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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