2. Computational Neuroanatomy John Ashburner john@fil.ion.ucl.ac.uk zSmoothing zRigid registration zSpatial normalisation zSegmentation zMorphometry zSmoothing zRigid registration zSpatial normalisation zSegmentation zMorphometry

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

4. Contents zSmoothing zRigid Registration zSpatial normalisation zSegmentation zMorphometry

5. Smoothing Before convolutionConvolved with a circleConvolved with a Gaussian Each voxel after smoothing effectively becomes the result of applying a weighted region of interest (ROI).

6. Smoothing zWhy smooth? yPotentially increase sensitivity yInter-subject averaging yIncrease validity of SPM zSmoothing is a convolution with a Gaussian kernel Gaussian convolution is separable

7. Contents zSmoothing zRigid Registration yRigid-body transforms yOptimisation & objective functions yInterpolation zSpatial normalisation zSegmentation zMorphometry

8. Within-subject Registration zAssumes there is no shape change, and motion is rigid-body  Used by [Realign] and [Coregister] functions zThe steps are: zRegistration - i.e. Optimising the parameters that describe a rigid body transformation between the source and reference images zTransformation - i.e. Re-sampling according to the determined transformation

9. Affine Transforms zRigid-body transformations are a subset zParallel lines remain parallel zOperations can be represented by: x 1 = m 11 x 0 + m 12 y 0 + m 13 z 0 + m 14 y 1 = m 21 x 0 + m 22 y 0 + m 23 z 0 + m 24 z 1 = m 31 x 0 + m 32 y 0 + m 33 z 0 + m 34 zOr as matrices: y=mx

10. 2D Affine Transforms zTranslations by t x and t y yx 1 = x 0 + t x yy 1 = y 0 + t y  Rotation around the origin by  radians  x 1 = cos(  ) x 0 + sin(  ) y 0  y 1 = -sin(  ) x 0 + cos(  ) y 0 zZooms by s x and s y : yx 1 = s x x 0 yy 1 = s y y 0 zShear zx 1 = x 0 + h y 0 zy 1 = y 0

11. 2D Affine Transforms zTranslations by t x and t y yx 1 = 1 x 0 + 0 y 0 + t x yy 1 = 0 x 0 + 1 y 0 + t y  Rotation around the origin by  radians  x 1 = cos(  ) x 0 + sin(  ) y 0 + 0  y 1 = -sin(  ) x 0 + cos(  ) y 0 + 0 zZooms by s x and s y : yx 1 = s x x 0 + 0 y 0 + 0 yy 1 = 0 x 0 + s y y 0 + 0 zShear zx 1 = 1 x 0 + h y 0 + 0 zy 1 = 0 x 0 + 1 y 0 + 0

12. 3D Rigid-body Transformations zA 3D rigid body transform is defined by: y3 translations - in X, Y & Z directions y3 rotations - about X, Y & Z axes zThe order of the operations matters TranslationsPitch about x axis Roll about y axis Yaw about z axis

13. Voxel-to-world Transforms zAffine transform associated with each image yMaps from voxels (x=1..N x, y=1..N y, z=1..N z ) to some world co-ordinate system. e.g., xScanner co-ordinates - images from DICOM toolbox xT&T/MNI coordinates - spatially normalised zRegistering image B (source) to image A (target) will update B’s vox-to-world mapping yMapping from voxels in A to voxels in B is by xA-to-world using M A, then world-to-b using M B -1 x M B -1 M A

14. Left- and Right-handed Coordinate Systems zAnalyze™ files are stored in a left-handed system zTalairach & Tournoux uses a right-handed system zMapping between them requires a flip yAffine transform with a negative determinant

15. Optimisation zOptimisation involves finding some “best” parameters according to an “objective function”, which is either minimised or maximised zThe “objective function” is often related to a probability based on some model Value of parameter Objective function Most probable solution (global optimum) Local optimum

16. Objective Functions for Image Registration zIntra-modal yMean squared difference (minimise) yNormalised cross correlation (maximise) yEntropy of difference (minimise) zInter-modal (or intra-modal) yMutual information (maximise) yNormalised mutual information (maximise) yEntropy correlation coefficient (maximise) yAIR cost function (minimise)

17. Mean-squared Difference zMinimising mean-squared difference works for intra-modal registration (realignment) zSimple relationship between intensities in one image, versus those in the other yAssumes normally distributed differences

18. Gauss-Newton Optimisation zWorks best for least- squares zMinimum is estimated by fitting a quadratic at each iteration

19. Mutual Information zUsed for between-modality registration zDerived from joint histograms  MI=  ab P(a,b) log 2 [P(a,b)/( P(a) P(b) )] yRelated to entropy: MI = -H(a,b) + H(a) + H(b) Where H(a) = -  a P(a) log 2 P(a) and H(a,b) = -  a P(a,b) log 2 P(a,b)

20. Image Transformations zImages are re-sampled. An example in 2D: for y 0 =1..n y0, % loop over rows for x 0 =1..n x0, % loop over pixels in row x 1 = t x (x 0,y 0,q) % transform according to q y 1 = t y (x 0,y 0,q) if 1  x 1  n x1 & 1  y 1  n y1 then, % voxel in range f 1 (x 0,y 0 ) = f 0 (x 1,y 1 ) % assign re-sampled value end % voxel in range end % loop over pixels in row end % loop over rows zWhat happens if x 1 and y 1 are not integers?

21. zNearest neighbour yTake the value of the closest voxel zTri-linear yJust a weighted average of the neighbouring voxels yf 5 = f 1 x 2 + f 2 x 1 yf 6 = f 3 x 2 + f 4 x 1 yf 7 = f 5 y 2 + f 6 y 1 Simple Interpolation

22. B-spline Interpolation B-splines are piecewise polynomials A continuous function is represented by a linear combination of basis functions 2D B-spline basis functions of degrees 0, 1, 2 and 3 Nearest neighbour and trilinear interpolation are the same as B-spline interpolation with degrees 0 and 1.

23. Contents zSmoothing zRigid Registration zSpatial normalisation yAffine registration yNonlinear registration yRegularisation zSegmentation zMorphometry

24. Spatial Normalisation - Reasons zInter-subject averaging yIncrease sensitivity with more subjects xfixed-effects analysis yExtrapolate findings to the population as a whole xmixed-effects analysis zStandard coordinate system ye.g. Talairach & Tournoux space

25. Spatial Normalisation - Objective zWarp the images such that functionally homologous regions from different subjects are as close together as possible yProblems: xNo exact match between structure and function xDifferent brains are organised differently xComputational problems (local minima, not enough information in the images, computationally expensive) zCompromise by correcting gross differences followed by smoothing of normalised images

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

27. EPI T2 T1Transm PDPET 305T1 PD T2 SS Template Images“Canonical” images A wider range of contrasts can be registered 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. Spatial Normalisation - Templates T1PD PET

28. Spatial Normalisation - Affine zThe first part is a 12 parameter affine transform y3 translations y3 rotations y3 zooms y3 shears zFits overall shape and size zAlgorithm simultaneously minimises yMean-squared difference between template and source image ySquared distance between parameters and their expected values (regularisation)

29. 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 yMean squared difference between template and source image ySquared distance between parameters and their known expectation

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

31. Contents zSmoothing zRigid Registration zSpatial normalisation zSegmentation yGaussian mixture model yIncluding prior probability maps yIntensity non-uniformity correction zMorphometry

32. Segmentation - Mixture Model zIntensities are modelled by a mixture of K Gaussian distributions, parameterised by: xmeans xvariances xmixing proportions zCan be multi-spectral yMultivariate Gaussian distributions

33. Segmentation - Priors zOverlay prior belonging probability maps to assist the segmentation yPrior probability of each voxel being of a particular type is derived from segmented images of 151 subjects xAssumed to be representative yRequires initial registration to standard space

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

35. Segmentation - Algorithm zResults contain some non-brain tissue zRemoved automatically using morphological operations yerosion yconditional dilation

36. Contents zSmoothing zRigid Registration zSpatial normalisation zSegmentation zMorphometry yVolumes from deformations yVoxel-based morphometry

37. TemplateWarpedOriginal Relative volumes from Jacobian determinants Morphometry

38. Very hard to define a one-to- one mapping of cortical folding

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

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

41. 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 References