Image Registration + Other Stuff

Transcription

1 Image Registration + Other Stuff John Ashburner Pre-processing Overview fmri time-series Motion Correct Anatomical MRI Coregister m11 m 21 m 31 m12 m13 m14 m 22 m 23 m 24 m 32 m 33 m 34 1 Template Estimate Spatial Norm Deformation Statistics or whatever Smoothed Smooth Spatially normalised Contents * Preliminaries * Smooth * Rigid-Body and Affine Transformations * Optimisation and Objective Functions * Transformations and Interpolation * Intra-Subject Registration * Inter-Subject Registration * VBM 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

2 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 t x and t y * x 1 = x + t x * y 1 = y + t y * Rotation around the origin by Θ radians * x 1 = cos(θ) x + sin(θ) y * y 1 = -sin(θ) x + cos(θ) y * Zooms by s x and s y * x 1 = s x x * y 1 = s y y *Shear *x 1 = x + h y *y 1 = y 2D Affine Transforms * Translations by t x and t y * x 1 = 1 x + y + t x * y 1 = x + 1 y + t y * Rotation around the origin by Θ radians * x 1 = cos(θ) x + sin(θ) y + * y 1 = -sin(θ) x + cos(θ) y + * Zooms by s x and s y : * x 1 = s x x + y + * y 1 = x + s y y + *Shear *x 1 = 1 x + h y + *y 1 = x + 1 y + 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 X trans 1 Y trans Zt rans 1 cos Φ sin Φ sin Φ cos Φ Pitch about x axis cos Θ sin Θ 1 1 sin Θ cos Θ Roll about y axis cos Ω sin Ω 1 sin Ω cos Ω 1 Yaw about z axis 1

3 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 coordinate 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 M A, then world-to-b using M B -1 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 * M B -1 M A 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 Objective function Most probable solution (global optimum) Local optimum Local 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) Value of parameter

4 Simple Interpolation * 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 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, 1, 2 and 3 Nearest neighbour and trilinear interpolation are the same as B-spline interpolation with degrees and 1. Contents * Preliminaries * Intra-Subject Registration * Realign * Mean-squared difference objective function * Residual artifacts and distortion correction * Coregister * Inter-Subject Registration * VBM Mean-squared Difference * Minimising mean-squared difference works for intramodal registration (realignment) * Simple relationship between intensities in one image, versus those in the other * Assumes normally distributed differences

5 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 Movement by Distortion Interaction of fmri *Subject disrupts B field, rendering it inhomogeneous * distortions in phase-encode direction *Subject moves during EPI time series *Distortions vary with subject orientation *shape varies * Functions of the estimated motion parameters can be modelled as confounds in subsequent analyses Movement by distortion interaction Correcting for distortion changes using Unwarp Estimate reference from mean of all scans. 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. Unwarp time series. Δϕ +Δθ B ϕ B θ Andersson et al, 21

6 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) log 2 [P(a,b)/( P(a) P(b) )] * Related to entropy: MI = -H(a,b) + H(a) + H(b) Contents * Preliminaries * Intra-Subject Registration * Inter-Subject Registration * Normalise * Affine Registration * Nonlinear Registration * Regularisation * Segment * DARTEL * VBM * Where H(a) = - a P(a) log 2 P(a) and H(a,b) = - a P(a,b) log 2 P(a,b)

7 Spatial Normalisation - Procedure * Minimise mean squared difference from template image(s) T2 T1 Transm T1 35 EPI PD PET PD T2 SS 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. Affine registration Non-linear registration 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

8 Spatial Normalisation - Overfitting Without regularisation, the non-linear spatial normalisation can introduce unnecessary warps. Target image Non-linear registration using regularisation. (χ 2 = 32.7) Affine registration. (χ 2 = 472.1) Non-linear registration without regularisation. (χ 2 = 287.3) Contents * Preliminaries * Intra-Subject Registration * Inter-Subject Registration * Normalise * Segment * Gaussian mixture model * Intensity non-uniformity correction * Deformed tissue probability maps * DARTEL * VBM Segmentation Mixture of Gaussians (MOG) * 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 * 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

9 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. Tissue Probability Maps * Tissue probability maps (TPMs) are used instead of the proportion of voxels in each Gaussian as the prior. Corrupted image Bias Field Corrected image 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.

10 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. I K γ kbik( α) E = log ρi( β) K i= 1 k = 1 j= 1 γ jbij( α) 1 2πσ 2 k ( ρi( β) yi μk ) exp 2 2σk 2 Steepest Descent Start Spatially normalised BrainWeb phantoms (T1, T2 and PD) Optimum Tissue probability maps of GM and WM Alternate between optimising different groups of parameters Cocosco, Kollokian, Kwan & Evans. BrainWeb: Online Interface to a 3D MRI Simulated Brain Database. NeuroImage 5(4):S425 (1997)

11 Extensions for SPM8 * Additional tissue classes * Grey matter, white matter, CSF, skull, scalp. * Multi-channel Segmentation * More detailed nonlinear registration * More robust initial affine registration * Extra tissue class maps can be generated Image Intensity Distributions (T1-weighted MRI) Belonging Probabilities

12 Small deformation approximation

13 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. * ½u T Hu * A balance between the two terms.

14 Likelihood Model * 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) = Σ j Σ k t jk log(μ k (ϕ j )) t μ ϕ individual GM, WM and background template GM, WM and background deformation Prior Model Simultaneous registration of GM to GM and WM to WM Subject 1 Grey matter White matter Subject 2 Grey matter White matter Grey matter White matter Template Grey matter White matter Grey matter White matter Subject 4 Subject 3 Template Iteratively generated from 471 subjects Began with rigidly aligned tissue probability maps Used an inverse consistent formulation Initial Average After a few iterations Final template

15 Grey matter average of 452 subjects affine Grey matter average of 471 subjects Initial GM images Warped GM images

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18 Contents * Preliminaries * Intra-Subject Registration * Inter-Subject Registration * VBM Volumetry Spatial Normalisation T1-Weighted MRI Grey Matter Original Registered Template All subjects data are aligned with a common reference compare like with like

19 Preserving Tissue Volumes Smoothing 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 Warped without correction Warped with correction Jacobian Determinants Tissue class images are warped such that the volumes of tissue are preserved 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. New Segment Generate low resolution GM and WM images for each subject ( DARTEL imported ). Generate full resolution GM map for each subject.

20 Run DARTEL (create Templates) Simultaneously align DARTEL imported GM and WM for all subjects. Generates templates and parameterisations of relative shapes. Normalise to MNI Space Use shape parameterisations to generate smoothed Jacobian scaled and spatially normalised GM images for each subject. Possible Explanations for VBM Findings Mis-classify Mis-register Folding Thinning Thickening Mis-register Mis-classify 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. IPMI 95 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: (2). * Andersson et al. Modeling geometric deformations in EPI time series. Neuroimage 13: (21). * Ashburner & Friston. Unified Segmentation. NeuroImage 26: (25). * Ashburner. A Fast Diffeomorphic Image Registration Algorithm. NeuroImage 38: (27).