1. Chart Packing Heuristic
Xun Shi

2. Problem Statement
Finding the least area to pack all the texture charts for a 3D models.
Optimal packing
Extract texture

3. Outline Bin packing and its heuristics Algs Chart packing Algs
Non-overlapping bounding rectangle Heuristic
Bruno’s Heuristic
APTAS Heuristic

4. Bin packing problem
Objects of different volumes must be packed into a finite number of bins with capacity V in a way that minimizes the number of bins used.
NP-hard;
Several Heuristic Algs;
Applications: chart packing, linear packing, packing by weight, etc.

5. Heuristic Algs First Fit Decreasing (FFD) Best Fit Decreasing
first sort the items by volume, then insert into the first bin with sufficient space;
With sorting: 11/9 OPT + 1 bins;
Without sorting: 17/10 OPT + 2 bins;
A More efficient Alg: 71/60 OPT + 1 bins;
Best Fit Decreasing

6. Chart Packing Problem
given a set of polygons, find a non-overlapping placement of the polygons that the enclosing rectangle is of minimum area.
Tangram

7. Chart Packing Problem
Ancient China

8. Chart Packing Problem
given a set of polygons, find a non-overlapping placement of the polygons that the enclosing rectangle is of minimum area.
Tangram
HOW??

9. ALG 1: Packing by non-overlapping bounding rectangle
However, due to the arbitrary shapes…

10. ALG 2: Bruno’s Heuristic
Rotate the charts: maximum diameter of the charts are oriented vertically
Sort the charts in decreasing order

11. ALG 2: Bruno’s Heuristic cont.
Insert the charts one by one

12. ALG 2: Bruno’s Heuristic cont.
Using horizontal function to approximate chart position;
Rely on experience
Not proved !!

13. ALG 2: Bruno’s Heuristic cont.
Save space
Waste space

14. APTAS for charts packing
APTAS: asymptotic polynomial time approximation scheme
de la Vega and Leuker, 1980
For every fixed ε> 0 algorithm outputs a solution of size (1+ε)OPT + 1 in time polynomial in n.

15. APTAS for charts packing cont.
Instance: Multiset of real numbers
Solution: A partition of I into subsets
Optimal Solution: A solution with a minimum number of subsets k.

16. APTAS for charts packing cont.
Two constructive reductions
Reduce BP into Large Set BP[ ], with the threshold
Also have Small Set
Reduce Large Set BP[ ] into BP[ , m], all are chosen from a set of at most m different values

17. APTAS for charts packing cont.
Large Set: BP[ , m] can be solved by dynamic programming in polynomial of n.
k ≤(1+ ) OPT
Small Set: Greedy insert

18. APTAS for charts packing cont.
Running time: dominated for packing Large Set
k≤(1+2 ) OPT + 1 for sufficiently small ( ≤½)

19. Reference
B. Lévy, S. Petitjean, N. Ray, J. MaillotLeast, “Squares Conformal Maps for Automatic Texture Atlas Generation”, ACM Press, 2002
P. Sander, J. Snyder, S. Gortler, H. Hoppe, “Texture mapping progressive meshes” In SIGGRAPH 01 Conf. Proc ACM Press, 2001.
U. Zwick, E. Reshef, “Bin Packing”, course note, 1997/8
del la Vega, G. S. Lueker, “Bin packing can be solved within 1+ε in linear time”, Combinatiorica, 1(4): , 1981

20. THANK YOU VERY MUCH!!