SA2014.SIGGRAPH.ORG SPONSORED BY

SA2014.SIGGRAPH.ORG SPONSORED BY Approximate Pyramidal Shape Decomposition Ruizhen Hu Honghua Li Hao Zhang Daniel Cohen-Or Simon Fraser UniversityZhejiang UniversityTel Aviv University

SA2014.SIGGRAPH.ORG SPONSORED BY Pyramidal shape 3 Terrain 2.5D Heightfield

SA2014.SIGGRAPH.ORG SPONSORED BY Pyramidal shape 4

SA2014.SIGGRAPH.ORG SPONSORED BY Various applications 5

SA2014.SIGGRAPH.ORG SPONSORED BY Various applications 6 3D layered printing

SA2014.SIGGRAPH.ORG SPONSORED BY Various applications 7 3D layered printing Molding/casting

SA2014.SIGGRAPH.ORG SPONSORED BY Various applications 8 3D layered printing Molding/casting Point inside/outside testing

SA2014.SIGGRAPH.ORG SPONSORED BY Various applications 9 3D layered printing Molding/casting Point inside/outside testing

SA2014.SIGGRAPH.ORG SPONSORED BY However, 10 Many common objects are not pyramidal… Pyramidal decomposition

SA2014.SIGGRAPH.ORG SPONSORED BY Pyramidal decomposition Exact pyramidal decomposition ◦Minimum number of pyramidal parts 11 NP-hard NP-hard [Fekete and Mitchell 2001]

SA2014.SIGGRAPH.ORG SPONSORED BY Pyramidal decomposition Approximate pyramidal decomposition ◦Parts being only approximately pyramidal ◦Still aiming to produce as few parts as possible 12

SA2014.SIGGRAPH.ORG SPONSORED BY Key idea 13 Exact Cover Problem {1234} …

SA2014.SIGGRAPH.ORG SPONSORED BY Key idea 14 {1234} … Exact Cover Problem

SA2014.SIGGRAPH.ORG SPONSORED BY Key idea 15 {1234} … Over-segmentation Candidate parts by merging segments Exact Cover Problem

SA2014.SIGGRAPH.ORG SPONSORED BY Key idea 16 {1234} … Pyramidal decomposition Pyramidal Final decomposition Over-segmentation Candidate parts by merging segments Exact Cover Problem

SA2014.SIGGRAPH.ORG SPONSORED BY Find an appropriate over-segmentation of the shape Build an appropriate set of candidate pyramidal parts Main challenge 17 Infinite possible pyramidal parts

SA2014.SIGGRAPH.ORG SPONSORED BY Intuition Large pyramidal parts ◦Minimum-sized decompositions ◦Easier to generate 18

SA2014.SIGGRAPH.ORG SPONSORED BY Intuition Large pyramidal parts May not be enough

SA2014.SIGGRAPH.ORG SPONSORED BY Intuition Large pyramidal parts May not be enough Combination of intersections

SA2014.SIGGRAPH.ORG SPONSORED BY Intuition Large pyramidal parts May not be enough Combination of intersections …

SA2014.SIGGRAPH.ORG SPONSORED BY Intuition Large pyramidal parts May not be enough Combination of intersections …

SA2014.SIGGRAPH.ORG SPONSORED BY Algorithm overview 23 Set Collection of subsets Exact cover Intersections

SA2014.SIGGRAPH.ORG SPONSORED BY Cell generation Base sampling ◦Sample a set of directions ◦Sample a set of bases along each direction 24

SA2014.SIGGRAPH.ORG SPONSORED BY Cell generation Base sampling Base selection ◦Capacity measure 25 Total pyramidal areas Number of decomposed parts

SA2014.SIGGRAPH.ORG SPONSORED BY Cell generation Base sampling Base selection ◦Capacity measure 26

SA2014.SIGGRAPH.ORG SPONSORED BY Cell generation Base sampling Base selection ◦Capability measure ◦Base voting 27

SA2014.SIGGRAPH.ORG SPONSORED BY Cell generation Base sampling Base selection Cell formation ◦Points voted for the same set of bases 28 Estimation of intersections

SA2014.SIGGRAPH.ORG SPONSORED BY Block construction To further reduce the number of candidate parts ◦Cluster cells into blocks ◦Find the boundaries 29

SA2014.SIGGRAPH.ORG SPONSORED BY Candidate parts & decomposition Parts are formed by merging blocks Decomposition can be found by solving ECP 30 Blocks Decompositions Candidate parts

SA2014.SIGGRAPH.ORG SPONSORED BY Results Datasets ◦149 2D shapes ◦80 3D shapes 31

SA2014.SIGGRAPH.ORG SPONSORED BY Results 2D shapes 32

SA2014.SIGGRAPH.ORG SPONSORED BY Results 3D shapes 33

SA2014.SIGGRAPH.ORG SPONSORED BY Evaluations 2-pyramidality ◦25 2D shapes & 30 3D shapes 34

SA2014.SIGGRAPH.ORG SPONSORED BY Evaluations Comparison to greedy approach ◦2-pyramidality (2D shapes) 35 Ours: Greedy:

SA2014.SIGGRAPH.ORG SPONSORED BY Evaluations Comparison to human judgement ◦30 users & 20 2D shapes 36 OursUser decomposition

SA2014.SIGGRAPH.ORG SPONSORED BY 3D Printing 37

SA2014.SIGGRAPH.ORG SPONSORED BY Conclusion A relatively new and unexplored shape property. The first construction algorithm for approximate pyramidal decomposition. The key insight is to turn the problem into a exact cover problem. The bottom-up clustering-based approach is both general and versatile. 38

SA2014.SIGGRAPH.ORG SPONSORED BY Limitation & further work Realistic waste estimation for 3D printing Pyramidalization ◦Minimal modification to maximally reduce the size of its pyramidal decomposition 39

SA2014.SIGGRAPH.ORG SPONSORED BY Set Collection of subsets Exact cover Intersections Thank you! Approximate Pyramidal Shape Decomposition Ruizhen Hu Honghua Li Hao Zhang Daniel Cohen-Or