1. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Simple and Fast Polygonal Simplification using Intergral Absolute Mean Curvature Graduate School of Software Dongseo University, Busan Woo Jong Lim, Hyo Min Ahn and Byung Gook Lee Nam Woo Kim, d5302010@dongseo.ac.kr Korea Game Society, 1~3 February 2005, Muju

2. Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Polygonal simplification 3 차원 공간에서 다각형 (polygon) 으로 이루어진 메쉬 (mesh) 모델을 간단하게 변형시키는 것. 최대한 원 모델 (original model) 의 모양을 유지하면서 다각형의 수를 줄이는 것이 목적.

3. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Simplification methods Vertex removal Edge collapse Triangle collapse Order V = # V er t ex E = # E d ge F = # F ace

4. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Vertex removal 점 (vertex) 을 제거하여 메쉬 수를 줄이는 기법. [1] William J. Schroeder, Jonathan A. Zarge, William E. Lorensen : Decimation of Decimation of triangle meshes, SIGGRAPH 1992 : 65-70. [2] P. A. Hinker, Charles D. Hansen : Geometric Optimization, IEEE Visualization 1993 : 189-195. V = V ¡ 1 ; E = E ¡ 3 ; F = F ¡ 2

5. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Edge collapse 선 (edge) 을 제거하여 메쉬 수를 줄이는 기법 Edge =. Preserve location, New location [3] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, W. Stuetzle. Mesh optimization, ACM SIGGRAPH 1993, 19-26. [4] H. Hoppe. Progressive meshes, ACM SIGGRAPH 1996, 99-108. [5] J. Popovic, H. Hoppe Progressive simplicial complexes, ACM SIGGRAPH 1997, 217-224. [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 T.L. Kunii, Genova, Italy, June 28-July 2,1993. [7] Greg Turk, Re-Tiling Polygonal Surfaces, Computer Graphics, Vol. 26, No. 2, SIGGRAPH 92, July 1992, pp. 55-64. f u ; v g V = V ¡ 1 ; E = E ¡ 3 ; F = F ¡ 2

6. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Triangle collapse 삼각면 (triangle) 을 제거하여 메쉬 수를 줄이는 기법. Preserve location, New location [8] Hamann, B. 1994, A data reduction scheme for triangulated surfaces, Computer Aided Geometric Design 11(2), Elsevier, pp. 197-214. V = V ¡ 2 ; E = E ¡ 6 ; F = F ¡ 4

7. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Stan Melax’s Edge cost Stan Melax 가 제안한 edge collapse 방법 방향성과 cost 값을 이용, 두 개의 점 중에서 Cost 가 낮은 점으로 이동하여 축 약하는 방법 [9] Stan Melax, A simple, Fase, and effective polygon reduction algorithm, Game developer november 1998. u v

8. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Stan Melax’s Edge cost (cont.) = 점 를 포함하는 면의 집합 = 점 를 포함하는 면의 집합 = 를 포함하는 면의 집합 = 를 포함하는 면의 집합 T u T uv e f u ; v g u T u T uv u uv

9. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Curvature in 2D Measurement of bending Two criteria The curvature of a straight line is zero The curvature of a straight line is zero The curvature of a circle The curvature of a circle r r 1 r

10. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Curvature in 3D Curve of intersection of a plane containing the normal to the surface at P n P

11. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Geometric curvatures Principal curvature Mean curvature Gaussian curvature n P H = k m i n + k max 2 P : k m i n ; k max K = k m i n k max

12. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Integral absolute mean curvature [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. 276-283, Beijing,China, October 2002. k ~ e i k j ¯ i j j ¹ H j = Z s j H j = 1 4 n X i = 1 k ~ e i kj ¯ i j ~ n i = ~ e i £ ~ e i+ 1 k ~ e i £ ~ e i+ 1 k ; ~ e n + 1 = ~ e 1 ¯ i = \ ( ~ n i ; ~ n i + 1 ) ; ~ n n + 1 = ~ n 1

13. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Edge cost & IAMC Euler formula Stan Melax ‘s Edge cost Stan Melax 의 Cost 적용 시 계산 횟수 : Stan Melax 의 Cost 적용 시 계산 횟수 : IAMC (Integral Absolute Mean Curvature) Integral Absolute Mean Curvature 적용 시 계산 횟수 : Integral Absolute Mean Curvature 적용 시 계산 횟수 : V < E V ¡ E + F = 22 £ E V

14. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Cost of IAMC Edge cost 공식을 Integral Absolute Mean Curvature 에 적용하여 재구성한 식 Cost 값이 낮은 것 우선 중간점 New Location 방법 cos t ( e ( u ; v )) = k u ¡ v k £ © max ( IAMC ( u ) ; IAMC ( v )) £ j IAMC ( u ) ¡ IAMC ( v ) j ª

15. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Conclusion Cost of Integral Absolute Mean Curvature Cost of Integral Absolute Mean Curvature

16. Korea Game Society, 1~3 February 2005, Muju Simple and Fast Polygonal Simplification using IAMC, Kim Nam Woo Reference [1]William J. Schroeder, Jonathan A. Zarge, William E. Lorensen : Decimation of triangle meshes, SIGGRAPH 1992: 65-70. [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, ACM SIGGRAPH 1993, 19-26. [4] H. Hoppe. Progressive meshes, ACM SIGGRAPH 1996, 99-108. [5] J. Popovic, H. Hoppe Progressive simplicial complexes, ACM SIGGRAPH 1997, 217-224. [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 T.L. Kunii, Genova, Italy, June 28-July 2,1993. [7] Greg Turk, Re-Tiling Polygonal Surfaces, Computer Graphics, Vol. 26, No. 2, (SIGGRAPH 92), July 1992, pp. 55- 64. [8] Hamann, B., A data reduction scheme for triangulated surfaces, Computer Aided Geometric Design 11(2), Elsevier, 1994, pp. 197-214. [9] Stan Melax, A simple, Fast, and effective polygon reduction algorithm, Game developer november 1998. [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.