New Computation of Normal Vector and Curvature - WSEAS · New Computation of Normal Vector and...

New Computation of Normal Vector and Curvature

Hyoung-Seok Kim Department of Multimedia Engineering, Dong-Eui University

995 Eomgwangro, Busanjingu, Busan, 614-714, KOREA [email protected]

Ho-Sook Kim

Division of Advanced Computer Information, Dong-Eui Institute of Technology 152 Yangji 5 ro, Busanjingu, Busan, 614-715, KOREA

[email protected]

Abstract: - The local geometric properties such as curvatures and normal vectors play important roles in analyzing the local shape of objects. The result of the geometric operations such as mesh simplification and mesh smoothing is dependent on how to compute the normal vectors and the curvatures of vertices, because there are no exact definitions of the normal vector and the discrete curvature in meshes. Therefore, the discrete curvature and normal vector estimation play the fundamental roles in the fields of computer graphics and computer vision. In this paper, we propose new methods for computing normal vector and curvature well, which are more intuitive than the previous methods. Our normal vector computation algorithm is able to compute the normal vectors more accurately and is available to meshes of arbitrary topology. It is due to the properties of local conformal mapping and the mean value coordinates. Secondly, we point out the fatal error of the previous discrete curvature estimations, and then propose a new discrete sectional-curvature estimation to be able to overcome the error. The method is based on the parabola interpolation and the geometric properties of Bezier curve. It is confirmed by experiment that the normal vector and the curvature generated by our algorithm are more accurate than that of the previous methods. Key-Words: - Normal vector, curvature, local geometric property, mesh segmentation 1 Introduction The complicated objects dealt in the field of computer graphics may be obtained by acquiring 3D position information by using 3D scanners, or generated by operating 3D modeling tools such as 3D Studio MAX or Maya. In order to efficiently deal with the objects, they may be represented by a form of mesh which includes the position information of vertices and the connectivity of them. According to the demand of users, the geometric operations such as mesh simplification and smoothing are applied to the objects[1,2,3]. In order to accomplish such geometric operations efficiently, we have to find out rightly the local properties of the mesh. In general, normal vector and curvature play the important roles as the local properties in the field of differential geometry such as curve and surface [19,20]. So the users want to use the same properties in the discrete space such as polygon and mesh as they were. However, there is not the exact formula of such properties because mesh is a discrete type. Therefore, we need a more exact measure of the local properties. Especially, the normal vector of vertices plays an important role in the rendering process which makes much account of reality.

2 Previous Works 2.1 Normal Vector There are two approaches in the computation of normal vectors [4,5]. The first approach is to approximate the surface interpolating a vertex and compute the normal vector of the surface at the vertex. The result of this approach is tightly dependent on the configuration of the 1-ring neighborhood vertices and the degree of approximated surface. The second approach is to combine the normal vectors of faces adjacent to a vertex. Gouroud proposed the first algorithm of this approach that uses the average ( GN ) of the normal vectors of adjacent faces to the vertex in a triangular mesh [6]. This method is simple so that it is fast. However, it never uses the geometric information of the neighborhood of the vertex so that we can not apply a variety of geometric operations directly. Moreover, this method may bring the different normal vectors according to the change of topology as shown in Figure 1.

WSEAS TRANSACTIONS on COMPUTERS Hyoung-Seok Kim, Ho-Sook Kim

ISSN: 1109-2750 1661 Issue 10, Volume 8, October 2009

Fig.1 The change of normal vector

according to the topology Thurmer et al. proposed a more improved method that is based on the angle-weight ( TN ) in order to resolve the topology-dependent problem of Gouroud’ method [7]. This method uses only the interior angle without the information such as the area of adjacent faces and the length of incident edges. It is not considered that this method prefectly reflect the geometric properties.

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Taubin [8] and Max [9] proposed area-weight normal vector ( MN ) algorithms in 1995 and 1999. The normal vector of a vertex by these methods is generated when we approximate a surface interpolating the vertex and represent the surface with Taylor series. The area-weight is induced when the normal vector of the surface is converted to a mesh. It is not considered that they derived a more exact normal vector because the parameter used in their Taylor’s series is not geodesic so that the error of approximation is large. In order to improve the area-weighted normal vector computation, Chen [10]

proposed a new area-weighted normal vector ( CN ) computation the weight of which is the value of the area of an adjacent face divided by the length of the line segment between the given vertex and the center of the face. The normal vertex of a vertex v is defined by

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of mass of if , respectively. This normal vector is affected by the non-adjacent vertices as well as the two adjacent vertices so that this method may cause inaccurate normal vectors. Figure 2 shows an example that two faces have the same local geometric property but the different weights.

Fig. 2: Two faces with the same local

geometric property and the different weights

In this paper, we propose a new normal vector computation method which considers the interior angle and the length of adjacent edges simultaneously. Our method applies conformal mapping to the 1-ring neighborhood vertices of a given vertex, and then computes the mean value coordinates of the vertex with respect to the middle points between the mapped adjacent vertices and the Origin. Finally, the normal vector of the vertex is represented by linear combination of the edge normal vectors with the mean value coordinates. The edge normal vector is the average of the normal vectors of two adjacent faces. It is well-known that the conformal mapping reflects the local geometric properties well and the mean value coordinate is continuous over the entire domain. Our experimental results show that our method can compute the more exact normal vectors than the previous algorithms. Moreover, we may resolve the problem of phong shading that the intensity at a point is dependent on how to choose a coordinate system of world

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The normal vector ( CMN ) of our method may be regarded as that it reflects the local geometric property better than the previous methods because it is generated by applying the conformal mapping and the mean value coordinates. In general, the geometric property of a vertex in a 3D mesh has a tendency to resemble the property of the surface interpolating the 1-ring neighborhood and the vertex. The edges of the 1-ring neighborhood are approximated to the surface better than the faces because of the interpolating condition. So our method is very intuitive. 4 Curvature Computation First of all, we review the circle-based discrete curvature estimation adopted in the previous methods of estimating the discrete curvature of meshes. This method utilizes the Taylor series of a curve and adopts the distance between two vertices as parameter of the curve. It makes the trajectory of points with the same curvature be a circle. So, in this paper we call this method as C-discrete curvature. 4.1 C-Discrete Curvature Formula Most of the previous discrete curvature estimation directly compute the sectional discrete curvature whether they use the tensor of curvature or Laplacian Operator [13,14,17]. All of them compute the curvature by using the Taylor series of a curve interpolating two vertices. Let p and pi be a given vertex and its adjacent vertex, respectively and let g(s) be a continuous curve passing through the two vertices: g(0) = p, ipsg =)( . Then the curve may be represented by its Taylor series:

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N · g s p s N · T s N · N)

Because N ⋅T = 0 and N ⋅N = 1, the above equation may be changed to the following simple equation:

κ p2 N · g s p

s The previous methods define the C-discrete curvature κC (p) by assuming that the parameter of the series is the distance between two vertices:

κC p2 N · g s p

|g s p|

Now, we point out the fatal error of the c-Discrete method. First of all, we find out the set of points with the same discrete curvature α. For the convenience of computation, we assume that p=(0,0), pi = (x, y) and N = (0, -1). Then the formula of the C-discrete curvature of p is as follows:

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|g s p|2y

x yα

Figure 7. The set of points with curvature α

Then, x2 + ( y + 1/α)2 = 1/α2The trajectory is the circle of center (0, -1/α) and radius 1/α as shown in Figure 7. That is, whenever the adjacent vertex pi is on the radius of the circle though it makes the different interior angle with the given vertex p, the sectional C-discrete curvature of p to the direction ppi is the same as that of the circle. In general, the interior angle of a triangle is sharper than those of square and octagon. So, we may consider that the curvature of a vertex in triangles is greater than that of other regular polygons (see Figure 3). The result of the C-discrete curvature estimation breaks the general concepts. Moreover, there is another drawback of this method that the range of the value is restricted. If ||ppi|| >1,



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of the last polygon has a sharper angle than the others. In order to solve this imperfection, we propose the symmetric parabola-based discrete curvature estimation using a symmetric parabola. 4.1 SP-Discrete Curvature Formula Let N be the unit vector bisecting the interior angle of B: N = ( BA/||BA|| + BC/||BC|| ) / || BA/||BA|| + BC/||BC|| ||. Then, we can consider two right-angled triangles AA'B and CC'B as shown in Figure 13. One has the width of length vA and the height of hA , the other has the width of length vC and the height of hC:

Figure 13. Polygon and its Parabola

hA = N ⋅ BA, vA = || BA - (N ⋅BA) N ||, hC = N ⋅ BC, vC = || BC - (N ⋅BC) N ||. Then, we can consider the right-angled triangle MM'B with the width vm = (vA+ vC)/2 and the height hm = (hA + hC)/2. Therefore, we define the symmetric P-type discrete curvature of the vertex B as κSP(B)≡ (2 hm)/vm

2.

5 Experimental Results 5.1 Normal Vector Figure 14 shows a variety of polyhedrons which are approximations of a unit sphere. The upper polyhedrons are generated by recursively subdividing a regular tetrahedron, and the lower ones are generated by using the parameters of longitude and latitude. The upper cases are more irregular than the lower cases because the triangles of a different size may be appear, so that the error of the normal vector of a vertex in recursively generated polyhedrons may be larger than the others. Table 2 and 3 show the mean error of normal vectors of a vertex in the approximated polyhedral of the unit sphere. In order to measure the exactness of the normal vectors, we used the following error energy

><−= NNvvE ii ,1)( ,

where iNv is the unit normal vector of a vertex iV of the approximated sphere and N is the normal vector of the unit sphere.

Fig. 14: Approximated Spehres

Table 2. Mean error for mesh generated by recursive method No. Of vertices

10 34 130 514 2050

]2[GN 0.00E-08 7.25E-03 1.48E-03 3.41E-04 7.00E-05

]3[TN 0.00E-08 3.57E-03 6.98E-04 1.35E-04 2.50E-05]4[MN 0.00E-08 1.42E-02 2.89E-03 6.47E-04 1.31E-04]6[CN 0.00E-08 7.42E-04 3.57E-04 9.30E-05 2.00E-05

CMN 0.00E-08 5.22E-04 1.41E-04 3.10E-05 6.00E-06

Table 3. Mean error for mesh generated by parametric method

No. of Vertices

23 50 122 262 578

]2[GN 4.03E-03 7.86E-04 2.22E-04 5.20E-05 2.40E-05

]3[TN 1.41E-03 5.54E-04 1.11E-04 6.00E-05 6.00E-06]4[MN 8.87E-03 1.96E-03 5.88E-04 6.30E-04 6.30E-05

]6[CN 1.77E-03 1.11E-03 1.11E-04 5.00E-05 5.00E-06

CMN 7.89E-04 4.63E-04 5.70E-05 1.60E-05 2.00E-06

According to the result of experiment, the normal vector computation proposed by Chen [10] is better than the other previous methods. However, the exactness of our method is two times than that of Chen. So we argue that our method is the best of all. Figure 15 shows the results of rendering generated by Phong shading which are based on the GN method, CN method, and CMN method. The positions of camera and light are the same so that the highlighted region should symmetrically appear near the vertex positioned in the center of the rendering. Figure 14(a) is the wire-frame of a cube. It has irregular topology. The vertex positioned in



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All of them use the circle-based discrete curvature to compute the sectional curvature. In this paper, we point out the fatal error of C-type discrete curvature computation, and proposed the parabola-based discrete curvature computation to resolve the problem. Our method may be the basis of normal vector estimation and segmentation of meshes. References: [1] Dennis Maier, Jürgen Hesser, Reinhard Männer, Fast

and Accurate Closest Point Search on Triangulated Surfaces and its Application to Head Motion Estimation, Wseas Transactions on Computers, Issue 4, Vol. 2, pp 874-878, 2003

[2] Jifang Li and Renfang Wang, Robust Denoising of Point-Sampled Surfaces Wseas Transactions on Computers, Issue 1, Vol. 8, pp 153-162, 2009

[3] Ravi S. Gautam and Sachin B. Patkar, Fitting convex surface to scattered data points with applications to medical imaging, Wseas Transactions on Computers, Issue 1, Vol. 3, pp 50-56, 2004

[4] Hyoungseok Kim and Hosook Kim, Computing Vertex Normals from Aribitrary Meshes, Proceeding of the 9th WSEAS Int. Conference on Signal Processing, Computational Geometry & Artificial Vision(ISCGAV’09), pp 68-72, 2009

[5] Shuangshuang Jin, Robert R. Lewis, and David West, A Comparison of Algorithms for Vertex Normal Computation, The Visual Computer, Vol. 21, No. 1-2, pp 71-82, 2005

[6] Henri Gouraud, Continuous Shading of Curved Surfaces, IEEE Transactions of Computers, Vol. 20, No. 6, pp 623-629, 1971

[7] Grit Thurmer and Charles A. Wuthrich, Computing vertex normals from polygonal facets, Journal of Graphics Tools, 3(1), pp 43-46, 1998

[8].G. Taubin, Estimating the tensor of curvature of a surface from a polyhedral approximation, Proceeding of the Fifth International Conference on Computer Vision, pp 902-907, 1995

[9] N Max, Weights for computing vertex normals from facet normals, Journal of Graphics Tools, 4(2), pp 1-6, 1999

[10] Sheng-Gwo Chen and Jyh-Yang Wu, Estimating normal vectors and curvatures by centroid weights, Computer Aided Geometric Design 21(5), p 447-458, 2004

[11] Michael S. Floater, Mean value coordinates, Comp. Aided Geom. Design Vol. 20, p 19-27, 2003.

[12] Sylvain Petitjean, A Survey of Methods for Recovering Quadrics in Triangle Meshes, ACM Computing Surveys, Vol. 34, No. 2, pp 211—262 (2002).

[13] Mark Meyer, Mathieu Desbrun, Peter Schroder, and Alan H. Barr, Discrete Differential Geometry Operators for Triangulated 2-Manifolds, Proc of VisMath, (2002)

[14] J. Goldfeather and V. Interrante, A novel cubic-order algorithm for approximating principal direction vectors, ACM Transaction on Graphics, Vol . 23, No. 1, pp 45—63 (2004)

[15] Holger Theisel, Christian Rossl, Rhaleb Zayer, and Hans-Peter Seidel, Normal Based Estimation of the Curvature Tensor for Triangular Meshes, Proc. of Pacific Graphics'04, pp 288—297 (2004)

[16] Tatiana Surazhsky, Evgeny Magid, Octavian Soldea, Gershon Elber, and Ehud Rivlin, A Comparison of Gaussian and Mean Curvatures Estimation Methods on Triangular Meshes, Proc. of 2003 IEEE International Conference on Robotics and Automation, (2003)

[17] Gabriel Taubin, Estimating the Tensor of Curvature of a Surface from Polyhedral Approximation, Proc. of the Fifth International Conference on Computer Vision, (1995)

[18] Kouki Watanabe and Alexander G. Belyaev, Detection of Salient Curvatures Features on Polygonal Surfaces, Computer Graphics Forum, Vol. 20, No. 3, (2001)

[19] David Cohen-Steiner and Jean-Marie Morvan, Restricted Delaunay Triangulations and Normal Cycle, ACM Symposium of Computational Geometry, (2003)

[20] M. Garland and P. S. Heckbert, Surface simplification using quadratic error metrics, SIGGRAPH ‘97, pp 209—216 (1997)



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