1. Volumetric Intersubject Registration John Ashburner Wellcome Department of Imaging Neuroscience, 12 Queen Square, London, UK.

2. Intersubject registration for fMRI *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

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

4. Overview *Part I: General Inter-subject registration *Spatial transformations *Affine *Global nonlinear *Local nonlinear *Objective functions for registration *Likelihood Models *Mean squared difference *Information Theoretic measures *Prior Models *Part II: The Segmentation Method in SPM5

5. Image Registration zRegistration - i.e. Optimise the parameters that describe a spatial transformation between the source and reference (template) images zTransformation - i.e. Re-sample according to the determined transformation parameters

6. A Mapping from one image to another Need x, y and z coordinates in one image that correspond to those of another

7. Affine Transforms *Rigid-body transformations are a subset *Parallel lines remain parallel *Operations can be represented by: x’ = m 11 x + m 12 y + m 13 z + m 14 y’ = m 21 x + m 22 y + m 23 z + m 24 z’ = m 31 x + m 32 y + m 33 z + m 34 *Or as matrices: Y=Mx

8. 2D Affine Transforms *Translations by t x and t y *x’ = x + t x *y’ = y + t y *Rotation around the origin by  radians *x’ = cos(  ) x + sin(  ) y *y’ = -sin(  ) x + cos(  ) y *Zooms by s x and s y *x’ = s x x *y’ = s y y *Shear *x’ = x + h y *y’ = y

9. 2D Affine Transforms *Translations by t x and t y *x’ = 1 x + 0 y + t x *y’ = 0 x + 1 y + t y *Rotation around the origin by  radians *x’ = cos(  ) x + sin(  ) y + 0 *y’ = -sin(  ) x + cos(  ) y + 0 *Zooms by s x and s y : *x’ = s x x + 0 y + 0 *y’ = 0 x + s y y + 0 *Shear *x’ = 1 x + h y + 0 *y’ = 0 x + 1 y + 0

10. Polynomial Basis Functions As used by Roger Woods’ AIR Software

11. Cosine Transform Basis Functions As used by SPM software

12. SPM Spatial Normalisation Non-linear registration *Begin with affine registration *Refine with some non-linear registration Affine registration

13. Accuracy of Automated Volumetric Inter-subject Registration Hellier et al. Inter subject registration of functional and anatomical data using SPM. MICCAI'02 LNCS 2489 (2002) Hellier et al. Retrospective evaluation of inter-subject brain registration. MIUA (2001)

14. Local Basis Functions *More detailed deformations use lots of basis functions with local support. *Local support means that the basis functions are mostly all zero *Faster computations

15. Simple addition of displacements Notice that there is no longer a one-to-one mapping

16. Generating large one-to-one deformations The principle behind the one-to-one mappings of viscous fluid registration Y 2 = Y 1  Y 1 Y 3 = Y 1  Y 2 Y1Y1 Y 4 = Y 1  Y 3 Y 5 = Y 1  Y 4 Y 6 = Y 1  Y 5 Y 7 = Y 1  Y 6 Y 8 = Y 1  Y 7

17. Faster to repeatedly square the deformation Y1Y1 Y 2 = Y 1  Y 1 Y 4 = Y 2  Y 2 Y 8 = Y 4  Y 4 Note that this is analogous to computing a matrix exponential (c.f. Lie Groups and exponential mappings)

18. Y 16 = Y 8  Y 8

19. One-to-One Mappings *One-to-one mappings break down beyond a certain scale *The concept of a single “best” mapping may be meaningless at higher resolution Pictures taken from http://www.messybeast.com/freak-face.htm

20. 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 Value of parameter Objective function Most probable solution (global optimum) Local optimum

21. Bayes Rule *Most registration procedures maximise a joint probability of the deformation (warp) and the images (data). *P(Warp,Data) = P(Warp | Data) x P(Data) = P(Data | Warp) x P(Warp) *In practice, this can be by minimising *-log P(Warp,Data) = -log P(Data | Warp) -log P(Warp) Likelihood Prior

23. Mean Squared Difference Objective Function *Assumes one image is a warped version of the other with Gaussian noise added… *P(f i |t) = (2  2 ) -1/2 exp(-(f i -g i (t)) 2 /(2  2 )) so *-log P(f i |t) = (f i -g i (t)) 2 /(2  2 ) + 1/2 log(2  2 ) *Assumes that voxels are independent... *P(f 1,f 2,…,f N,...) = P(f 1 ) P(f 2 ) … P(f N ) so *-log P(f 1,f 2,…,f N ) = ((f 1 -g 1 (t)) 2 + (f 2 -g 2 (t)) 2 +…+ (f N -g N (t)) 2 )/(2  2 ) + 1/2 N log(2  2 )

24. Information Theoretic Approaches *Used when there is no simple relationship between intensities in one image and those of another

25. Joint Probability Density *Intensities in one image predict those of another. *Joint probability often represented by a histogram.

26. Mutual Information *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) *H(a) = -  a P(a) log P(a) da *H(a,b) = -  a P(a,b) log P(a,b) da

27. More Joint Probabilities 4x256 Joint Histograms

28. Joint Probabilities generated from Tissue Probability Maps Rather than using an image of discrete values, multiple images showing which voxels are in which class can be used. These can be constructed from an average of many subjects 4x256 Joint Histogram

29. Priors enforce “smooth” deformations *Membrane Energy *Bending Energy *Linear Elastic Energy

30. Priors enforce “smooth” deformations *The form of prior determines how the deformations behave in regions with no matching information

31. Overview *Part I: General Inter-subject registration *Part II: The Segmentation Method in SPM5 *Modelling intensities by a Mixture of Gaussians *Bias correction *Tissue Probability Maps to assist the segmentation *Warping the tissue probability maps to match the image

32. Traditional View of Pre-processing *Brain image processing is often thought of as a pipeline procedure. *One tool applied before another etc... *For example… Original Image Skull Strip Non-uniformity Correct Classify Brain Tissues Extract Brain Surfaces

33. Segmentation in SPM5 *Uses a generative model, which involves: *Mixture of Gaussians (MOG) *Bias Correction Component *Warping (Non-linear Registration) Component y1y1 c1c1   y2y2 y3y3 c2c2 c3c3     CC  CC yIyI cIcI Ashburner & Friston. Unified Segmentation. NeuroImage 26:839-851 (2005).

34. Gaussian Probability Density *If intensities are assumed to be Gaussian of mean  k and variance  2 k, then the probability of a value y i is:

35. Non-Gaussian Probability Distribution *A non-Gaussian probability density function can be modelled by a Mixture of Gaussians (MOG): Mixing proportion - positive and sums to one

36. Belonging Probabilities Belonging probabilities are assigned by normalising to one.

37. Mixing Proportions *The mixing proportion  k represents the prior probability of a voxel being drawn from class k - irrespective of its intensity. *So:

38. Non-Gaussian Intensity Distributions *Multiple Gaussians per tissue class allow non-Gaussian intensity distributions to be modelled. *E.g. accounting for partial volume effects

39. Probability of Whole Dataset *If the voxels are assumed to be independent, then the probability of the whole image is the product of the probabilities of each voxel: *A maximum-likelihood solution can be found by minimising the negative log-probability:

40. Modelling a Bias Field *A bias field is included, such that the required scaling at voxel i, parameterised by , is  i (  ). *Replace the means by  k /  i (  ) *Replace the variances by (  k /  i (  )) 2

41. Modelling a Bias Field *After rearranging... ()() y y ()y () y1y1 c1c1   y2y2 y3y3 c2c2 c3c3     CC  CC yIyI cIcI

42. Tissue Probability Maps *Tissue probability maps (TPMs) are used instead of the proportion of voxels in each Gaussian as the prior. 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.

43. “Mixing Proportions” *Tissue probability maps for each class are included. *The probability of obtaining class k at voxel i, given weights  is then: y1y1 c1c1   y2y2 y3y3 c2c2 c3c3     CC  CC yIyI cIcI

44. Deforming the Tissue Probability Maps *Tissue probability images are deformed according to parameters . *The probability of obtaining class k at voxel i is then: y1y1 c1c1   y2y2 y3y3 c2c2 c3c3     CC  CC yIyI cIcI

45. The Extended Model *By combining the modified P(c i =k|  ) and P(y i |c i =k,  ), the overall objective function (E) becomes: The Objective Function

46. 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.

47. Steepest Descent Start Optimum Alternate between optimising different groups of parameters

48. Schematic of optimisation Repeat until convergence… Hold , ,  2 and  constant, and minimise E w.r.t.  - Levenberg-Marquardt strategy, using dE/d  and d 2 E/d  2 Hold , ,  2 and  constant, and minimise E w.r.t.  - Levenberg-Marquardt strategy, using dE/d  and d 2 E/d  2 Hold  and  constant, and minimise E w.r.t. ,  and  2 -Use an Expectation Maximisation (EM) strategy. end

49. Levenberg-Marquardt Optimisation *LM optimisation is used for nonlinear registration (  ) and bias correction (  ). *Requires first and second derivatives of the objective function (E). *Parameters  and  are updated by *Increase to improve stability (at expense of decreasing speed of convergence).

50. Expectation Maximisation is used to update ,  2 and  *For iteration (n), alternate between: *E-step: Estimate belonging probabilities by: *M-step: Set  (n+1) to values that reduce:

51. Regularisation *Some bias fields and warps are more probable (a priori) than others. *Encoded using Bayes rule (for a maximum a posteriori solution): *Prior probability distributions modelled by a multivariate normal distribution. *Mean vector   and    *Covariance matrix   and    *-log[P(  )] = (  -    T   -1 (    + const *-log[P(  )] = (  -    T   -1 (    + const

52. Tissue probability maps of GM and WM Spatially normalised BrainWeb phantoms (T1, T2 and PD) Cocosco, Kollokian, Kwan & Evans. “BrainWeb: Online Interface to a 3D MRI Simulated Brain Database”. NeuroImage 5(4):S425 (1997)

53. Summary *Part I: General Inter-subject registration *Spatial transformations *Affine *Global nonlinear *Local nonlinear *Objective functions for registration *Likelihood Models *Mean squared difference *Information Theoretic measures *Prior Models *Part II: The Segmentation Method in SPM5 *Modelling intensities by a Mixture of Gaussians *Bias correction *Tissue Probability Maps to assist the segmentation *Warping the tissue probability maps to match the image

54. References *Friston et al. Spatial registration and normalisation of images. Human Brain Mapping 3:165-189 (1995). *Collignon et al. Automated multi-modality image registration based on information theory. IPMI’95 pp 263-274 (1995). *Ashburner et al. Incorporating prior knowledge into image registration. NeuroImage 6:344-352 (1997). *Ashburner & Friston. Nonlinear spatial normalisation using basis functions. Human Brain Mapping 7:254-266 (1999). *Thévenaz et al. Interpolation revisited. IEEE Trans. Med. Imaging 19:739-758 (2000). *Andersson et al. Modeling geometric deformations in EPI time series. Neuroimage 13:903-919 (2001). *Ashburner & Friston. Unified Segmentation. NeuroImage in press (2005).

57. Spare slides

58. Very hard to define a one-to-one mapping of cortical folding Use only approximate registration.

59. Smooth Before convolutionConvolved with a circleConvolved with a Gaussian Smoothing is done by convolution. Each voxel after smoothing effectively becomes the result of applying a weighted region of interest (ROI).

60. 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 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 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 * M B -1 M A

61. 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

62. Transforming an image *Images are re-sampled. An example in 2D: for y=1..n y % loop over rows for x=1..n x % loop over pixels in row x’= t x (x,y,a) % transform according to a y’= t y (x,y,a) if 1  x’  n x & 1  y’  n y then % voxel in range f (x,y) = f’(x’,y’) % assign re-sampled value end % voxel in range end % loop over pixels in row end % loop over rows *What happens if x’ and y’ are not integers?

63. *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 Simple Interpolation

64. 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.

65. Inverse

66. 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

67. 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

68. A Growing Trend *Larger and more complex models are being produced to explain brain imaging data. *Bigger and better computers *allow more powerful models to be used *More experience among software developers *Older and wiser *More engineers - rather than e.g. psychiatrists & biochemists *This presentation is about combining various preprocessing procedures for anatomical images into a single generative model.

69. Another example (for VBM)

70. Bias Correction helps Registration *MRI images are corrupted by a smooth intensity non- uniformity (bias). *Image intensity non-uniformity artefact has a negative impact on most registration approaches. *Much better if this artefact is corrected. Image with bias artefact Corrected image

71. Bias Correction helps Segmentation *Similar tissues no longer have similar intensities. *Artefact should be corrected to enable intensity-based tissue classification.

72. Registration helps Segmentation *SPM99 and SPM2 require tissue probability maps to be overlaid prior to segmentation.

73. Segmentation helps Bias Correction *Bias correction should not eliminate differences between tissue classes. *Can be done by *make all white matter about the same intensity *make all grey matter about the same intensity *etc *Currently fairly standard practice to combine bias correction and tissue classification

74. Segmentation helps Registration Original MRI Template Grey Matter Segment Affine register Tissue probability maps Deformation Affine Transform Spatial Normalisation - estimation Spatial Normalisation - writing Spatially Normalised MRI A convoluted method using SPM2