Image Registration John Ashburner * * * * Smooth Realign Normalise Segment With slides by Chloe Hutton and Jesper Andersson Overview of SPM Analysis Motion Correction Statistical Parametric Map Design matrix fMRI time-series Smoothing General Linear Model Spatial Normalisation Anatomical Reference Parameter Estimates Contents * Preliminaries * * * * Smooth Rigid-Body and Affine Transformations Optimisation and Objective Functions Transformations and Interpolation * Intra-Subject Registration * Inter-Subject Registration Smooth Smoothing is done by convolution. Each voxel after smoothing effectively becomes the result of applying a weighted region of interest (ROI). Before convolution Convolved with a circle Convolved with a Gaussian 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 Shear x1 = 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 Shear x1 = 1 x0 + h y0 + 0 y1 = 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 1 0 0 0 0 0 1 0 0 1 0 0 Xtrans 1 Ytrans 0 Ztrans 0 1 0 Translations 0 0 cos Φ sinΦ sinΦ cos Φ 0 0 Pitch about x axis 0 cos Θ 0 0 0 sinΘ 1 0 0 sinΘ 1 0 0 cos Θ 0 0 Roll about y axis 0 cos Ω 0 sinΩ 0 0 1 0 sinΩ 0 cos Ω 0 0 1 0 0 Yaw about z axis 0 0 0 1 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 Objective function Most probable solution (global optimum) Local optimum Local optimum Value of parameter 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 range end % loop over pixels in row end % loop over rows * What happens if x1 and y1 are not integers? Simple Interpolation * 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 B-spline Interpolation A continuous function is represented by a linear combination of basis functions B-splines are piecewise polynomials 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 phaseencode 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 reference from mean of all scans. Estimate new distortion fields for each image: Estimate movement parameters. • estimate rate of change of field with respect to the current estimate of movement parameters in pitch and roll. Unwarp time series. + B0 B0 Andersson et al, 2001 Contents * Preliminaries * Intra-Subject Registration * Realign * Coregister * Mutual Information objective function * Inter-Subject Registration Inter-modal 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. 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 * Minimise mean squared difference from template image(s) Affine registration Non-linear registration T2 T1 Transm T1 305 EPI PD PET PD T2 Template Images “Canonical” images PET A wider range of contrasts can be registered to a linear combination of template images. T1 PD SS Spatial normalisation can be weighted so that nonbrain voxels do not influence the result. Similar weighting masks can be used for normalising lesioned brains. Spatial Normalisation - Templates 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-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 Spatial Normalisation - Overfitting Without regularisation, the non-linear Template spatial image normalisation can introduce unnecessary Non-linear warps. registration using regularisation. (2 = 302.7) Affine registration. (2 = 472.1) Non-linear registration without regularisation. (2 = 287.3) 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 mk and variance s2k, 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 gk 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 b, is ri(b). * Replace the means by mk/ri(b) * Replace the variances by (sk/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 included. * The probability of obtaining class k at voxel i, given weights g 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 g and parameters a 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 Descent Start Optimum Alternate between optimising different groups of parameters Schematic of optimisation Repeat until convergence… Hold g, m, s2 and a constant, and minimise E w.r.t. b - Levenberg-Marquardt strategy, using dE/db and d2E/db2 Hold g, m, s2 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. g, m and s2 -Use an Expectation Maximisation (EM) strategy. end Levenberg-Marquardt Optimisation * LM optimisation is used for nonlinear registration (a) and bias correction (b). * 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). Expectation Maximisation is used to update m, s2 and g * 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 ma and mb Covariance matrix Sa and Sb -log[P(a)] = (a-ma)TSa-1(ama) + const -log[P(b)] = (b-mb)TSb-1(bmb) + 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. Joint Probability Histogram See D'Agostino, Maes, Vandermeulen & P. Suetens. “Non-rigid Atlas-to-Image Registration by Minimization of ClassConditional Image Entropy”. Proc. MICCAI 2004. LNCS 3216, 2004. Pages 745-753. Background voxels excluded 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. Spatially normalised BrainWeb phantoms (T1, T2 and PD) Tissue probability maps of GM and WM 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).
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