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).