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
Estimating 3D Distributions
• Macroscopic statistics of a 2D image are
related to,
but not equal to the statistics of a 3D
volume
– Distributions of Spheres
– Distributions for Other Particles
– Managing Multiple Particle Types
Distributions of Spheres
• d max：maximum diameter
• Establish a relationship between
– the size distribution of 2D circles
(as the number of circles per unit area)
– the size distribution of 3D spheres
(as the number of spheres per unit volume)
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}
Densities NV ,
K ij
N A are related by the values K ij
Relative probabilities：
j
n
－a sphere in the j th histogram bin with diameter
－a profile in the i th histogram bin with diameter
(i  1)
i
d 
n
n
Recovering Sphere Distributions
Note that the profile source is ambiguous
For the following examples, n = 4
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
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
• Polygon mesh：random orientation
• Render
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
Scale Factor
• Scale factor s ：to relate the size of particle
P to the size of the particles in input image
s  Aimg /APmax
– profile maximum area
• Aimg ：input image
• APmax：particle P
• Mean caliper diameter
H  sH P
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.
Managing Multiple Particle Types
• particle type：i
•
•
•
•
mean caliper diameter：H i
representative matrix：K
i
distribution： NVi
probability that a particle
is type i ：P( i )
N A   ( H i K i NVi )
i
N A   ( H i K i P(i) NV )
i
  ( H i K i P(i ))NV
i
• total particle density：
NV   NVi
1


NV   ( H i K i P(i))  N A
 i

Reconstructing the Volume
• Particle Positions
• Color
• Adding Fine Detail
Particle Position - Annealing
• Populate the volume with all of the particles,
ignoring overlap
• Perform simulated annealing to resolve
collision
– Repeatedly searches for all collision
(in the x, y, z directions)
– Relaxes particle positions to reduce
interpenetration
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
Prior Work – Revisited
Input
Heeger & Bergen ’95
Our result
Results- Testing Precision
Input
distribution
Estimated
distribution
Result- Comparison
Collection of Particle Shapes
• Can’t predict exact
particle shapes
• Unable to count
small profiles
• Limited to fewer
profile observation
Calculations error
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