Lapped Textures

1. Lapped Textures Princeton University Princeton University Microsoft Research Emil Praun Adam Finkelstein Hugues Hoppe

2. Goal mesh geometry ? textured surface “example” image

3. Goal • Little user effort • No apparent seams • No obvious periodicity • Low distortion • Local texture control • Anisotropy

4. Previous 2D Texture Synthesis Histogram equalization [Heeger ’96] Laplacian block shuffling [de Bonet ’97] Pixel template matching [Efros ’99] [Wei ’00]

5. Previous 2D Texture Synthesis Histogram equalization [Heeger ’96] Laplacian block shuffling [de Bonet ’97] Pixel template matching [Efros ’99] [Wei ’00] Random pasting of image blocks [Xu ’00]

6. Previous 3D Texturing • Volumetric textures: • Noise functions [Perlin ’85, Worley ’96] • Solid textures by example [Ghazanfarpour ’96] • Synthesizing texture on a surface: • Reaction-diffusion [Turk ’91, Witkin ’91] • Cellular textures [Fleischer ’95] • Covering surface with triangular tiles [Neyret ’99]

7. Approach texture patch surface

8. Key Idea: Patch Pasting texture patch “lapped textures” surface

9. Anisotropic vs. Isotropic images

10. PROCESS

11. Algorithm • Cut texture patches from input texture • [Specify texture direction & scale on mesh] • REPEAT • Pick texture patch T, mesh point P • Grow surface patch S around P to size(T) • Flatten S over T; paste T on mesh • Update mesh coverage info • UNTIL mesh is covered

12. Algorithm texture patch surface

13. Algorithm texture patch surface

14. Algorithm texture patch surface

15. Algorithm texture patch surface

16. Issues • Texture patch creation • Specifying direction field • Surface patch growth • Patch parametrization • Face coverage estimation • Texture storage and rendering

17. Issues • Texture patch creation • Specifying direction field • Surface patch growth • Patch parametrization • Face coverage estimation • Texture storage and rendering

18. Texture Patch Creation

19. Less Structure  Splotch

20. Issues • Texture patch creation • Specifying direction field • Surface patch growth • Patch parametrization • Face coverage estimation • Texture storage and rendering

21. Direction Field: User-specified

22. Direction Field: User-specified • Angle - Only for anisotropic patches. • Can be set for only small number of faces – Gaussian radial basis function is used for interpolation. • Since the normal lies within the plane of the face we get an orthogonal basis : • S = T x N. (N – normal to the plane).

23. Direction Field:Local to Patch

24. Issues • Texture patch creation • Specifying direction field • Surface patch growth • Patch parametrization • Face coverage estimation • Texture storage and rendering

25. Patch Growth • 1. Choose a random point on a non fully covered triangle. • 2. Add triangles in a BFS order to cover the texture patch. • 3. Stop when the patch is fully coverd. • Condintions: • The surface patch homeomorphic to a disk. • The distortion should not exceed a given limit. • We add edge only if it is within the texture patch.

26. Patch Growth

27. Patch Growth

28. Patch Growth

29. Patch Growth

30. Patch Growth

31. Patch Growth

32. Patch Growth

33. Patch Growth

34. Polygonal Hulls • Start w/ boundary pixels, edges between neighbors • Conservatively simplify polygons

35. Issues • Texture patch creation • Specifying direction field • Surface patch growth • Patch parametrization • Face coverage estimation • Texture storage and rendering

36. Align Patch to Direction Field texture patch surface

37. Tangential Vector Field

38. Optimizing the Parametrization Least squares best match to unit axes Sparse linear system. No explicit fairness functional

39. ^ (T) t ^ s (S) Optimizing the Parametrization  (A) A (B) T S B C (C) face in 3D 2D texture space

40. Optimizing the Parametrization • For each mesh face f ={A,B,C} the vector T lies within the face plain. • Barycentric coordinates: • T = A + B + C and  +  +  = 0.

41. ^ (T) t ^ s (S) Optimizing the Parametrization  (A) A (B) T S B C (C) face in 3D 2D texture space ^ dT = (A) + (B) + (C) - t We wish to minimize :

42. Result of Optimization

43. Issues • Texture patch creation • Specifying direction field • Surface patch growth • Patch parametrization • Face coverage estimation • Texture storage and rendering

44. Coverage estimation Render patch triangles Flag covered triangles Remember 1 pixel per uncovered triangle • off-screen buffer

45. Coverage estimation Render patch triangles Flag covered triangles Remember 1 pixel per uncovered triangle • off-screen buffer

46. Coverage estimation Render patch triangles Flag covered triangles Remember 1 pixel per uncovered triangle • off-screen buffer

47. Issues • Texture patch creation • Specifying direction field • Surface patch growth • Patch parametrization • Face coverage estimation • Texture storage and rendering

48. Texture Storage and Rendering • Method 1: Texture Atlas • Pre-composite into a global texture map. • -- OR -- • Method 2: Runtime pasting • Composite at run-time using hardware

49. Texture Atlas • Texture Atlas is a set of charts {1,…, k}. • Each chart j is a mapping from a subset U j of the mesh to the Euclidean plane (the texture plane). • {U1,…, Uk} is a cover of the surface. • If i  j then j and i don’t overlap except on the edges.

50. Texture Atlas • Patches of triangles with similar normals • 2D packing problem for arbitrary polygons