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
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