1. Image Registration Carlton CHU
Smooth
Realign
Normalise
Segment
DARTEL
With slides by John Ashburner, Chloe Hutton and Jesper Andersson

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

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 Shear 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 Shear 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. 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

11. 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)

12. Transformation 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?

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

25. 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

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

27. 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.
PET
A wider range of contrasts can be registered to a linear combination of template images. T1 PD
Spatial Normalisation - Templates

28. 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)

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
Mean squared difference between template and source image
Squared distance between parameters and their known expectation

30. 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)

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

32. 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

33. Gaussian Probability Density
If intensities are assumed to be Gaussian of mean mk and variance s2k, then the probability of a value yi is:

34. 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

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

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

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

38. 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:

39. Modelling a Bias Field
A bias field is included, such that the required scaling at voxel i, parameterised by b, is ri(b).
Replace the means by mk/ri(b)
Replace the variances by (sk/ri(b))2

40. Modelling a Bias Field
After rearranging... y
r(b)
y r(b)

41. 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.

42. “Mixing Proportions”
Tissue probability maps for each class are included.
The probability of obtaining class k at voxel i, given weights g is then:

43. Deforming the Tissue Probability Maps
Tissue probability images are deformed according to parameters a.
The probability of obtaining class k at voxel i, given weights g and parameters a is then:

44. The Objective Function
The Extended Model
By combining the modified P(ci=k|q) and P(yi|ci=k,q), the overall objective function (E) becomes:
The Objective Function

45. 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.

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

47. Schematic of optimisation
Repeat until convergence…
Hold g, m, s2 and a constant, and minimise E w.r.t. b
- Levenberg-Marquardt strategy, using dE/db and d2E/db2
Hold g, m, s2 and b constant, and minimise E w.r.t. a
- Levenberg-Marquardt strategy, using dE/da and d2E/da2
Hold a and b constant, and minimise E w.r.t. g, m and s2
-Use an Expectation Maximisation (EM) strategy.
end

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

49. Expectation Maximisation is used to update m, s2 and g
For iteration (n), alternate between:
E-step: Estimate belonging probabilities by:
M-step: Set q(n+1) to values that reduce:

50. 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 ma and mb
Covariance matrix Sa and Sb
-log[P(a)] = (a-ma)TSa-1(a-ma) + const
-log[P(b)] = (b-mb)TSb-1(b-mb) + const

51. Initial Affine Registration
The procedure begins with a Mutual Information affine registration of the image with the tissue probability maps. MI is computed from a 4x256 joint probability histogram.
See D'Agostino, Maes, Vandermeulen & P. Suetens. “Non-rigid Atlas-to-Image Registration by Minimization of Class-Conditional Image Entropy”. Proc. MICCAI LNCS 3216, Pages
Joint Probability Histogram
Background voxels excluded

52. Background Voxels are Excluded
An intensity threshold is found by fitting image intensities to a mixture of two Gaussians. This threshold is used to exclude most of the voxels containing only air.

53. 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)

54. Image Registration Figure out how to warp one image to match another
Normally, all subjects’ scans are matched with a common template

55. Current SPM approach Only about 1000 parameters.
Unable model detailed deformations

56. 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
Large deformation
approximation
Small deformation approximation

57. 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 1
t=0

58. Flow Field

59. 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

60. Grey matter average of 452 subjects – affine

61. Grey matter average of 471 subjects

62. Subject 1
Subject 2

63. VBM results of Patients with Alzheimer’s Disease
Both sets were smoothed with only 3mm FWHM Gaussian kernel, (FWE p>0.05)
Conventional SPM5 normalized image
DARTEL normalized image

64. VBM results of Patients with Huntington’s Disease
Both sets were smoothed with only 6mm FWHM Gaussian kernel, (FWE p>0.05)
Conventional SPM5 normalized image
DARTEL normalized image

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 in press (2005).
Ashburner. A fast diffeomorphic image reigstration algorithm. NeuroImage in press (2007).