Computational Neuroanatomy

Transcription

1 Computational Neuroanatomy John Ashburner Smoothing Motion Correction Between Modality Co-registration Spatial Normalisation Segmentation Morphometry

2 Overview fmri time-series kernel Design matrix Statistical Parametric Map Motion correction smoothing General Linear Model Spatial normalisation Parameter Estimates anatomical reference

3 Smoothing Why Smooth? Potentially increase signal to to noise. Inter-subject averaging. Increase validity of of SPM. In SPM, smoothing is is a convolution with a Gaussian kernel. Kernel defined in in terms of FWHM (full width at half maximum). Gaussian convolution is separable Gaussian smoothing kernel

4 Reasons for Motion Correction Subjects will always move in in the scanner. movement may be related to to the tasks performed. When identifying areas in in the brain that appear activated due to to the subject performing a task, it it may not be possible to to discount artefacts that have arisen due to to motion. The sensitivity of of the analysis is is determined by the amount of residual noise in in the image series, so movement that is is unrelated to to the task will add to to this noise and reduce the sensitivity. The Steps in Motion Correction registration - i.e. determining the 6 parameters that describe the rigid body transformation between each image and a reference image. transformation - i.e. re- sampling each image according to the determined transformation parameters.

5 Registration Determine the rigid body transformation that minimises the sum of of squared difference between images. Rigid body transformation is is defined by: 3 translations - in in X, X, Y & Z directions. 3 rotations - about X, X, Y & Z axes. Operations can be represented as affine transformation matrixes: x = m, x, + m,2 y,2 + m,3 z,3 + m,4,4 y = m 2, x 2, + m 2,2 y 2,2 + m 2,3 z 2,3 + m 2,4 2,4 z = m 3, x 3, + m 3,2 y 3,2 + m 3,3 z 3,3 + m 3,4 3,4 Xtrans Ytrans Ztrans Rigid body transformations parameterised by: Translations Pitch Roll Yaw cos( Φ) sin( Φ) sin( Φ) cos( Φ) cos( Θ) sin( Θ) sin( Θ) cos( Θ ) cos( Ω) sin( Ω ) sin( Ω ) cos( Ω )

6 Transformation One if the simplest resampling methods is trilinear interpolation. Other methods include nearest neighbour resampling, and various forms of sinc interpolation using different numbers of neighbouring voxels. v v4 d d2 v2 v3 d3 d4 Residual Errors from PET Incorrect attenuation correction because transmission scan no 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 Ghosts (and other artefacts) in in the images do do not not move according to to the the same rigid body rules as as the the subject Slices are not acquired simultaneously rapid movements not not accounted for for by by rigid body model fmri images are distorted rigid body model does not not model these types of of distortion Spin excitation history effects variations in in residual magnetisation Functions of of the the estimated motion parameters can can be be used as as confounds in in subsequent analyses.

7 Between Modality Co-registration Not based on simply minimising mean squared difference between images. A three step approach is is used instead. ) 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.

8 First Step - Affine Registrations. Requires template images of of same modalities. Both Both images are are registered - - using 2 2 parameter affine transformations - - to to their their corresponding templates by by minimising the the mean mean squared difference. Only the the rigid-body transformation parameters differ differ between the the two two registrations. This This gives: rigid rigid body mapping between the the images. affine mappings between the the images and and the the templates. Second Step - Segmentation. Mixture Model cluster analysis to to classify MR image (or (or images) as as GM, WM & CSF. Additional information is is obtained from a priori probability images, which are are overlaid using previously determined affine transformations. Third Step - Registration of of Partitions. Grey and white matter partitions are registered using a rigid body transformation. Simultaneously minimise sum of squared difference.

9 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: Σ i Σ j h ij log h ij Σ k h ik Σ l h lj PET T weighted MRI

10 Spatial normalisation Inter-subject averaging extrapolate findings to to the population as a whole increase activation signal above that obtained from single subject increase number of of possible degrees of of freedom allowed in 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 no exact match between structure and function different brains are are organised differently computational problems (local minima, not not enough information in in the the images, computationally expensive) Compromise by correcting for gross differences followed by smoothing of of normalised images

11 Spatial Normalisation Determine the spatial transformation that minimises the sum of squared difference between an image and a linear combination of one or more templates. Original image Spatially normalised 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

12 Affine versus affine and non-linear spatial normalisation Six affine registered images. Six basis function registered images

13 T2 T Transm T 35 EPI PD PET PD T2 SS Template Images Canonical images A wider range of different contrasts can be normalised by registering 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.

14 Bayesian Formulation Bayes rule states: p(q e) p(e q) p(q) p(q e) is is the a posteriori probability of of parameters q given errors e. e. p(e q) is is the likelihood of of observing errors e given parameters q. q. p(q) is is the a priori probability of parameters q. q. Maximum a posteriori (MAP) estimate maximises p(q e). Maximising p(q e) is is equivalent to to minimising the Gibbs potential of of the posterior distribution (H(q e), where H(q e) -log p(q e)). The posterior potential is 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 is based upon the sum of of squared difference between the images. The prior potential (H(q) -log p(q)) penalises unlikely deformations.

15 Spatial Normalisation - Spatial Normalisation - affine affine The first part of spatial normalisation is a The first part of spatial normalisation is a 2 parameter 2 parameter Affine Affine Transformation Transformation 3 translations 3 translations 3 rotations 3 rotations 3 zooms 3 zooms 3 shears 3 shears Empirically generated priors Ω Ω Ω Ω Θ Θ Θ Θ Φ Φ Φ Φ ) cos( ) sin( ) sin( ) cos( ) cos( ) sin( ) sin( ) cos( ) cos( ) sin( ) sin( ) cos( Z Y X trans trans trans YZ XZ XY Z Y X shear shear shear zoom zoom zoom 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.

16 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 (p T C - p). p T C - p describes the membrane energy of the deformations. 2 2 u membraneenergy = λ = = x ji i j k ki 2

17 Without the Bayesian formulation, the non-linear spatial normalisation can introduce unnecessary warping into the spatially normalised images. Template image Affine Registration. (χ 2 = 472.) Non-linear registration using regularisation. (χ 2 = 32.7) Non-linear registration without regularisation. (χ 2 = 287.3)

18 Mixture Model cluster analysis to to classify MR image (or (or images) as as GM, WM & CSF. Additional information is is obtained from prior probability images,, which are are overlaid. Assumes that each MRI voxel is is one of of a number of of distinct tissue types (clusters). Each cluster has has a (multivariate) normal distribution. Segmentation. A smooth intensity modulating function can be be modelled by by a linear combination of of DCT basis functions..

19 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 be automatically. removed using morphological operations (erosion followed by by conditional dilation).

20 Morphometric Measures Voxel-by-voxel where are the differences between the populations? produce an an SPM of of regional differences Univariate - e.g., Voxel- Based Morphometry Multivariate - e.g., Tensor- Based Morphometry Volume based is is there a difference between the populations? Multivariate - e.g., Deformation-Based Morphometry MANCOVA & CCA

21 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.

22 Morphometric approaches based on deformation fields Deformation-based Morphometry looks at absolute displacements. Tensor-based Morphometry looks at local shapes

23 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.

24 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.

25 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: (U m -I)/m when m~=, and log(u) if m==.

26 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.

27 References Friston et al (995): Spatial registration and normalisation of images. Human Brain Mapping 3(3):65-89 Ashburner & Friston (997): Multimodal image coregistration and partitioning - a unified framework. NeuroImage 6(3):29-27 Collignon et al (995): Automated multi-modality image registration based on information theory. IPMI 95 pp Ashburner et al (997): Incorporating prior knowledge into image registration. NeuroImage 6(4): Ashburner et al (999): Nonlinear spatial normalisation using basis functions. Human Brain Mapping 7(4): Ashburner & Friston (2): Voxel-based morphometry - the methods. To appear in NeuroImage.