1. Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe

2. Background Modeling geometry with (triangular) meshes regular / iregular meshes current tecniques form semi-regular meshes typically: cut into disks parametrize each disk each disk is regular all disks are iregular network

3. Advantages of Regularity Improved compression (implicit connectivity) Reduce tangent non-uniformity Better start for hirarchial resolution

4. Introduction Remesh into a completely regular structure geometry as 2D array of points more arrays for “surface signals”: –normals –colours –texture encode as RGB picture

7. The General Algorithm Cut the mesh parametrize into an n*n square [x,y,z] [R,G,B] encode using image compression algorithms other attributes - additional images implicit parametrization

8. Challenges Find a “good cut” (form a disk-like surface) Avoid “cracks in the cut” along the boundary Evenly distributing parametrization lossy compression - cut fusing

9. Creating The Geometry Image

10. The Cut Cut  = set of edges M = Original meshM’ = New mesh

11. mates Split Split (non boundary) edges form “open cut”  ’  ’ is a directed loop = edge of M’

12. Split - cont. Valence k==>k replications valence  2==>“cut node”

13. Cut Path Path between 2 ordered cut nodes of  ’ mate to cut path

14. Parametrization (piecewise) linear map  : [n * n] => [v  M’] resample mesh at grid triangle interpolation

15. Parametrization

16. Boundary parametrization Map  ’ ==> boundary of square cut nodes => on grid cut mates => same length

17. Boundary parametrization - cont. How is it done? –Allocate proportional length –redistribute

18. Boundary parametrization - cont. Avoid degeneracies: –no full triangle on 1 side (split middle) –break edges over corners (add vertex)

19. Interior parametrization Geometric stretch parametrization –minimize spacing on surface (parameters distribute evenly) –P.V. Sander, J. Snyder, S.J. Gortler and H. Hoppe. Texture mapping progressive meshes. Also serves as metric for mesh

20. Cutting

21. Initial Cut Fact: –genus 0 ==> 1 edge cut –genus g ==> 2g generator loops cut (that form basis to surface’s fundamental group)

22. Find the cut - retractor algorithm Remove a seed triangle Repeatedly remove an edge adjacent to exactly 1 triangle and the triangle Repeatedly remove ‘dandling’ edges (vertices with degree 1 + the edge) do not change boundary (if exists)

23. Original mesh

24. Remove seed triangle

26. Remove edges and triangles

28. Iterate

57. Done: no triangles left (no faces)

58. Remove dandling edges

59. Iterate

68. We are left with this:

69. Initial Cut Result of the algorithm: –generator loops (genus > 0) –one vertex (genus 0) if 1 vertex : add 2 more (adjacent)

70. Iterated Cut Augmentation Parametrize on unit circle (Floater) –M. Floater. Parametrization and smooth approximation of surface triangulations. Find vertex with highest geometric stretch Find shortest path to boundary Add path to cut Stop when geometric stretch increases for genus 0: disregard original cut

80. Putting it all together

87. Rendering

94. Compression

97. Again this Crack!

98. Topological Sideband Table of nodes in  for each node, record: –valence (k) –k coordinate pairs (s,t) Deduce  ’ matching from the table “Fuse cut” using matchings