Geometric Compression Through Topological Surgery

1. Geometric Compression Through Topological Surgery Gabriel Taubin (IBM) Jarek Rossignac (Georgia Tech)

2. Problem Statement • Want to send mesh across network • Compress Losslessly • Want to save CPU time • Decompress on load • Want to get triangle strips • Save on vertex reads + transforms

3. Basic Idea • Use properties of planar graphs to compress • Extend to graphs 3d meshes using hacks • Lossily drop bits left & right out of vertex position since vertex pos dominates size • Use reasonably good predictor to “guess” location

4. Compressing Topology • Start with meshes having Euler Characteristic 2 (Deformable to sphere)

5. Compressing Topology • Want to cut it into a planar mesh with lots of long triangle strips

6. Compressing Topology • A MST on vertices indicates some interesting cut lines

7. Compressing Topology • Notice how the edge boundary graph has 2 edges for each MST cut line

8. Compressing Topology • Splaying our faithful Euler-2 characteristic mesh looks as follows

9. Marching edge Compressing Topology • Note that in the flattened shape a binary graph can represent edges

10. Compressing Topology • If we save these two graphs & 1 bit per marching edge graph as to how it fits to MST, we’ve got topology of Euler-2 meshes

11. Not all cuts are equal • But some will earn you a prize from l’Institut Paul Bocuse in Lyons • Others…

12. Not all cuts are equal

13. Why cuts are not equal • Vertex edge graph stored as: • Integer: runlength • Bit: leaf? • Bit: more runs starting from here? • Longer triangle strips • Amortize-out 2 starting vertices • Faster rendering

14. 1st cut: spanning tree construction • Use edge length and run MST algorithm • Disaster with a capital D

15. Actual attempt to make MST • Distance from edge midpoint to tree root

16. Modification to improve result • 2 pass algorithm where only non-branching edges are allowed on 1st pass

17. Concentric Rings Method • Choose a root, order by concentric rings

18. Connect concentric layers • with triangle that minimize branches

19. Final Mesh

20. Concentric Rings applied to bone • Generating concentric rings makes for much longer triangle runs

21. B=f(A)+B G=f(A,B,C,D)+G C=f(A,B)+c H=f(A,B,C,D,G)+H B D=f(A,B,C)+D E=f(A,B,C,D)+E C F=f(A,B,C,D,E)+F D F E G H Encoding Vertex Position • Vertex Spanning Tree offers mechanism for vertex position prediction • Build a prediction function based on samples from parents in run A • Store per-vertex fixed-point delta from given predictor

22. Vertex Position Predictor • Assume a linear predictor f(v0…vK)=b0v0+…+bkvk • Thus estimate b0…bk by minimizing the least square error over all vertices of depth n>k for given model

23. Dealing with real meshes • Jump Edges • Ugly hack to deal with non-spheres • Read paper for “juicy” details • Making sure that edges line up after quantization + prediction

24. Results • They don’t all add up • Feels like they use the red herring called • ASCII VRML • To boost their results (sometimes literally)

25. Results: (what a) Crock • About as lossless as a spam stock investment 12 bits per coord 10 bits per coord 8 bits per coord original

26. Results: Crocodile

27. Results:Architecture

28. 17332 vertices x 12+34404 triangles x 12 = 248,832 80,056 x 19.63 = 1,571,499 80,056 /0.1658 = 482,864 Results:Architecture

29. Their stated results • Parsing a file in compressed binary is 20x faster than in ascii • So is transcribing it from stone tablets • Writing uncompressed binary is 30x faster than writing in ascii • Are they using printf or something?! • Unless you’re writing aligned data, this bitwise stuff should kill your perf if you understand what you are doing, and it should be no slower than 2x-4x (ASCII is about 2-4x bigger)

30. Their stated results • Decompression reconstructs 60-90Ktris/s • Takes same time to construct scene in VRML as compressing it with their algo • If they would only compare to something not-terrible • Writing scene in compressed form is 10x faster than in uncompressed. • If you’re writing to disk or network, sure

31. The cool result • Optimality analysis of their algorithm • Number of triangulations of simply connected polygon of n+2 vertices • If enumerated and an index into triangulation is used, encoding requires log2(ceil(Cn)) bits. As n->inf expr-> 2

32. The cool result

33. The cool result • Optimal fixed-length encoding 2bits per vertex • On some examples they do better • They have some evidence of better performance than this fixed length scheme • on highly tesselated models • Like a bunny subdivided 2 times to have 38,000 polys (~1.5 bits per tri) • Potentially like a modern game model? • Around 2.5 for other models

34. Triangle Mesh Compression Costa Touma, Craig Gotsman Technion - Israel Institute of Technology

35. Insight • In polygonal mesh that is orientable • Vertices incident on any mesh vertex may be ordered • Separation property • If you cut out a ring of a mesh, it separates it into 2 disjoint meshes: • set of vertices inside (may be empty) • set outside • Connectivity can be encoded by degree

36. Definitions • Vertex cycle • Cyclic sequence of vertices along tri edges • Active list • Vertex cycle at “wavefront” of encoding. Mesh divided into portion of mesh encoded and portion not encoded • Focus • Vertex in active list being processed

37. Step 0: Make object closed

38. Step 1: Pick tri for active list Specify node degrees:“add 6, add 7, add 4”

39. Continue working with focus • Add 4, Add 8

40. And now focus is done • Add 5, Add 5

41. Active edge added twice: split • Add 5, Split 5

42. Algorithm • Start with triangle, active list of 3 vertices • Add n vertices (clockwise order) adjacent to current focus to active list with “add <n>” command • If active list intersects itself, it is split to two active list with a “split <offset>” command

43. Merge case • This deals with genus 1+ objects (torus,etc) • If first free edge of active vertex is in active list of vertex, 2 possibilities • It is on current active list • Split <offset> • It is on active list already split • Merge <index> <offset>

44. Better prediction of vertex position • Have previous triangle • Compute plane equation • Assume next triangle is coplanar • Encode error term • Simple, elegant, low error • Use codebook for most common errors

45. Results • For regular meshes • Get around .2 bits per vertex for topology • With RLE • Increasing quantization from 8-10 bits, size goes up by 30-40% (!?)

46. Results

47. Decompression • Leaf triangle reconstructed from root id

48. Decompression

49. Decompression

50. Decompression