stereological techniques for solid textures rob jagnow mit julie dorsey yale university holly...

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

Estimating 3D DistributionsEstimating 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 SpheresDistributions 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)

maxd

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

Group profiles and particles into n binsaccording to diameter

}1{),( niiN A }1{),( niiNV

Particle densities =

Profile densities =

Densities , are related by the values ijKANVN

Relative probabilities ：

－ a sphere in the j th histogram bin with diameter － a profile in the i th histogram bin with diameter

ijK

n

id

n

i

)1(

n

j

Recovering Sphere DistributionsRecovering Sphere Distributions

Note that the profile source is ambiguous

For the following examples, n = 4

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:

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

• Polygon mesh ： random orientation

• Render

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

Scale FactorScale 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

Pmaximg /AAs

PmaxA

imgA

PHsH

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,

Managing Multiple Particle TypesManaging Multiple Particle Types

• particle type ： i• mean caliper diameter ：• representative matrix ：• distribution ：• probability that a particle

is type i ： P( i )• total particle density ：

iH

iK

ViN

i

ViiiA NKHN )(

i

ViiA NiPKHN ))((

ViV NN

Vi

ii NiPKH ))((

Ai

iiV NiPKHN1

))((

Reconstructing the VolumeReconstructing the Volume

• Particle Positions

• Color

• Adding Fine Detail

Particle Position - AnnealingParticle 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 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 volume

Prior Work – RevisitedPrior Work – Revisited

Input Heeger & Bergen ’95 Our result

Results- Testing PrecisionResults- Testing Precision

Inputdistribution

Estimateddistribution

Result- ComparisonResult- Comparison

Collection of Particle ShapesCollection of Particle Shapes

• Can’t predict exact particle shapes

• Unable to count small profiles

• Limited to fewer profile observation

Calculations error

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