SPM5 Segmentation

A Growing Trend

•Larger and more complex models are being produced to explain brain imaging data.•Bigger and better computers

•allow more powerful models to be used

•More experience among software developers•Older and wiser

•More engineers - rather than e.g. psychiatrists & biochemists

•This presentation is about combining various preprocessing procedures for anatomical images into a single generative model.

Traditional View of Pre-processing

•Brain image processing is often thought of as a pipeline procedure.

•One tool applied before another etc...•For example…

OriginalImage

SkullStrip

Non-uniformityCorrect

Classify BrainTissues

Extract BrainSurfaces

Another example (for VBM)

Bias Correction Informs Registration

•MRI images are corrupted by a smooth intensity non-uniformity (bias).

•Image intensity non-uniformity artefact has a negative impact on most registration approaches.

•Much better if this artefact is corrected.Image

with bias artefact

Corrected image

Bias Correction Informs Segmentation

•Similar tissues no longer have similar intensities.

•Artefact should be corrected to enable intensity-based tissue classification.

Registration Informs Segmentation•SPM99 and SPM2 require tissue probability

maps to be overlaid prior to segmentation.

Segmentation Informs Bias Correction

•Bias correction should not eliminate differences between tissue classes.

•Can be done by•make all white matter about the same intensity

•make all grey matter about the same intensity

•etc

•Currently fairly standard practice to combine bias correction and tissue classification

Segmentation Informs Registration

Original MRITemplate

Grey MatterSegment

Affine register

Priors Deformation

Affine Transform

Spatial Normalisation

- estimation

Spatial Normalisation

- writing

Spatially Normalised

MRI

Unified Segmentation

•The solution to this circularity is to put everything in the same Generative Model.•A MAP solution is found by repeatedly alternating

among classification, bias correction and registration steps.

•The Generative Model involves:•Mixture of Gaussians (MOG)

•Bias Correction Component

•Warping (Non-linear Registration) Component

Generative Model

y1c1

y2

y3

c2

c3

C

CyIcI

Gaussian Probability Density

•If intensities are assumed to be Gaussian of mean k and variance 2

k, then the probability of a value yi is:

Non-Gaussian Probability Distribution

•A non-Gaussian probability density function can be modelled by a Mixture of Gaussians (MOG):

Mixing proportion - positive and sums to one

Belonging Probabilities

Belonging probabilities are assigned by normalising to one.

Mixing Proportions

•The mixing proportion k represents the prior probability of a voxel being drawn from class k - irrespective of its intensity.

•So:

Non-Gaussian Intensity Distributions

•Multiple Gaussians per tissue class allow non-Gaussian intensity distributions to be modelled.

Probability of Whole Dataset

•If the voxels are assumed to be independent, then the probability of the whole image is the product of the probabilities of each voxel:

•It is often easier to work with negative log-probabilities:

Modelling a Bias Field

•A bias field is included, such that the required scaling at voxel i, parameterised by , is i().

•Replace the means by k/i()

•Replace the variances by (k/i())2

Modelling a Bias Field•After rearranging:

()y y ()

Tissue Probability Maps

•Tissue probability maps (TPMs) are used instead of the proportion of voxels in each Gaussian as the prior.

ICBM Tissue Probabilistic Atlases. These tissue probability maps are kindly provided by the International Consortium for Brain Mapping, John C. Mazziotta and Arthur W. Toga.

“Mixing Proportions”• Tissue probability maps for

each class are included.

• The probability of obtaining class k at voxel i, given weights is then:

Deforming the Tissue Probability Maps• Tissue probability

images are deformed according to parameters .

• The probability of obtaining class k at voxel i, given weights and parameters is then:

The Extended Model

• By combining the modified P(ci=k|) and P(yi|ci=k,), the overall objective function (E) becomes:

The Objective Function

Optimisation

•The “best” parameters are those that minimise this objective function.

•Optimisation involves finding them.

•Begin with starting estimates, and repeatedly change them so that the objective function decreases each time.

Steepest DescentStart

Optimum

Alternate between optimising different

groups of parameters

Schematic of optimisation

Repeat until convergence...

Hold , , 2 and constant, and minimise E w.r.t. - Levenberg-Marquardt strategy, using dE/d and d2E/d2

Hold , , 2 and constant, and minimise E w.r.t. - Levenberg-Marquardt strategy, using dE/d and d2E/d2

Hold and constant, and minimise E w.r.t. , and 2

-Use an Expectation Maximisation (EM) strategy.

end

Levenberg-Marquardt Optimisation

•LM optimisation is used for the nonlinear registration and bias correction components.

•Requires first and second derivatives of the objective function (E).

•Parameters and are updated by

•Increase to improve stability (at expense of decreasing speed of convergence).

Expectation Maximisation is used to update , 2 and

•For iteration (n), alternate between:•E-step: Estimate belonging probabilities by:

•M-step: Set (n+1) to values that reduce:

Linear Regularisation

•Some bias fields and distortions are more probable (a priori) than others.

•Encoded using Bayes rule:

•Prior probability distributions can be modelled by a multivariate normal distribution.• Mean vector and

• Covariance matrix and

• -log[P()] = (-T-1( + const

Initial Affine Registration

The procedure begins with a Mutual Information affine registration of the image with the tissue probability maps. MI is computed from a 4x256 joint probability histogram.

See D'Agostino, Maes, Vandermeulen & P. Suetens. “Non-rigid Atlas-to-Image Registration by Minimization of Class-Conditional Image Entropy”. Proc. MICCAI 2004. LNCS 3216, 2004. Pages 745-753. Background voxels

excluded

Joint Probability Histogram

Background Voxels are Excluded

An intensity threshold is found by fitting image intensities to a mixture of two Gaussians. This threshold is used to exclude most of the voxels containing only air.

Tissue probability maps of GM

and WM

Spatially normalised BrainWeb phantoms

(T1, T2 and PD)

Cocosco, Kollokian, Kwan & Evans. “BrainWeb: Online Interface to a 3D MRI Simulated Brain Database”. NeuroImage 5(4):S425 (1997)

Bayes Rule

• y - the data

• - a theory, model, or set of parameters

• P(|y) - probability of given y (posterior probability)

• P(y|) - probability of y given (likelihood)

• P() - probability of (prior probability)

• P(y) - probability of y (evidence)

• P(,y) - probability of and y (joint probability)