1. Modeling and Rendering of Weathered Stone SIGGRAPH 1999 Julie Dorsey Alan Edelman Henrik Wann Jensen Justin Legakis Hans Kohling Pedersen M.I.T. Andrea Rowan February 16, 2001

2. Outline n Problem description n Previous Work n System – Slab data structure – Stone weathering model – Light scattering n Results n Successes / Problems

3. Problem Description n Visually represent the weathering of stone n Chemical weathering - erosion by water, pollutants

4. Processes to Model n Movement of water – Porous stone n Dissolution/recrystallization of minerals – Oxides of Carbon, Sulfur, Nitrogen n Chemical transformation of minerals – Affects stone’s appearance n Deposition of atmospheric pollution – Airborne pollution or acid rain

5. How is This Model Unique? n Volume Monitoring – Slab data structure n Simulation – Water flow – Transport/Dissolution of minerals – Surface erosion n Subsurface scattering of light

6. Previous Work n Volume modeling – Voxels (Kaufman et al. [16]) n High storage + calculation requirements – Shells (Udupa et al. [34]) n Set of voxels near surface boundary n Axis-aligned n Subsurface Light Scattering – Dorsey et al. [7], Hanrahan et al. [12] n Assume homogeneous layers of surface

7. Previous Work n Weathering effects – 2D effects n Water flow (Dorsey et al. [7],[8]) n Watercolors (Curtis et al. [6]) – Erosion of fractal terrains (Musgrave et al. [20]) n Drop water on surface, let it run down surface collecting and depositing minerals n Doesn’t account for different minerals/rocks

8. System Architecture n Input – Polyhedral mesh – Water maps – Mineral deposit maps n Voxelizer n Quarry n Weathering Simulator n Polygonizer n Renderer

9. Voxels n Store mineral properties n 3-D stone density function s n No stone present – s = 0 n Decay index d – Tendancy to erode to clay

10. Slabs n Groups of Voxels n Surface-Aligned n 8-cornered n Separated by bilinear patches n Slab edges are average of area normals

11. Quarry n Rendering of Unweathered Stone n Combination of solid 3D procedural textures n Noise function – Mineral patterns of granite – Veins of marble

12. Weathering Simulation n 2-D stone surface – Stone meets outside environment – Water evaporates from stone n 3-D Weathered interior – Grows during wet cycles n Interior moist/dry front – Internal boundary

13. Travel of Moisture n Darcy’s law shows fluid speed in stone: v = -K/  (  p -  g)  v = velocity of front of fluid (calculated) K = permeability of stone (input constant)  = viscosity, or resistance to flow (input constant) p = pressure of water on surface (varying)  = density of water (input constant) g = gravity (input constant)

14. Travel of Moisture n Location of front at any time t: dp/dt = -  ·(  p) = -  2 p -  ·  p  = porosity, or the ratio: volume of empty space/volume of mass in stone (input constant) p = pressure of water on surface (varying)

15. Travel of Moisture n Location of moisture evolved through time with loop: – Solve dp/dt with current pressure p n Internal front n External surface pressure (varies as go from wet to dry seasons) – Update front location with Darcy’s law (showing v of front)

16. Dissolution/Recrystallization n Dissolution calculated at internal front: dC i /dt = - k i (m i - C i ) C i = Concentration of dissolved mineral in the water (Calculated) k i = Solubility of the mineral m (input constant) m i = Saturated level (puts limit on dissolution) (Calculated) i = Mineral index

17. Mineral movement n Convective-diffusion equation:  /  t(  C i ) + v·  (  C i ) =  ·(  D i  C i )  = porosity (input constant) C i = Concentration of dissolved mineral in the water (calculated) v = velocity from pressure gradient (calculated) D i = Diffusivity of mineral (input constant)

18. Mineral movement n Minerals form crust on surface n Green’s theorem preserves total mass n Decay index (d) of each voxel is continuously modified as minerals are dissolved/deposited.

19. Numerical Calculations n Finite Difference Schemes – Solves gradient problem n Slabs can be trapezoidal – Laplacian (  2 ) calculation is complicated

20. Light Scattering n Stone contains transparent crystal grains n Must consider subsurface scattering of light

21. Light Scattering n Mie scattering (back & forward!) – Light hits a particle or a molecule whose diameter is >= the wavelength of the light

22. Light Scattering n Scattered Radiance L s = L d + L i – L d = Radiance from direct illumination n Shadow ray from light source – L i = Radiance from indirect illumination n Photon map estimate (Photons emitted from light sources)

23. Results n Simulations: – Quad 250 MHz R10000 SGI n Renderings: – Dual 400 MHz Pentium II PC with Linux

24. Sphinx n 2.2 million triangles n 281 slabs, 32 3 voxels each n Simulation - 24 hours n Rendering - 80 minutes

25. Sphinx

26. Sandstone Column n 100,000 triangles n 240 slabs, 32 3 voxels each n Simulation - 4 hours n Rendering - 30 minutes

27. Sandstone Column

28. Successes n Scientifically-based model with few hacks! n Realistic looking results n Good framework for diversity of effects – Easy to implement salt-water erosion

29. Problems n Slabs are edited by hand to fix overlapping n Slow computation time – Can’t interactively weather the stone! n Limited by lack of complete scientific knowledge

30. References n Dorsey et al. Modeling and Rendering of Weathered Stone. SIGGRAPH Conference Proceedings, 1999. n Musgrave et al. The Synthesis and Rendering of Eroded Fractal Terrains. Computer Graphics, July 1989. n Udupa et al. Shell Rendering. IEEE Computer Graphics and Applications, November 1993.