Stereological Techniques for Solid Textures

1
Stereological Techniquesfor Solid Textures

• Rob Jagnow
• MIT

Julie Dorsey Yale University
Holly Rushmeier Yale University
2
Objective
Given a 2D slice through an aggregate material,
create a 3D volume with a comparable appearance.
3
Real-World Materials

• Concrete
• Asphalt
• Terrazzo
• Igneous
• minerals
• Porous
• materials

4
Independently Recover

• Particle distribution
• Color
• Residual noise

5
In Our Toolbox
The study of 3D properties based on 2D
observations.
6
Prior Work Texture Synthesis

• 2D 2D
• 3D 3D

Efros Leung 99

• Procedural Textures

7
Prior Work Texture Synthesis
Input
Heeger Bergen, 95
8
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

9
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

10
Distributions of Spheres

• 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)

11
Recovering Sphere Distributions
Profile density (number of circles per unit
area)
Particle density (number of spheres per unit
volume)
Mean caliper particle diameter
12
Recovering Sphere Distributions
Group profiles and particles into n
bins according to diameter
13
Recovering Sphere Distributions
Note that the profile source is ambiguous
For the following examples, n 4
14
Recovering Sphere Distributions
How many profiles of the largest size?

Probability that particle NV(j) exhibits
profile NA(i)
15
Recovering Sphere Distributions
How many profiles of the smallest size?




Probability that particle NV(j) exhibits
profile NA(i)
16
Recovering Sphere Distributions
Putting it all together

17
Recovering Sphere Distributions
Some minor rearrangements

Maximum diameter
Normalize probabilities for each column j
18
Recovering Sphere Distributions
K is upper-triangular and invertible
For spheres, we can solve for K analytically
for
otherwise
19
Other Particle Types
We cannot classify arbitrary particles by d/dmax
Instead, we choose to use
Approach Collect statistics for 2D profiles and
3D particles
20
Profile Statistics
Segment input image to obtain profile densities
NA.
Input
Segmentation
Bin profiles according to their area,
21
Particle Statistics

• Polygon meshrandom orientation
• Render

22
Particle Statistics
Look at thousands of random slices to obtain H
and K
Example probabilities of for simple
particles
probability
23
Scale Factor

• Scale factor s to relate the size of particle P
  to the size of the particles in input image
• profile maximum area
• input image
• particle P
• Mean caliper diameter

24
Recovering Particle Distributions
Just like before,
Use NV to populate a synthetic volume.
25
Managing Multiple Particle Types

• particle typei
• mean caliper diameter
• representative matrix
• distribution
• probability that a particleis type i P( i )
• total particle density

26
Reconstructing the Volume

• Particle Positions
• Color
• Adding Fine Detail

27
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

28
Recovering Color
Select mean particle colors from segmented
regions in the input image
Input
Mean Colors
Synthetic Volume
29
Recovering Noise
How can we replicate the noisy appearance of the
input?
-

Input
Mean Colors
Residual
30
Putting it all together
Input
Synthetic volume
31
Prior Work Revisited
Input
Heeger Bergen 95
Our result
32
Results- Testing Precision
Input distribution
Estimated distribution
33
Result- Comparison
34
Collection of Particle Shapes

• Cant predict exact particle shapes
• Unable to count small profiles
• Limited to fewer profile observation
• Calculations error

35
Results Physical Data
Physical Model
Heeger Bergen 95
Our Method
36
Results
Input
Result
37
Results
Input
Result
38
Summary

• Particle distribution
• Stereological techniques
• Color
• Mean colors of segmented profiles
• Residual noise
• Replicated using Heeger Bergen 95

39
Future Work

• Automated particle construction
• Extend technique to other domains and anisotropic
  appearances
• Perceptual analysis of results