1. Approximate Convex Decomposition of Polygons Reporter: Hong guang Zhou Math Dept. ZJU May 17th, 2007 Jyh-Ming Lien Nancy M. Amato Computational Geometry: Theory & Applications, 2005

2. Comparison:

3. Outline: Introduction Previous work The Frame of work Measuring concavity Implementation detail Experimental results Conclusions

4. Introduction About Author: Jyh-Ming Lien Taiwanese Ph.D. Student Parasol Lab, Dept. of Computer Science Texas A&M University neilien@cs.tamu.edu AS a postdoc in the CITRIS Tele-Immersion lab at UC Berkeley, 2006CITRIS Tele-Immersion lab As an assistant professor in the Dept. of Compute Science at the George Mason University. Jan,2007George Mason University

5. Introduction About Paper: propose a new strategy: Approximate convex decomposition (ACD) decompose a polygon, containing zero or more holes, into “ approximate convex ” pieces.

6. Introduction Features of the approach (ACD) is : Similar benefits as convex components Applies to simple polygon, with or without holes Provides a mechanism to focus on key features Produces a hierarchical representation of convex decomposition of various levels of approximation The resulting decomposition is smaller, computed more efficiently.

7. Previous work (1) Decomposition is a technique commonly used to break complex models into sub-models that are easier to handle. Convex decomposition is most important, because many algorithms perform more efficiently on convex objects, It has applications in many areas. Drawback: It can be costly to construct. It can result in an unmanageable number of components.

8. Previous work (2) Siddiqi and Kimia use curvature and region information to decomposition. Simmons and Sequin proposed decomposition using an axial shape graph,a weighted Medial Axis. Dey et al. proposed decomposition using Delaunay triangles of sampled points on the polygon boundary.

9. Preliminaries (1) Hp : the smallest convex set enclosing P. (the convex hull of a polygon P) P is convex, if P = Hp. Notches: Vertices of P which are not vertices of Hp. have internal angles greater than 180.

10. Preliminaries (2) Bridges (p): convex hull edges which connect two non-adjacent vertices of a Po. Bridges (p): aHp \ aP. Pockets (p): maximal chains of non convex hull edges Of P. Pockets (p) : aP \ aHp.

11. Preliminaries (3)

12. The Frame of ACD (1) Approximate convex decomposition (ACD) Input : A polygon P and toleranceδ Output: A decomposition of P, {Ci}, max {concave (Ci) } ≤δ 1: Let a point x ∈ aP, s.t concave (P) = concave (x) 2: if concave (P) < δ,then return P. 3: else {Ci} = Resolve (P, x). 4: Applied recursively to all new components {Ci} until max {concave (Ci) } ≤δ.

13. The Frame of ACD (2) A divide-and-conquer strategy to decompose P into a set of δ- approximate convex pieces.

14. Measuring Concavity Concave (p) = max {concave (x)},x ∈ a P. The definition is better identifiers of important features than summing concavities.

15. Measuring Concavity External Boundary Concavity: x ∈ a Po Straight Line Concavity (SL-Concavity) Shortest Path Concavity (SP-Concavity) Hybrid Concavity (H-Concavity) Hole Boundary Concavity: x ∈ a Pi, (i>0)

16. SL-Concavity concavity (x): the straight-line distance from x to its associated bridge β. Pro: Simple, Efficient Con: Does not reflect intuitive definition of concavity. sometimes not accurate

17. SP-Concavity A shortest path from x to bridge line segment β,lies entirely in the area Pρ(enclosed by bridge and pocket). Pro: accurately, apply to many cases Con: slowly, do not intuitional understand

18. SP-Concavity Split Pρinto polygon A,B,C. Construct two visibility trees T1,T2,rooted at β-, β+, respectively, to all vertices in A,C. Compute Π(v, β-,β+), from T1,T2 in A,C. Compute an ordered set V β +,in B from T1,T2. For each pair (V i, V j ) ∈ V β + do For i<k<j do Π(V k, β) = min((Π(V k, V i ) + Π(V i, β) ), ((Π(V k, V j ) + Π(V j, β) ).

19. SP-Concavity

20. H-Concavity Has the advantages of both methods (SL, SP). Measure the SL-concavity If concave (p) > δ,then Done Else if n i * n β <0, (bridge) Measure the SP-concavity if concave (p) > δ,then Done else return

21. H-Concavity

22. Hole Boundary Concavity The maximum dist (p, cw (p)): the diameter of P i. The pair (p, cw (p)): the antipodal pair of the hole P i. Represent an important feature: p (cw (p)) will have maximum concavity on aP i, after cw (P) (p) is resolved.

23. Hole Boundary Concavity The medial axis (MA) of Hole P i forms a tree. Approximate p, cw (p) as the two points at maximum Distance in the tree. The Principal Axis (PA) of P i : a line l, ∑dist (x, l) < ∑ dist (x, k), k ≠l, x: all vertices of P i. Approximate p, cw (p) as the two points of P i, in two Extreme directions on PA.

24. Hole Boundary Concavity Compute the antipodal pair, p i and cw (p i ) for each hole P i, 1≤i ≤k. Hole P i is resolved :a diagonal is added between p i and aPo. Concave (P i ) = concave (Po)+ concave (x)+ dist (p i, aPo)+ dist (pi, cw(p i )) Concave (p) = concave (Po)+ concave (x)+ dist (p i, aPo) Concave (x) = concave (p)+ dist (p, x)

25. Resolving Concave Features A polygon P, a notch r of P If r ∈ aPo, then Find x to Decompose P, a diagonal rx. Else (r ∈ aP i, i>0) Resolved P i, (add a diagonal rx, where x is the closest vertex to r.) Merge into a new polygon NP. calculate concave (x) again (x ∈ NP).

26. Choose a diagonal Choose a x in polygon P, r is a notch. rx is a diagonal in polygon P. x ∈ aPo, α, βare user defined scalars, α= 0.1, β= 1. Pick x,when f is highest.

27. Results:

31. Conclusion: A new approach to decompose Extend to 3D As alternative Shape descriptors

32. Thank you Questions ?