korea game society, 1~3 february 2005, muju simple and fast polygonal simplification using iamc, kim...

Korea Game Society, 1~3 February 2005, Korea Game Society, 1~3 February 2005, MujuMuju Simple and Fast Polygonal Simplification using IAMC, Kim Nam WooSimple and Fast Polygonal Simplification using IAMC, Kim Nam Woo

Simple and Fast Simple and Fast Polygonal Simplification usingPolygonal Simplification usingIntergral Absolute Mean Intergral Absolute Mean CurvatureCurvature

Graduate School of SoftwareGraduate School of SoftwareDongseo University, BusanDongseo University, Busan

Woo Jong Lim, Hyo Min Ahn and Byung Gook Woo Jong Lim, Hyo Min Ahn and Byung Gook LeeLee

Nam Woo Kim, [email protected] Woo Kim, [email protected]

Korea Game Society, 1~3 February 2005, Korea Game Society, 1~3 February 2005, MujuMuju


Polygonal simplificationPolygonal simplification

33 차원 공간에서 다각형차원 공간에서 다각형 (polygon)(polygon) 으로 이루어진 메쉬으로 이루어진 메쉬 (mesh) (mesh) 모델을 모델을 간단하게 변형시키는 것간단하게 변형시키는 것 ..

최대한 원 모델최대한 원 모델 (original model)(original model) 의 모양을 유지하면서 다각형의 수를 의 모양을 유지하면서 다각형의 수를 줄이는 것이 목적줄이는 것이 목적 ..


Simplification methodsSimplification methods

Vertex removalVertex removal

Edge collapseEdge collapse

Triangle collapseTriangle collapse

OrderOrder

V = #Vertex

E = #Edge

F = #F ace


Vertex removalVertex removal점점 (vertex)(vertex) 을 제거하여 메쉬 수를 줄이는 기법을 제거하여 메쉬 수를 줄이는 기법

..

[1] William J. Schroeder, Jonathan A. Zarge, William E. Lorensen : Decimation of [1] William J. Schroeder, Jonathan A. Zarge, William E. Lorensen : Decimation of Decimation of triangle meshes, SIGGRAPH 1992 : 65-70.Decimation of triangle meshes, SIGGRAPH 1992 : 65-70.

[2] P. A. Hinker, Charles D. Hansen : Geometric Optimization, IEEE Visualization [2] P. A. Hinker, Charles D. Hansen : Geometric Optimization, IEEE Visualization 1993 : 189-195.1993 : 189-195.

V = V ¡ 1;E = E ¡ 3;F = F ¡ 2


Edge collapseEdge collapse선선 (edge)(edge) 을 제거하여 메쉬 수를 줄이는 기법을 제거하여 메쉬 수를 줄이는 기법

Edge =Edge =

..

Preserve location, New locationPreserve location, New location

[3] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, W. Stuetzle. Mesh optimization, [3] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, W. Stuetzle. Mesh optimization, ACM ACM SIGGRAPH 1993SIGGRAPH 1993, 19-26., 19-26.

[4] H. Hoppe. Progressive meshes, [4] H. Hoppe. Progressive meshes, ACM SIGGRAPH 1996ACM SIGGRAPH 1996, 99-108. , 99-108. [5] J. Popovic, H. Hoppe Progressive simplicial complexes, [5] J. Popovic, H. Hoppe Progressive simplicial complexes, ACM SIGGRAPH 1997ACM SIGGRAPH 1997, 217-, 217-

224.224.[6] Jarek Rossignac and Paul Borrel. Multi-resolution 3D approximations for rendering [6] Jarek Rossignac and Paul Borrel. Multi-resolution 3D approximations for rendering

complex scenes, In Geometric Modeling in Computer Graphics, pp. 455-465, Springer complex scenes, In Geometric Modeling in Computer Graphics, pp. 455-465, Springer Verlag, Eds. B.Falcidieno and T.L. Kunii, Genova, Italy, June 28-July 2,1993.Verlag, Eds. B.Falcidieno and T.L. Kunii, Genova, Italy, June 28-July 2,1993.

[7] Greg Turk, Re-Tiling Polygonal Surfaces, [7] Greg Turk, Re-Tiling Polygonal Surfaces, Computer GraphicsComputer Graphics, Vol. 26, No. 2, , Vol. 26, No. 2, SIGGRAPH 92, July 1992, pp. 55-64.SIGGRAPH 92, July 1992, pp. 55-64.

fu;vg

V = V ¡ 1;E = E ¡ 3;F = F ¡ 2


Triangle collapseTriangle collapse삼각면삼각면 (triangle)(triangle) 을 제거하여 메쉬 수를 줄이는 기법을 제거하여 메쉬 수를 줄이는 기법

..

Preserve location, New locationPreserve location, New location

[8] Hamann, B. 1994, A data reduction scheme for triangulated surfaces, Computer Aided [8] Hamann, B. 1994, A data reduction scheme for triangulated surfaces, Computer Aided Geometric Design 11(2), Elsevier, pp. 197-214.Geometric Design 11(2), Elsevier, pp. 197-214.

V = V ¡ 2;E = E ¡ 6;F = F ¡ 4


Stan Melax’s Edge costStan Melax’s Edge costStan MelaxStan Melax 가 제안한 가 제안한 edge collapse edge collapse 방법방법

방향성과 방향성과 costcost 값을 이용값을 이용 , , 두 개의 점 중에서두 개의 점 중에서 CostCost 가 낮은 점으로 가 낮은 점으로 이동하여 축약하는 방법이동하여 축약하는 방법

[9] Stan Melax, A simple, Fase, and effective polygon reduction algorithm, Game [9] Stan Melax, A simple, Fase, and effective polygon reduction algorithm, Game developer november 1998.developer november 1998.

u v


Stan Melax’s Edge cost Stan Melax’s Edge cost (cont.)(cont.)

= = 점 를 포함하는 면의 집합점 를 포함하는 면의 집합

= = 를 포함하는 면의 집합를 포함하는 면의 집합

}}2)..1{(min{max),(cos

normalnnormalfvuvutTuvnTuf

Tu

Tuv efu;vg

u

Tu Tuv

u u v


Curvature in 2DCurvature in 2D

Measurement of bendingMeasurement of bending

Two criteriaTwo criteria The curvature of a straight line is zeroThe curvature of a straight line is zero The curvature of a circleThe curvature of a circle

r r

1r


Curvature in 3DCurvature in 3D

Curve of intersection of a plane containing the normal to the Curve of intersection of a plane containing the normal to the surface at Psurface at P

n

P


Geometric curvaturesGeometric curvatures

Principal curvaturePrincipal curvature

Mean curvatureMean curvature

Gaussian curvatureGaussian curvature

n

PH = km i n +km a x

2

P : kmin ;kmax

K = kminkmax


Integral absolute Integral absolute mean curvaturemean curvature

[10] Sun-Jeong Kim, Soo-Kyun Kim, and Chang-Hun Kim, Discrete Differential Error [10] Sun-Jeong Kim, Soo-Kyun Kim, and Chang-Hun Kim, Discrete Differential Error Metric for Surface Simplification, In Proceedings of Pacific Graphics 2002, pp. Metric for Surface Simplification, In Proceedings of Pacific Graphics 2002, pp. 276-283, Beijing,China, October 2002.276-283, Beijing,China, October 2002.

k~ei kj¯ i j

j ¹Hj =Z

sjHj =

14

nX

i=1

k~eikj¯i j

~ni = ~ei £ ~ei + 1k~ei £ ~ei + 1k ;~en+1 = ~e1

¯ i = \ (~ni ;~ni+1) ;~nn+1 = ~n1


Edge cost & IAMCEdge cost & IAMC

Euler formulaEuler formula

Stan Melax ‘s Stan Melax ‘s Edge costEdge cost

Stan Melax Stan Melax 의 의 Cost Cost 적용 시 계산 횟수 적용 시 계산 횟수 ::

IAMC (Integral Absolute Mean Curvature)IAMC (Integral Absolute Mean Curvature)

Integral Absolute Mean Curvature Integral Absolute Mean Curvature 적용 시 계산 횟수 적용 시 계산 횟수 : :

V < E

V ¡ E +F = 2

2£ E

V


Cost of IAMCCost of IAMC

Edge cost Edge cost 공식을 공식을 Integral Absolute Mean CurvatureIntegral Absolute Mean Curvature 에 적용하여 에 적용하여 재구성한 식재구성한 식

Cost Cost 값이 낮은 것 우선값이 낮은 것 우선

중간점 중간점 New Location New Location 방법방법

cost(e(u;v)) = ku¡ vk£©max(I AM C(u); I AM C(v))£ jI AM C(u)¡ I AM C(v)j

ª


ConclusionConclusion

Cost of Integral Absolute Mean CurvatureCost of Integral Absolute Mean Curvature


ReferenceReference[1][1] William J. Schroeder, Jonathan A. Zarge, William E. Lorensen : Decimation of triangle meshes, William J. Schroeder, Jonathan A. Zarge, William E. Lorensen : Decimation of triangle meshes,

SIGGRAPH 1992: 65-70.SIGGRAPH 1992: 65-70.

[2] P. A. Hinker, Charles D. Hansen : Geometric Optimization, IEEE Visualization 1993: 189-195.[2] P. A. Hinker, Charles D. Hansen : Geometric Optimization, IEEE Visualization 1993: 189-195.

[3] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, W. Stuetzle. Mesh optimization[3] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, W. Stuetzle. Mesh optimization,, ACM SIGGRAPH ACM SIGGRAPH 19931993, 19-26., 19-26.

[4] H. Hoppe. Progressive meshes[4] H. Hoppe. Progressive meshes,, ACM SIGGRAPH 1996ACM SIGGRAPH 1996, 99-108., 99-108.

[5] J. Popovic, H. Hoppe Progressive simplicial complexes[5] J. Popovic, H. Hoppe Progressive simplicial complexes, , AACM SIGGRAPH 1997CM SIGGRAPH 1997, 217-224., 217-224.

[6] Jarek Rossignac and Paul Borrel. Multi-resolution 3D approximations for rendering complex scenes, [6] Jarek Rossignac and Paul Borrel. Multi-resolution 3D approximations for rendering complex scenes, In Geometric Modeling in Computer Graphics, pp. 455-465, Springer Verlag, Eds. B.Falcidieno and In Geometric Modeling in Computer Graphics, pp. 455-465, Springer Verlag, Eds. B.Falcidieno and T.L. Kunii, Genova, Italy, June 28-July 2,1993.T.L. Kunii, Genova, Italy, June 28-July 2,1993.

[7] Greg Turk, Re-Tiling Polygonal Surfaces, [7] Greg Turk, Re-Tiling Polygonal Surfaces, Computer GraphicsComputer Graphics, Vol. 26, No. 2, (SIGGRAPH 92), July , Vol. 26, No. 2, (SIGGRAPH 92), July 1992, pp. 55-64.1992, pp. 55-64.

[8] Hamann, B. , A data reduction scheme for triangulated surfaces, Computer Aided Geometric Design [8] Hamann, B. , A data reduction scheme for triangulated surfaces, Computer Aided Geometric Design 11(2), Elsevier, 1994, pp. 197-214.11(2), Elsevier, 1994, pp. 197-214.

[9] Stan Melax, A simple, Fast, and effective polygon reduction algorithm, Game developer november [9] Stan Melax, A simple, Fast, and effective polygon reduction algorithm, Game developer november 1998.1998.

[10] Sun-Jeong Kim, Soo-Kyun Kim, and Chang-Hun Kim, Discrete Differential Error Metric for Surface [10] Sun-Jeong Kim, Soo-Kyun Kim, and Chang-Hun Kim, Discrete Differential Error Metric for Surface Simplification, In Proceedings of Pacific Graphics 2002, Beijing,China, October 2002, pp. 276-283.Simplification, In Proceedings of Pacific Graphics 2002, Beijing,China, October 2002, pp. 276-283.

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