1 Prior Work Stereology Prior Work Stereology Prior Work - - PDF document

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Stereological Techniques for Solid Textures

Rob Jagnow MIT Julie Dorsey Yale University Holly Rushmeier Yale University

Given a 2D slice through an aggregate material, create a 3D volume with a comparable appearance.

Objective Objective Real-World Materials Real-World Materials

• Concrete
• Asphalt
• Terrazzo
• Igneous

minerals

• Porous

materials

Independently Recover… Independently Recover…

• Particle distribution
• Color
• Residual noise

Stereology (ster'e-ol' -je) e The study of 3D properties based on 2D observations.

In Our Toolbox… In Our Toolbox…

Prior Work – Texture Synthesis Prior Work – Texture Synthesis

• 2D 2D
• 3D 3D

Efros & Leung ’99

• 2D 3D

– Heeger & Bergen 1995 – Dischler et al. 1998 – Wei 2003

Heeger & Bergen ’95 Wei 2003

• Procedural Textures

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Prior Work – Texture Synthesis Prior Work – Texture Synthesis

Input Heeger & Bergen, ’95

Prior Work – Stereology 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 Recovering Sphere Distributions

A

N H

V

N

= Profile density (number of circles per unit area) = Mean caliper particle diameter = Particle density (number of spheres per unit volume)

V A

N H N =

The fundamental relationship

• f stereology:

Recovering Sphere Distributions Recovering Sphere Distributions

} 1 { ), ( n i i N A ≤ ≤

Group profiles and particles into n bins according to diameter

} 1 { ), ( n i i NV ≤ ≤

Particle densities = Profile densities = For the following examples, n = 4

Recovering Sphere Distributions Recovering Sphere Distributions

Note that the profile source is ambiguous

Recovering Sphere Distributions Recovering Sphere Distributions

How many profiles of the largest size?

) 4 (

A

N ) 4 (

V

N

44

K

=

ij

K

= Probability that particle NV(j) exhibits profile NA(i)

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Recovering Sphere Distributions Recovering Sphere Distributions

How many profiles of the smallest size?

) 1 (

A

N ) 4 (

V

N

11

K

= + + +

12

K

13

K

14

K ) 3 (

V

N ) 2 (

V

N ) 1 (

V

N

= Probability that particle NV(j) exhibits profile NA(i)

ij

K Recovering Sphere Distributions Recovering Sphere Distributions

Putting it all together…

A

N

V

N K

=

Recovering Sphere Distributions Recovering Sphere Distributions

Some minor rearrangements… =

max

d K

A

N

V

N n j K

n i ij

/

1

=

∑

=

Normalize probabilities for each column j: = Maximum diameter

max

d

Recovering Sphere Distributions Recovering Sphere Distributions

V A

KN d N

max

=

For spheres, we can solve for K analytically:

( )

⎪ ⎩ ⎪ ⎨ ⎧ − − − − ⋅ = ) 1 ( / 1

2 2 2 2

i j i j n Kij

K is upper-triangular and invertible

for

i j ≥

• therwise

A V

N K d N

1 max

1

−

=

Solving for particle densities:

Testing precision Testing precision

Input distribution Estimated distribution

Other Particle Types Other Particle Types

We cannot classify arbitrary particles by d/dmax Instead, we choose to use

max

/ A A Approach: Collect statistics for 2D profiles and 3D particles Algorithm inputs:

+

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Profile Statistics Profile Statistics

Segment input image to obtain profile densities NA. Bin profiles according to their area,

max

/ A A Input Segmentation

Particle Statistics Particle Statistics

Look at thousands of random slices to obtain

H and K

Example probabilities of for simple particles

max

/ A A

p r

• b

a b i l i t y

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 sphere cube long ellipsoid flat ellipsoid

A/Amax probability

Recovering Particle Distributions Recovering Particle Distributions

Just like before,

V A

KN H N = Use NV to populate a synthetic volume.

A V

N K H N

1

1

−

=

Solving for the particle densities,

Recovering Color Recovering Color

Select mean particle colors from segmented regions in the input image

Input Mean Colors Synthetic Volume

Recovering Noise 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 with noise

Putting it all together Putting it all together

Input Synthetic volume

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Prior Work – Revisited Prior Work – Revisited

Input Heeger & Bergen ’95 Our result

Results – Physical Data Results – Physical Data

Physical Model Heeger & Bergen ’95 Our Method

Results Results

Input Result

Results Results

Input Result

Summary Summary

• Particle distribution

– Stereological techniques

• Color

– Mean colors of segmented profiles

• Residual noise

– Replicated using Heeger & Bergen ’95

Future Work Future Work

• Automated particle construction
• Extend technique to other domains and

anisotropic appearances

• Perceptual analysis of results

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Thanks to… Thanks to…

• Maxwell Planck, undergraduate assistant
• Virginia Bernhardt
• Bob Sumner
• John Alex