1. Olga Sorkine and Daniel Cohen-Or Tel-Aviv University Warped textures for UV mapping encoding

2. Texture mapping encoding 3D mesh : geometry + topology Textures Texture mapping function: each vertex has associated (u,v) coordinates in the texture image.

3. Texture mapping encoding 3D mesh : geometry + topology Textures Texture mapping function: each vertex has associated (u,v) coordinates in the texture image.

4. Texture mapping encoding 3D mesh : geometry + topology Textures Texture mapping function: each vertex has associated (u,v) coordinates in the texture image.

5. Texture mapping encoding 3D mesh : geometry + topology Textures Texture mapping function: each vertex has associated (u,v) coordinates in the texture image. uvuv

6. The problem: Enumerating the (u,v) coordinates is an explicit mapping (space- inefficient). We look for implicit representation of the texture mapping function.

7. How to compress UV-coordinates without compressing them?

8. The idea: warp the original textures in order to embed the texture mapping inside.

9. Encoding - example Flattening Apply texture to flattened triangles Original mesh Original texture Textured mesh (rendered) Flattened mesh Warped texture

10. Decoding - example FlatteningOriginal mesh Restored textured mesh (rendered) Flattened mesh Warped texture Fitting texture to each triangle

11. Embedding requirements Distortion of each triangle is minimal Flat meshes (patches) produced can be fitted into rectangular images with minimal void area Small number of patches - to minimize seams between textures

12. Non-distorting embedding Unfolding with zero distortion: Embed each triangle separately or peel triangle strips from the mesh.

13. Unfolding with zero distortion Artifacts in mip-mapping (due to fragmentation) Large “void” areas

14. Small distortion tolerance Allow to stretch the triangles by some bounded factor. Enables to create large continuous flat patches.

15. Embedding algorithm Take a seed triangle, flatten it as is. Proceed to flatten neighbours in BFS order If a triangle is distorted above threshold, discard it from current patch When no triangles can be flattened in the current patch, start a new one.

16. Growing a patch Seed triangle

17. Growing a patch

22. Embedding a vertex In 3D In 2D - unfold each triangle independently Combine the three positions so that the distortion is minimal x z

23. Distortion metric e1e1 e2e2 e3e3 e' 1 e’ 2 e’ 3 L i = max{e i, e’ i } l i = min{e i, e’ i } i = 1, 2, 3 distortion = max{L 1 / l 1, L 2 / l 2, L 3 / l 3 } distortion  1; distortion = 1  the triangles are isometric

24. Some results The twisted loop model Two of the five patches produced

25. The warped textures The two patches created Cone model rendered

26. Advantages of implicit UV-mapping representation Only the mapped parts of texture images are used in the warped atlas With smart packing, the size of new warped textures isn’t significantly larger than the original images, and no (u,v) data is needed  total size of texture mapping representation is smaller.

27. Olga Sorkine and Daniel Cohen-Or Tel-Aviv University Warped textures for UV mapping encoding