1. Smooth Geometry Images Frank Losasso, Hugues Hoppe, Scott Schaefer, Joe Warren

2. Overview Provide a simple representation using a single uniform bi-cubic B-spline Multiple PatchesSingle Patch

3. Geometry Images Sample arbitrary surface using a regular 2D grid Connectivity is implicit [Gu et al 2002]

4. cut parametrize

5. cut sample

6. cut [r,g,b] = [x,y,z] render store [Gu et al 2002]

7. Supports surfaces of arbitrary genus But, boundary has complicated topology – requires sideband a a’ a a’ General cut [Gu et al 2002]

8. Our Approach: X-Cut Special type of cut curve Make a X-cut centered at x Unfold domain into a square image Creates simple boundary symmetries

9. The X-Cut

10. Spherical Remeshing demo image I domain D sphere S mesh M [Praun and Hoppe 2003]

11. How To Obtain Control Points B-spline is approximating  should not directly sample surface Instead, use least-squares fitting: 

12. How To Obtain Smoothness Application of boundary rules Pad image to recreate 1-ring around all vertices

13. How To Obtain Smoothness All vertices regular, except boundary midpoints generally not C 1 C1C1C1C1 Add a simple linear constraint

14. Bi-cubic Subdivision on GPU 4 operators, stored as fragment programs bilinear subdivision mesh averaging limit tangent repeat

15. 4 operators, stored as fragment programs Bi-cubic Subdivision on GPU bilinear subdivision mesh averaging limit tangent repeat

16. Rendering “Interpret as vertex array” (OpenGL extension) Render using triangles

17. Discrete Subdivision Levels Both sub-sampling and subdivision are easily implemented Original Image …… SubsampledSubdivided …… (2 k +1) x (2 k +1)

18. Continuous Subdivision Levels Prevent ‘popping’ when changing subdivision levels gkgk g k+1

19. mesh averaging Continuous Subdivision Levels Prevent ‘popping’ when changing subdivision levels gkgk +  1-  g k+  g k+1 linear subdivision bilinear subdivision

20. Real-Time Demo

21. Displacement Mapping 33x33 257x257 GPU GPU scalar displacement map CPU simulation

22. Performance Results

23. Drawbacks and Limitations Genus cannot be >0 Surface rippling

24. Summary Closed smooth surface using single patch Stored as geometry image Simple and efficient GPU evaluation Continuous level-of-detail Displacement mapping 5x59x9

25. Eye candy 65x6533x3365x65