stereological techniques for solid textures
DESCRIPTION
Julie Dorsey Yale University. Holly Rushmeier Yale University. Stereological Techniques for Solid Textures. Rob Jagnow MIT. Objective. Given a 2D slice through an aggregate material, create a 3D volume with a comparable appearance. Real-World Materials. Concrete Asphalt Terrazzo - PowerPoint PPT PresentationTRANSCRIPT
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Stereological Techniquesfor Solid Textures
Rob Jagnow
MIT
Julie Dorsey
Yale University
Holly Rushmeier
Yale University
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Given a 2D slice through an aggregate material, create a 3D volume with a comparable appearance.
ObjectiveObjective
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Real-World MaterialsReal-World Materials
• Concrete
• Asphalt
• Terrazzo
• Igneous
minerals
• Porous
materials
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Independently Recover…Independently Recover…
• Particle distribution
• Color
• Residual noise
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Stereology (ster'e-ol' -je)
e
The study of 3Dproperties based on2D observations.
In Our Toolbox…In Our Toolbox…
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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
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Prior Work – Texture SynthesisPrior Work – Texture Synthesis
Input Heeger & Bergen, ’95
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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
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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:
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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
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Recovering Sphere DistributionsRecovering Sphere Distributions
Note that the profile source is ambiguous
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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)
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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
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Recovering Sphere DistributionsRecovering Sphere Distributions
Putting it all together…
AN VNK
=
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Recovering Sphere DistributionsRecovering Sphere Distributions
Some minor rearrangements…
= maxd KAN VN
njKn
iij /
1
Normalize probabilities for each column j:
= Maximum diametermaxd
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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:
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Testing precisionTesting precision
Inputdistribution
Estimateddistribution
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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:
+
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Profile StatisticsProfile Statistics
Segment input image to obtain profile densities NA.
Bin profiles according to their area, max/ AA
Input Segmentation
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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
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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,
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Recovering ColorRecovering Color
Select mean particle colors fromsegmented regions in the input image
Input Mean ColorsSyntheticVolume
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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
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without noise
Putting it all togetherPutting it all together
Input
Synthetic volumewith noise
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Prior Work – RevisitedPrior Work – Revisited
Input Heeger & Bergen ’95 Our result
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Results – Physical DataResults – Physical Data
PhysicalModel
Heeger &Bergen ’95
Our Method
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ResultsResultsInput Result
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ResultsResults
Input Result
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SummarySummary
• Particle distribution– Stereological techniques
• Color– Mean colors of segmented profiles
• Residual noise– Replicated using Heeger & Bergen ’95
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Future WorkFuture 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