1. Image Registration John Ashburner

2. Pre-processing Overview
Statistics or whatever
fMRI time-series
Template
Anatomical MRI
Smoothed
Estimate Spatial Norm
Motion Correct
Smooth
Coregister
Spatially normalised
Deformation

3. Contents Preliminaries Intra-Subject Registration
Smooth
Rigid-Body and Affine Transformations
Optimisation and Objective Functions
Transformations and Interpolation
Intra-Subject Registration
Inter-Subject Registration

4. Smooth Smoothing is done by convolution.
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. Image Registration
Registration - i.e. Optimise the parameters that describe a spatial transformation between the source and reference (template) images
Transformation - i.e. Re-sample according to the determined transformation parameters

6. 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

7. 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

8. 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

9. 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 voxel-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

10. Left- and Right-handed Coordinate Systems
NIfTI format files are stored in either a left- or right-handed system
Indicated in the header
Talairach & Tournoux uses a right-handed system
Mapping between them sometimes requires a flip
Affine transform has a negative determinant

11. Optimisation Image registration is done by 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

12. Objective Functions Intra-modal Inter-modal (or 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)

13. Simple Interpolation Nearest neighbour Tri-linear
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

14. 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.

15. Contents Preliminaries Intra-Subject Registration
Realign
Mean-squared difference objective function
Residual artifacts and distortion correction
Coregister
Inter-Subject Registration

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. Residual Errors from aligned fMRI
Re-sampling can introduce interpolation errors
especially tri-linear interpolation
Gaps between slices can cause aliasing artefacts
Slices are not acquired simultaneously
rapid movements not accounted for by rigid body model
Image artefacts may not move according to a rigid body model
image distortion
image dropout
Nyquist ghost
Functions of the estimated motion parameters can be modelled as confounds in subsequent analyses

18. Movement by Distortion Interaction of fMRI
Subject disrupts B0 field, rendering it inhomogeneous
distortions in phase-encode direction
Subject moves during EPI time series
Distortions vary with subject orientation
shape varies

19. Movement by distortion interaction

20. Correcting for distortion changes using Unwarp
Estimate reference from mean of all scans.
Estimate new distortion fields for each image:
estimate rate of change of field with respect to the current estimate of movement parameters in pitch and roll.
Unwarp time series.
Estimate movement parameters. 
+
Andersson et al, 2001

21. Contents Preliminaries Intra-Subject Registration
Realign
Coregister
Mutual Information objective function
Inter-Subject Registration

22. Inter-modal registration
Match images from same subject but different modalities:
anatomical localisation of single subject activations
achieve more precise spatial normalisation of functional image using anatomical image.

23. 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)

24. Contents Preliminaries Intra-Subject Registration
Inter-Subject Registration
Normalise
Affine Registration
Nonlinear Registration
Regularisation
Segment
DARTEL

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

26. Spatial Normalisation - Templates
Transm T1
305 PD T2 SS
EPI PD
PET
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.
Spatial Normalisation - Templates

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)
Target
image
Non-linear
registration
without
regularisation.
(2 = 287.3)
Non-linear
registration
using
regularisation.
(2 = 302.7)

30. Contents Preliminaries Intra-Subject Registration
Inter-Subject Registration
Normalise
Segment
Gaussian mixture model
Intensity non-uniformity correction
Deformed tissue probability maps
DARTEL

31. Segmentation
Segmentation in SPM5 also estimates a spatial transformation that can be used for spatially normalising images.
It uses a generative model, which involves:
Mixture of Gaussians (MOG)
Bias Correction Component
Warping (Non-linear Registration) Component

32. Mixture of Gaussians (MOG)
Classification is based on a Mixture of Gaussians model (MOG), which represents the intensity probability density by a number of Gaussian distributions.
Frequency
Image Intensity

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

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

35. Modelling a Bias Field
A bias field is modelled as a linear combination of basis functions.
Corrected image
Corrupted image
Bias Field

36. 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.

37. Deforming the Tissue Probability Maps
Tissue probability images are deformed so that they can be overlaid on top of the image to segment.

38. 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.

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

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

41. Contents Preliminaries Intra-Subject Registration
Inter-Subject Registration
Normalise
Segment
DARTEL
Flow field parameterisation
Objective function
Template creation
Examples

42. Small deformation approximation
A one-to-one mapping
Many models simply add a smooth displacement to an identity transform
One-to-one mapping not enforced
Inverses approximately obtained by subtracting the displacement
Not a real inverse
Small deformation approximation

43. φ(1)(x) = ∫ u(φ(t)(x))dt u is a flow field to be estimated
DARTEL
Parameterising the deformation
φ(0)(x) = x
φ(1)(x) = ∫ u(φ(t)(x))dt
u is a flow field to be estimated
Scaling and squaring is used to generate deformations. 1
t=0

44. Scaling and squaring example

45. Forward and backward transforms

46. Registration objective function
Simultaneously minimize the sum of:
Likelihood component
Drives the matching of the images.
Multinomial assumption
Prior component
A measure of deformation roughness
Regularises the registration.
½uTHu
A balance between the two terms.

47. Likelihood Model log p(t|μ,ϕ) = ΣjΣk tjk log(μk(ϕj))
Current DARTEL model is multinomial for matching tissue class images.
Template represents probability of obtaining different tissues at each point.
log p(t|μ,ϕ) = ΣjΣk tjk log(μk(ϕj))
t – individual GM, WM and background
μ – template GM, WM and background
ϕ – deformation

48. Prior Model

49. Simultaneous registration of GM to GM and WM to WM
Grey matter
White matter
Subject 1
Grey matter
White matter
Subject 3
Grey matter
White matter
Grey matter
White matter
Template
Grey matter
White matter
Subject 2
Subject 4

50. Template Initial Average Iteratively generated from 471 subjects
Began with rigidly aligned tissue probability maps
Used an inverse consistent formulation
After a few iterations
Final template

51. Grey matter average of 452 subjects – affine

52. Grey matter average of 471 subjects

53. Initial
GM images

54. Warped
GM images

64. VBM Pre-processing
Use Segment button for characterising intensity distributions of tissue classes.
DARTEL import, to generate rigidly aligned grey and white matter maps.
DARTEL warping to generate “modulated” warped grey matter.
Smooth.
Statistics.

65. References
Friston et al. Spatial registration and normalisation of images. Human Brain Mapping 3: (1995).
Collignon et al. Automated multi-modality image registration based on information theory. IPMI’95 pp (1995).
Ashburner et al. Incorporating prior knowledge into image registration. NeuroImage 6: (1997).
Ashburner & Friston. Nonlinear spatial normalisation using basis functions. Human Brain Mapping 7: (1999).
Thévenaz et al. Interpolation revisited. IEEE Trans. Med. Imaging 19: (2000).
Andersson et al. Modeling geometric deformations in EPI time series. Neuroimage 13: (2001).
Ashburner & Friston. Unified Segmentation. NeuroImage 26: (2005).
Ashburner. A Fast Diffeomorphic Image Registration Algorithm. NeuroImage 38: (2007).