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Statistical Parametric Mapping (SPM) 1. Talk I: Spatial Pre-processing 2. Talk II: General Linear Model 3. Talk III:Statistical Inference 3. Talk IV: Experimental Design. Spatial Preprocessing & Computational Neuroanatomy With thanks to:

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Statistical Parametric Mapping (SPM) 1. Talk I: Spatial Pre-processing

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Statistical parametric mapping spm 1 talk i spatial pre processing

Statistical Parametric

Mapping (SPM)

1. Talk I: Spatial Pre-processing

2. Talk II: General Linear Model

3. Talk III:Statistical Inference

3. Talk IV: Experimental Design


Statistical parametric mapping spm 1 talk i spatial pre processing

Spatial Preprocessing &

Computational Neuroanatomy

With thanks to:

John Ashburner, Jesper Andersson


Overview

Statistical Parametric Map

Design matrix

fMRI time-series

kernel

Motion

correction

Smoothing

General Linear Model

Spatial

normalisation

Parameter Estimates

Standard

template

Overview


Statistical parametric mapping spm 1 talk i spatial pre processing

Overview

1. Realignment (motion correction)

2. Normalisation (to stereotactic space)

3. Smoothing

4. Between-modality Coregistration

5. Segmentation (to gray/white/CSF)

6. Morphometry (VBM/DBM/TBM)


Statistical parametric mapping spm 1 talk i spatial pre processing

Overview

1. Realignment (motion correction)

2. Normalisation (to stereotactic space)

3. Smoothing

4. Between-modality Coregistration

5. Segmentation (to gray/white/CSF)

6. Morphometry (VBM/DBM/TBM)


Reasons for motion correction

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 - determining the 6 parameters that describe the rigid body transformation between each image and a reference image

    • 2. Transformation (reslicing) - re-sampling each image according to the determined transformation parameters


1 registration

Squared Error

Rigid body transformations parameterised by:

Translations

Pitch

Roll

Yaw

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 affinetransformation 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


1 registration1

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 *.mat file updated for each volume (do not have to be resliced)

  • Slice-timing correction can be performed before or after realignment (depending on acquisition)


Statistical parametric mapping spm 1 talk i spatial pre processing

Nearest Neighbour

Linear

Full sinc (no alias)

Windowed sinc

2. Transformation (reslicing)

  • 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 in SPM2, nth-order “b-splines”


Statistical parametric mapping spm 1 talk i spatial pre processing

Residual Errors after Realignment

  • Interpolation errors, especially with tri-linear interpolation and small-window sinc

  • PET:

    • Incorrect attenuation correction because scans are no longer aligned with transmission scan (a transmission scan is often acquired to give a map of local positron attenuation)

  • fMRI (EPI):

    • 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


Statistical parametric mapping spm 1 talk i spatial pre processing

New in

SPM2

Field-map

Distorted image

Corrected image

Unwarp

  • Echo-planar images (EPI) contain distortions owing to field inhomogenieties (susceptibility artifacts, particularly 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”


Statistical parametric mapping spm 1 talk i spatial pre processing

New in

SPM2

Estimated derivative fields

Pitch

 +



B0 

Roll

Unwarp

  • 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)…


Statistical parametric mapping spm 1 talk i spatial pre processing

New in

SPM2

(0th-order term can be determined from fieldmap)

+ error

+ 

=

+ 

B0{i}

fi

f1

B0

-

+2

1

i

i

+ ... +5

+ ...

i

Unwarp


Statistical parametric mapping spm 1 talk i spatial pre processing

New in

SPM2

No correction

Correction by covariation

Correction by Unwarp

tmax=13.38

tmax=5.06

tmax=9.57

Unwarp

Example: Movement correlated with design


Statistical parametric mapping spm 1 talk i spatial pre processing

Overview

1. Realignment (motion correction)

2. Normalisation (to stereotactic space)

3. Smoothing

4. Between-modality Coregistration

5. Segmentation (to gray/white/CSF)

6. Morphometry (VBM/DBM/TBM)


Reasons for normalisation

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?)

  • 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 variabilility accommodated by subsequent spatial smoothing


Summary

Summary

Spatially normalised

Original image

  • 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


Stage 1 full affine transformation

Rigid body

Rotation

Shear

Translation

Zoom

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)


Six affine registered images

Six affine registered images

Insufficieny of Affine-only normalisation

Six affine + nonlinear registered


Stage 2 nonlinear warps

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)


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)

Bayesian Constraints

Without the Bayesian formulation, the non-linear spatial normalisation can introduce unnecessary warping into the spatially normalised images


Bayesian constraints1

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) (Gibbs potential)

  • 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”


Bayesian constraints2

Empirically generated priors

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)


Statistical parametric mapping spm 1 talk i spatial pre processing

Overview

1. Realignment (motion correction)

2. Normalisation (to stereotactic space)

3. Smoothing

4. Between-modality Coregistration

5. Segmentation (to gray/white/CSF)

6. Morphometry (VBM/DBM/TBM)


Reasons for smoothing

FWHM

Gaussian smoothing kernel

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)


Statistical parametric mapping spm 1 talk i spatial pre processing

Overview

1. Realignment (motion correction)

2. Normalisation (to stereotactic space)

3. Smoothing

4. Between-modality Coregistration

5. Segmentation (to gray/white/CSF)

6. Morphometry (VBM/DBM/TBM)


Between modality co registration

T1

Transm

T2

PD

PET

EPI

Between Modality Co-registration

  • Because different modality images have different properties (e.g., relative intensity of gray/white matter), cannot simply minimise difference between images

  • Two main approaches:

    I. Via Templates:

    1) Simultaneous affine registrations between each image and same-modality template

    2) Segmentation into grey and white matter

    3) Final simultaneous registration of segments

    II. Mutual Information

  • Useful, for example, to display functional results (EPI) onto high resolution anatomical image (T1)


3 registration of partitions

Between Modality Co-registration: I. Via Templates

3. Registration of Partitions

1. Affine Registrations

  • ‘Mixture Model’ cluster analysis to classify MR image as GM, WM & CSF

  • Additional information is obtained from a priori probability images - see later

  • Both images are registered - using 12 parameter affine transformations - to their corresponding templates...

  • … but only the rigid-body transformation parameters allowed to differ between the two registrations

  • This gives:

    • rigid body mapping between the images

    • affine mappings between the images and the templates

2. Segmentation

  • Grey and white matter partitions are registered using a rigid body transformation

  • Simultaneously minimise sum of squared difference


Between modality coregistration ii mutual information

New in

SPM2

Another way is to maximise the Mutual Information in the 2D histogram (plot of one image against other)

For histograms normalised to integrate to unity, the Mutual Information is:

SiSj hij log hij

Sk hikSl hlj

PET

T1 MRI

Between Modality Coregistration: II. Mutual Information


Statistical parametric mapping spm 1 talk i spatial pre processing

Overview

1. Realignment (motion correction)

2. Normalisation (to stereotactic space)

3. Smoothing

4. Between-modality Coregistration

5. Segmentation (to gray/white/CSF)

6. Morphometry (VBM/DBM/TBM)


Image segmentation

Priors:

Intensity histogram

fit by multi-Gaussians

.

Image:

GM

WM

CSF

Brain/skull

Image Segmentation

  • 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 fromprior probability images, which are overlaid

  • Additional correction for intensity inhomogeniety possible


Statistical parametric mapping spm 1 talk i spatial pre processing

Overview

1. Realignment (motion correction)

2. Normalisation (to stereotactic space)

3. Smoothing

4. Between-modality Coregistration

5. Segmentation (to gray/white/CSF)

6. Morphometry (VBM/DBM/TBM)


Morphometry computational neuroanatomy

Original

Template

Spatial Normalisation

Normalised

Deformation field

TBM

VBM

DBM

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


Statistical parametric mapping spm 1 talk i spatial pre processing

Spatially

normalised

Segmented

grey matter

Original

image

Smoothed

SPM

Voxel-Based Morphometry (VBM)

A voxel by voxel statistical analysis is used to detect regional differences in the amount of grey matter between populations

“Optimised” VBM involves segmenting images before normalising, so as to normalise gray matter / white matter / CSF separately...


Statistical parametric mapping spm 1 talk i spatial pre processing

T1 image

template

Optimised VBM

Affine registration

Affine transform

priors

Segmentation & Extraction

STATS

volume

smooth

Spatial normalisation

Modulation

Apply deformation

Segmentation & extraction

Normalised T1

STATS

concentration

smooth


Statistical parametric mapping spm 1 talk i spatial pre processing

Grey matter volume

loss with age

superior parietal

pre and post central

insula

cingulate/parafalcine

VBM Examples: Aging


Statistical parametric mapping spm 1 talk i spatial pre processing

Females > Males

Males > Females

L superior temporal sulcus

R middle temporal gyrus

intraparietal sulci

mesial temporal

temporal pole

anterior cerebellar

VBM Examples: Sex Differences


Statistical parametric mapping spm 1 talk i spatial pre processing

Right frontal and left occipital petalia

VBM Examples: Brain Asymmetries


Statistical parametric mapping spm 1 talk i spatial pre processing

Vector field

Tensor field

Morphometry on deformation fields: DBM/TBM

Deformation-based Morphometry

looks at absolute displacements

Tensor-based Morphometry

looks at local shapes


Statistical parametric mapping spm 1 talk i spatial pre processing

Deformation

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

Deformation-based Morphometry (DBM)


Statistical parametric mapping spm 1 talk i spatial pre processing

Non-linear warps of sex differences characterised by canonical variates analysis

Mean differences (mapping from an average female to male brain)

DBM Example: Sex Differences


Statistical parametric mapping spm 1 talk i spatial pre processing

Original

Warped

Template

Relative volumes

Strain tensor

Tensor-based morphometry

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.

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.


Statistical parametric mapping spm 1 talk i spatial pre processing

References

Friston et al (1995): Spatial registration and normalisation of images.Human Brain Mapping 3(3):165-189

Ashburner & Friston (1997): Multimodal image coregistration and partitioning - a unified framework.NeuroImage 6(3):209-217

Collignon et al (1995): Automated multi-modality image registration based on information theory.IPMI’95 pp 263-274

Ashburner et al (1997): Incorporating prior knowledge into image registration.NeuroImage 6(4):344-352

Ashburner et al (1999): Nonlinear spatial normalisation using basis functions.Human Brain Mapping 7(4):254-266

Ashburner & Friston (2000): Voxel-based morphometry - the methods.NeuroImage 11:805-821


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