated for the new mesh. Instead we can also interpolate G (v, v ) simultaneously during subdivision. The latter is better in that subdividing ensures the same vertex is deformed to the same position when checking edge lengths. 4.5 Mesh Segmentation Sometimes we had to restrict deformation in a limited region rather than the whole mesh. The special part to be deformed may be designated arbitrarily by the user, so we first segment the mesh as desired. The simplest way is to limit the deformation in a region according to topological distance D (v, v ) or G (v, v ) with a specified threshold ε d. The boundary constraint is necessary indeed. There is another intelligent and perceptual way. We adopt Zhang s method [16], which uses markers to find out where to slit the mesh. He introduces makercontrolled image segmentation into 3D space. The boundary is likely where it should be according to Minima Rule, a human perception phenomenon. In our application, the handle points are rather good markers indicating different separate parts. To split the mesh properly, some additional markers are needed. The three parts of the bunny model in Fig. 1 is first segmented in this way and deformed under boundary constraint. 5 Results and Discussion The deforming and modeling method presented in the previous sections is convenient for surface design and animation. The hand model in Fig. 1 bends the index finger first and the thumb subsequently, accompanied with other fingers

9 9 curved slightly. Fig. 7 gives more examples. There is no difference between surface inside and outside, for the trajectory is the deforming direction (Fig. 7(a)). In Fig. 7(b), linear scaling can be achieved using a unit index although the handle point is restricted to some limited locations and the global parameterization function is a bit different. Moreover, the reverse deformation, i.e., from Fig. 7(b) right shape to left shape with a reverse trajectory, is similar due to the linearity. Furthermore, we can limit the deformation effect in small regions with interest index larger than one, and the operations can be carried out one by one. The dolphin in Fig. 7(c) changes its pose, first tail, and second head, using multiple curve deformation. The eagle in Fig. 7(d) waves its wings simultaneously and lays down its tail for landing. Euclidean distance is used for it is completely made up of separate polygons. The interest indices in Fig. 7(c) and 7(d) are the same respectively. (a) (b) (c) (d) Fig. 7. Deformed objects. (a)sphere: α = 7. (b)spring: α = 1. (c)dolphin: α = 2. (d)eagle: α = 1.8

10 10 For interactive and real-time applications, starting from a given surface, the user may use the mouse to pick a handle point and drag to some locations. We can fit the hand-drawn path to a smooth spline curve, used as the trajectory in our method. The distance values and sweeping parameters are determined once the handle point is selected. The deformation process is fast because the transformation of surface is linear with the number of total vertices, and the fitting algorithm may first reduce the path to a few points and then fit them. The proposed method is suitable for animations. Note that in Equation 1, the parameter t is between 0 and 1. If we introduce a time factor s to control t s upper bound, the continuously deformed surfaces at various s make a series of animation frames. Accompanied with the handle point moving along the trajectory, the surface deforms gradually. The animation process can be expressed to Equation 11: S(s) = S(st), s [ 0, 1 ] (11) 6 Conclusions and Future Works We have presented a novel surface deformation method based on swept volume. The primary objective is to integrate sweeping modeling techniques with surface deformation methodology for the interactive editing and direct manipulation of the objective surface. The proposed method is easy to exert pulling and pushing effect on the surface. With multiple curve support, the input surface can be deformed symmetrically to rather complicated shapes under several constrains. Mesh segmentation and adaptive subdivision may restrict the deformation in a certain region and make the output surface smoother. Moreover, it can also be used for computer animations. However, for the sweeping parameter defined in Equation 5, it s not continuous around the handle point when α > 1 and around the farthest point when α < 1 (Fig. 5). We would consider a smoother function in the future. References 1. A.H. Barr. Global and local deformation of solid primitives. In: Computer Graphics Proceedings, Annual Conference Series, vol. 18(3), ACM SIGGRAPH, 1984, pp T.W. Sederberg and S.R. Parry. Free-form deformation of solid geometric models. In: Computer Graphics Proceedings, Annual Conference Series, vol. 20(4), ACM SIGGRAPH, 1986, pp W.H. Hsu, J.F. Hughes and H. Kaufman. Direct manipulation of free-form deformations. In: Computer Graphics Proceedings, Annual Conference Series, vol. 26(2), ACM SIGGRAPH, 1992, pp S. Coquillart. Extended free-form deformation: a sculpting tool for 3D geometric modeling. In: Computer Graphics Proceedings, Annual Conference Series, vol. 24(4), ACM SIGGRAPH, 1990, pp F. Lazarus, S. Coquillart and P. Jancène. Axial deformations: an intuitive deformation technique. Computer-Aided Design, vol. 26(8), 1994, pp

11 11 6. D. Zorin, P. Schröder, and W. Sweldens. Interactive multiresolution mesh editing. In: Computer Graphics Proceedings, Annual Conference Series, vol. 31(4), ACM SIGGRAPH, 1997, pp I.M. Boier-Martin, R. Ronfard, and F, Bernardini. Detail-preserving variational surface design with multiresolution constraints. In: Proceedings of Shape Modeling International (SMI) 2004, pp , Genova, Italy, June G. Wang, J. Sun and X. Hua. The sweep-envelope differential equation algorithm for general deformed swept volumes. Computer Aided Geometric Design, vol. 17(5), 2000, pp G. Wang, J. Sun. Shape control of swept surface with profiles. Computer-Aided Design, vol. 33(12), 2001, pp K. Weinert, S. Du, P. Damm, and M. Stautner. Swept volume generation for the simulation of machining processes. International Journal of Machine Tools & Manufacture, 44 (2004), pp B.K. Choi and C. Lee. Sweep surfaces modeling via coordinate transformations and blending. Computer-Aided Design, vol. 22(2), 1990, pp L. Piegel and W. Tiller. The NURBS Book, Springer Verlag, Berlin, A. Angelidis, G. Wyvill and M.P. Cani. Sweepers: swept user-defined tools for modeling by deformation. International Conference on Shape Modeling and Applications 2004 (SMI 04), pp N. Dyn, D. Levin and J.A. Gregory. A butterfly subdivision scheme for surface interpolation with tension control. ACM Transactions on Graphics, vol. 9(2), 1990, pp D. Zorin, P. Schröder, and W. Sweldens. Interpolating subdivision for meshes with arbitrary topology. In: Computer Graphics Proceedings, Annual Conference Series, vol. 30, ACM SIGGRAPH, 1996, pp C. Zhang, N. Zhang, C. Li, and G. Wang. Marker-controlled perception-based mesh segmentation. In: Proceedings of the Third International Conference on Image and Graphics (ICIG 2004), Dec.18-20, 2004, Hong Kong, China, pp Appendix Algorithm II: UpdateDistance(v) // Algorithm II: UpdateDistance(v) // init the FIFO queue Q Q.push(v) // loop while size(q)>0 qc=size(q) for each v i in Q for each v j, direct neighbor of v i nd=d[v i ]+ v j v i if (nd<d[v j ]) d[v j ]=nd Q.push(v j ) // push Q.pop(qc) // pop the first qc elements Note that the Q here is another local FIFO queue rather than the one in Algorithm I.