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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
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.
∑=n
iG Nfn
N1
1,
∑
∑=
n
i
n
i
T
Nf
NfN
1
1 ,
∑
∑=
n
ii
n
ii
M
Nff
NffN
1
1
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
∑
∑=
n
ii
n
ii
C
Nf
NfN
1
1
ω
ω, vg
f
i
ii −=ω
where if and ig are the area of face if and the center
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
N
N
WSEAS TRANSACTIONS on COMPUTERS Hyoung-Seok Kim, Ho-Sook Kim
ISSN: 1109-2750 1662 Issue 10, Volume 8, October 2009
spacthe w 2.2 The imposegmof efno cmethcurvneigGoldsche1-rindepehas apprand meththat methcurvtheoof cWatprincal proper
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uadratic that bx locally. se of a cubicount vertex hese curvatuIf the one-rina oscillated not resemble
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WSEAS TRANSACTIONS on COMPUTERS Hyoung-Seok Kim, Ho-Sook Kim
ISSN: 1109-2750 1663 Issue 10, Volume 8, October 2009
Floasatis
Figuvaluvalu7-gofigurconcfigurthe bcoorwhetnot tthe ris vdiscoimpointer
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ure 5 shows ue coordinateue coordinateon, which isres are the rcave polygonres are genebarycentric crdinate systether or not a the polygon iresult generavery confusontinuous. Tortant role inrpolation.
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the coefficienove equation
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e of excellencolor meansct to the veraxis. The lefa convex polly. The uppe mean valueespectively. s the continude or outsideconcave. On
arycentric coe value of ty of the coeff application
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ction, we innormal vect
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ntroduce outor of a verte assume that
mV,,L . The
1+Δ= iiVVV , ctor of a facl vector of th
by iNe . In ect to the adj
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,
∑=
= m
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: (a) 3D Mesh
ector based oconformal m
,
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∑
=
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ur algorithmtex based ont a vertex Ven the vertex
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to the 1wn in the Figun of the 1t of the confoices V and V
i , respecti
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h (b) 2D polyg
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∑=
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value y the
WSEAS TRANSACTIONS on COMPUTERS Hyoung-Seok Kim, Ho-Sook Kim
ISSN: 1109-2750 1664 Issue 10, Volume 8, October 2009
PU k
kk
k −
+=
− )2
tan()2
tan( 1 αα
ω .
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:
)(!2
)0(!1
)0()0()( 32 sOsgsggsg +′′
+′
+=
)()(2
)0()( 32 sOspNsTgsg +⋅⋅+⋅+= κ
where T and N are the unit tangent vector of the unit normal vector, respectively, and κ(p) is the sectional curvature of g(s) at point p to the direction ppi. By applying the inner product to both sides with N, we may get the following equation:
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:
κC p2 N · g s p
|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,
WSEAS TRANSACTIONS on COMPUTERS Hyoung-Seok Kim, Ho-Sook Kim
ISSN: 1109-2750 1665 Issue 10, Volume 8, October 2009
thenwe hshapanglFigu
Figuof thpolyBy Cvaludiffepoinsquapoininterhas awe mThercurv
4.2. In thestimformestimof povertebetw
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crete Curva
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WSEAS TRANSACTIONS on COMPUTERS Hyoung-Seok Kim, Ho-Sook Kim
ISSN: 1109-2750 1666 Issue 10, Volume 8, October 2009
Hencurv
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vature bling three
nd the screte lygon vertex
WSEAS TRANSACTIONS on COMPUTERS Hyoung-Seok Kim, Ho-Sook Kim
ISSN: 1109-2750 1667 Issue 10, Volume 8, October 2009
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
WSEAS TRANSACTIONS on COMPUTERS Hyoung-Seok Kim, Ho-Sook Kim
ISSN: 1109-2750 1668 Issue 10, Volume 8, October 2009
the cwherrectashouthe rFigurealihighthe o
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Horizontal Distance
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WSEAS TRANSACTIONS on COMPUTERS Hyoung-Seok Kim, Ho-Sook Kim
ISSN: 1109-2750 1669 Issue 10, Volume 8, October 2009
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
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WSEAS TRANSACTIONS on COMPUTERS Hyoung-Seok Kim, Ho-Sook Kim
ISSN: 1109-2750 1670 Issue 10, Volume 8, October 2009