National Alliance for Medical Image Computing Segmentation Foundations Easy Segmentation –Tissue/Air (except bone in MR) –Bone in CT

National Alliance for Medical Image Computing http://na-mic.org

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


Statistical Classification

• Probabilistic model of intensity as a function of (tissue) class

• Intensity data

• Prior model

Classification ofvoxels

[Duda, Hart 78][MRI: MikeVannier late 80s]


Measurement Model

• Characterize sensor

p(x|tissue class J)

probabilitydensity

intensity Tissue class conditional modelof signal intensity

mean for tissue J


A bit of notation…

• Estimate by finding the one that maximizes the function f


Maximum Likelihood (ML) Estimation

• Estimate parameters to maximize probability of observed data conditioned on parameters .

• yo : observed data• p(y|) : Measurement Model• Model Parameters


Example

intensity

p(x|gray matter)

p(x|white matter)


Example - revisited

white matter

threshold

gray matter


Multiple Sclerosis

PDw T2w

Provided by S Warfield


Dual Echo MRI Feature Space

csfseverelesions

gmwmair

T2

Inte

nsit

y

PD Intensity


Detail• MS Lesions are “graded

phenomenon” in MRI, and can be anywhere on the curve

gmwm

lesionscsf

healthymild

severe


Multiple Sclerosis

PDw T2w Segmentation



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


Multiple Sclerosis

PDwT2w

kNN 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


ML Estimation – with missing data

• x : missing data (true labeling)

• y0 : observed intensities

• : (parameters of) bias field


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


EM Algorithm

• General exponential family• Iterate to convergence:

E step:

M step:


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


Estimate intensity correctionusing residuals based on current posteriors.

Compute tissue posteriors using current intensity correction.

M-Step

E-Step

EM-Segmentation

Provided by T Kapur


EM Segmentation…

PD, T2 Data

Seg Resultw/o EM

Seg ResultWith EM


EM Segmentation…

External Surface of Brain


EM Segmentation…

WM Surface with EM WM Surface w/o EM


EM Segmentation: MS Example

Data provided by Charles Guttmann

PD T2


EM Segmentation: MS Example

Seg w/o EM 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


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


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.


Prior Models: Ising Model

• Ising Model relaxes spatial independence assumption

• Voxels depend conditionally on (only) their neighbors

• More probable to agree with neighbor


Define the Neighborhood

2nd Order Lattice

26 Neighbors

1st Order Lattice

6 Neighbors

Reduce calculation cost => use 1st order LatticeNeighbors = {East, South, West, North, Up, Down}

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.)


Some ResultsEM EM-MF

Provided by T Kapur


More Results

Noisy MRI EM Segmentation EM-MF Segmentation

Provided by T Kapur


Posterior Probabilities (EM)

Whitematter

Graymatter

Provided by T Kapur


Posterior Probabilities (EM-MF)

Whitematter

Graymatter

Provided by T Kapur


Segmentation of 31 Structures

Kilian Pohl PhD (defense several weeks ago)



Lower Front

Provided by Kilian Pohl



Superior Temporal Gyrus

Provided by Kilian Pohl


•The End

Documents

National Alliance for Medical Image Computing Segmentation Foundations Easy Segmentation –Tissue/Air (except bone in MR) –Bone in CT