Transcript Stereological Techniques for Solid Textures

Stereological Techniques
for Solid Textures
Rob Jagnow
MIT
Julie Dorsey
Yale University
Holly Rushmeier
Yale University
Objective
Given a 2D slice through an aggregate material,
create a 3D volume with a comparable appearance.
Real-World Materials
•
•
•
•
Concrete
Asphalt
Terrazzo
Igneous
minerals
• Porous
materials
Independently Recover…
• Particle distribution
• Color
• Residual noise
In Our Toolbox…
e
Stereology (ster'e-ol' -je)
The study of 3D
properties based on
2D observations.
Prior Work – Texture Synthesis
• 2D 2D
• 3D 3D
• Procedural Textures
Efros & Leung ’99
• 2D 3D
– Heeger & Bergen 1995
– Dischler et al. 1998
– Wei 2003
Heeger & Bergen ’95
Wei 2003
Prior Work – Texture Synthesis
Input
Heeger & Bergen, ’95
Prior Work – Stereology
• Saltikov 1967
Particle size distributions from section measurements
• Underwood 1970
Quantitative Stereology
• Howard and Reed 1998
Unbiased Stereology
• Wojnar 2002
Stereology from one of all the possible angles
Recovering Sphere Distributions
N A = Profile density
(number of circles per unit area)
NV = Particle density
(number of spheres per unit volume)
H = Mean caliper particle diameter
The fundamental relationship
of stereology:
N A  H NV
Recovering Sphere Distributions
Group profiles and particles into n bins
according to diameter
Particle densities = N A (i), {1  i  n}
Profile densities = NV (i), {1  i  n}
For the following examples, n = 4
Recovering Sphere Distributions
Note that the profile source is ambiguous
Recovering Sphere Distributions
How many profiles of the largest size?
=
N A (4)
K ij
K 44 NV (4)
= Probability that particle NV(j) exhibits profile NA(i)
Recovering Sphere Distributions
How many profiles of the smallest size?
=
+
N A (1) K11NV (1)
K ij
+
K12 NV (2)
+
K13 NV (3)
K14
NV (4)
= Probability that particle NV(j) exhibits profile NA(i)
Recovering Sphere Distributions
Putting it all together…
=
NA
K
NV
Recovering Sphere Distributions
Some minor rearrangements…
N A = d max
K
NV
d max = Maximum diameter
Normalize probabilities for each column j:
n
K
i 1
ij
 j/n
Recovering Sphere Distributions
N A  d max KNV
K is upper-triangular and invertible
For spheres, we can solve for K analytically:
1 / n 
K ij  


j 2  (i  1) 2 
0
j2  i2

for j  i
otherwise
Solving for particle densities: NV  1
d max
K 1 N A
Testing precision
Input
distribution
Estimated
distribution
Other Particle Types
We cannot classify arbitrary particles by d/dmax
Instead, we choose to use
A / Amax
Algorithm inputs:
+
Approach: Collect statistics for 2D profiles and 3D particles
Profile Statistics
Segment input image to obtain profile densities NA.
Input
Segmentation
Bin profiles according to their area,
A / Amax
Particle Statistics
Look at thousands of
random slices to obtain
H and K
0.45
sphere
cube
long ellipsoid
flat ellipsoid
0.4
0.35
probability
0.3
Example probabilities of
A / Amax for simple particles
0.25
0.2
0.15
probability
0.1
0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
A/Amax
0.7
0.8
0.9
1
Recovering Particle Distributions
Just like before, N A  H KNV
Solving for the particle densities,
NV  1
H
K 1 N A
Use NV to populate a synthetic volume.
Recovering Color
Select mean particle colors from
segmented regions in the input image
Input
Mean Colors
Synthetic
Volume
Recovering Noise
How can we replicate the noisy appearance of the input?
Input
=
Mean Colors
Residual
The noise residual is less
structured and responds well to
Heeger & Bergen’s method
Synthesized Residual
Putting it all together
Input
Synthetic volume
without
noise
with noise
Prior Work – Revisited
Input
Heeger & Bergen ’95
Our result
Results – Physical Data
Physical
Model
Heeger &
Bergen ’95
Our
Method
Results
Input
Result
Results
Input
Result
Summary
• Particle distribution
– Stereological techniques
• Color
– Mean colors of segmented profiles
• Residual noise
– Replicated using Heeger & Bergen ’95
Future Work
• Automated particle construction
• Extend technique to other domains and
anisotropic appearances
• Perceptual analysis of results
Thanks to…
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•
•
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Maxwell Planck, undergraduate assistant
Virginia Bernhardt
Bob Sumner
John Alex