Computational Neuroanatomy John Ashburner [email protected] • • • • • • Smoothing Motion Correction Between Modality Co-registration Spatial Normalisation Segmentation Morphometry Overview Statistical Parametric Map fMRI time-series kernel Design matrix Motion correction smoothing General Linear Model Parameter Estimates Spatial normalisation anatomical reference Smoothing • Why Smooth? – Potentially increase signal to noise. – Inter-subject averaging. – Increase validity of SPM. • In SPM, smoothing is a convolution with a Gaussian kernel. • Kernel defined in terms of FWHM (full width at half maximum). Gaussian convolution is separable Gaussian smoothing kernel Smoothing Smoothing is done by convolving with a 3D Gaussian - defined by its full width at half maximum (FWHM) 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 Reasons for Motion Correction The Steps in Motion Correction • Subjects will always move in the scanner. – movement may be related to the tasks performed. • When identifying areas in the brain that appear activated due to the subject performing a task, it may not be possible to discount artefacts that have arisen due to motion. • The sensitivity of the analysis is determined by the amount of residual noise in the image series, so movement that is unrelated to the task will add to this noise and reduce the sensitivity. • registration - i.e. determining the 6 parameters that describe the rigid body transformation between each image and a reference image. • transformation - i.e. resampling each image according to the determined transformation parameters. Registration • Determine the rigid body transformation that minimises the sum of squared difference between images. • Rigid body transformation is defined by: – 3 translations - in X, Y & Z directions. – 3 rotations - about X, Y & Z axes. • Operations can be represented as affine transformation matrixes: x1 = m1,1x0 + m1,2y0 + m1,3z0 + m1,4 y1 = m2,1x0 + m2,2y0 + m2,3z0 + m2,4 z1 = m3,1x0 + m3,2y0 + m3,3z0 + m3,4 Rigid body transformations parameterised by: Translations 1 0 0 0 1 0 0 0 1 0 0 0 X trans Y trans Pitch 1 0 Zt rans 0 1 0 0 Roll 0 cos( ) sin( ) sin( ) cos( ) 0 0 0 0 cos( ) 0 0 sin( 1 0 ) 0 Yaw sin( ) 1 0 0 cos( ) 0 0 0 cos( ) sin( ) 0 0 0 sin( ) cos( ) 0 0 0 0 1 0 1 0 0 0 1 0 Residual Errors from PET Transformation • Incorrect attenuation correction because transmission scan no longer aligned with emission scans. Residual Errors from fMRI • • Gaps between slices can cause aliasing artefacts Re-sampling can introduce errors – especially tri-linear interpolation • d1 One if the simplest resampling methods is trilinear interpolation. v1 – do not move according to the same rigid body rules as the subject v2 d3 d4 v4 Ghosts (and other artefacts) in the images d2 • – rapid movements not accounted for by rigid body model v3 • Other methods include nearest neighbour resampling, and various forms of sinc interpolation using different numbers of neighbouring voxels. Slices are not acquired simultaneously fMRI images are distorted – rigid body model does not model these types of distortion • Spin excitation history effects – variations in residual magnetisation Functions of the estimated motion parameters can be used as confounds in subsequent analyses. Between Modality Co-registration • Not based on simply minimising mean squared difference between images. • A three step approach is used instead. 1) Simultaneous affine registrations between each image and template images of same modality. 2) Partitioning of images into grey and white matter. 3) Final simultaneous registration of image partitions. Rigid registration between high resolution structural images and echo planer functional images is a problem. Results are only approximate because of spatial distortions of EPI data. First Step - Affine Registrations. • • • • Requires template images of same modalities. Both images are registered - using 12 parameter affine transformations - to their corresponding templates by minimising the mean squared difference. Only the rigid-body transformation parameters differ between the two registrations. This gives: – rigid body mapping between the images. – affine mappings between the images and the templates. Second Step - Segmentation. • • ‘Mixture Model’ cluster analysis to classify MR image (or images) as GM, WM & CSF. Additional information is obtained from a priori probability images, which are overlaid using previously determined affine transformations. Third Step Registration of Partitions. •Grey and white matter partitions are registered using a rigid body transformation. •Simultaneously minimise sum of squared difference. Between Modality Coregistration using Mutual Information An alternative between modality registration method available within SPM99 maximises Mutual Information in the 2D histogram. For histograms normalised to integrate to unity, the Mutual Information is defined by: SiSj hij log hij Sk hik Sl hlj PET T1 weighted MRI Spatial normalisation • Inter-subject averaging – extrapolate findings to the population as a whole – increase activation signal above that obtained from single subject – increase number of possible degrees of freedom allowed in statistical model • Enable reporting of activations as co-ordinates within a known standard space – e.g. the space described by Talairach & Tournoux • Warp the images such that functionally homologous regions from the different subjects are as close together as possible – Problems: • no exact match between structure and function • different brains are organised differently • computational problems (local minima, not enough information in the images, computationally expensive) • Compromise by correcting for gross differences followed by smoothing of normalised images Spatial Normalisation Original image Spatially normalised Determine the spatial transformation that minimises the sum of squared difference between an image and a linear combination of one or more templates. Begins with an affine registration to match the size and position of the image. Spatial Normalisation Followed by a global non-linear warping to match the overall brain shape. Uses a Bayesian framework to simultaneously maximise the smoothness of the warps. Template image Deformation field Affine versus affine and non-linear spatial normalisation Six affine registered images. Six basis function registered images T2 T1 Transm T1 305 EPI PD PET PD T2 Template Images A wider range of different contrasts can be normalised by registering to a linear combination of template images. SS “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. Bayesian Formulation • Bayes rule states: p(q|e) p(e|q) p(q) – p(q|e) is the a posteriori probability of parameters q given errors e. – p(e|q) is the likelihood of observing errors e given parameters q. – p(q) is the a priori probability of parameters q. • Maximum a posteriori (MAP) estimate maximises p(q|e). • Maximising p(q|e) is equivalent to minimising the Gibbs potential of the posterior distribution (H(q|e), where H(q|e) -log p(q|e)). • The posterior potential is the sum of the likelihood and prior potentials: H(q|e) = H(e|q) + H(q) + c – The likelihood potential (H(e|q) -log p(e|q)) is based upon the sum of squared difference between the images. – The prior potential (H(q) -log p(q)) penalises unlikely deformations. Spatial Normalisation - affine • The first part of spatial normalisation is a 12 parameter Affine Transformation – – – – 3 translations 3 rotations 3 zooms 3 shears Empirically generated priors 1 0 0 0 0 0 X trans 1 0 0 0 0 0 1 0 Y trans cos( ) sin( ) 0 1 Z trans sin( ) cos( ) 0 0 1 0 0 cos( ) 0 0 0 sin( ) 1 0 0 0 sin( ) 1 0 0 cos( ) 0 0 cos( ) 0 sin( ) 0 0 1 0 0 sin( ) 0 cos( ) 0 0 0 1 0 0 0 1 0 X zoom 0 0 0 0 0 Y zoom 0 0 Z zoom 0 0 0 1 0 0 0 0 1 0 0 XY shear XZ shear 1 YZ shear 0 0 1 0 0 0 1 Find the parameters that minimise the sum of squared difference between the image and template(s) - and also the square of the number of standard deviations away from the expected parameter values. Spatial Normalisation - Non-linear • Deformations consist of a linear combination of smooth basis images. • These are the lowest frequency basis images of a 3-D discrete cosine transform (DCT). • Can be generated rapidly from a separable form. • Algorithm simultaneously minimises – Sum of squared difference between template and object image . – Squared distance between the parameters and their known expectation (pTC0-1 p). • pTC0-1 p describes the membrane energy of the deformations. membrane energy = i u ji x j 1 k 1 ki 2 2 2 Without the Bayesian formulation, the non-linear spatial normalisation can introduce unnecessary warping into the spatially normalised images. Affine Registration. (2 = 472.1) Template image Non-linear registration using regularisation. (2 = 302.7) Non-linear registration without regularisation. (2 = 287.3) Segmentation. • • • • ‘Mixture Model’ cluster analysis to classify MR image (or images) as GM, WM & CSF. Additional information is obtained from prior probability images, which are overlaid. Assumes that each MRI voxel is one of a number of distinct tissue types (clusters). Each cluster has a (multivariate) normal distribution. . • A smooth intensity modulating function can be modelled by a linear combination of DCT basis functions. More than one image can be used to produce a multi-spectral classification. The segmented images contain a little non-brain tissue, which can be automatically . removed using morphological operations (erosion followed by conditional dilation). Morphometric Measures • Voxel-by-voxel – where are the differences between the populations? – produce an SPM of regional differences • Univariate - e.g., VoxelBased Morphometry • Multivariate - e.g., TensorBased Morphometry • Volume based – is there a difference between the populations? • Multivariate - e.g., Deformation-Based Morphometry MANCOVA & CCA Voxel-Based Morphometry Preparation of images for each subject Original image Spatially normalised Partitioned grey matter Smoothed A voxel by voxel statistical analysis is used to detect regional differences in the amount of grey matter between populations. Morphometric approaches based on deformation fields Deformation-based Morphometry looks at absolute displacements. Tensor-based Morphometry looks at local shapes Deformation-based morphometry Deformation fields ... Remove positional and size information - leave shape Parameter reduction using principal component analysis (SVD). Multivariate analysis of covariance used to identify differences between groups. Canonical correlation analysis used to characterise differences between groups. Sex Differences using Deformation-based Morphometry Non-linear warps pertaining to sex differences characterised by canonical variates analysis (above), and mean differences (below, mapping from an average female to male brain). In the transverse and coronal sections, the left side of the brain is on the left side of the figure. Tensor-based morphometry Original Warped Template If the original Jacobian matrix is donated by A, then this can be decomposed into: A = RU, where R is an orthonormal rotation matrix, and U is a symmetric matrix containing only zooms and shears. Relative volumes Strain tensor Strain tensors are defined that model the amount of distortion. If there is no strain, then tensors are all zero. Generically, the family of Lagrangean strain tensors are given by: (Um-I)/m when m~=0, and log(U) if m==0. High dimensional warping Millions of parameters are needed for more precise image registration….. Takes a very long time Relative volumes of brain structures can be computed from the determinants of the deformation fields Data From the Dementia Research Group, London, UK. References Friston et al (1995): Spatial registration and normalisation of images. Human Brain Mapping 3(3):165-189 Ashburner & Friston (1997): Multimodal image coregistration and partitioning - a unified framework. NeuroImage 6(3):209-217 Collignon et al (1995): Automated multi-modality image registration based on information theory. IPMI’95 pp 263-274 Ashburner et al (1997): Incorporating prior knowledge into image registration. NeuroImage 6(4):344-352 Ashburner et al (1999): Nonlinear spatial normalisation using basis functions. Human Brain Mapping 7(4):254-266 Ashburner & Friston (2000): Voxel-based morphometry - the methods. NeuroImage 11:805-821

© Copyright 2024 Paperzz