1. Statistical Parametric
Mapping (SPM)
Talk I: Spatial Pre-processing & Morphometry
Talk II: General Linear Model
Talk III: Experimental Design & Efficiency
Talk IV: EEG/MEG

2. Spatial Pre-processing & Morphometry
Rik Henson
With thanks to: John Ashburner, Matthew Brett, Karl Friston

3. Overview fMRI time-series Statistical Parametric Map Smoothing kernel
General Linear Model
Design matrix
Motion
correction
Parameter Estimates
Spatial
normalisation
Standard
template

4. Overview 1. Realignment (within-modality)
2. Between-modality Coregistration
3. Normalisation (to stereotactic space)
4. Smoothing
5. Unified Segmentation & Normalisation
6. Morphometry (VBM/DBM/TBM)

5. Overview 1. Realignment (within-modality)
- Spatial (movement correction)
- Unwarping (movement-by-distortion)
- Temporal (slice-timing correction)
2. Between-modality Coregistration
3. Normalisation (to stereotactic space)
4. Smoothing
5. Unified Segmentation & Normalisation
6. Morphometry (VBM/DBM/TBM)

6. Spatial Realignment: Reasons for Motion Correction
Subjects will always move in the scanner
The sensitivity of the analysis depends on the residual noise in the image series, so movement that is unrelated to the subject’s task will add to this noise and hence realignment will increase the sensitivity
However, subject movement may also correlate with the task…
…in which case realignment may reduce sensitivity (and it may not be possible to discount artefacts that owe to motion)
Realignment (of same-modality images from same subject) involves two stages:
1. Registration - estimate the 6 parameters that describe the rigid body transformation between each image and a reference image
2. Reslicing - re-sample each image according to the determined transformation parameters

7. Rigid body transformations parameterised by:
1. Registration
Determine the rigid body transformation that minimises the sum of squared difference between images
Rigid body transformation is defined by:
3 translations - in X, Y & Z directions
3 rotations - about X, Y & Z axes
Operations can be represented as affine transformation matrices:
x1 = m1,1x0 + m1,2y0 + m1,3z0 + m1,4
y1 = m2,1x0 + m2,2y0 + m2,3z0 + m2,4
z1 = m3,1x0 + m3,2y0 + m3,3z0 + m3,4
Squared Error
Translations
Pitch
Roll
Yaw
Rigid body transformations parameterised by:

8. 1. Registration Iterative procedure (Gauss-Newton ascent)
Additional scaling parameter
Nx6 matrix of realignment parameters written to file (N is number of scans)
Orientation matrices in header of image file (data not changed until reslicing)

9. 2. Reslicing
Nearest Neighbour
Linear
Application of registration parameters involves re-sampling the image to create new voxels by interpolation from existing voxels
Interpolation can be nearest neighbour (0-order), tri-linear (1st-order), (windowed) fourier/sinc, or nth-order “b-splines”
Full sinc (no alias)
Windowed sinc

10. Residual Errors after Spatial Realignment
Interpolation errors, especially with tri-linear interpolation and small-window sinc
Ghosts (and other artefacts) in the image (which do not move as a rigid body)
Rapid movements within a scan (which cause non-rigid image deformation)
Spin excitation history effects (residual magnetisation effects of previous scans)
Interaction between movement and local field inhomogeniety, giving non-rigid distortion…
Before realignment
After realignment

11. Overview 1. Realignment (within-modality)
- Spatial (movement correction)
- Unwarping (movement-by-distortion)
- Temporal (slice-timing correction)
2. Between-modality Coregistration
3. Normalisation (to stereotactic space)
4. Smoothing
5. Unified Segmentation & Normalisation
6. Morphometry (VBM/DBM/TBM)

12. Unwarping
EPIs contain distortions owing to field inhomogenieties (susceptibility artifacts, specifically in phase-encoding direction)
They can be “undistorted” by use of a field-map (available in the “FieldMap” SPM toolbox)
(Note that susceptibility artifacts that cause drop- out are more difficult to correct)
However, movement interacts with the field inhomogeniety (presence of object affects B0), ie distortions change with position of object in field
This movement-by-distortion can be accommodated during realignment using “unwarp”
Field-map
Distorted image
Corrected image

13. Unwarping Pitch  +  B0  Roll Estimated derivative fields
One could include the movement parameters as confounds in the statistical model of activations
However, this may remove activations of interest if they are correlated with the movement
Better is to incorporate physics knowledge, eg to model how field changes as function of pitch and roll (assuming phase-encoding is in y-direction)…
… using Taylor expansion (about mean realigned image):
Iterate: 1) estimate movement parameters (, ), 2) estimate deformation fields, 1) re-estimate movement …
Fields expressed by spatial basis functions (3D discrete cosine set)…
Pitch
Estimated derivative fields
 + 
B0 
Roll

14. Unwarping tmax=13.38 tmax=5.06 tmax=9.57
Example: Movement correlated with design
tmax=13.38
No correction
tmax=5.06
Correction by covariation
tmax=9.57
Correction by Unwarp

15. Overview 1. Realignment (within-modality)
- Spatial (movement correction)
- Unwarping (movement-by-distortion)
- Temporal (slice-timing correction)
2. Between-modality Coregistration
3. Normalisation (to stereotactic space)
4. Smoothing
5. Unified Segmentation & Normalisation
6. Morphometry (VBM/DBM/TBM)

16. Temporal Realignment (Slice-Timing Correction)
Most functional MRI uses Echo-Planar Imaging (EPI)
Each plane (slice) is typically acquired every 3mm
normally axial…
… requiring ~32 slices to cover cortex (40 to cover cerebellum too)
(actually consists of slice-thickness, eg 2mm, and interslice gap, eg 1mm, sometimes expressed in terms of “distance factor”)
(slices can be acquired contiguously, eg [ ], or interleaved, eg [ ])
Each plane (slice) takes about ~60ms to acquire…
…entailing a typical TR for whole volume of 2-3s
Volumes normally acquired continuously (though sometimes gap so that TR>TA)
2-3s between sampling the BOLD response in the first slice and the last slice
(a problem for transient neural activity; less so for sustained neural activity)

17. Temporal Realignment (Slice-Timing Correction)
Bottom Slice
Top Slice
SPM{t}
TR=3s
…the “Slice-timing Problem”
(Henson et al, 1999)
Slices acquired at different times, yet model is the same for all slices
(slice-specific models would have problems when reslice, eg normalise)
Two main solutions:
Temporal interpolation of data
(to a specified reference slice) … but less good for longer TRs
2. More general basis set (e.g., with temporal derivatives) … but inference more difficult
Interpolated
SPM{t}
Derivative
SPM{F}

18. Temporal Interpolation
10 slice volume
Voxel in slice 1
Voxel in slice 5
Interpolated time course
Estimated value at time of first slice

19. Temporal Realignment (Slice-Timing Correction)
Interpolation artefacts often greater the more the interpolation, so choose middle slice in time as reference slice
(but don’t forget to synchronise model with the middle of the scan!)
Should any slice-time correction (temporal realignment) be done before or after motion correction (spatial realignment)?
Well, depends on slice order (sequential or interleaved), and amount of movement…
Movement of more than one slice (eg >3mm) would cause temporal interpolation of data from different tissue-types (markedly different values, eg brain boundary), increasing interpolation artefacts…

20. Interpolation Artifacts
Synch Interpolation can spread artefact across time (scans)

21. Temporal Realignment (Slice-Timing Correction)
Interpolation artefacts often greater the more the interpolation, so choose middle slice in time as reference slice
(but don’t forget to synchronise model with the middle of the scan!)
Should any slice-time correction (temporal realignment) be done before or after motion correction (spatial realignment)?
Well, depends on slice order (sequential or interleaved), and amount of movement…
Movement of more than one slice (eg >3mm) would cause temporal interpolation of data from different tissue-types (markedly different values, eg brain boundary), increasing interpolation artefacts…
…which may be than greater timing errors resulting from movement correction first (e.g, only ~TR/N for contiguous, or TR/2 for interleaved)
Probably best to perform slice-time correction AFTER movement correction, particularly for contiguous acquisition

22. Overview 1. Realignment (within-modality)
- Spatial (movement correction)
- Unwarping (movement-by-distortion)
- Temporal (slice-timing correction)
2. Between-modality Coregistration
3. Normalisation (to stereotactic space)
4. Smoothing
5. Unified Segmentation & Normalisation
6. Morphometry (VBM/DBM/TBM)

23. Between Modality Co-registration
Useful, for example, to display functional results (EPI) onto high resolution structural image (T1)…
…indeed, necessary if spatial normalisation is determined by T1 image
Because different modality images have different properties (e.g., relative intensity of gray/white matter), cannot simply minimise difference between images
Therefore, use Mutual Information as cost function, rather than squared differences…
EPI T2 T1
Transm PD
PET

24. Between Modality Co-registration

25. Between Modality Coregistration: Mutual Information
Mutual Information apparent from 2D histogram (plot of one image against other)
For histograms normalised to integrate to unity, the Mutual Information is:
SiSj hij log (hij)
Sk hik Sl hlj
PET
T1 MRI

26. Overview 1. Realignment (within-modality)
- Temporal (slice-timing correction)
- Spatial (movement correction)
- Unwarping (movement-by-distortion)
2. Between-modality Coregistration
3. Normalisation (to stereotactic space)
4. Smoothing
5. Unified Segmentation & Normalisation
6. Morphometry (VBM/DBM/TBM)

27. Reasons for Normalisation
Inter-subject averaging
extrapolate findings to the population as a whole
increase statistical power above that obtained from single subject
Reporting of activations as co-ordinates within a standard stereotactic space
e.g. the space described by Talairach & Tournoux
Label-based approaches: Warp the images such that defined landmarks (points/lines/surfaces) are aligned
but few readily identifiable landmarks (and manually defined?)
sulcal matching may help for cortex, but not deep-brain structures
Intensity-based approaches: Warp to images to maximise some voxel-wise similarity measure
eg, squared error, assuming intensity correspondence (within-modality)
Normalisation constrained to correct for only gross differences; residual variability accommodated by subsequent spatial smoothing

28. Spatial Normalisation
Summary
Original image
Spatially normalised
Determine transformation that minimises the sum of squared difference between an image and a (combination of) template image(s)
Two stages:
1. affine registration to match size and position of the images
2. non-linear warping to match the overall brain shape
Uses a Bayesian framework to constrain affine and warps
Spatial Normalisation
Template
image
Deformation field

29. Stage 1. Full Affine Transformation
The first part of normalisation is a 12 parameter affine transformation
3 translations
3 rotations
3 zooms
3 shears
Better if template image in same modality (eg because of image distortions in EPI but not T1)
Rotation
Translation
Zoom
Shear
Rigid body

30. Six affine registered images
Insufficiency of Affine-only normalisation
Six affine registered images
Six affine + nonlinear registered

31. Stage 2. Nonlinear Warps
Deformations consist of a linear combination of smooth basis images
These are the lowest frequency basis images of a 3-D discrete cosine transform
Brain masks can be applied (eg for lesions)

32. Bayesian Constraints
Template
image
Affine Registration
(2 = 472.1)
Non-linear
registration
with
regularisation
(2 = 302.7)
Non-linear
registration
without
regularisation
(2 = 287.3)
Without the Bayesian formulation, the non-linear spatial normalisation can introduce unnecessary warping into the spatially normalised images

33. Bayesian Constraints
Using Bayes rule, we can constrain (“regularise”) the nonlinear fit by incorporating prior knowledge of the likely extent of deformations:
p(p|e)  p(e|p) p(p) (Bayes Rule)
p(p|e) is the a posteriori probability of parameters p given errors e
p(e|p) is the likelihood of observing errors e given parameters p
p(p) is the a priori probability of parameters p
For Maximum a posteriori (MAP) estimate, we minimise (taking logs):
H(p|e)  H(e|p) + H(p)
H(e|p) (-log p(e|p)) is the squared difference between the images (error)
H(p) (-log p(p)) constrains parameters (penalises unlikely deformations)
 is “regularisation” hyperparameter, weighting effect of “priors”

34. Bayesian Constraints Algorithm simultaneously minimises:
Sum of squared difference between template and object
Squared distance between the parameters and their expectation
Bayesian constraints applied to both:
1) affine transformations
based on empirical prior ranges
2) nonlinear deformations
based on smoothness constraint (minimising membrane energy)
Empirically generated priors

35. DARTEL: Diffeomorphic Registration (SPM8)
Nonlinear warps do not guarantee a 1-to-1 mapping between object and target image...
...not “diffeomorphic”...
...so inverting this mapping (eg un-normalising a brain) is not precise
Better approach is to construct a flow field, so one image can slowly “flow” into another => DARTEL

36. DARTEL: Diffeomorphic Registration (SPM8)
Grey matter average of 452 subjects
Affine
Grey matter average of 471 subjects
DARTEL

37. Overview 1. Realignment (within-modality)
- Temporal (slice-timing correction)
- Spatial (movement correction)
- Unwarping (movement-by-distortion)
2. Between-modality Coregistration
3. Normalisation (to stereotactic space)
4. Smoothing
5. Unified Segmentation & Normalisation
6. Morphometry (VBM/DBM/TBM)

38. Reasons for Smoothing
Potentially increase signal to noise (matched filter theorem)
Inter-subject averaging (allowing for residual differences after normalisation)
Increase validity of statistics (more likely that errors distributed normally)
Kernel defined in terms of FWHM (full width at half maximum) of filter - usually ~16-20mm (PET) or ~6-8mm (fMRI) of a Gaussian
Ultimate smoothness is function of applied smoothing and intrinsic image smoothness (sometimes expressed as “resels” - RESolvable Elements)
Gaussian smoothing kernel
FWHM

39. Overview 1. Realignment (within-modality)
- Temporal (slice-timing correction)
- Spatial (movement correction)
- Unwarping (movement-by-distortion)
2. Between-modality Coregistration
3. Normalisation (to stereotactic space)
4. Smoothing
5. Unified Segmentation & Normalisation
6. Morphometry (VBM/DBM/TBM)

40. Image Segmentation (in general)
Partition into gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF)
‘Mixture Model’ cluster analysis used, which assumes each voxel is one of a number of distinct tissue types (clusters), each with a (multivariate) normal distribution
Further Bayesian constraints from prior probability images, which are overlaid
Additional correction for (low-spatial frequency) intensity inhomogeniety… .
Intensity histogram
fit by multi-Gaussians
Priors:
Image:
Brain/skull
CSF WM GM

41. Image Segmentation (in practice (SPM5))
Intensity-based partitioning of image is complicated by spatial image intensity inhomogenieties…
…but correction of image inhomogeniety is complicated by different spatial distributions of different tissue-types
Original
Corrected

42. Image Segmentation (in practice (SPM5))
Intensity-based partitioning of image is aided by coregistration with prior tissue-type probabilities in template space….
…but coregistration of an image with template space (ie normalisation) is aided by separating different tissue-types
(eg, might want normalisation to be based on gray-matter only)

43. Image Segmentation (in practice (SPM5))
These interdependencies prevent traditional serial-stage processing (e.g, inhomogeniety-correction -> normalisation -> segmentation)
Need a single, unified generative model (solved iteratively)…

44. Image Segmentation (in practice (SPM5))
Initial coregistration with templates using (4x256) Mutual Information
Then Repeat until convergence:
1. Estimate field inhomogenieties (using discrete cosine set)
2. Estimate tissue classifications (using Mixture of Gaussians)
3. Estimate new warps for coregistration with templates
Result is not only segmentation, but superior normalisation to standard normalisation…
…and more robust to lesions
(Though aided by manual repositioning close to template space)

45. Overview 1. Realignment (within-modality)
- Temporal (slice-timing correction)
- Spatial (movement correction)
- Unwarping (movement-by-distortion)
2. Between-modality Coregistration
3. Normalisation (to stereotactic space)
4. Smoothing
5. Unified Segmentation & Normalisation
6. Morphometry (VBM/DBM/TBM)

46. Morphometry (Computational Neuroanatomy)
Voxel-by-voxel: where are the differences between populations?
Univariate: e.g, Voxel-Based Morphometry (VBM)
Multivariate: e.g, Tensor-Based Morphometry (TBM)
Volume-based: is there a difference between populations?
Multivariate: e.g, Deformation-Based Morphometry (DBM)
Continuum:
perfect normalisation => all information in Deformation field (no VBM differences)
no normalisation => all in VBM
Spatial Normalisation
Original
Template
Normalised
Deformation field
VBM
TBM
DBM

47. Voxel-Based Morphometry (VBM)
A voxel by voxel statistical analysis is used to detect regional differences in the amount of grey matter between populations
Original
image
Spatially
normalised
Segmented
grey matter
Smoothed
SPM
Voxel intensities can be further modulated by determinant of Jacobean of warps: i.e,
whether want to compare graymatter density (unmodulated) or volume (modulated)

48. cingulate/parafalcine
VBM Examples: Aging
Grey matter volume
loss with age
superior parietal
pre and post central
insula
cingulate/parafalcine

49. VBM Examples: Brain Asymmetries
Right frontal and left occipital petalia

50. Morphometry on deformation fields: DBM/TBM
Deformation-based Morphometry
looks at absolute displacements
Tensor-based Morphometry
looks at local shapes
Vector field
Tensor field

51. Deformation-based Morphometry (DBM)
fields
...
Remove positional and size information - leave shape
Parameter reduction using principal component analysis (SVD)
Multivariate analysis of covariance used to identify differences between groups
Canonical correlation analysis used to characterise differences between groups

52. DBM Example: Sex Differences
Non-linear warps of sex differences characterised by canonical variates analysis
Mean differences (mapping from an average female to male brain)

53. Tensor-based morphometry
Template
Warped
Original
If the original Jacobian matrix is donated by A, then this can be decomposed into: A = RU, where R is an orthonormal rotation matrix, and U is a symmetric matrix containing only zooms and shears.
Relative volumes
Strain tensor
Strain tensors are defined that model the amount of distortion. If there is no strain, then tensors are all zero. Generically, the family of Lagrangean strain tensors are given by: (Um-I)/m when m~=0, and log(U) if m==0.

54. References
Friston et al (1995): Spatial registration and normalisation of images. Human Brain Mapping 3(3):
Ashburner & Friston (1997): Multimodal image coregistration and partitioning - a unified framework. NeuroImage 6(3):
Collignon et al (1995): Automated multi-modality image registration based on information theory. IPMI’95 pp
Ashburner et al (1997): Incorporating prior knowledge into image registration. NeuroImage 6(4):
Ashburner et al (1999): Nonlinear spatial normalisation using basis functions. Human Brain Mapping 7(4):
Ashburner & Friston (2000): Voxel-based morphometry - the methods. NeuroImage 11:

56. SPM5 requirements… Workstation Matlab 6.5 - 7.1
developed on Sun Solaris UNIX
Solaris, Linux, Mac & Windows supported
other UNIX
disk & memory…
Matlab
no special “toolboxes” required
ANSII C Compiler
to compile external C–mex routines
ready for Solaris, Linux, & Windows
NIfTI/Analyze format images
conversion program
extend SPM
Internet access
…for SPMweb & the discussion list
Plenty of time!

57. SPM documentation… peer reviewed literature
SPM course notes, Human Brain Function &
SPM manual
algorithm descriptions, code annotations, pseudo-code
online help & function descriptions
&
SPM5 Manual

58. SPM
Online
Bibliography

59. some SPM internet resources…
SPMweb site
SPM discussion list
FIL neuroscience resources links
SPM Wiki
MRC-CBU imagers

60. SPMweb… Introduction to SPM SPM distribution: SPM99, SPM2, SPM5
Documentation & Bibliography
SPM discussion list
SPM short course
Example data sets
SPM extensions

61. SPM – email discussion list
Web home page
Archives, archive searches, membership lists, instructions
Subscribe
join spm Firstname Lastname
Participate & learn
Monitored by SPMauthors
Usage queries, theoretical discussions, bug reports, patches, techniques, &c…

62. (0th-order term can be determined from fieldmap)
Unwarping
(0th-order term can be determined from fieldmap)
B0{i}
B0 =
+ 
+ 
+ error - f1 fi  1
+2 5
+ ... i 