Transcript Statistical Parametric Mapping Institute of Neurology

Segmentation
C. Phillips, Institut Montefiore, ULg, 2006
Definition
In image analysis, segmentation is the partition
of a digital image into multiple regions (sets of
pixels), according to some criterion.
The goal of segmentation is typically to locate
certain objects of interest which may be depicted
in the image.
Segmentation criteria can be arbitrarily complex,
and take into account global as well as local
criteria. A common requirement is that each region
must be connected in some sense.
A simple example of segmentation is thresholding
a grayscale image with a fixed threshold t: each
pixel p is assigned to one of two classes, P0 or P1,
depending on whether I(p) < t or I(p) ≥ t.
t=.5
Example: medical imaging...
How to fix the threshold ?
Goal of brain image segmentation
Split the head volume into its « main »
components:
•
•
•
•
gray matter (GM)
white matter (WM)
cerebrol-spinal fluid (CSF)
the rest/others
• (tumour)
Segmentation approaches
Manual segmentation:
an operator classifies the voxels manually
Segmentation approaches
Semi-automatic segmentation:
an operator defines a set of parameters,
that are passed to an algorithm
Example:
threshold at
t=200
Segmentation approaches
Automatic segmentation:
no operator intervention
 objective and reproducible
Intensity based segmentation
Model the histogram of the image !
Segmentation - Mixture Model
• Intensities are modelled by a mixture of K
Gaussian distributions, parameterised by:
– means i
– variances  i
– mixing proportions  i
1
1
2
2
3
3
Segmentation - Algorithm
1
Starting estimates for
belonging probabilities
Compute Gaussian
parameters from
belonging probabilities
1
2
2
3
3
Compute belonging
probabilities from
Gaussian parameters
No
Converged ?
Yes
STOP
Segmentation - Problems
Noise & Partial volume effect
Segmentation - Problems
Intensity bias field
MR images are corrupted by a smooth
intensity non-uniformity (bias).
Image with
bias
artefact
Corrected
image
Segmentation - Priors
Overlay prior belonging probability maps to
assist the segmentation
– Prior probability of each voxel being of a
particular type is derived from segmented
images of 151
subjects
• Assumed to be
representative
– Requires initial
registration to
standard space.
Unified approach:
segmentation-correction-registration
• Bias correction informs segmentation
• Registration informs segmentation
• Segmentation informs bias correction
• Bias correction informs registration
• Segmentation informs registration
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
Gaussian Probability Density
• If intensities are assumed to be
Gaussian of mean k and variance 2k,
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
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 b, is ri(b).
• Replace the means by k/ri(b)
• Replace the variances by (k/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 available.
• 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 a.
• The probability of
obtaining class k
at voxel i, given
weights  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.
Schematic of optimisation
Repeat until convergence...
Hold , , 2 and a constant, and minimise E w.r.t. b
- Levenberg-Marquardt strategy, using dE/db and d2E/db2
Hold , , 2 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. ,  and
2
-Use an Expectation Maximisation (EM) strategy.
end
(Iterated Conditional Mode)
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).
EM is used to update , 2 and 
For iteration (n), alternate between:
– E-step: Estimate belonging probabilities by:
– M-step: Set q(n+1) to values that reduce:
Voxels are assumed independent!
Hidden Markov Random Field
Voxels are NOT independent:
GM voxels are surrounded by other GM voxels,
at least on one side.
Model the intensity and classification of the image
voxels by 2 random field:
• a visible field y for the intensities
• a hidden field c for the classifications
Modify the cost function E:
And, at each voxel, the 6 neighbouring voxels are used
to to build Umrf, imposing local spatial constraints.
Hidden Markov Random Field
T1 image
T2 image
Hidden Markov Random Field
T1 & T2: MoG + hmrf
T1 only: MoG only
White matter
Hidden Markov Random Field
T1 & T2: MoG + hmrf
T1 only: MoG only
Gray matter
Hidden Markov Random Field
T1 & T2: MoG + hmrf
T1 only: MoG only
CSF
Perspectives
•Multimodal segmentation :
1 image is good but 2 is better !
Model the joint histogram using multidimensional normal distributions.
•Tumour detection :
contrasted images to modify the prior images
automatic detection of outliers ?
Thank you for your attention !