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Smooth Geometry Images

Smooth Geometry Images. Frank Losasso, Hugues Hoppe, Scott Schaefer, Joe Warren. Overview. Provide a simple representation using a single uniform bi-cubic B-spline. Multiple Patches. Single Patch. Geometry Images. [Gu et al 2002]. Sample arbitrary surface using a regular 2D grid

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Smooth Geometry Images

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  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 Patches Single Patch

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

  4. [Gu et al 2002] cut parametrize

  5. [Gu et al 2002] cut sample

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

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

  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 mesh M sphere S domain D image I [Praun and Hoppe 2003] demo

  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. Add a simple linear constraint C1 How To Obtain Smoothness • All vertices regular, except boundary midpoints generally not C1

  14. Bi-cubic Subdivision on GPU 4 operators, stored as fragment programs limit bilinearsubdivision meshaveraging tangent repeat

  15. 4 operators, stored as fragment programs Bi-cubic Subdivision on GPU limit bilinearsubdivision meshaveraging tangent repeat

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

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

  18. Continuous Subdivision Levels • Prevent ‘popping’ when changing subdivision levels gk gk+1

  19. Continuous Subdivision Levels • Prevent ‘popping’ when changing subdivision levels linearsubdivision 1- +  gk+ bilinearsubdivision meshaveraging gk gk+1

  20. Real-Time Demo

  21. Displacement Mapping simulation CPU GPU 33x33 GPU scalardisplacementmap 257x257

  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 5x5 9x9

  25. Eye candy 65x65 65x65 33x33

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