1. Unwarping

2. Assumptions of statistical tests in functional imaging
In order to assign an observed response to a particular brain structure, or cortical area, the data must conform to a known anatomical space.
In order to combine data from different scans from the same subject, or data from different subjects it is necessary that they conform to the same anatomical frame of reference.
Voxel-based analyses assume that the data from a particular voxel all derive from the same part of the brain.  Violations of this assumption will introduce artifactual changes in the voxel values that may obscure changes, or differences, of interest. E.g. if movement of the subject in the scanner pushes a voxel from an area of low to high signal, this may register as a false-positive ‘activation’.

3. Pre-processing steps to cope with violations of these assumptions
All scans must conform to the same anatomical frame of reference: Realign the data to 'undo' the effects of subject movement during the scanning session. 
Data must conform to a known anatomical space: After realignment the data are then transformed using linear or nonlinear warps into a standard anatomical space. 
Finally, the data are usually spatially smoothed before entering the analysis proper.

4. Pre-processing steps to cope with violations of these assumptions
All scans must conform to the same anatomical frame of reference: Realign the data to 'undo' the effects of subject movement during the scanning session. 
!UNWARP!
Data must conform to a known anatomical space: After realignment the data are then transformed using linear or nonlinear warps into a standard anatomical space. 
Finally, the data are usually spatially smoothed before entering the analysis proper.

5. Errors after realignment
After realignment, there can be residual errors in images for a number of reasons.
The residual variance can be dealt with by assuming that it is related to subject movement.
One way is to account for subject movement in the design matrix of the analysis proper, by including the movement parameters estimated from re-alignment as covariates.

6. Covarying for movement-related errors after realignment
However, this may remove activations of interest if they are correlated with movement
tmax=13.38
No correction
tmax=5.06
Correction by covariation

7. Problems with covariation
Covariation using movement parameters assumes only rigid deformation of the image between scans.
BUT: images are sampled according to gradients of the magnetic field B, in 3 image dimensions.
ω = γB
ω = resonant frequency B = magnetic field strength
By applying a gradient field across B0, B varies according to position. There will only be one position at which 1H spins are precessing at a particular resonant frequency, so can assign resultant signal to this location.
Signals are assigned in 3 image dimensions by applying field gradients across B0 in 3 dimensions.
y z B0+Gzz
B B B0
B0-Gxx x B0+Gxx
B0-Gzz

8. Non-rigid deformation
Knowing the location at which 1H spins will precess at a particular frequency and thus where the signal comes from is dependent upon correctly assigning a particular field strength to a particular location.
If the field B0 is homogeneous, then the image is sampled according to a regular grid and voxels can be localised to the same bit of brain tissue over subsequent scans by realigning, this is because the same transformation is applied to all voxels between each scan.
If there are inhomogeneities in B0, then different deformations will occur at different points in the field over different scans, giving rise to non-rigid deformation.
B0 Expect field strength to be B0
here, so H atoms with signal associated
with resonant frequency ω0 to be located
here.
In fact, because of inhomogeneity, they are

9. Field inhomogeneities
Due to microscopic gradients or variations in magnetic field strengths that occur at interfaces of substances of different magnetic susceptibility. E.g., metallic material (ferromagnetic) and the human body (diamagnetic).
Also occurs close to tissue-air and tissue-bone interfaces such as around frontal sinuses.
Field inhomogeneities have the effect that locations on the image are ‘deflected’ with respect to the real object.
A deformation field indicates the directions and magnitudes of location deflections throughout the FOV with respect to the real object.
igl.stanford.edu/~torsten/ct-dsa.html

10. Movement-by-inhomogeneity interactions
Field inhomogeneities change with the position of the object in the field, so there can be non-rigid, as well as rigid distortion over subsequent scans.
The movement-by-inhomogeneity interaction can be observed by changes in the deformation field over subsequent scans.
The amount of distortion is proportional to the absolute value of the field inhomogeneity and the data acquisition time. EPI is particularly sensitive to the effects of magnetic field inhomogeneities because it has long TR

11. Controlling for movement-by-inhomogeneity interactions
One solution is to explicitly measure field inhomogeneity by use of a field-map (available in the “FieldMap” SPM toolbox). A field map then has to be generated for each scan in the time-series.
Measurement of field-maps is complicated by noise, and rapid loss of signal towards the edges of the object.
In practice, rather than generating a statistical field map for every image in the EPI data set, can compute how the statistical maps are warped over subsequent scans and then unwarp the statistical map itself in order to make accurate identification of activated areas.
Computing how the images are warped over subsequent scans requires knowing how the deformation fields change with displacement of the subject, i.e. the derivatives of B with respect to displacement of the subject.

12. Principles of UNWARP
Given the derivative of the field with respect to subject movement, and the movement parameters estimated from realignment, can predict the non-rigid deformation in the scan series.
In practice, we know the non-rigid deformation (in terms of extra variance after realignment) and the subject movement (movement parameters) so we can estimate the derivatives of the field B0 with respect to subject movement – thus estimate how the field is warped over the time series and ‘undo’ this using UNWARP.

13. Movements modelled in UNWARP
Translations and rotations in plane perpendicular to B0 will not affect B0, so only need to model derivatives of B0 with respect to rotations out of perpendicular plane, i.e. pitching and rolling , . x B0 y 
 z
In UNWARP there is a (default) option to re-estimate the movement parameters with each unwarp iteration. This is recommended by John Ashburner. It is computed via a series of iterations; 1. estimate movement parameters (, ), 2. estimate deformation fields,  B0, 3. re-estimate movement with new model of magnetic field B0

14. Modelling changes in B0
The field B0, which changes as a function of displacement , , can be modelled by the first two terms of a Taylor expansion
B0(, ) = B0 (, ) + [(δB0/ δ)  + (δB0/ δ ) ]
The ‘static’ deformation field, Changes in the deformation field with
Which is the same throughout subject movement. Estimated via iteration
The time series. Procedure in UNWARP.
Calculated using ‘Fieldmap’ in
SPM
It is possible to model the next term in the Taylor expansion as well, i.e. the second derivative of B with respect to , , but this is not necessary.

15. Applying the deformation field to the image
Once the deformation field has been
modelled over time, the time-variant
field is applied to the image.
effect of sampling a regular object over a curved surface.
The image is therefore re-sampled
assuming voxels, corresponding to
the same bits of brain tissue, occur
at different locations over time.

16. Advantages of incorporating this in pre-processing
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.
tmax=13.38
No correction
tmax=5.06
Correction by covariation
tmax=9.57
Correction by Unwarp

17. UNWARP: Benefits and Limitations
Although for small movements a limited portion of the total variance is removed, the susceptibility-by-movement interaction effects are quite localised to "problem" areas. For a subset of voxels in e.g. frontal-medial and orbitofrontal cortices and parts of the temporal lobes the reduction can be quite dramatic (>90%).
However, UNWARP only tackles one source of variance after re-alignment, and other errors may arise from:
- Susceptibility-dropout-by-movement interaction: Field inhomogeneities can also cause signal loss due to through-plane dephasing (which will not be rephased by encoding gradients that are all in-plane).
- Spin-history effects: The signal will depend on how much longitudinal magnetisation has recovered (through T1 relaxation) since it was last excited (short TR→low signal). If the subject moves in the direction of increasing slice number between one excitation and the next, then the effective TR will be longer (resulting in increasing signal intensity).
- Slice-to-vol effects: The rigid-body model that is used by most motion-correction (e.g. SPM) methods assume that any movement will occur between scans. However there is also movement within scans – leading to further apparent shape changes.

18. Summary
Movement-by-inhomogeneity interactions can be accommodated during realignment using “unwarp” in SPM5
WARNING!! UNWARP can be computationally intensive, and therefore take a long time!

19. References
Jezzard, P. and Clare, S Sources of distortion in functional MRI data. Human Brain Mapping, 8:80-85
Andersson JLR, Hutton C, Ashburner J, Turner R, Friston K (2001) Modelling geometric deformations in EPI time series. Neuroimage 13:
John Ashburner’s slides
Paul Tofts’ MRI Physics Course at the IoN (slides not yet on the web – TBA)