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Image Registration
John Ashburner* Smooth* Realign* Normalise* Segment
With slides by Chloe Hutton and Jesper AnderssonWith slides by Chloe Hutton and Jesper Andersson
Overview of SPM Analysis
MotionCorrection
Smoothing
SpatialNormalisation
General Linear Model
Statistical Parametric MapfMRI time-series
Parameter Estimates
Design matrix
Anatomical Reference
Contents* Preliminaries
* Smooth* Rigid-Body and Affine Transformations* Optimisation and Objective Functions* Transformations and Interpolation
* Intra-Subject Registration* Inter-Subject 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).
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
2D Affine Transforms* Translations by tx and ty
* x1 = x0 + tx
* y1 = y0 + ty
* Rotation around the origin by radians* x1 = cos() x0 + sin() y0
* y1 = -sin() x0 + cos() y0
* Zooms by sx and sy
* x1 = sx x0
* y1 = sy y0
Shearx1 = x0 + h y0
y1 = y0
2D Affine Transforms* Translations by tx and ty
* x1 = 1 x0 + 0 y0 + tx
* y1 = 0 x0 + 1 y0 + ty
* Rotation around the origin by radians* x1 = cos() x0 + sin() y0 + 0
* y1 = -sin() x0 + cos() y0 + 0
* Zooms by sx and sy:
* x1 = sx x0 + 0 y0 + 0
* y1 = 0 x0 + sy y0 + 0
Shearx1 = 1 x0 + h y0 + 0y1 = 0 x0 + 1 y0 + 0
3D Rigid-body Transformations
* A 3D rigid body transform is defined by:* 3 translations - in X, Y & Z directions* 3 rotations - about X, Y & Z axes
* The order of the operations matters
1000
0100
00cossin
00sincos
1000
0cos0sin
0010
0sin0cos
1000
0cossin0
0sincos0
0001
1000
Zt100
Y010
X001
rans
trans
trans
ΩΩ
ΩΩ
ΘΘ
ΘΘ
ΦΦ
ΦΦ
Translations Pitchabout x axis
Rollabout y axis
Yawabout z axis
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
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
Objective Functions
* Intra-modal* Mean squared difference (minimise)* Normalised cross correlation (maximise)* Entropy of difference (minimise)
* Inter-modal (or intra-modal)* Mutual information (maximise)* Normalised mutual information (maximise)* Entropy correlation coefficient (maximise)* AIR cost function (minimise)
Transformation* Images are re-sampled. An example in 2D:
for y0=1..ny0 % loop over rows
for x0=1..nx0 % loop over pixels in row
x1 = tx(x0,y0,q) % transform according to q
y1 = ty(x0,y0,q)
if 1x1 nx1 & 1y1ny1 then % voxel in range
f1(x0,y0) = f0(x1,y1) % assign re-sampled value
end % voxel in rangeend % loop over pixels in row
end % loop over rows
* What happens if x1 and y1 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.
Contents* Preliminaries* Intra-Subject Registration
* Realign* Mean-squared difference objective function* Residual artifacts and distortion correction
* Coregister
* Inter-Subject Registration
Mean-squared Difference
* Minimising mean-squared difference works for intra-modal registration (realignment)
* Simple relationship between intensities in one image, versus those in the other* Assumes normally distributed differences
Residual Errors from aligned fMRI* Re-sampling can introduce interpolation errors
* especially tri-linear interpolation
* Gaps between slices can cause aliasing artefacts* Slices are not acquired simultaneously
* rapid movements not accounted for by rigid body model
* Image artefacts may not move according to a rigid body model* image distortion* image dropout* Nyquist ghost
* Functions of the estimated motion parameters can be modelled as confounds in subsequent analyses
Movement by Distortion Interaction of fMRI•Subject disrupts B0 field, rendering it inhomogeneous
=> distortions in phase-encode direction
•Subject moves during EPI time series•Distortions vary with subject orientation
=> shape varies
Movement by distortion interaction
Correcting for distortion changes using Unwarp
Estimate movement parameters.
Estimate new distortion fields for each image:
• estimate rate of change of field with respect to the current estimate of movement parameters in pitch and roll.
Estimate reference from mean of all scans.
Unwarp time series.
0B 0B
+
Andersson et al, 2001
Contents* Preliminaries* Intra-Subject Registration
* Realign* Coregister
* Mutual Information objective function
* Inter-Subject Registration
•Match images from same subject but different modalities:
–anatomical localisation of single subject activations
–achieve more precise spatial normalisation of functional image using anatomical image.
Inter-modal registration
Mutual Information
* Used for between-modality registration* Derived from joint histograms
* MI=ab P(a,b) log2 [P(a,b)/( P(a) P(b) )]* Related to entropy: MI = -H(a,b) + H(a) + H(b)
* Where H(a) = -a P(a) log2P(a) and H(a,b) = -a P(a,b) log2P(a,b)
Contents* Preliminaries* Intra-Subject Registration* Inter-Subject Registration
* Normalise* Affine Registration* Nonlinear Registration* Regularisation
* Segment
Spatial Normalisation - Reasons
* 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
Spatial Normalisation - Procedure
Non-linear registration
* Minimise mean squared difference from template image(s)
Affine registration
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
Spatial Normalisation - Affine* The first part is a 12 parameter
affine transform* 3 translations* 3 rotations* 3 zooms* 3 shears
* Fits overall shape and size
* Algorithm simultaneously minimises* Mean-squared difference between template and source image* Squared distance between parameters and their expected values
(regularisation)
Spatial Normalisation - Non-linearDeformations consist of a linear combination of smooth basis functions
These are the lowest frequencies of a 3D discrete cosine transform (DCT)
Algorithm simultaneously minimises* Mean squared difference between
template and source image * Squared distance between parameters
and their known expectation
Algorithm simultaneously minimises* Mean squared difference between
template and source image * Squared distance between parameters
and their known expectation
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
Contents* Preliminaries* Intra-Subject Registration* Inter-Subject Registration
* Normalise* Segment
* Gaussian mixture model* Intensity non-uniformity correction* Deformed tissue probability maps
Segmentation
* Segmentation in SPM5 also estimates a spatial transformation that can be used for spatially normalising images.
* It uses a generative model, which 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
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 ()
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 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
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)
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).