Computational Neuroanatomy John Ashburner [e-mail]

1
Computational NeuroanatomyJohn
Ashburnerjohn_at_fil.ion.ucl.ac.uk

• Smoothing
• Rigid registration
• Spatial normalisation
• Segmentation
• Morphometry

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Overview of SPM Analysis
Statistical Parametric Map
Design matrix
fMRI time-series
Motion Correction
Smoothing
General Linear Model
Parameter Estimates
Spatial Normalisation
Anatomical Reference
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Contents

• Smoothing
• Rigid Registration
• Spatial normalisation
• Segmentation
• Morphometry

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Smoothing
Each voxel after smoothing effectively becomes
the result of applying a weighted region of
interest (ROI).
Before convolution
Convolved with a circle
Convolved with a Gaussian
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Smoothing

• Why smooth?
• Potentially increase sensitivity
• Inter-subject averaging
• Increase validity of SPM
• Smoothing is a convolution with a Gaussian kernel

Gaussian convolution is separable
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Contents

• Smoothing
• Rigid Registration
• Rigid-body transforms
• Optimisation objective functions
• Interpolation
• Spatial normalisation
• Segmentation
• Morphometry

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Within-subject Registration

• Assumes there is no shape change, and motion is
  rigid-body
• Used by Realign and Coregister functions
• The steps are
• Registration - i.e. Optimising the parameters
  that describe a rigid body transformation between
  the source and reference images
• Transformation - i.e. Re-sampling according to
  the determined transformation

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Affine Transforms

• 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
• ymx

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2D Affine Transforms

• Translations by tx and ty
• x1 x0 tx
• y1 y0 ty
• Rotation around the origin by ? radians
• x1 cos(?) x0 sin(?) y0
• y1 -sin(?) x0 cos(?) y0
• Zooms by sx and sy
• x1 sx x0
• y1 sy y0

• Shear
• x1 x0 h y0
• y1 y0

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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

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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
• The order of the operations matters

Translations
Pitch about x axis
Roll about y axis
Yaw about z axis
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Voxel-to-world Transforms

• Affine transform associated with each image
• Maps from voxels (x1..Nx, y1..Ny, z1..Nz) to
  some world co-ordinate system. e.g.,
• Scanner co-ordinates - images from DICOM toolbox
• TT/MNI coordinates - spatially normalised
• Registering image B (source) to image A (target)
  will update Bs vox-to-world mapping
• Mapping from voxels in A to voxels in B is by
• A-to-world using MA, then world-to-b using MB-1
• MB-1 MA

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Left- and Right-handed Coordinate Systems

• Analyze files are stored in a left-handed system
• Talairach Tournoux uses a right-handed system
• Mapping between them requires a flip
• Affine transform with a negative determinant

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Optimisation

• Optimisation involves finding some best
  parameters according to an objective function,
  which is either minimised or maximised
• The objective function is often related to a
  probability based on some model

Most probable solution (global optimum)
Objective function
Local optimum
Local optimum
Value of parameter
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Objective Functions for Image Registration

• Intra-modal
• Mean squared difference (minimise)
• Normalised cross correlation (maximise)
• Entropy of difference (minimise)
• Inter-modal (or intra-modal)
• Mutual information (maximise)
• Normalised mutual information (maximise)
• Entropy correlation coefficient (maximise)
• AIR cost function (minimise)

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Mean-squared Difference

• Minimising mean-squared difference works for
  intra-modal registration (realignment)
• Simple relationship between intensities in one
  image, versus those in the other
• Assumes normally distributed differences

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Gauss-Newton Optimisation

• Works best for least-squares
• Minimum is estimated by fitting a quadratic at
  each iteration

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Mutual Information

• Used for between-modality registration
• Derived from joint histograms
• MI ?ab P(a,b) log2 P(a,b)/( P(a) P(b) )
• Related to entropy MI -H(a,b) H(a) H(b)
• Where H(a) -?a P(a) log2P(a) and H(a,b) -?a
  P(a,b) log2P(a,b)

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Image Transformations

• Images are re-sampled. An example in 2D
• for y01..ny0, loop over rows
• for x01..nx0, loop over pixels in row
• x1 tx(x0,y0,q) transform according to q
• y1 ty(x0,y0,q)
• if 1?x1? nx1 1?y1?ny1 then, voxel in range
• f1(x0,y0) f0(x1,y1) assign re-sampled value
• end voxel in range
• end loop over pixels in row
• end loop over rows
• What happens if x1 and y1 are not integers?

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Simple Interpolation

• Nearest neighbour
• Take the value of the closest voxel
• Tri-linear
• Just a weighted average of the neighbouring
  voxels
• f5 f1 x2 f2 x1
• f6 f3 x2 f4 x1
• f7 f5 y2 f6 y1

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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.
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Contents

• Smoothing
• Rigid Registration
• Spatial normalisation
• Affine registration
• Nonlinear registration
• Regularisation
• Segmentation
• Morphometry

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Spatial Normalisation - Reasons

• Inter-subject averaging
• Increase sensitivity with more subjects
• fixed-effects analysis
• Extrapolate findings to the population as a whole
• mixed-effects analysis
• Standard coordinate system
• e.g. Talairach Tournoux space

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Spatial Normalisation - Objective

• Warp the images such that functionally homologous
  regions from different subjects are as close
  together as possible
• Problems
• No 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

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Spatial Normalisation - Procedure

• Minimise mean squared difference from template
  image(s)

Non-linear registration
Affine registration
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Spatial Normalisation - Templates
T1
Transm
T2
305
T1
T2
PD
SS
PD
PET
EPI
Template Images
Canonical images
Spatial normalisation can be weighted so that
non-brain voxels do not influence the
result. Similar weighting masks can be used for
normalising lesioned brains.
PET
A wider range of contrasts can be registered to a
linear combination of template images.
T1
PD
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Spatial Normalisation - Affine

• The first part is a 12 parameter affine transform
• 3 translations
• 3 rotations
• 3 zooms
• 3 shears
• Fits overall shape and size

• Algorithm simultaneously minimises
• Mean-squared difference between template and
  source image
• Squared distance between parameters and their
  expected values (regularisation)

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Spatial Normalisation - Non-linear
Deformations consist of a linear combination of
smooth basis functions These are the lowest
frequencies of a 3D discrete cosine transform
(DCT)

• Algorithm simultaneously minimises
• Mean squared difference between template and
  source image
• Squared distance between parameters and their
  known expectation

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Spatial Normalisation - Overfitting
Without regularisation, the non-linear spatial
normalisation can introduce unnecessary warps.
Affine registration. (?2 472.1)
Template image
Non-linear registration without regularisation. (?
2 287.3)
Non-linear registration using regularisation. (?2
302.7)
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Contents

• Smoothing
• Rigid Registration
• Spatial normalisation
• Segmentation
• Gaussian mixture model
• Including prior probability maps
• Intensity non-uniformity correction
• Morphometry

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Segmentation - Mixture Model

• Intensities are modelled by a mixture of K
  Gaussian distributions, parameterised by
• means
• variances
• mixing proportions
• Can be multi-spectral
• MultivariateGaussiandistributions

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Segmentation - Priors

• Overlay prior belonging probability maps to
  assist the segmentation
• Prior probability of each voxel being of a
  particular type is derived from segmented images
  of 151subjects
• Assumed to berepresentative
• Requires initialregistration tostandard space

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Segmentation - Bias Correction

• A smooth intensity modulating function can be
  modelled by a linear combination of DCT basis
  functions

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Segmentation - Algorithm

• Results contain some non-brain tissue
• Removed automaticallyusing morphologicaloperatio
  ns
• erosion
• conditional dilation

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Contents

• Smoothing
• Rigid Registration
• Spatial normalisation
• Segmentation
• Morphometry
• Volumes from deformations
• Voxel-based morphometry

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Morphometry
Template
Warped
Original
Relative volumes from Jacobian determinants
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Very hard to define a one-to-one mappingof
cortical folding
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Early Late
Difference
Data from the Dementia Research Group, Queen
Square.
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Pre-processing for Voxel-Based Morphometry (VBM)
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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.IPMI95 pp 263-274 Ashburner
et al (1997) Incorporating prior knowledge into
image registration.NeuroImage 6(4)344-352 Ashbu
rner 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
11805-821