Local Modification of Subdivision Surfaces Based on Curved Mesh

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1 Local Modification of Subdivision Surfaces Based on Curved Mesh Yoshimasa Tokuyama Tokyo Polytechnic University Kouichi Konno Iwate University Junji Sone Tokyo Polytechnic University R.P.C. Janaka Rajapakse Tokyo Polytechnic University ac.jp Abstract In this paper, to take an advantage of the features of both Catmull-Clark subdivision surface and Gregory patch, we propose a local modification method that generates a curved mesh from a polygon mesh with the same topology using a Catmull-Clark subdivision rules, modifies the generated curved mesh and recreate the polygon mesh by applying the modification result. One of the characteristics of our method is that modification of some vertices or edges affects only their adjacent surfaces. Also, the surfaces within a specified range can be modified freely with maintaining the continuity between the surfaces and their adjacent surfaces outside the specified area. Applying modification of a curved mesh to a polygon mesh enables local modification or intuitive modification that are rather difficult to perform for a polygon mesh. Key Words Subdivision Surface, Limit Point, Curved Mesh, Gregory Patch, Interpolation, Local Modification 1 Introduction For computer-graphics (CG) animation, shape models of humans, animals, and other characters are important. These shapes have been represented with Bezier, B-spline, NURBS, or other parametric surfaces. However, it is still difficult to connect surfaces smoothly, and complex processing to modify the shapes smoothly is necessary. After a surface is created so as to connect to other surface smoothly, modifying locally a part of the surface may destroy the continuity between surfaces, and therefore further adjustment of continuity is required. To solve this problem, subdivision surfaces are popularly used in recent years. Subdivision surface is a method to create smooth surfaces by dividing a polygon mesh and applying weight to the polygon mesh repeatedly. Doo et al. [1] and Catmull et al. [2] constructed the basic theory and Loop [3] proposed a method for triangular meshes. All the Loop subdivision surfaces are applied to triangular meshes, while Catmull-Clark subdivision surfaces are not limited about target objects. With this merit, a lot of 3D CG systems adopt Catmull-Clark subdivision surfaces. Subdivision surfaces are versatile since they can be applied to polygon meshes of arbitrary topology. In the process of modeling with subdivision surfaces, however, designers must observe and confirm both the rendering image of a model and its mesh image after subdivision as they operate shape modification such as adding, deleting or moving vertices, edges or faces. At this moment, to obtain a desired shape, its polygon shape must be modified through a trial-and-error process. In the modeling method with subdivision surfaces, however, the system creates smooth surfaces automatically from a mesh of arbitrary topology, so that it is impossible for a designer to model a surface shape directly. This means that it is difficult to obtain a desired surface shape easily and intuitively. Besides, moving a vertex on a polygon mesh modifies not only the shape around the vertex, but also the shape in a wide range [4]. Since the range is automatically determined, it is difficult to specify arbitrary range of surfaces to be modified. Several methods of local modification using subdivision surfaces have been suggested. Khodakovsky et al. [5] suggest the method to create a feature curve interactively on an arbitrary spot of a Loop subdivision surface and to modify the shape around the feature curve so that a feature shape is generated along the feature curve. Biermann et al. [6] suggest the method to create a feature curve interactively on an arbitrary spot of a multiresolution subdivision surface so that a sharp shape is generated along the feature curve. Sederberg et al. [7] suggest nonuniform subdivision surface NURBS, which enables nonuniform knot insertion against uniform knot insertion of Catmull-Clark subdivision surface. This features creation of limit subdivision surface shape containing a sharp spot by arranging knot intervals on vertices or edges. Zorin et al. [8] suggest the multiresolution edit algorithm on Loop subdivision surfaces. Local modification of shapes is enabled by moving a group of vertices on a triangular mesh. In these local modification methods, flexible shape modification is possible without regard to topology. If, however, the range of modification is regarded, it is necessary to consider topology. Another method is proposed to model complex freeform surface shape, with which the boundary curves of a surface are specified to create a curved mesh and the mesh area within the boundary is interpolated. After interpola-

2 tion, the designer evaluates the created surface. Then the designer adds curves to the mesh, modifies the shape of curves, deletes unnecessary curves [9]. In addition to the basic modification operation as mentioned above, curved mesh modeling enables local modification with topological structure or modification range considered. This kind of modeling method enables to modify a curved mesh directly with intuition so as to achieve a desired shape. Chiyokura et al. proposed Gregory patch [10] and rational boundary Gregory patch [11] as a surface representation for smooth interpolation of an irregular curved mesh. For Gregory patch and rational boundary Gregory patch, it is possible to define the cross boundary derivative of each of u and v parametric directions separately. The cross boundary derivative is the first derivative vector function in the direction that crosses a boundary curve. Due to this feature, if boundary curves of an area are fixed, adjacent surfaces can be connected with G 1 -continuity after interpolation, although an irregular curved mesh is specified. We suggest a method of local modification that combines the merits of the two methods, Catmull-Clark subdivision surface and Gregory patch on a curved mesh. To be concrete, one method creates a curved mesh from a polygon mesh and the other recreates the polygon mesh from the curved mesh. With these methods, a curved mesh can be modified as desired using local modification operations and then polygon mesh is recreated by applying the modification result. By using the method described in this paper, when the vertices or the edges of a curved mesh are modified, the shape of only the surfaces connecting the vertices or the edges is modified. In addition, as a range on a curved mesh is specified, the shape of the surfaces in the range can be modified as desired with maintaining the continuity with the outer surfaces. Besides, applying the modification on a curved mesh to a polygon mesh enables local modification that is difficult on a curved mesh or intuitive modification. 2 Method for Creating a Curved Mesh The method for creating a curved mesh of a subdivision surface presented in this paper is composed of the following procedures: 1. Subdivide each edge of an initial mesh twice to obtain five vertices. 2. Create a cubic Bezier curve that fits in with the limit points on the five vertices [12]. 3. Modify the control points of the fitting curve using the normal vectors on the limit points to fit in with the limit points again [13]. For details of the method of how to create a curved mesh, refer to [14]. Figure 1 shows the initial mesh of a CG character (width 65, height 100, depth 50). Figure 1 shows the curved mesh created using our method, having the same topology as the initial mesh. Figure 1. Initial control meshes of a dog character Curved meshes after subdivision 3 Gregory Patch Interpolation Based on Control Vector A curved mesh represents boundaries of subdivision surfaces. Each area is enclosed with cubic Bezier curves. If an area of the mesh is interpolated with bicubic Gregory patches by using the method of Chiyokura et al. [10], cross boundary derivatives are defined for each of u and v parameters separately. Due to this feature, surfaces can be created so that G 1 -continuity between adjacent surfaces are maintained. As shown in Figure 2, however, the center of the surfaces on triangular areas around the ears are protuberant. Besides, as Figure 2 shows, the quadrilateral surfaces around the neck are undulating. In the former case of the surfaces around the ears, when an N-sided area is interpolated, inner curves are created between the center of the corresponding area and the midpoint of each edge. The N-sided area is divided into multiple rectangular areas, each of which is interpolated with a bicubic Gregory patch. Upon creation of the inner curves, the center of the N-sided area is decided from the cross boundary derivative at the midpoint of each boundary curve. Because of this feature, the shape of the created inner curve depends on the shape of boundary curves, and therefore the interpolated surface may be likely to result in protuberant shape. In the latter case of the surfaces around the neck, an undulating surface is generated because the inner control points of a Gregory patch corresponding to each edge of the quadrilateral depend largely on the tangent vectors of edges that connect to each of the edges of the quadrilateral. To solve the former problem, we propose to create inner curves in an N-sided area by using the information of the limit point on a vertex obtained by subdivision. To solve the latter problem, we propose to add a control vector to an edge by using the information of the limit point on a vertex obtained by subdivision. Konno et al. have been studied the concept of control vector for the surface interpolation method [15]. As Figure

3 Figure 2. Surfaces around ears Surfaces around neck Figure 4. patch Inner curves and control points of a Gregory v u Figure 3. Control vector based on the first order partial derivative vector 2. Create a cubic Bezier curve from the five points by using the method described in section 2, step A plane is defined from the limit point at the center point of the N-sided area of the initial mesh and the unit normal vector at the limit point. Next, the tangent vector of the Bezier curve at the start point is projected to the plane. For the end point of the inner curve, the same processing is applied to modify the tangent vector of the Bezier curve. Then, a Bezier curve is recreated by using the method described in section 2, step 3. Figure 4 shows the inner curves created in a triangular area and the control points of three bicubic Gregory patches interpolated in the sub-areas. 3 shows, if control vector is added to each edge, the inner control points of a Gregory patch are calculated so that partial derivative vector at each surface boundary become the same as the control vector. Because of this feature, control vectors largely affects the shape of an interpolating surface. If control vectors are added, an interpolating surface is represented with a biquartic Gregory patch expressed by equation. 3.1 Creating inner curves for an N-sided area If the initial polygon mesh contains a non-quadrilateral area (N-sided area), the N-sided area is divided into the N number of quadrilateral areas by creating inner curves, and then each of the created area is interpolated with a bicubic Gregory patch. In this paper, inner curves of an N-sided area are created in the following procedure: 1. The start point of an inner curve is set to the limit point at the center point of an N-sided area of the initial mesh [14][16]. The end point of the inner curve is set to the midpoint of each edge of the N-sided area of a curved mesh. Three sampling points between the start and end points are set to the limit points obtained by three times of subdivision of the initial mesh. 3.2 Creating control vectors of a quadrilateral First-order control vectors are added to both sides of each edge of a quadrilateral. The vectors are used when the quadrilateral is interpolated with a biquartic Gregory patch. In this paper, the parameter on the edge to which the firstorder control vector is added is set to 0.5. The method to create control vectors are composed of the steps below: 1. On both sides of each edge of a quadrilateral area, unit tangent vectors are created in the following procedure: A cubic Bezier curve is created. The start and end points of the Bezier curve are set at the position of parameter 0.5 on two opposing edges (q 10 and q 14 of Figure 5). Three sampling points between the start and end points are set to the limit points of three inner points (q 11, q 12 and q 13 of Figure 5) obtained by twice of subdivision of the quadrilateral. Create a cubic Bezier curve from the five points by using the method described in section 2, step 2. (c) A plane is defined from the limit point at parameter 0.5 of an edge and the unit normal vector at the limit point. Then, the unit tangent vectors of

4 q20 q21 q22 q23 q24 q15 q16 q17 q18 q19 q10 q11 q12 q13 q14 q5 q6 q7 q8 q9 q0 q1 q2 q3 q4 Figure 5. Sample points for generating control vectors Figure 6. Shaded image of curved mesh surfaces Shaded image of subdivision surfaces the Bezier curve at the start and end points are projected to the plane. 2. Since the unit tangent vectors are created separately on both sides of each edge of the quadrilateral, G 1 - continuity may not be satisfied. If not, modify the unit tangent vectors to be G 1 -continuous to obtain a smooth surface. To be concrete, a difference vector between the unit tangent vectors on both sides are obtained and the unit tangent vectors are projected on the difference vector. 3. Based on the unit tangent vectors created in step 2, a cubic Bezier curve is recreated between opposing edges of each area by using the method described in section 2, step The first-order derivative vectors on both end points of the cubic Bezier curve are set as the first-order control vectors used for interpolating the area with a biquartic Gregory patch. 3.3 Determining central control point of a biquartic Gregory patch A quadrilateral area is interpolated with a biquartic Gregory patch. On account of the condition of G 1 -continuity, the control points are already determined in the quadrilateral area except the central control point. The central control point is the degree of freedom and it cannot be determined by the G 1 -continuity conditions. In this paper, the central control point is determined in the following procedures: 1. Create a curve in the direction of u = 0.5 on a surface as a cubic Bezier curve. Both end points of the cubic Bezier curve are the points on the parameter 0.5 of the boundary curves on both sides. The inner control points are determined as the first-order partial derivative vector at the parameter 0.5 calculated from a cross boundary derivative is set as the first-order derivative vector of the cubic Bezier curve. 2. Elevate the degree of the cubic Bezier curve to quartic. Among the five control points, set the center one for A. 3. With using the methods of steps 1 and 2, create a curve in the direction of v = 0.5 on a surface as a cubic Bezier curve, and elevate the degree to quartic. Among the five control points, set the center one for B. 4. Set the mid point between control points A and B for the control point at the center of the biquartic Gregory patch. 4 Evaluation of Curved Mesh Figure 6 shows the shading image after interpolation with Gregory patches. Since the mesh shape obtained after the third subdivision is very similar to the limit subdivision surfaces, the image after the third subdivision is shown in Figure 6. As Figures 6 and are compared, the shape of surfaces on the curved mesh obtained after interpolation with Gregory patches is very similar to the shape of Catmull-Clark subdivision surfaces. Besides, the maximum distance is between a Bezier curve and a limit point of fitting; and the maximum distance is between a Gregory patch on a quadrilateral of a polygon mesh and the limit points of nine inner vertices, obtained after the second subdivision of the quadrilateral. Therefore, it is found that Figures 6 and are similar to each other in both shape and distance. We have examined a lot of cases to found that approximation precision of our method is satisfactory. 5 Modification Based on Curved Mesh Curved meshes are advantageous since the shape can be modified intuitively. To use this advantage, we present a method to reconstruct a polygon mesh from the resultant shape of modification of a curved mesh. Figure 7 outlines this method. The procedures of this method are as follows:

5 Modification M 0 M 1 M 2 M 3 CAD data Modification Curved mesh Figure 8. Moving a vertex in curved mesh Shaded image Figure 7. Outline of our method 1. Subdivide an initial polygon mesh M 0 once to obtain polygon mesh M Subdivide polygon mesh M 1 once to obtain polygon mesh M Using the limit points of each vertex of M 2, create a curved mesh corresponding to the initial polygon mesh M Modify the curved mesh. 5. Apply the modification result of the curved mesh to polygon mesh M Subdivide polygon mesh M 2 once to obtain polygon mesh M 3. Or use least square method to obtain the coordinates of the vertices of M 1 or M 0 from M 2. The modification procedures are described in more detail through a cube shown in Figure Create a curved mesh corresponding to the initial polygon mesh with using the limit points on vertices of polygon mesh M 2 of Figure 7. At the same time, correspond the surfaces of the curved mesh to the limit points of vertices of M 2. A surface created in an N- sided area corresponds to nine limit points and the other surface corresponds to twenty-five limit points. In addition, calculate (u, v) parameters of the corresponding limit points in Gregory patch information of each surface. 2. Modify the vertices, the edges and the control vectors of the edges of the curved mesh. When the vertices are moved, the edges and the surfaces connecting to the vertices are also modified. When the edges or the control vectors are modified, the surfaces on both sides of each edge are modified. Figure 8 shows the curved mesh created after one of the vertices of the curved mesh is moved. 3. Calculate the coordinates of the new limit points using (u, v) parameters of the corresponding limit points in Gregory patch information of each surface. (c) Figure 9. Generated curved mesh Curved mesh modified (c) Polygon mesh M 2 regenerated 4. The following equation represents the relation between the coordinates of the vertices of M 2 and those of the limit points: Ax = b (1) where x represents the coordinates of the vertices of M 2, b represents the coordinates of the limit points and A represents the square matrix created [16]. The coordinates of the vertices can be obtained from the coordinates of the limit points using equation (2). Since A is a sparse matrix, equation (2) is solved using conjugate gradient method [17]. x = A 1 b (2) In the procedure described above, the result of modification of a curved mesh is applied to polygon mesh M 2. Figure 8 shows the reconstructed polygon mesh M 2. Althouge the least square method can be used to obtain the coordinates of the vertices of M 1 or M 0 from M 2 [16], the effect of local modification may fade. In the following, the effect of local modification based on M 2 is evaluated. 6 Practical Examples and Consideration Figure 9 shows the result of local modification where the shape of the left hand of Figure 9 is modified. Figure 9 (c) shows the reconstructed polygon mesh M 2. To

6 Figure 10. Shaded image of curved mesh Shaded image of M 3 compare the shapes, the shaded image of the curved mesh after modification is shown in Figure 10 and that after subdividing once the polygon mesh of M 2 (polygon mesh M 3 after subdividing three times if subdivided from the initial mesh) is shown in Figure 10. It is found that the modified spot of Figure 10 is applied to Figure 10. Besides, the shape that is not related to the modification is not modified. Accordingly, our method enables intuitive local modification for curved meshes and the modification can be applied to the corresponding polygon mesh. 7 Conclusion In this study, we suggested a method of local modification of subdivision surfaces based on curved meshes. As the modification for a curved mesh is applied to the corresponding polygon mesh, intuitive local modification is enabled, which is difficult to perform on a polygon mesh. Our future subject is to enhance our modeling method to be more flexible so that a surface model of CAD is converted to a polygon mesh of subdivision surfaces. Acknowledgement This research was supported by High-Tech Research Center Project for Private Universities: matching fund subsidy from MEXT (Ministry of Education, Culture, Sports, Science and Technology), References [1] D. Doo and M. Sabin, Analysis of the behaviour of recursive division surface near extraordinary points, Computer aided Design, Vol.10, No.6, pp , [2] E. Catmull and J. Clark, Recursively generated B- spline surfaces on arbitrary topological meshes, Computer aided Design, Vol.10, No.6, pp , [3] C. T. Loop, Smooth subdivision surfaces based on triangles, Master s thesis Department of Mathematics, University of Utash, August [4] J. Feng, J. Shao, X. Jin, Q. Peng, A. R. Forrest, Multiresolution free-form deformation with subdivision surface of arbitrary topology, The Visual Computer, Vol.22, pp.28-42, [5] A. Khodakovsky and P. Schröder, Fine level feature editing for subdivision surfaces, Proceedings of the fifth ACM symposium on Solid modeling and applications, pp , [6] H. Biermann, I. Martin, D. Zorin, F. Bernardini, Sharp features on multiresolution subdivision surfaces, Graph Models, Vol.64, No.2, pp.61-77, [7] T. W. Sederberg, J. Zheng, D. Sewell, M. Sabin, Nonuniform recursive subdivision surfaces, Proceedings of SIGGRAPH 1998, pp , [8] D. Zorin, P. Schröder, W. Sweldens, Interactive multiresolution mesh editing, Proceedings of SIG- GRAPH 1997, pp , [9] K. Konno, Y. Tokuyama, and H. Chiyokura, A G 1 connection around complicated curve meshes using C 1 NURBS Boundary Gregory Patches, Computer Aided Design, Vol. 33, No.4, pp , [10] H. Chiyokura and F. Kimura, Design of solids with free-form surfaces, Computer Graphics, Vol.17, pp , [11] H. Chiyokura, T. Takamura, K. Konno, T. Harada, G 1 surface interpolation over irregular meshes with rational curves, In: Farin, G. (ed) NURBS for Curve and Surface Design. SIAM, Philadelphia, pp.15-34, [12] L. Piegl and W. Tiller, The NURBS Book, Springer- Verlag, [13] J. Hoschek, Approximate conversion of spline curves. Computer Aided Design, Vol.4, pp.59-66, [14] Y. Tokuyama, Y. Yoshii, K. Konno, J. Sone, Curved Mesh Generation Based on Limit Subdivision and Gregory Patch Interpolation, The Journal of the Institute of Image Electronics Engineers of Japan, Vol.35, No.6, pp , [15] K. Konno, T. Takamura, and H. Chiyokura, A New Control Method for Free-Form Surfaces with Tangent Continuity and its Applications, Scientific Visualizations of Physical Phenomena, N.M. PatriKalakis ed., Springer-Verlag, Heidelberg, pp , [16] M. Halstead, M. Kass, T. DeRose, Efficient, Fair Interpolation using Catmull-Clark Surfaces, Proceedings of SIGGRAPH 1993, pp.35-44, [17] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in C, Cambridge University Press, 1992.

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