Stereological Techniquesfor 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.

ObjectiveObjective

Real-World MaterialsReal-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 3Dproperties based on2D observations.

In Our Toolbox…In Our Toolbox…

Prior Work – Texture SynthesisPrior Work – Texture Synthesis

• 2D 2D

• 3D 3DEfros & Leung ’99

• 2D 3D– Heeger & Bergen 1995– Dischler et al. 1998– Wei 2003

Heeger & Bergen ’95

Wei 2003

• Procedural Textures

Prior Work – Texture SynthesisPrior Work – Texture Synthesis

Input Heeger & Bergen, ’95

Prior Work – StereologyPrior Work – Stereology

• Saltikov 1967Particle size distributions from section measurements

• Underwood 1970Quantitative Stereology

• Howard and Reed 1998Unbiased Stereology

• Wojnar 2002Stereology from one of all the possible angles

Recovering Sphere DistributionsRecovering Sphere Distributions

AN

H

VN

= Profile density (number of circles per unit area)

= Mean caliper particle diameter

= Particle density (number of spheres per unit volume)

VA NHN

The fundamental relationshipof stereology:

Recovering Sphere DistributionsRecovering Sphere Distributions

}1{),( niiN A

Group profiles and particles into n binsaccording to diameter

}1{),( niiNV Particle densities =

Profile densities =

For the following examples, n = 4

Recovering Sphere DistributionsRecovering Sphere Distributions

Note that the profile source is ambiguous

Recovering Sphere DistributionsRecovering Sphere Distributions

How many profiles of the largest size?

)4(AN )4(VN44K

=

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

Recovering Sphere DistributionsRecovering Sphere Distributions

How many profiles of the smallest size?

)1(AN )4(VN11K

= + + +12K 13K 14K)3(VN)2(VN)1(VN

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

Recovering Sphere DistributionsRecovering Sphere Distributions

Putting it all together…

AN VNK

=

Recovering Sphere DistributionsRecovering Sphere Distributions

Some minor rearrangements…

= maxd KAN VN

njKn

iij /

1

Normalize probabilities for each column j:

= Maximum diametermaxd

Recovering Sphere DistributionsRecovering Sphere Distributions

VA KNdN max

For spheres, we can solve for K analytically:

0

)1(/1 2222 ijijnK ij

K is upper-triangular and invertible

for ij otherwise

AV NKdN 1

max

1 Solving for particle densities:

Testing precisionTesting precision

Inputdistribution

Estimateddistribution

Other Particle TypesOther Particle Types

We cannot classify arbitrary particles by d/dmax

Instead, we choose to use max/ AA

Approach: Collect statistics for 2D profiles and 3D particles

Algorithm inputs:

+

Profile StatisticsProfile Statistics

Segment input image to obtain profile densities NA.

Bin profiles according to their area, max/ AA

Input Segmentation

Particle StatisticsParticle Statistics

Look at thousands of random slices to obtain H and K

Example probabilities of for simple particlesmax/ AA

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

spherecubelong ellipsoidflat ellipsoid

A/Amax

pro

ba

bili

ty

Recovering Particle DistributionsRecovering Particle Distributions

Just like before, VA KNHN

Use NV to populate a synthetic volume.

AV NKH

N 11

Solving for the particle densities,

Recovering ColorRecovering Color

Select mean particle colors fromsegmented regions in the input image

Input Mean ColorsSyntheticVolume

Recovering NoiseRecovering NoiseHow 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

without noise

Putting it all togetherPutting it all together

Input

Synthetic volumewith noise

Prior Work – RevisitedPrior Work – Revisited

Input Heeger & Bergen ’95 Our result

Results – Physical DataResults – Physical Data

PhysicalModel

Heeger &Bergen ’95

Our Method

ResultsResultsInput Result

ResultsResults

Input Result

SummarySummary

• Particle distribution– Stereological techniques

• Color– Mean colors of segmented profiles

• Residual noise– Replicated using Heeger & Bergen ’95

Future WorkFuture Work

• Automated particle construction

• Extend technique to other domains and anisotropic appearances

• Perceptual analysis of results

Thanks to…Thanks to…

• Maxwell Planck, undergraduate assistant

• Virginia Bernhardt

• Bob Sumner

• John Alex