Image Registration John Ashburner

Pre-processing Overview Statistics or whatever fMRI time-series Template Anatomical MRI Smoothed Estimate Spatial Norm Motion Correct Smooth Coregister Spatially normalised Deformation

Contents Preliminaries Intra-Subject Registration 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 Translations Pitch about x axis Roll about y axis Yaw about 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 NIfTI format files are stored in either a left- or right-handed system Indicated in the header Talairach & Tournoux uses a right-handed system Mapping between them sometimes requires a flip Affine transform has a negative determinant

Optimisation Image registration is done by 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 Most probable solution (global optimum) Objective function Local optimum Local optimum Value of parameter

Objective Functions Intra-modal Inter-modal (or 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)

Simple Interpolation Nearest neighbour Tri-linear 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 2D B-spline basis functions of degrees 0, 1, 2 and 3 B-splines are piecewise polynomials 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 reference from mean of all scans. 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. Unwarp time series. Estimate movement parameters.  + 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 DARTEL

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

Spatial Normalisation - Templates Transm T1 305 PD T2 SS EPI PD PET Template Images “Canonical” 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

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 spatial normalisation can introduce unnecessary warps. Affine registration. (2 = 472.1) Target image Non-linear registration without regularisation. (2 = 287.3) Non-linear registration using regularisation. (2 = 302.7)

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

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

Mixture of Gaussians (MOG) Classification is based on a Mixture of Gaussians model (MOG), which represents the intensity probability density by a number of Gaussian distributions. Frequency Image Intensity

Belonging Probabilities Belonging probabilities are assigned by normalising to one.

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

Modelling a Bias Field A bias field is modelled as a linear combination of basis functions. Corrected image Corrupted image Bias Field

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.

Deforming the Tissue Probability Maps Tissue probability images are deformed so that they can be overlaid on top of the image to segment.

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.

Alternate between optimising different groups of parameters Steepest Descent Start Optimum Alternate between optimising different groups of parameters

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)

Contents Preliminaries Intra-Subject Registration Inter-Subject Registration Normalise Segment DARTEL Flow field parameterisation Objective function Template creation Examples

Small deformation approximation 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

φ(1)(x) = ∫ u(φ(t)(x))dt u is a flow field to be estimated DARTEL Parameterising the deformation φ(0)(x) = x φ(1)(x) = ∫ u(φ(t)(x))dt u is a flow field to be estimated Scaling and squaring is used to generate deformations. 1 t=0

Scaling and squaring example

Forward and backward transforms

Registration objective function Simultaneously minimize the sum of: Likelihood component Drives the matching of the images. Multinomial assumption Prior component A measure of deformation roughness Regularises the registration. ½uTHu A balance between the two terms.

Likelihood Model log p(t|μ,ϕ) = ΣjΣk tjk log(μk(ϕj)) Current DARTEL model is multinomial for matching tissue class images. Template represents probability of obtaining different tissues at each point. log p(t|μ,ϕ) = ΣjΣk tjk log(μk(ϕj)) t – individual GM, WM and background μ – template GM, WM and background ϕ – deformation

Prior Model

Simultaneous registration of GM to GM and WM to WM Grey matter White matter Subject 1 Grey matter White matter Subject 3 Grey matter White matter Grey matter White matter Template Grey matter White matter Subject 2 Subject 4

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

Grey matter average of 452 subjects – affine

Grey matter average of 471 subjects

Initial GM images

Warped GM images

VBM Pre-processing Use Segment button for characterising intensity distributions of tissue classes. DARTEL import, to generate rigidly aligned grey and white matter maps. DARTEL warping to generate “modulated” warped grey matter. Smooth. Statistics.

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 26:839-851 (2005). Ashburner. A Fast Diffeomorphic Image Registration Algorithm. NeuroImage 38:95-113 (2007).