Volumetric Intersubject Registration
John AshburnerWellcome Department of Imaging Neuroscience,
12 Queen Square, London, UK.
Intersubject registration for fMRI
* Inter-subject averaging* Increase sensitivity with more subjects
* Fixed-effects analysis
* Extrapolate findings to the population as a whole* Mixed-effects analysis
* Standard coordinate system* e.g., Talairach & Tournoux space
Typical overview of fMRI analysis
MotionCorrection
Smoothing
SpatialNormalisation
General Linear Model
Statistical Parametric MapfMRI time-series
Parameter Estimates
Design matrix
Anatomical Reference
Overview* Part I: General Inter-subject registration
* Spatial transformations* Affine* Global nonlinear* Local nonlinear
* Objective functions for registration* Likelihood Models
* Mean squared difference* Information Theoretic measures
* Prior Models
* Part II: The Segmentation Method in SPM5
Image Registration
Registration - i.e. Optimise the parameters that describe a spatial transformation between the source and reference (template) images
Transformation - i.e. Re-sample according to the determined transformation parameters
A Mapping from one image to another
Need x, y and z coordinates in one image that correspond to those of another
Affine Transforms* Rigid-body transformations are a subset* Parallel lines remain parallel * Operations can be represented by:
x’ = m11x + m12y + m13z + m14
y’ = m21x + m22y + m23z + m24
z’ = m31x + m32y + m33z + m34
* Or as matrices:Y=Mx
1
z
y
x
1000
mmmm
mmmm
mmmm
1
z
y
x
34333231
24232221
14131211
'
'
'
2D Affine Transforms* Translations by tx and ty
* x’ = x + tx
* y’ = y + ty
* Rotation around the origin by radians* x’ = cos() x + sin() y* y’ = -sin() x + cos() y
* Zooms by sx and sy
* x’ = sx x
* y’ = sy y
*Shear*x’ = x + h y*y’ = y
2D Affine Transforms* Translations by tx and ty
* x’ = 1 x + 0 y + tx
* y’ = 0 x + 1 y + ty
* Rotation around the origin by radians* x’ = cos() x + sin() y + 0* y’ = -sin() x + cos() y + 0
* Zooms by sx and sy:
* x’ = sx x + 0 y + 0
* y’ = 0 x + sy y + 0
*Shear*x’ = 1 x + h y + 0*y’ = 0 x + 1 y + 0
Polynomial Basis Functions
216
1514
2131211
ya
xyaya
xaxaa'x
226
2524
2232221
ya
xyaya
xaxaa'y
As used by Roger Woods’ AIR Software
Cosine Transform Basis Functions
As used by SPM software
SPM Spatial Normalisation
Non-linear registration
* Begin with affine registration* Refine with some non-linear registration
Affine registration
Accuracy of Automated Volumetric Inter-subject Registration
Sulcal misregistration
0
2
4
6
8
10
12
A D M P R SPM2Method
Dis
tanc
e (
mm
)
Hellier et al. Inter subject registration of functional and anatomical data using SPM. MICCAI'02 LNCS 2489 (2002)Hellier et al. Retrospective evaluation of inter-subject brain registration. MIUA (2001)
Local Basis Functions* More detailed
deformations use lots of basis functions with local support.
* Local support means that the basis functions are mostly all zero* Faster computations
Simple addition of displacements
Notice that there is no longer a one-to-one mapping
Generating large one-to-one deformations
The principle behind the one-to-one mappings of viscous fluid registration
Y2 = Y1 Y1 Y3 = Y1 Y2Y1 Y4 = Y1 Y3
Y5 = Y1 Y4 Y6 = Y1 Y5 Y7 = Y1 Y6 Y8 = Y1 Y7
Faster to repeatedly square the deformation
Y1 Y2 = Y1 Y1
Y4 = Y2 Y2 Y8 = Y4 Y4
Note that this is analogous to computing a matrix exponential (c.f. Lie Groups and exponential mappings)
Y16 = Y8 Y8
One-to-One Mappings* One-to-one
mappings break down beyond a certain scale
* The concept of a single “best” mapping may be meaningless at higher resolution Pictures taken from
http://www.messybeast.com/freak-face.htm
Optimisation
* Optimisation involves finding some “best” parameters according to an “objective function”, which is either minimised or maximised
* The “objective function” is often related to a probability based on some model
Value of parameter
Objective function
Most probable solution (global
optimum)Local optimumLocal optimum
Bayes Rule* Most registration procedures maximise a joint
probability of the deformation (warp) and the images (data).* P(Warp,Data) = P(Warp | Data) x P(Data) = P(Data | Warp) x P(Warp)
* In practice, this can be by minimising* -log P(Warp,Data) = -log P(Data | Warp) -log P(Warp)
Likelihood Prior
Mean Squared Difference Objective Function* Assumes one image is a warped version of the
other with Gaussian noise added…* P(fi|t) = (22)-1/2 exp(-(fi-gi(t))2/(22))
so
* -log P(fi |t) = (fi-gi(t))2 /(22) + 1/2 log(22)
* Assumes that voxels are independent...* P(f1,f2,…,fN,...) = P(f1) P(f2) … P(fN)
so
* -log P(f1,f2,…,fN)
= ((f1-g1(t))2 + (f2-g2(t))2 +…+ (fN-gN(t))2)/(22)
+ 1/2 N log(22)
Information Theoretic Approaches* Used when there is no simple relationship
between intensities in one image and those of another
Joint Probability Density* Intensities in one image predict those of another.* Joint probability often represented by a
histogram.
Mutual Information
* MI=ab P(a,b) log2 [P(a,b)/( P(a) P(b) )]* Related to entropy: MI = -H(a,b) + H(a) + H(b)
* H(a) = -a P(a) log P(a) da
* H(a,b) = -a P(a,b) log P(a,b) da
More Joint Probabilities
4x256 Joint Histograms
4x256 Joint Histograms
Joint Probabilities generated from Tissue Probability Maps
Rather than using an image of discrete values, multiple images showing which voxels are in which class can be used.
These can be constructed from an average of many subjects
4x256 Joint Histogram
Priors enforce “smooth” deformations* Membrane Energy
* Bending Energy
* Linear Elastic Energy
dxx
)(y2
)(Plog2
31i
i
x
Y
dxxx
)(y2
)(Plog
2
31i
31j
ji
2
xY
dxx
)(y
x)(y
4x
)(y
x)(y
2)(Plog
2
i
j
j
i
j
j31i
31j
i
i
xxxxY
Priors enforce “smooth” deformations* The form of prior determines how the
deformations behave in regions with no matching information
Overview* Part I: General Inter-subject registration* Part II: The Segmentation Method in SPM5
* Modelling intensities by a Mixture of Gaussians* Bias correction* Tissue Probability Maps to assist the segmentation* Warping the tissue probability maps to match the
image
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
Segmentation in SPM5
* Uses a generative model, which involves:* Mixture of Gaussians (MOG)* Bias Correction Component* Warping (Non-linear Registration) Component
y1c1
y2
y3
c2
c3
C
CyIcI
Ashburner & Friston. Unified Segmentation. NeuroImage 26:839-851 (2005).
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
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.* E.g. accounting for partial volume effects
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:
* A maximum-likelihood solution can be found by minimising the negative log-probability:
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 ()
y1c1
y2
y3
c2
c3
C
CyIcI
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:
y1c1
y2
y3
c2
c3
C
CyIcI
Deforming the Tissue Probability Maps* Tissue probability images
are deformed according to parameters .
* The probability of obtaining class k at voxel i is then:
y1c1
y2
y3
c2
c3
C
CyIcI
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 optimisationRepeat 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 nonlinear registration
() and bias correction ().* 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:
Regularisation* Some bias fields and warps are more probable (a
priori) than others.* Encoded using Bayes rule (for a maximum a
posteriori solution):
* Prior probability distributions modelled by a multivariate normal distribution.* Mean vector and
* Covariance matrix and
* -log[P()] = (-T-1( + const
* -log[P()] = (-T-1( + const
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)
Summary* Part I: General Inter-subject registration
* Spatial transformations* Affine* Global nonlinear* Local nonlinear
* Objective functions for registration* Likelihood Models
* Mean squared difference* Information Theoretic measures
* Prior Models
* Part II: The Segmentation Method in SPM5* Modelling intensities by a Mixture of Gaussians* Bias correction* Tissue Probability Maps to assist the segmentation* Warping the tissue probability maps to match the image
References* Friston et al. Spatial registration and normalisation of images.
Human Brain Mapping 3:165-189 (1995).* Collignon et al. Automated multi-modality image registration based on
information theory. IPMI’95 pp 263-274 (1995).* Ashburner et al. Incorporating prior knowledge into image registration.
NeuroImage 6:344-352 (1997).* Ashburner & Friston. Nonlinear spatial normalisation using basis
functions.Human Brain Mapping 7:254-266 (1999).
* Thévenaz et al. Interpolation revisited.IEEE Trans. Med. Imaging 19:739-758 (2000).
* Andersson et al. Modeling geometric deformations in EPI time series.Neuroimage 13:903-919 (2001).
* Ashburner & Friston. Unified Segmentation.NeuroImage in press (2005).
Spare slides
Very hard to define a one-to-one mappingof cortical
folding
Use only approximat
e registration
.
Smooth
Before convolution Convolved with a circleConvolved with a Gaussian
Smoothing is done by convolution.
Each voxel after smoothing effectively becomes the result of applying a weighted region of interest (ROI).
Voxel-to-world Transforms* Affine transform associated with each image
* Maps from voxels (x=1..nx, y=1..ny, z=1..nz) to some world co-ordinate system. e.g.,
* Scanner co-ordinates - images from DICOM toolbox* T&T/MNI coordinates - spatially normalised
* Registering image B (source) to image A (target) will update B’s voxel-to-world mapping* Mapping from voxels in A to voxels in B is by
* A-to-world using MA, then world-to-B using MB-1
* MB-1 MA
Left- and Right-handed Coordinate Systems
* Analyze™ files are stored in a left-handed system* Talairach & Tournoux uses a right-handed system* Mapping between them requires a flip
* Affine transform with a negative determinant
Transforming an image* Images are re-sampled. An example in 2D:
for y=1..ny % loop over rows
for x=1..nx % loop over pixels in row
x’= tx(x,y,a) % transform according to a
y’= ty(x,y,a)
if 1x’ nx & 1y’ny then % voxel in range
f (x,y) = f’(x’,y’) % assign re-sampled value
end % voxel in rangeend % loop over pixels in row
end % loop over rows
* What happens if x’ and y’ are not integers?
* Nearest neighbour* Take the value of the
closest voxel
* Tri-linear* Just a weighted
average of the neighbouring voxels
* f5 = f1 x2 + f2 x1
* f6 = f3 x2 + f4 x1
* f7 = f5 y2 + f6 y1
Simple Interpolation
B-spline Interpolation
B-splines are piecewise polynomials
A continuous function is represented by a linear combination of basis
functions
2D B-spline basis functions of degrees 0, 1,
2 and 3
Nearest neighbour and trilinear interpolation are the same as B-spline interpolation with degrees 0 and 1.
Inverse
EPI
T2 T1 Transm
PD PET
305T1
PD T2 SS
Template Images “Canonical” images
A wider range of contrasts can be registered to a linear combination of template images.
Spatial normalisation can be weighted so that non-brain voxels do not influence the result.
Similar weighting masks can be used for normalising lesioned brains.
Spatial Normalisation - Templates
T1 PD
PET
Templateimage
Affine registration.(2 = 472.1)
Non-linearregistration
withoutregularisation.(2 = 287.3)
Non-linearregistration
usingregularisation.(2 = 302.7)
Without regularisation, the non-linear spatial normalisation can introduce unnecessary warps.
Spatial Normalisation - Overfitting
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.
Another example (for VBM)
Bias Correction helps 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 helps Segmentation* Similar tissues no
longer have similar intensities.
* Artefact should be corrected to enable intensity-based tissue classification.
Registration helps Segmentation* SPM99 and SPM2 require tissue probability
maps to be overlaid prior to segmentation.
Segmentation helps 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 helps Registration
Original MRITemplate
Grey MatterSegment
Affine register
Tissue probability maps
Deformation
Affine Transform
Spatial Normalisation- estimation
Spatial Normalisation- writing
Spatially NormalisedMRI
A convoluted method using SPM2
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