Transcript Course

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…
Original
Image
Skull
Strip
Classify Brain
Tissues
Non-uniformity
Correct
Extract Brain
Surfaces
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
Template
Spatially Normalised
MRI
Original MRI
Affine register
Affine Transform
Segment
Priors
Grey Matter
Spatial Normalisation
- estimation
Spatial Normalisation
- writing
Deformation
Unified Segmentation
• The
solution to this circularity is to
everything in the same Generative Model.
•A
put
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
g
a0
a
Ca
c1
y1
m
c2
y2
s2
c3
y3
b
cI
yI
Cb
b0
Gaussian Probability Density
• If intensities are assumed to be Gaussian of mean
mk and variance s2k, 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 gk 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 nonGaussian 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 logprobabilities:
Modelling a Bias Field
•A
bias field is included, such that the required
scaling at voxel i, parameterised by b, is ri(b).
• Replace the means by mk/ri(b)
• Replace the variances by (sk/ri(b))2
Modelling a Bias Field
• After rearranging:
y
r(b)
y r(b)
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 g is then:
Deforming the Tissue Probability Maps
• Tissue probability
•
images are
deformed according
to parameters a.
The probability of
obtaining class k at
voxel i, given
weights g and
parameters a is
then:
The Extended Model
•
By combining the modified P(ci=k|q) and P(yi|ci=k,q), 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 Descent
Start
Optimum
Alternate between
optimising different groups
of parameters
Schematic of optimisation
Repeat until convergence...
Hold g, m, s2 and a constant, and minimise E w.r.t. b
- Levenberg-Marquardt strategy, using dE/db and d2E/db2
Hold g, m, s2 and b constant, and minimise E w.r.t. a
- Levenberg-Marquardt strategy, using dE/da and d2E/da2
Hold a and b constant, and minimise E w.r.t. g, m and s2
-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 a and b are updated by
• Increase
l to improve stability (at expense of
decreasing speed of convergence).
Expectation Maximisation is used to
update m, s2 and g
• For iteration (n), alternate between:
• E-step: Estimate belonging probabilities by:
• M-step: Set q(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
ma and mb
• Covariance matrix Sa and Sb
• -log[P(a)] = (a-ma)TSa-1(a-ma) + 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.
Joint Probability Histogram
See D'Agostino, Maes, Vandermeulen &
P. Suetens. “Non-rigid Atlas-to-Image
Registration by Minimization of ClassConditional Image Entropy”. Proc.
MICCAI 2004. LNCS 3216, 2004. Pages
745-753.
Background voxels
excluded
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.
Spatially
normalised
BrainWeb
phantoms (T1,
T2 and PD)
Tissue
probability
maps of GM
and WM
Cocosco, Kollokian, Kwan & Evans. “BrainWeb: Online Interface to a 3D MRI Simulated Brain Database”. NeuroImage 5(4):S425 (1997)
Bayes Rule
•
•
y
q
• P(q|y)
• P(y|q)
• P(q)
• P(y)
• P(q,y)
- the data
- a theory, model, or set of parameters
- probability of q given y (posterior probability)
- probability of y given q (likelihood)
- probability of q (prior probability)
- probability of y (evidence)
- probability of q and y (joint probability)