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Image Registration Carlton CHU *Smooth *Realign *Normalise *Segment *DARTEL With slides by John Ashburner, Chloe Hutton and Jesper Andersson

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Overview of SPM Analysis Motion Correction Smoothing Spatial Normalisation General Linear Model Statistical Parametric Map fMRI time-series Parameter Estimates Design matrix Anatomical Reference

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Contents *Preliminaries *Smooth *Rigid-Body and Affine Transformations *Optimisation and Objective Functions *Transformations and Interpolation *Intra-Subject Registration *Inter-Subject Registration

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Smooth Before convolutionConvolved 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).

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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

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2D Affine Transforms *Translations by t x and t y *x 1 = x 0 + t x *y 1 = y 0 + t y *Rotation around the origin by radians *x 1 = cos( ) x 0 + sin( ) y 0 *y 1 = -sin( ) x 0 + cos( ) y 0 *Zooms by s x and s y *x 1 = s x x 0 *y 1 = s y y 0 *Shear *x 1 = x 0 + h y 0 *y 1 = y 0

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2D Affine Transforms *Translations by t x and t y *x 1 = 1 x y 0 + t x *y 1 = 0 x y 0 + t y *Rotation around the origin by radians *x 1 = cos( ) x 0 + sin( ) y *y 1 = -sin( ) x 0 + cos( ) y *Zooms by s x and s y : *x 1 = s x x y *y 1 = 0 x 0 + s y y *Shear *x 1 = 1 x 0 + h y *y 1 = 0 x y 0 + 0

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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 TranslationsPitch about x axis Roll about y axis Yaw about z axis

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Voxel-to-world Transforms *Affine transform associated with each image *Maps from voxels (x=1..n x, y=1..n y, z=1..n z ) 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 Bs voxel-to-world mapping *Mapping from voxels in A to voxels in B is by *A-to-world using M A, then world-to-B using M B -1 * M B -1 M A

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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 optimum

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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)

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Transformation *Images are re-sampled. An example in 2D: for y 0 =1..n y0 % loop over rows for x 0 =1..n x0 % loop over pixels in row x 1 = t x (x 0,y 0,q) % transform according to q y 1 = t y (x 0,y 0,q) if 1 x 1 n x1 & 1 y 1 n y1 then % voxel in range f 1 (x 0,y 0 ) = f 0 (x 1,y 1 ) % assign re-sampled value end % voxel in range end % loop over pixels in row end % loop over rows *What happens if x 1 and y 1 are not integers?

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*Nearest neighbour *Take the value of the closest voxel *Tri-linear *Just a weighted average of the neighbouring voxels *f 5 = f 1 x 2 + f 2 x 1 *f 6 = f 3 x 2 + f 4 x 1 *f 7 = f 5 y 2 + f 6 y 1 Simple Interpolation

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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.

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Contents *Preliminaries *Intra-Subject Registration *Realign *Mean-squared difference objective function *Residual artifacts and distortion correction *Coregister *Inter-Subject Registration

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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

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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

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Movement by Distortion Interaction of fMRI *Subject disrupts B 0 field, rendering it inhomogeneous => distortions in phase- encode direction *Subject moves during EPI time series *Distortions vary with subject orientation => shape varies

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Movement by distortion interaction

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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. + Andersson et al, 2001

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Contents *Preliminaries *Intra-Subject Registration *Realign *Coregister *Mutual Information objective function *Inter-Subject Registration

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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

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Mutual Information *Used for between-modality registration *Derived from joint histograms *MI= ab P(a,b) log 2 [P(a,b)/( P(a) P(b) )] *Related to entropy: MI = -H(a,b) + H(a) + H(b) *Where H(a) = - a P(a) log 2 P(a) and H(a,b) = - a P(a,b) log 2 P(a,b)

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Contents *Preliminaries *Intra-Subject Registration *Inter-Subject Registration *Normalise *Affine Registration *Nonlinear Registration *Regularisation *Segment

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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

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Spatial Normalisation - Procedure Non-linear registration *Minimise mean squared difference from template image(s) Affine registration

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EPI T2 T1Transm PDPET 305T1 PD T2 SS Template ImagesCanonical 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 T1PD PET

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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)

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Spatial Normalisation - Non-linear Deformations 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

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Template image Affine registration. ( 2 = 472.1) Non-linear registration without regularisation. ( 2 = 287.3) Non-linear registration using regularisation. ( 2 = 302.7) Without regularisation, the non-linear spatial normalisation can introduce unnecessary warps. Spatial Normalisation - Overfitting

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Contents *Preliminaries *Intra-Subject Registration *Inter-Subject Registration *Normalise *Segment *Gaussian mixture model *Intensity non-uniformity correction *Deformed tissue probability maps

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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

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Gaussian Probability Density *If intensities are assumed to be Gaussian of mean k and variance 2 k, then the probability of a value y i is:

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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

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Belonging Probabilities Belonging probabilities are assigned by normalising to one.

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Mixing Proportions *The mixing proportion k represents the prior probability of a voxel being drawn from class k - irrespective of its intensity. *So:

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Non-Gaussian Intensity Distributions *Multiple Gaussians per tissue class allow non-Gaussian intensity distributions to be modelled. *E.g. accounting for partial volume effects

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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:

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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

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Modelling a Bias Field *After rearranging... ( ) y y ( )

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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.

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Mixing Proportions *Tissue probability maps for each class are included. *The probability of obtaining class k at voxel i, given weights is then:

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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:

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The Extended Model *By combining the modified P(c i =k| ) and P(y i |c i =k, ), the overall objective function (E) becomes: The Objective Function

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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.

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Steepest Descent Start Optimum Alternate between optimising different groups of parameters

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Schematic of optimisation Repeat until convergence… Hold,, 2 and constant, and minimise E w.r.t. - Levenberg-Marquardt strategy, using dE/d and d 2 E/d 2 Hold,, 2 and constant, and minimise E w.r.t. - Levenberg-Marquardt strategy, using dE/d and d 2 E/d 2 Hold and constant, and minimise E w.r.t., and 2 -Use an Expectation Maximisation (EM) strategy. end

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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).

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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:

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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

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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 LNCS 3216, Pages Background voxels excluded Joint Probability Histogram

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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.

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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)

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Image Registration *Figure out how to warp one image to match another *Normally, all subjects scans are matched with a common template

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Current SPM approach *Only about 1000 parameters. *Unable model detailed deformations

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A one-to-one mapping *Many models simply add a smooth displacement to an identity transform *One-to-one mapping not enforced *Inverses approximately obtained by subtracting the displacement *Not a real inverse Small deformation approximation Large deformation approximation

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DARTEL *Parameterising the deformation *φ (0) (x) = x *φ (1) (x) = u ( φ (t) (x) ) dt *u is a flow field to be estimated t=0 1

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Flow Field

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Template Initial Average After a few iterations Final template Iteratively generated from 471 subjects Began with rigidly aligned tissue probability maps Used an inverse consistent formulation

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Grey matter average of 452 subjects – affine

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Grey matter average of 471 subjects

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Subject 1 Subject 2

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VBM results of Patients with Alzheimers Disease Conventional SPM5 normalized image DARTEL normalized image Both sets were smoothed with only 3mm FWHM Gaussian kernel, (FWE p>0.05)

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VBM results of Patients with Huntingtons Disease Conventional SPM5 normalized image DARTEL normalized image Both sets were smoothed with only 6mm FWHM Gaussian kernel, (FWE p>0.05)

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References *Friston et al. Spatial registration and normalisation of images. Human Brain Mapping 3: (1995). *Collignon et al. Automated multi-modality image registration based on information theory. IPMI95 pp (1995). *Ashburner et al. Incorporating prior knowledge into image registration. NeuroImage 6: (1997). *Ashburner & Friston. Nonlinear spatial normalisation using basis functions. Human Brain Mapping 7: (1999). *Thévenaz et al. Interpolation revisited. IEEE Trans. Med. Imaging 19: (2000). *Andersson et al. Modeling geometric deformations in EPI time series. Neuroimage 13: (2001). *Ashburner & Friston. Unified Segmentation. NeuroImage in press (2005). *Ashburner. A fast diffeomorphic image reigstration algorithm. NeuroImage in press (2007).

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