Transcript Segmentation Foundations • Easy Segmentation • Feasible Segmentation

Segmentation Foundations
• Easy Segmentation
– Tissue/Air (except bone in MR)
– Bone in CT
• Feasible Segmentation
– White Matter/Gray Matter: MRI
– M.S. White Matter Lesions: MRI
National Alliance for Medical Image Computing
http://na-mic.org
Statistical Classification
• Probabilistic model of
intensity as a function
of (tissue) class
• Intensity data
• Prior model
[Duda, Hart 78][MRI: MikeVannier late 80s]
National Alliance for Medical Image Computing
http://na-mic.org
Classification of
voxels
Measurement Model
• Characterize sensor
p(x|tissue class J)
intensity
probability
density
Tissue class conditional model
of signal intensity
mean for tissue J
National Alliance for Medical Image Computing
http://na-mic.org
A bit of notation…
• Estimate  by finding the one that
maximizes the function f
National Alliance for Medical Image Computing
http://na-mic.org
Maximum Likelihood (ML)
Estimation
• Estimate parameters to maximize
probability of observed data conditioned on
parameters .
• yo : observed data
• p(y|) : Measurement Model
•  : Model Parameters
National Alliance for Medical Image Computing
http://na-mic.org
Example
p(x|gray matter)
p(x|white matter)
intensity
National Alliance for Medical Image Computing
http://na-mic.org
Example - revisited
gray matter
white matter
threshold
National Alliance for Medical Image Computing
http://na-mic.org
Multiple Sclerosis
PDw
National Alliance for Medical Image Computing
http://na-mic.org
T2w
Provided by S Warfield
T2 Intensity
Dual Echo MRI Feature Space
csf
severe
lesions
air
wm
gm
PD Intensity
National Alliance for Medical Image Computing
http://na-mic.org
Detail
• MS Lesions are “graded
phenomenon” in MRI, and can be
anywhere on the curve
severe
csf
lesions
mild
healthy
wm
National Alliance for Medical Image Computing
http://na-mic.org
gm
Multiple Sclerosis
PDw
T2w
National Alliance for Medical Image Computing
http://na-mic.org
Segmentation
Provided by S Warfield
Maximum A-Posteriori (MAP)
Estimation
• Estimate parameters to maximize
posterior probability model parameters
conditioned on observed data
• Use Baye’s rule – ignore denominator
• p() : Prior Model
National Alliance for Medical Image Computing
http://na-mic.org
Provided by S Warfield
Multiple Sclerosis
PDw
T2w
kNN
National Alliance for Medical Image Computing
http://na-mic.org
SVC
Background: Intensity
Inhomogeneities in MRI
• MRI signal derived from RF
signals…
• Intra Scan Inhomogeneities
– “Shading” … from coil imperfections
– interaction with tissue?
• Inter Scan Inhomogeneities
– Auto Tune
– Equipment Upgrades
National Alliance for Medical Image Computing
http://na-mic.org
ML Estimation – with missing data
• x : missing data (true labeling)
• y0 : observed intensities
•  : (parameters of) bias field
National Alliance for Medical Image Computing
http://na-mic.org
ML Estimation – EM Approach
• E []: Expected value under p(x|yo, )
• Take expectation of objective function with respect
to the missing data, conditioned on everything we
know
• x : missing data (true labeling)
• y0 : observed intensities
•  : (parameters of) bias field
National Alliance for Medical Image Computing
http://na-mic.org
EM Algorithm
• General exponential family
• Iterate to convergence:
M step:
E step:
National Alliance for Medical Image Computing
http://na-mic.org
EM Algorithm: Example
• Measurement Model
– Tissue intensity properties with bias
correction
• Missing Data
– Unknown true classification
• Prior Models
– Tissue Frequencies
– Intensity Correction is Low Frequency
• ML estimate of bias
National Alliance for Medical Image Computing
http://na-mic.org
EM-Segmentation
E-Step
Compute tissue posteriors
using current intensity
correction.
Estimate intensity correction
using residuals based on
current posteriors.
M-Step
National Alliance for Medical Image Computing
http://na-mic.org
Provided by T Kapur
EM Segmentation…
Seg Result
w/o EM
Seg Result
With EM
PD, T2 Data
National Alliance for Medical Image Computing
http://na-mic.org
EM Segmentation…
External Surface of Brain
National Alliance for Medical Image Computing
http://na-mic.org
EM Segmentation…
WM Surface with EM
National Alliance for Medical Image Computing
http://na-mic.org
WM Surface w/o EM
EM Segmentation: MS
Example
PD
National Alliance for Medical Image Computing
http://na-mic.org
T2
Data provided by Charles Guttmann
EM Segmentation: MS
Example
Seg w/o EM
National Alliance for Medical Image Computing
http://na-mic.org
Seg with EM
Prior Probability Models
• Simple: Frequency of Tissues
• More Interesting:
– Powerful Mechanism for Incorporating
Domain Knowledge into Segmentation
• Tissue properties
• Relative Location of Structures
• Atlases
National Alliance for Medical Image Computing
http://na-mic.org
Prior Model Example: EM-MF
Segmentation
• Tina Kapur PhD thesis
• EM Segmentation, augmented with
– Ising prior of tissue homogeneity
• Solved with Mean Field Approxomation
– Prior on relative position of organs
• Spatially Conditioned Models
National Alliance for Medical Image Computing
http://na-mic.org
Prior Models: Ising Model
• Ising Model can capture the phenomenon
of piecewise-homogeneity.
• Initially used in Statistical Physics to
model the magnetic domains in
Ferromagnetism.
• Used in Medical Image Processing to
model the piecewise-homogeneity of
Tissue.
National Alliance for Medical Image Computing
http://na-mic.org
Prior Models: Ising Model
• Ising Model relaxes spatial independence
assumption
• Voxels depend conditionally on (only)
their neighbors
• More probable to agree with neighbor
National Alliance for Medical Image Computing
http://na-mic.org
Define the Neighborhood
1st Order Lattice
2nd Order Lattice
6 Neighbors
26 Neighbors
Reduce calculation cost => use 1st order Lattice
Neighbors = {East, South, West, North, Up, Down}
National Alliance for Medical Image Computing
http://na-mic.org
Provided by K Pohl
Potts Model
• Potts model generalizes Ising model
so that each lattice site takes on
several values (more than two).
• Frequently used to model tissues
(e.g. White Matter, Gray Matter, CSF,
Fat, Air, etc.)
National Alliance for Medical Image Computing
http://na-mic.org
Some Results
EM
National Alliance for Medical Image Computing
http://na-mic.org
EM-MF
Provided by T Kapur
More Results
Noisy MRI
EM Segmentation
National Alliance for Medical Image Computing
http://na-mic.org
EM-MF Segmentation
Provided by T Kapur
Posterior Probabilities (EM)
White
matter
National Alliance for Medical Image Computing
http://na-mic.org
Gray
matter
Provided by T Kapur
Posterior Probabilities (EMMF)
White
matter
National Alliance for Medical Image Computing
http://na-mic.org
Gray
matter
Provided by T Kapur
Segmentation of 31
Structures
Kilian Pohl PhD (defense several weeks ago)
National Alliance for Medical Image Computing
http://na-mic.org
Segmentation of 31
Structures
Lower Front
National Alliance for Medical Image Computing
http://na-mic.org
Provided by Kilian Pohl
Segmentation of 31
Structures
Superior Temporal Gyrus
National Alliance for Medical Image Computing
Provided by Kilian Pohl
http://na-mic.org
•The End
National Alliance for Medical Image Computing
http://na-mic.org