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
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
–

Dissolution/recrystallization of minerals
–

Oxides of Carbon, Sulfur, Nitrogen
Chemical transformation of minerals
–

Porous stone
Affects stone’s appearance
Deposition of atmospheric pollution
–
Airborne pollution or acid rain
How is This Model Unique?

Volume Monitoring
–

Simulation
–
–
–

Slab data structure
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
–
–

3-D Weathered interior
–

Stone meets outside environment
Water evaporates from stone
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) = ·(DiCi)
 = 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.