Spatial preprocessing of fMRI data
Klaas Enno Stephan
Laboratory for Social and Neural Systrems Research
Institute for Empirical Research in Economics
University of Zurich
With many thanks for helpful slides to:
Functional Imaging Laboratory (FIL)
Wellcome Trust Centre for Neuroimaging
University College London
Ged Ridgway
John Ashburner
Meike Grol
Methods & models for fMRI data analysis in neuroeconomics
17 April 2010
Overview of SPM
Image time-series
Realignment
Kernel
Design matrix
Smoothing
General linear model
Statistical parametric map (SPM)
Statistical
inference
Normalisation
Gaussian
field theory
p <0.05
Template
Parameter estimates
Functional MRI (fMRI)
• Uses echo planar imaging (EPI) for
fast acquisition of T2*-weighted
images.
• Spatial resolution:
– 3 mm
(standard 1.5 T scanner)
– < 200 μm(high-field systems)
EPI
(T2*)
dropout
• Sampling speed:
– 1 slice: 50-100 ms
• Requires spatial pre-processing and
statistical analysis.
T1
Terminology of fMRI
subjects
sessions
runs
single run
volume
slices
TR = repetition time
time required to scan
one volume
voxel
Terminology of fMRI
Scan Volume:
Field of View
(FOV),
e.g. 192 mm
Axial slices
Slice thickness
e.g., 3 mm
Matrix Size
e.g., 64 x 64
In-plane resolution
192 mm / 64
= 3 mm
3 mm
3 mm
3 mm
Voxel Size
(volumetric pixel)
Standard space
The Talairach Atlas
The MNI/ICBM AVG152 Template
Why does fMRI require spatial preprocessing?
• Head motion artefacts during scanning
→ Realignment
• Problems of EPI acquisition: distortion and signal
dropouts
→ “Unwarping”
• Brains are quite different across subjects
→ Normalisation (“Warping”)
→ Smoothing
Realignment or “motion correction”
• Even small head movements can be a major problem:
– increase in residual variance
– data may get completely lost if sudden movements occur
during a single volume
– movements may be correlated with the task performed
• Therefore:
– always constrain the volunteer’s head
– instruct him/her explicitly to remain as calm as possible
– do not scan for too long – everyone will move after while !
minimising movements is one of the most important factors for
ensuring good data quality!
Realignment = rigid-body registration
• Assumes that all movements are those of a
rigid body, i.e. the shape of the brain does not
change
• Two steps:
Registration: optimising six parameters that
describe a rigid body transformation between the
source and a reference image
Transformation: re-sampling according to the
determined transformation
Linear (affine) transformations
• Rigid-body transformations are a subset
• Parallel lines remain parallel
• Operations can be represented by:
x1 = m11x0 + m12y0 + m13z0 + m14
y1 = m21x0 + m22y0 + m23z0 + m24
z1 = m31x0 + m32y0 + m33z0 + m34
• Or as matrices:
 x1  m 11
  
y
m
 1    21
 z 1  m 31
  
1  0
m 12
m 13
m 22
m 23
m 32
m 33
0
0
m 14   x 0 
  
m 24
y
 0
m 34   z 0 
  
1  1
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
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
• Non-commutative: the order of the operations matters
1


0


0

0
0
0
1
0
0
1
0
0
Xtrans 
1

 
Ytrans   0

 
Ztrans   0
 
1  0
Translations
0
0
cosΦ
sinΦ
 sinΦ
cosΦ
0
0
Pitch
about x axis
 cosΘ

 
0  0

 
0    sinΘ
 
1  0
0 
0
sinΘ
1
0
0
cosΘ
0
0
Roll
about y axis
 cosΩ

 
0    sinΩ

 
0  0
 
1  0
0 
sinΩ
0
cosΩ
0
0 
0
1
0
0
0
1
Yaw
about z axis
0 


Realignment
• Goal: minimise squared differences between source and
reference image
• Other methods available (e.g. mutual information)
A special case ...
• If a subject remained perfectly still during a fMRI study,
would realignment still be a good idea to perform?
• When could this issue be of practical relevance?
Coregistration
• also affine registration (like realignment)
• used to register a structural image to a (mean)
functional one
– allows more accurate anatomical localisation of activations
– must be done before spatial normalisation (if warping
parameters are estimated from the T1 image)
• examples in SPM8 Manual, Chapters 28, 29
Practical demonstration: realignment & coregistration
Joint and marginal histograms
intensity
frequency
frequency
intensity
Interpolation
• Nearest neighbour
– Take the value of the
closest voxel
• linear (2D: bilinear; 3D:
trilinear)
– Just a weighted average
of the neighbouring voxels
– f5 = f1 x2 + f2 x1
– f6 = f3 x2 + f4 x1
– f7 = f5 y2 + f6 y1
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.
Why does fMRI require spatial preprocessing?
• Head motion artefacts during scanning
→ Realignment
• Problems of EPI acquisition: distortion and signal
dropouts
→ “Unwarping”
• Brains are quite different across subjects
→ Normalisation (“Warping”)
→ Smoothing
Residual errors after realignment
• Resampling can introduce interpolation errors
• 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 included
as confound regressors in subsequent statistical analyses.
Movement by distortion interactions
• Subject disrupts B0 field,
rendering it inhomogeneous
→ distortions in phase-encode
direction
• Subject moves during EPI time
series
→ distortions vary with subject
orientation
→ shape of imaged brain
varies
Andersson et al. 2001, NeuroImage
Movement by distortion interaction
Movement by distortion interactions
after head rotation
original deformations
deformations after realignment
mismatch in deformations
Andersson et al. 2001, NeuroImage
Different strategies for correcting movement
artefacts
• liberal control:
realignment only
• moderate control:
realignment + “unwarping”
• strict control:
realignment + inclusion of realignment parameters in
statistical model
Why does fMRI require spatial preprocessing?
• Head motion artefacts during scanning
→ Realignment
• Problems of EPI acquisition: distortion and signal
dropouts
→ “Unwarping”
• Brains are quite different across subjects
→ Normalisation (“Warping”)
→ Smoothing
Individual brains differ in size, shape and folding
Spatial normalisation: why necessary?
• Inter-subject averaging
– Increase sensitivity with more subjects
• Fixed-effects analysis
– Extrapolate findings to the population as a whole
• Random / mixed-effects analysis
• Make results from different studies comparable
by bringing them into a standard coordinate
system
– e.g. MNI space
Spatial normalisation: objective
• Warp the images such that functionally corresponding
regions from different subjects are as close together
as possible
• Problems:
– Not always exact match between structure and function
– Different brains are organised differently
– Computational problems (local minima, not enough
information in the images, computationally expensive)
• Compromise by correcting gross differences followed
by smoothing of normalised images
Spatial normalisation: affine step
• The first part is a 12 parameter
affine transform
–
–
–
–
3 translations
3 rotations
3 zooms
3 shears
• Fits overall shape and size
Spatial normalisation: non-linear step
Deformations consist of a linear combination
of smooth basis functions.
These basis functions result from a 3D
discrete cosine transform (DCT).
Spatial normalisation: Bayesian regularisation
Deformations consist of a linear
combination of smooth basis functions

set of frequencies from a 3D
discrete cosine transform.
Find maximum a posteriori (MAP) estimates: simultaneously minimise
– squared difference between template and source image
– squared difference between parameters and their priors
Deformation parameters
MAP: log p ( | y )  log p ( y |  )  log p ( )  log p ( y )
“Difference” between template
and source image
Squared distance between parameters and
their expected values (regularisation)
Spatial normalisation: overfitting
Without
regularisation,
Template
the non-linear
image
spatial
normalisation
can introduce
unnecessary
Non-linear
warps.
registration
using
regularisation.
(2 = 302.7)
Affine registration.
(2 = 472.1)
Non-linear
registration
without
regularisation.
(2 = 287.3)
Segmentation
GM and WM segmentations
overlaid on original images
Structural image, GM and
WM segments, and brainmask (sum of GM and WM)
Segmentation & normalisation
• Circular relationship between segmentation & normalisation:
– Knowing which tissue type a voxel belongs to helps normalisation.
– Knowing where a voxel is (in standard space) helps segmentation.
• Build a joint generative model:
– model how voxel intensities result from mixture of tissue type distributions
– model how tissue types of one brain have to be spatially deformed to match
those of another brain
• Using a priori knowledge about the parameters:
adopt Bayesian approach and maximise the posterior probability
Ashburner & Friston 2005, NeuroImage
Unified segmentation with tissue class priors
• Goal: for each voxel, compute
probability that it belongs to a
particular tissue type, given its
intensity
p (tissue | intensity)
p (intensity | tissue) ∙ p (tissue)
• Likelihood model:
Intensities are modelled by a
mixture of Gaussian distributions
representing different tissue
classes (e.g. GM, WM, CSF).
• Priors are obtained from tissue
probability maps (segmented
images of 151 subjects).
Ashburner & Friston 2005, NeuroImage
Normalisation options in practice
• Conventional normalisation:
– either warp functional scans to EPI template directly
– or coregister structural scan to functional scans and then
warp structural scan to T1 template; then apply these
parameters to functional scans (“Normalise: Write”)
• Unified segmentation:
– coregister structural scan to functional scans
– unified segmentation provides normalisation parameters
– apply these parameters to functional scans (“Normalise:
Write”)
Normalising structural image to T1 template
Estimate normalization parameters
(T1 -> T1 template)
T1 image
Apply the normalization
parameters to the T1 image
EPI time series
And apply the normalization
parameters to the (coregistered)
functional images
Normalising EPI images to EPI template
Calculate
mean of
EPI time
series
Estimate
normalization
parameters
(EPI -> EPI
template)
EPI time series
Apply the normalization
parameters to all EPI images
T1 image
Apply the normalization
parameters to the (coregistered)
T1 image
Smoothing
• Why smooth?
– increase signal to noise
– inter-subject averaging
– increase validity of Gaussian Random Field theory
• In SPM, smoothing is a convolution with a Gaussian kernel.
• Kernel defined in terms of FWHM (full width at half maximum).
Gaussian convolution is separable
Gaussian smoothing kernel
Smoothing
Smoothing is done by convolving with a 3D
Gaussian which is defined by its full width at half
maximum (FWHM).
Each voxel after smoothing effectively becomes
the result of applying a weighted region of
interest.
Before convolution
Convolved with a circle
Convolved with a Gaussian
Summary: spatial preprocessing steps
• Head motion artefacts during scanning
→ Realignment
• Problems of EPI acquisition: distortion and signal
dropouts
→ “Unwarping”
• Brains are quite different across subjects
→ Normalisation (“Warping”)
→ Smoothing
Thank you!
Supplementary slides
World space & voxel space
x W   2 xV  9 2
y W  2 y V -1 2 8
z W  2 zV -7 4
(44, 66, 36)
(4, 4, -2)
Voxel index
(45, 66, 35)
World coords
(2, 4, -4) mm
(45, 65, 35)
(2, 2, -4)
Changing coordinate systems
x W   2 xV  92
y W  2 y V - 128
z W  2z V - 74
By moving to 4D, one
can include
translations within a
single
matrix multiplication
xW

 yW
z
 W
  2
 
 0
  0
 
xW

 yW
z
 W
 1

  2
 
  0

  0
 
  0
 
0
0  x V 


0  y V  
2   z V 
0
0
2
0
0
2
0
0
0
2
 92

  128
  74






92   x V 


 128   y V 
 74   z V 




1   1 
These 4-by-4 “homogeneous matrices” are the currency of voxel-world mappings,
affine coreg. and realignment. In SPM5 & SPM8, they are stored in the .hdr files. In
SPM2, they are stored in .mat files.
To find inverse mappings, or results of concatenating multiple transformations, we
simply follow the rules of matrix algebra
Representing rotations
 x 1   cos( θ )
   
 y 1   sin( θ)
 sin( θ)   x 0 
  
cos( θ )   y 0 
 x 1   cos( θ )
  
 y 1    sin( θ )
z   0
 1 
 sin( θ )
cos( θ )
0
0  x 0 
 
0  y 0 
1   z 0 