Computational Neuroanatomy John Ashburner john@fil.ion.ucl.ac.uk

1. Computational NeuroanatomyJohn Ashburnerjohn@fil.ion.ucl.ac.uk Smoothing Rigid registration Spatial normalisation Segmentation Morphometry

2. Overview of SPM Analysis Statistical Parametric Map Design matrix fMRI time-series Motion Correction Smoothing General Linear Model Parameter Estimates Spatial Normalisation Anatomical Reference

3. Contents • Smoothing • Rigid Registration • Spatial normalisation • Segmentation • Morphometry

4. 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

5. Smoothing • Why smooth? • Potentially increase sensitivity • Inter-subject averaging • Increase validity of SPM • Smoothing is a convolution with a Gaussian kernel Gaussian convolution is separable

6. Contents • Smoothing • Rigid Registration • Rigid-body transforms • Optimisation & objective functions • Interpolation • Spatial normalisation • Segmentation • Morphometry

7. Within-subject Registration • Assumes there is no shape change, and motion is rigid-body • Used by [Realign] and [Coregister] functions • The steps are: • Registration - i.e. Optimising the parameters that describe a rigid body transformation between the source and reference images • Transformation - i.e. Re-sampling according to the determined transformation

8. Affine Transforms • Rigid-body transformations are a subset • Parallel lines remain parallel • Operations can be represented by: x1 = m11x0 + m12y0 + m13z0 + m14 y1 = m21x0 + m22y0 + m23z0 + m24 z1 = m31x0 + m32y0 + m33z0 + m34 • Or as matrices: y=mx

9. 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

10. 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

11. 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

12. 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 vox-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

13. 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

14. 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

15. Objective Functions for Image Registration • 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)

16. 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

17. Gauss-Newton Optimisation • Works best for least-squares • Minimum is estimated by fitting a quadratic at each iteration

18. 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)

19. Image Transformations • 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 1x1nx1 & 1y1ny1 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?

20. 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

21. 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.

22. Contents • Smoothing • Rigid Registration • Spatial normalisation • Affine registration • Nonlinear registration • Regularisation • Segmentation • Morphometry

23. 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

24. Spatial Normalisation - Objective • Warp the images such that functionally homologous regions from 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 gross differences followed by smoothing of normalised images

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

26. Spatial Normalisation - Templates T1 Transm T2 T1 305 T2 PD SS PD PET EPI 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. PET A wider range of contrasts can be registered to a linear combination of template images. T1 PD

27. 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)

28. 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

29. Spatial Normalisation - Overfitting Without regularisation, the non-linear spatial normalisation can introduce unnecessary warps. Affine registration. (2 = 472.1) Template image Non-linear registration without regularisation. (2 = 287.3) Non-linear registration using regularisation. (2 = 302.7)

30. Contents • Smoothing • Rigid Registration • Spatial normalisation • Segmentation • Gaussian mixture model • Including prior probability maps • Intensity non-uniformity correction • Morphometry

31. Segmentation - Mixture Model • Intensities are modelled by a mixture of K Gaussian distributions, parameterised by: • means • variances • mixing proportions • Can be multi-spectral • MultivariateGaussiandistributions

32. Segmentation - Priors • Overlay prior belonging probability maps to assist the segmentation • Prior probability of each voxel being of a particular type is derived from segmented images of 151subjects • Assumed to berepresentative • Requires initialregistration tostandard space

33. Segmentation - Bias Correction • A smooth intensity modulating function can be modelled by a linear combination of DCT basis functions

34. Segmentation - Algorithm • Results contain some non-brain tissue • Removed automaticallyusing morphologicaloperations • erosion • conditional dilation

35. Contents • Smoothing • Rigid Registration • Spatial normalisation • Segmentation • Morphometry • Volumes from deformations • Voxel-based morphometry

36. Morphometry Original Warped Template Relative volumes from Jacobian determinants

37. Very hard to define a one-to-one mappingof cortical folding

38. Early Late Difference Data from the Dementia Research Group, Queen Square.

39. Pre-processing for Voxel-Based Morphometry (VBM)

40. 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