Modeling and Rendering of Weathered Stone
SIGGRAPH 1999
Julie Dorsey Alan Edelman Henrik Wann Jensen Justin Legakis Hans Kohling Pedersen
M.I.T.
Andrea RowanFebruary 16, 2001
Outline
Problem description Previous Work System
– Slab data structure– Stone weathering model– Light scattering
Results Successes / Problems
Problem Description
Visually represent the weathering of stone
Chemical weathering - erosion by water, pollutants
Processes to Model
Movement of water– Porous stone
Dissolution/recrystallization of minerals– Oxides of Carbon, Sulfur, Nitrogen
Chemical transformation of minerals– Affects stone’s appearance
Deposition of atmospheric pollution– Airborne pollution or acid rain
How is This Model Unique?
Volume Monitoring– Slab data structure
Simulation– Water flow– Transport/Dissolution of minerals– Surface erosion
Subsurface scattering of light
Previous Work
Volume modeling– Voxels (Kaufman et al. [16])
High storage + calculation requirements
– Shells (Udupa et al. [34]) Set of voxels near surface boundary Axis-aligned
Subsurface Light Scattering– Dorsey et al. [7], Hanrahan et al. [12]
Assume homogeneous layers of surface
Previous Work
Weathering effects– 2D effects
Water flow (Dorsey et al. [7],[8]) Watercolors (Curtis et al. [6])
– Erosion of fractal terrains (Musgrave et al. [20])
Drop water on surface, let it run down surface collecting and depositing minerals
Doesn’t account for different minerals/rocks
System Architecture
Input– Polyhedral mesh– Water maps– Mineral deposit maps
Voxelizer Quarry Weathering Simulator Polygonizer Renderer
Voxels
Store mineral properties 3-D stone density function s No stone present
– s = 0 Decay index d
– Tendancy to erode to clay
Slabs
Groups of Voxels Surface-Aligned 8-cornered Separated by bilinear patches Slab edges are average of area normals
Quarry
Rendering of Unweathered Stone Combination of solid 3D procedural
textures Noise function
– Mineral patterns of granite– Veins of marble
Weathering Simulation
2-D stone surface– Stone meets outside environment– Water evaporates from stone
3-D Weathered interior– Grows during wet cycles
Interior moist/dry front– Internal boundary
Travel of Moisture
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)
Travel of Moisture
Location of front at any time t:
dp/dt = -·(p) = -2p - ·p
= porosity, or the ratio: volume of empty space/volume of
mass in stone (input constant)
p = pressure of water on surface (varying)
Travel of Moisture
Location of moisture evolved through time with loop:– Solve dp/dt with current pressure p
Internal front External surface pressure (varies as go from
wet to dry seasons)
– Update front location with Darcy’s law (showing v of front)
Dissolution/Recrystallization
Dissolution calculated at internal front:
dCi/dt = - ki(mi - Ci)
Ci = Concentration of dissolved mineral in the water (Calculated)
ki = Solubility of the mineral m (input constant)
mi = Saturated level (puts limit on dissolution) (Calculated)i = Mineral index
Mineral movement
Convective-diffusion equation:
/t(Ci) + v·(Ci) = ·(DiCi)
= porosity (input constant)
Ci = Concentration of dissolved mineral in the water (calculated)
v = velocity from pressure gradient (calculated)
Di = Diffusivity of mineral (input constant)
Mineral movement
Minerals form crust on surface Green’s theorem preserves total mass Decay index (d) of each voxel is
continuously modified as minerals are dissolved/deposited.
Numerical Calculations
Finite Difference Schemes– Solves gradient problem
Slabs can be trapezoidal– Laplacian (2) calculation is complicated
Light Scattering
Stone contains transparent crystal grains
Must consider subsurface scattering of light
Light Scattering
Mie scattering (back & forward!)– Light hits a particle or a molecule whose
diameter is >= the wavelength of the light
Light Scattering
Scattered Radiance
Ls = Ld+ Li
– Ld = Radiance from direct illumination Shadow ray from light source
– Li = Radiance from indirect illumination Photon map estimate (Photons emitted from light
sources)
Results
Simulations:– Quad 250 MHz R10000 SGI
Renderings:– Dual 400 MHz Pentium II PC with Linux
Sphinx
2.2 million triangles 281 slabs, 323 voxels each Simulation - 24 hours Rendering - 80 minutes
Sphinx
Sandstone Column
100,000 triangles 240 slabs, 323 voxels each Simulation - 4 hours Rendering - 30 minutes
Sandstone Column
Successes
Scientifically-based model with few hacks!
Realistic looking results Good framework for diversity of effects
– Easy to implement salt-water erosion
Problems
Slabs are edited by hand to fix overlapping
Slow computation time– Can’t interactively weather the stone!
Limited by lack of complete scientific knowledge
References
Dorsey et al. Modeling and Rendering of Weathered Stone. SIGGRAPH Conference Proceedings, 1999.
Musgrave et al. The Synthesis and Rendering of Eroded Fractal Terrains. Computer Graphics, July 1989.
Udupa et al. Shell Rendering. IEEE Computer Graphics and Applications, November 1993.