1. Spatial Processing Chris Rorden Spatial Registration Interpolation
Motion correction
Coregistration
Normalization
Interpolation
Spatial Smoothing
Advanced notes:
Spatial distortions of EPI scans
Image intensity distortions
Matrix mathematics

2. Why spatially register data?
Statistics computed individually for voxels.
Only meaningful if voxel examines same region across images.
Therefore, images must be in spatially registered with each other.

3. We use spatial registration to align images
Motion correction (realignment) adjusts for an individual’s head movements.
Coregistration aligns two images of different modalities from the same individual.
Normalization aligns images from different people.

4. Within-subject registration
With-in subject registrations
Assumption: same individual, so there should be a good linear solution.
Coregistration
Motion correction
Registration of the fMRI scans (across time)
Registration of fMRI scans with high resolution image.

5. Rigid Body Transforms Translation Rotation
By measuring and correcting for translations and rotations, we can adjust for an object’s movement in an image.

6. How many parameters? Each transform can be applied in 3 dimensions.
Therefore, if we correct for both rotation and translation, we will compute 6 parameters.
Translation
Rotation X Z
Pitch
Yaw
Roll Y

7. Motion Correction Motion correction aligns all in time series.
Translations and rotations only
‘rigid body registration’
Assumes brain size and shape identical across images.

8. Motion Correction Cost Function2
When aligned, Difference squared ~ 0 =
Target
Reslice 2
When unaligned, Difference squared > 0 =
Target
Reslice
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9. Motion correction cost function
Motion correction uses least squares to check if images are a good match (aka minimum sum of squares).
Smaller difference2 = better match (‘least squares’).
Iterative: moves image a bit at a time until match is worse.
Image 1
Image 2
Difference
Difference²

10. Local Minima Search algorithm is iterative:
move the image a little bit.
Test cost function
Repeat until cost function does not get better.
Search algorithm can get stuck at local minima: cost function suggests that no matter how the transformation parameters are changed a minimum has been reached
Value of Cost Function
Local Minimum
Global Minimum
Translation in X
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11. Coregistration is more complicated than motion correction
Rigid body not enough:
Size differs between images (must rescale: zooms).
fMRI scans often have spatial distortion not seen in other scans (must skew: shears).
Least squares cost function will fail: relative contrast of gray matter, white matter, CSF and air differences between images.

12. Affine Transforms (aka linear, geometric)
Zoom
Shear
Translation
Rotation

13. Coregistration
Coregistration is used to align images of different modalities from the same individual
Uses ‘mutual information cost function’: Note aligned images have neater histograms.
Uses entropy reduction instead of variance reduction as cost function.

14. Coregistration T1 image Coregistered FLAIR
Used within individual, so linear transforms should be sufficient
Typically 12 parameters (translation, rotation, zooms, shear each in 3 dimensions)
Though note that different MRI sequences create different non-linear distortions
T1 image
Coregistered FLAIR

15. Between-subject: Normalization
Allows inference about general population
Subject 1
Template
Subject 2
Average activation
Normalization

16. Why normalize? Stereotaxic coordinates analogous to longitude
Universal description for anatomical location
Allows other to replicate findings
Allows between-subject analysis: crucial for inference that effects generalize across humanity.

17. Popular MNI Template based on T1-weighted scans from 152 individuals.
Normalization
Normalization attempts to register scans from different people.
We align each persons brain to a template.
Template often created from multiple people (so it is fairly average in shape, size, etc).
We typically use template that is in the same modality as the image we want to normalize
Therefore, variance cost function.
If different groups use similar templates, they can talk in common coordinates.
Popular MNI Template based on T1-weighted scans from 152 individuals.

18. Coordinates - normalization
Different people’s brains look different
‘Normalizing’ adjusts overall size and orientation
Normalized Images
Raw Images

19. SPM uses modality specific template
MNI T1 template, plus custom templates
By default, FSL uses MNI T1 template for all modalities
Requires intra-modal cost functions
T T2* PET

20. Coordinates - Earth
For earth (2D surface) we use latitude and longitude
Origin for latitude is equator
Explicit: defined by axis of rotation
Origin for longitude is Greenwich.
Arbitrary: could be Paris
What is crucial is that we we agree on the same origin.
For the brain
left-right side explicit: Interhemispheric Fissure analogous to equator
How about Anterior-Posterior and Superior-Inferior? We need an origin for these coordinates.

21. Coordinates - Talairach
Anterior Commissure (AC) is the origin for neuroscience.
We measure distance from AC
57x-67x0 means ‘right posterior middle’.
Three values: left-right, posterior-anterior, ventral-dorsal

22. Coordinates - Talairach
Axis for axial plane is defined by anterior commissure (AC) and posterior commissure (PC).
Both are small regions that are clear to see on most scans. Y+ Y-
 PC Z+ Z-
 AC

23. Templates
Original Talairach-Tournoux atlas based on histological slices from one 69-year old woman.
Single brain may not be representative
No MRI scans from this woman
Modern templates were at some stage aligned to images from the Montreal Neurological Institute.
MNI space slightly different from T&T atlas (larger in every dimension).

24. Affine Transforms
Co-linear points remain co-linear after any affine transform.
Transform influences entire image.

25. Spatial Processing
Non-linear transforms can match features that could not aligned with affine transforms.
SPM uses basis functions.

26. Nonlinear functions and normalization
Scans from 6 people
Linear Only
Linear + Nonlinear

27. Recent algorithms
Affine transforms (e.g. FSL’s FLIRT): 12 degrees of freedom
Translation, Rotation, Scaling, Shear *3 dimensions
Nonlinear basis functions (SPM5) thousands of dof
Diffeomorphic algorithms (SPM8’s DARTEL, ANT) millions dof
Affine
template
DARTEL
template

28. Regularization penalizes bending energy
What is the best way to show graph points with a smooth line? Heavy regularization is a poor fit, heavy regularization causes local distortion
Heavy Regularization
Medium Regularization
Little Regularization

29. Regularization
Regularization is a parameter that you can adjust that influences non-linear normalization
Medium Regularization
Little Regularization

30. Spatial Processing
Affine Transforms are robust – they influence the entire brain
Note that non-linear functions can have local effects.
This can improve normalization
This can also lead to image distortion.
E.G. In stroke patients, the injured region may not match the intensity of the template…
Area of brain injury looks different on scan from stroke patient.
‘Perfect’ alignment with template will still have high cost function in injured region.

31. Normalization conducted on smoothed images.
Sulcal matching
Normalization conducted on smoothed images.
We are not trying to precisely match sulci (would cause local distortion).
Sulcal matching approximate
old images: DARTEL somewhat better
Post-normalization alignment of calcarine sulcus, precentral gyrus, superior temporal gyrus.

32. SPM and FSL normalize overall brain shape.
Alternatives
SPM and FSL normalize overall brain shape.
Individual sulci largely ignored.
What are different normalization strategies?
Sulci are crucial for some tasks (Herschl’s gyrus and hearing)
Perhaps less so for others (e.g. Amunts et. al 2004 with Broca’s variability)

33. Alternatives
SPM/FSL normalization will roughly match orientation and shape of head.
Good if function is localized to proportional part of brain
Poor if function is localized to specific sulci (e.g. early visual area V1 tied to calcarine fissure).
Alternatively, use sulci as cost function (Goebel et al., 2006).
Image below: mean sulcal position for 12 people after standard normalization (left) followed by sucal registration (middle).
Note: This technique improves sulcal alignment, but distorts cortical size.

34. Interpolation
Each lower image rotated 12º.
Left looks jagged, right looks smooth.
Different reslicing interpolation.

35. 1D Interpolation 
How do we estimate values that occur between discrete samples?
Three popular methods:
Nearest neighbor (box)
Linear (tent)
Spline/Sinc  
Weather analogy: if it was 25º at 9am, and 31º at 12am, what would you estimate the temperature was at 10am?

36. Interpolation 
How do we estimate values that occur between discrete samples?
Three popular methods:
Nearest neighbor (box)
Linear (tent)
Spline/Sinc  

37. Linear Interpolation
For neuroimaging we usually use linear interpolation.
Much more accurate than nearest neighbor.
There is some loss of high frequencies.
Since we spatially smooth data after spatial registration, we will lose high frequencies eventually.
1D Linear Interpolation
Weighted mean of 2 samples
2D Bilinear Interpolation
Weighted mean of 4 samples
3D Trilinear Interpolation
Weighted mean of 8 samples

38. Linear Interpolation – High Frequency Loss
Original
Linear interpolation loses high frequencies
Multiple successive resampling will lead to blurry image
Solution: Minimize number of times the data is resliced.
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39. Advanced Interpolation
Sinc interpolation can retain high frequency information.
Computation very time consuming (in theory, infinite extent)
FSL: Windowing options limit extent
SPM: Splines are used for rapid approximation
Not necessary if you will heavily blur your data with a broad smoothing kernel.
2D Sinc Function
1D Sinc Function
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40. Interpolation versus smoothing
Interpolation kernel is always 100% at 0 and 0% at all other integers.
This means that interpolated estimates always cross through control points (observations).
Smoothing kernels blur an observation with its neighbors.
Interpolation
Smoothing
Kernel
Observations

41. Spatial Smoothing
Each voxel is noisy. However, neighbors tend to show similar effect. Smoothing results in a more stable signal.
Smooth also helps statistics: smoothed data tends to be more ‘normal’ – fits our assumptions. Also, allows RFT thresholding (see Statistics lecture).
Gaussian Smoothing =

42. FWHM
Smoothing is a form of convolution: the output intensity based on weighted-influence of neighbors.
The most popular kernel is the gaussian function (a normal distribution).
The ‘full width half maximum’ adjusts the amount of gaussian smoothing.
FWHM is a measure of dispersion (like standard deviation or variance)
Large FWHMs lead to more blurry images.
For fMRI, we typically use a FWHM that is ~x2..x3 our original resolution (e.g. 8mm for 3x3x3mm data).
However, the FWHM tunes the size of region we will be best able to detect.
E.G. If you want to look for a brain region that is around 10mm diameter, use a 10mm FWHM.
Dispersion Differs

43. Spatial smoothing useful for between-subject analyses.
Spatial normalization is only approximate: smoothing minimizes individual sulcal variability.
Smoothing controls for variation in functional localization between people.
None mm mm mm

44. Smoothing Limits Inference
Consider a study that observes ‘increased’ activation for strong versus weak motor movements.
After smoothing we can not distinguish between:
Increased activation of the same population of neurons
Recruitment of more neighboring neurons.
Example: note that after smoothing broad low contrast looks line looks like focused high contrast line. 2D 1D

45. Smoothing Alternatives
Gaussian smoothing is great for ‘normal’ (Gaussian) noise: lots of small errors, very few outliers.
Gaussian poor for spike noise
Outlier contaminates neighbors
Alternatives if your data has spikes:
Median filters
FSL’s SUSAN
Gaussian Noise
Gaussian Smooth
Gaussian
Median Filter
Spike Noise

46. Inhomogeneity artifacts
The head distorts the magentic field.
Shimming attempts to make field level homogeneous.
Even after shimming, there will be varying field strengths.
Specifically, regions with large density changes (sinus/bone of frontal lobe).
This inhomogeneity leads to intensity and spatial distortion.

47. Spatial unwarping
We can measure field homogeneity.
This can be used to unwarp images (FSL’s B0 unwarping, SPM’s FieldMap).
Structural
Unwarped EPI
Raw EPI

48. Intensity unwarping
Motion correction creates a spatially stabilized image.
However, head motion also changes image intensity – some regions of the brain will appear brighter/darker.
SPM: EPI unwarping corrects for brightness changes (right)
FSL: You can add motion parameters to statistical model (FEAT stats page).
Problem: We will lose statistical power if head motion is task related, e.g. pitch head every time we press a button
Above: motion related image intensity changes.

49. Inhomogeneity also leads to variability in image intensity.
Bias correction
Inhomogeneity also leads to variability in image intensity.
Bias correct anatomical scans (e.g. SPM’s segmentation, N3).
Field homogeneity issues more severe with higher field strength.
Parallel Imaging (collecting MRI with multiple coils) can dramatically reduce effects.

50. Vectors Vectors Vectors have a direction and a length
The 2D vector [1,0] points East and has a length of 1
The 2D vector [0,1] points North and has a length of 1
The 2D vector [1,2] points North-East and has a length of 2.23
2D: two values [x,y], 3D: three values [x,y,z]
1, 2
0, 1
1, 0

51. 2D Rotation matrix
2D rotation matrix as two vectors (horizontal and vertical)
Can be described by four numbers
Original
Zoom
Rotation
Shear
Flip
-1, 0
0, 1
1, 0
0, 1
1, 0
0, 2
1, 0
1, 1
.7,-.7
.7, .7

52. 2D Transformation matrix
Transformation matrix is a rotation matrix plus translation values (encodes x,y origin of vectors)
9 values, final row is always (must have as many rows as columns).
Original
Translated
Translated+Zoom
1, 0, 0
0, 1, 0
0, 0, 1
1, 0, 1
0, 1, 0
0, 0, 1
2, 0, .2
0, 1, .2
0, 0, 1
All affine transforms can be combined into a single matrix

53. A 4x4 matrix, but the last row always ‘0 0 0 1’
3D Matrices are 4x4
We can generate 4x4 matrices that will allow us to work with 3D images.
A 4x4 matrix, but the last row always ‘ ’
fx = (x*i)+(y*j)+(z*k)+l
fy = (x*m)+(y*n)+(z*o)+p
fz = (x*q)+(y*r)+(z*s)+t i j k m n o l p q r s t 1

54. Matrices and 3D space 3D matrices work just like 2D matrices
The identity matrix still has 1’s along the diagonal
Translations are the values in the final column (constants)
Zooms are done by scaling values of a row.
Shears are values added to the relevant orthogonal value
Rotations use sine/cosine in dimensions of plane.
Our twelve numbers can store all of the possible rotations, shears, translations and scaling.
Simply multiply previous matrix with our transform (order crucial).