Practical Logarithmic Shadow Maps

1. Practical Logarithmic Shadow Maps Brandon Lloyd UNC-CH Naga Govindaraju UNC-CH David Tuft UNC-CH Steve Molnar Nvidia Dinesh Manocha UNC-CH

2. Motivation • Interactive shadow computation remains a challenge • increasing scene complexity • large, open environments • user expect high quality

3. Shadow maps • Simple two pass algorithm • Supports wide range of geometric representations • Cheap to render • Disadvantage:Aliasing artifacts at shadow edges

4. Increase sample density where needed by reparametrizing the shadow map Warping algorithms Light-space perspective shadow maps (LSPSMs) [Wimmer et al. 2004] Perspective shadow maps(PSMs) [Stamminger and Drettakis 2002] Trapezoidal shadow maps(TSMs) [Martin and Tan 2004]

5. Comparison of parameterizations 4x4 projection matrix Logarithmic

6. Main results • Logarithmic parameterization for directional and point lights • Error analysis for point lights • Hardware architecture enhancements for logarithmic rasterization

7. Outline • Related work • Aliasing error • Logarithmic parameterization • Hardware architecture • Results

8. Separate shadow maps for different parts of the scene Includes cascaded shadow maps Plural sunlight buffers [Tadamura et al. 1999,2001] Partitioning algorithms Adaptive shadow maps [Fernando et al. 2001; Lefohn et al. 2006] Tiled shadow maps [Arvo 2004]

9. Warping + partitioning • 4x4 matrix • Lixel for every pixel[Chong and Gortler 2004] • PSM with cube maps[Kozlov 2004] • Warping + various frustum partitioning schemes [Lloyd et al. 2006] • Dual paraboloid [Brabec et al. 2002]

10. Specialized representations Silhouette shadow maps[Sen et al. 2004] Irregular shadow maps[Johnson et al. 2004,2005; Aila and Laine 2004] • Solves the aliasing problem • GPU support requires major changes

11. Outline • Related work • Aliasing error • Logarithmic parameterization • Hardware architecture • Results

12. Aliasing error light shadow plane eye viewfrustum

13. wl wi wi wi wi wl ' ' ' ' ' ' light beam image beam Aliasing error

14. wl wi wi wl ' ' ' ' cos θi wl m= ≈ wi cos θl θl wl Perspectivealiasing θi wi cos θi Projectionaliasing cos θl Aliasing error light beam wl image beam wi

15. wi wi wl wl ' ' ' ' cos θi wl m= ≈ wi cos θl θl wl Perspectivealiasing θi wi cos θi Projectionaliasing cos θl Aliasing error light beam wl image beam wi

16. To eliminate perspective aliasing:wl ≤wi Light beam width depends on: Shadow map resolution Parameterization Controlling aliasing error light beam image beam wl wi

17. Outline • Related work • Aliasing error • Logarithmic parameterization • Directional light • Point light • Hardware architecture • Results

18. Parameterization light imageplane light light shadow map shadow map Standard Perspective warped

19. Directional light light 1. Light beam widths 2. Texel spacing function 1 t light image plane z 3. Parameterization wl wi imagebeam [Wimmer et al. 04]

20. Directional light With 4x4 matrix With logarithm Uniform beam width Linear z beam width Quadratic z beam width z

21. 8 10 6 10 4 Shadow maptexels / image texels 10 2 10 0 10 1 2 3 4 10 10 10 10 f / n Comparison Standard Perspective Logarithmic frustum fov = 60○

22. Point lights light view frustum

23. Point lights light light imageplane wl ~ z y z view frustum face

24. Point light spacing function - z y face ylight z z

25. Point light spacing function - z y face ylight z z

26. Point light spacing function - z y face ylight z z

27. Point lights – parameterization

28. Parameterization for exact function is complicated Fit two parabolas: Parameterization: Fitting the spacing function - z Point light spacing function 0 z

29. To NDC coords. vc = Pv vn= vc /wc To screen coordinates s = a0ln( a1xn + a2 ) + a3 t = b0ln( b1zn + b2 ) + b3 4x4 matrix + logarithm vc : [xc zc ycwc]T - clip coordinate vn : [xn zn yn 1]T - normalized device coordinate v : [x z y 1]T - world coordinate P : perspective or orthographic projection matrix a0-a3 : constants b0-b3 : constants

30. Parameterization summary Spacing functions from4x4 matrix + log Actual spacing functions Directional Point without log with log x: ortho. z: persp.

31. Outline • Related work • Aliasing error • Logarithmic parameterization • Hardware architecture • Results

32. Logarithmic parameterization viewfrustum Unwarped

33. Vertex program • Transform vertices on the GPU • Requires finely tesselated model • Increases burden on vertex processor • Adaptive tesselation complicated • Easier with DX10 geometry shaders

34. Brute force rasterization Render bounding primitive Transform fragments to original triangle Discard if not inside 10x slow down Computing bounding primitive is complicated Easier with DX10 geometry shaders Fragment program

35. Graphics pipeline vertex processor memory interface clip & back-face cull rasterizer fragment processor alpha, stencil, & depth tests depth compression color compression blending

36. Coverage determination use sign of edge distance equations Attribute interpolation depth, color, texture coordinates, etc. Rasterizing equations

37. Rasterizing equations Edge distance and attribute interpolation: Parameterization:

38. Incremental computation 1 MULT + 2 ADD per step

39. Depth compression First order Second order Compressed tiles shown in red

40. Feasibility • Current hardware trend • Computational power increasing rapidly • Bandwidth lags behind • Log shadow maps • Increase computation • Reduce memory/bandwidth consumption

41. Nonlinear rasterization • Configurable rasterizer • Other rasterization functions are possible • Might be useful for other effects • Reflections, refractions, caustics, general multi-view perspective [Hou et al. 06]

42. Outline • Previous work • Aliasing error • Logarithmic parameterization • Hardware implementation • Results

43. Results Power plant 13 Mtris Oil tanker 82 Mtris Town scene 59 Ktris

44. Results Standard Perspective warping Logarithmic

45. Results Perspective warping Logarithmic

46. Results Perspective warping Logarithmic

47. 8 10 k=1 6 10 Shadow map texels / image texels k=2 4 10 k=4 2 10 k=8 0 10 1 2 3 4 10 10 10 10 f / n Comparison to z-partitioning Perspective with kz-partitions Logarithmic

48. Results z-partitioning (k=4) Logarithmic

49. Results

50. Results Perspective warping Logarithmic