1. 1 Compressing Triangle Meshes Leila De Floriani, Paola Magillo University of Genova Genova (Italy) Enrico Puppo National Research Council Genova (Italy)

2. 2 Why Geometric Compression? o Availability of large geometric datasets in mechanical CAD, virtual reality, medical imaging, scientific visualization, geographic information systems, etc. o Need for mspeeding up transmission of geometric models mreducing the costs of memory and of auxiliary storage required by such models menhancing rendering performances: limitations on on-board memory and on data transfer speed

3. 3...Why Geometric Compression?... Compression methods aimed at two complementary tasks: mcompression of geometry: efficient encoding of numerical information attached to the vertices (position, surface normal, color, texture parameters) mcompression of mesh connectivity: efficient encoding of the mesh topology Compression methods developed for triangle meshes

4. 4 Compression of Connectivity o Two kinds of compression methods: mDirect methods: Goal: minimize the number of bits needed to encode connectivity m Progressive methods: Goal: an interrupted bitstream must provide a description of the whole object at a lower level of detail

5. 5 Our Proposal o Direct method: mSequence of triangles in a shelling order o Progressive Method: mSequence of edge swaps where destructive operator = vertex removal

6. 6 Sequence of Triangles in a Shelling Order o Method based on a shelling order: a sequence of all the triangles in the mesh with the property that the boundary of the set of triangles corresponding to any subsequence forms a simple polygon o A triangle mesh is shellable if it admits a shelling sequence o A shellable mesh is extendably shellable if any shelling sequence for a submesh can be completed to a shelling sequence for the whole mesh o The method works for every triangulated surface homeomorphic to a sphere or a disk o Encoding: four 2-bits codes per edge: SKIP, VERTEX, LEFT, RIGHT

7. 7...Sequence of Triangles in a Shelling Order... Algorithm o Start from an arbitrary triangle, whose boundary forms the initial polygon o Loop on the edges of the current polygon: for each edge e: mtry to add the triangle t adjacent to e and lying outside the polygon mif successful, update the current polygon min any case, send a code mwhen necessary, send a vertex o Each edge is examined at most once o Each vertex is sent just once

8. 8...Sequence of Triangles in a Shelling Order... Algorithm mif t brings a new vertex ==> VERTEX + vertex coordinates mif t does not exist or cannot be added ==> SKIP

9. 9...Sequence of Triangles in a Shelling Order... Algorithm mif t shares the polygon edge on the left of e ==> LEFT mif t shares the polygon edge on the right of e ==> RIGHT

10. 10...Sequence of Triangles in a Shelling Order... Properties of the Shelling Method o Every vertex is encoded only once o Compression and decompression algorithms: mwork in time linear in the size of the mesh mno numerical computation necessary mconceptually simple and easy to implement o Adjacencies between triangles are reconstructed directly from the sequence at no additional cost

11. 11...Sequence of Triangles in a Shelling Order... Cost Evaluation o In theory: mat most two bits of connectivity information for each edge m==> at most 6n bits for a mesh with n vertices o In practice: mless than 4.5n bits of connectivity

12. 12...Sequence of Triangles in a Shelling Order... Experimental Results (on TINs) Exp #vert #tri #code bits compress. bits /vert time(tri/s) U1 42943 85290 182674 4.2538 1.644(51879) U2 28510 56540 123086 4.3173 1.077(52483) U3 13057 25818 57316 4.3897 0.479(53899) U4 6221 12240 27180 4.3690 0.215(56930) A1 15389 30566 64678 4.2029 0.565(54099) A2 15233 30235 63958 4.1986 0.561(53894) A3 15515 30818 65210 4.2030 0.572(53877) A4 15624 31042 65520 4.1935 0.577(53798) B1 5297 10570 22392 4.2273 0.182(58076) B2 5494 10959 23468 4.2716 0.188(58292) B3 5397 10768 23060 4.2727 0.186(57892) B4 5449 10874 23136 4.2459 0.187(58149) U1--4: uniform resolution (in decreasing order) A1--4: one fourth of the area is at high resolution, the rest is coarse B1--4: one 16th of the area is at high resolution, the rest is coarse

13. 13...Sequence of Triangles in a Shelling Order... Properties o The method generalizes to surfaces with arbitrary genus o The algorithm automatically cuts the surface into simply connected patches with a small overhead o No additional control code required o Vertices belonging to more than one patch are repeated o Cost: in practice, less than 5.5n bits of connectivity

14. 14...Sequence of Triangles in a Shelling Order... Experimental Results (on 3D meshes) Mesh #vert #tri # #repeated #code bits patches vert bits /vert eight 766 1536 6 198 3856 5.0339 shape 2562 5120 1 0 10478 4.0897 cow 3078 5804 25 356 13984 4.5432 femur 3897 7798 5 124 18894 4.8483 pieta 3475 6976 15 468 17124 4.9278 skull 10950 22104 80 3242 58150 5.3105 bunny 34834 69451 3 323 146986 4.2196 fandisk 6475 12946 1 0 27298 4.2159 phone 33204 66287 3 12 149058 4.4891

15. 15...Sequence of Triangles in a Shelling Order... Experimental Results (on 3D meshes) whole mesh patch 1 patch 2 …. + other 4 patches with few triangles each

16. 16...Sequence of Triangles in a Shelling Order... Experimental Results (on 3D meshes) whole mesh patch 1 patch 2 …. + other 78 patches with few triangles each

17. 17 Sequence of Edge Swaps o Method based on the iterative removal of a vertex of bounded degree (less than a constant b) selected according to an error-based criterion: mthe vertex which causes the least increase in the approximation error is always chosen  The polygonal hole  left by removing vertex v is retriangulated  The inverse constructive operator inserts vertex v and recovers the previous triangulation of 

18. 18 Sequence of Edge Swaps o The old triangulation T is recovered from the new one T' by first splitting the triangle t of T' containing vertex v and then applying a sequence of edge swaps...Sequence of Edge Swaps... T T’

19. 19...Sequence of Edge Swaps... Sequence of Edge Swaps o Encoding: mfor each removed vertex v: Ga vertex w and an integer number indicating a triangle around w (they define the triangle t of T' containing v)  the packed sequence of edge swap which generates T from T' Vertex: w Triangle index: 0 Sequence of edge swaps T’ T

20. 20...Sequence of Edge Swaps... 1) Split triangle t into three triangles T’

21. 21...Sequence of Edge Swaps... 2) Swap edge indicated by number 2 around v

22. 22...Sequence of Edge Swaps... 3) Swap edge indicated by number 0 around v

23. 23...Sequence of Edge Swaps... 4) Swap edge indicated by number 2 around v T ==> swap sequence: 2 0 2

24. 24...Sequence of Edge Swaps... Cost Evaluation o For each removed vertex v: mlog n bits for one vertex reference mlog b bits for the index of a triangle mfor edge swap: Glog r bits for the index of the edge to swap, where r is the current number of triangles incident at v Gr is initially 3, and increases by one at each edge swap Gat the last swap, r is at most b-1 G==> less than log((b-1)!)-1 bits for the whole sequence of swap indexes o ==> n(log n +log b+ log((b-1)!)-1) bits of connectivity information mfor instance, for n=2 16 and b=2 3 ==> about 26.5*2 16 bits of connectivity

25. 25...Sequence of Edge Swaps... Properties o Adaptivity to LOD generation is good since vertices are removed by taking into account the accuracy of the resulting approximation o Unlike other methods (Hoppe, 1996; Snoeyink and van Kreveld, 1997), no specific retriangulation criterion is assumed o The criterion used in the retriangulation is encoded in the sequence of swaps o Coding and decoding algorithms with different error-driven selection criteria experimented in the context of multiresolution triangulations (De Floriani, Magillo, Puppo, IEEE Visualization 1997)