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Computational NeuroanatomyComputational Neuroanatomy
John AshburnerJohn [email protected]@fil.ion.ucl.ac.uk
Computational NeuroanatomyComputational Neuroanatomy
John AshburnerJohn [email protected]@fil.ion.ucl.ac.uk
• SmoothingSmoothing
• Motion CorrectionMotion Correction
• Between Modality Co-registrationBetween Modality Co-registration
• Spatial NormalisationSpatial Normalisation
• SegmentationSegmentation
• MorphometryMorphometry
• SmoothingSmoothing
• Motion CorrectionMotion Correction
• Between Modality Co-registrationBetween Modality Co-registration
• Spatial NormalisationSpatial Normalisation
• SegmentationSegmentation
• MorphometryMorphometry
OverviewOverviewOverviewOverview
Motioncorrection
smoothing
Spatialnormalisation
General Linear Model
Statistical Parametric MapfMRI time-series
Parameter Estimates
Design matrix
anatomical reference
kernel
SmoothingSmoothingSmoothingSmoothing
• Why Smooth?Why Smooth?– Potentially increase signal to noise.Potentially increase signal to noise.
– Inter-subject averaging.Inter-subject averaging.
– Increase validity of SPM.Increase validity of SPM.
• In SPM, smoothing is a convolution with a Gaussian kernel.In SPM, smoothing is a convolution with a Gaussian kernel.
• Kernel defined in terms of FWHM (full width at half maximum).Kernel defined in terms of FWHM (full width at half maximum).
• Why Smooth?Why Smooth?– Potentially increase signal to noise.Potentially increase signal to noise.
– Inter-subject averaging.Inter-subject averaging.
– Increase validity of SPM.Increase validity of SPM.
• In SPM, smoothing is a convolution with a Gaussian kernel.In SPM, smoothing is a convolution with a Gaussian kernel.
• Kernel defined in terms of FWHM (full width at half maximum).Kernel defined in terms of FWHM (full width at half maximum).
Gaussian convolution is separable Gaussian smoothing kernel
SmoothingSmoothingSmoothingSmoothing
Before convolution Convolved with a circle Convolved with a Gaussian
Smoothing is done by convolving with a 3D Gaussian- defined by its full width at half maximum (FWHM)
Each voxel after smoothing effectively becomes the result of applying a weighted region of interest (ROI).
Reasons for Motion CorrectionReasons for Motion CorrectionReasons for Motion CorrectionReasons for Motion Correction
• Subjects will always move in the Subjects will always move in the scanner.scanner.
– movement may be related to the movement may be related to the tasks performed.tasks performed.
• When identifying areas in the brain that When identifying areas in the brain that appear activated due to the subject appear activated due to the subject performing a task, it may not be performing a task, it may not be possible to discount artefacts that have possible to discount artefacts that have arisen due to motion.arisen due to motion.
• The sensitivity of the analysis is The sensitivity of the analysis is determined by the amount of residual determined by the amount of residual noise in the image series, so movement noise in the image series, so movement that is unrelated to the task will add to that is unrelated to the task will add to this noise and reduce the sensitivity.this noise and reduce the sensitivity.
• Subjects will always move in the Subjects will always move in the scanner.scanner.
– movement may be related to the movement may be related to the tasks performed.tasks performed.
• When identifying areas in the brain that When identifying areas in the brain that appear activated due to the subject appear activated due to the subject performing a task, it may not be performing a task, it may not be possible to discount artefacts that have possible to discount artefacts that have arisen due to motion.arisen due to motion.
• The sensitivity of the analysis is The sensitivity of the analysis is determined by the amount of residual determined by the amount of residual noise in the image series, so movement noise in the image series, so movement that is unrelated to the task will add to that is unrelated to the task will add to this noise and reduce the sensitivity.this noise and reduce the sensitivity.
• registrationregistration - i.e. - i.e. determining the 6 determining the 6 parameters that describe the parameters that describe the rigid body transformation rigid body transformation between each image and a between each image and a reference image.reference image.
• transformationtransformation - i.e. re- - i.e. re-sampling each image sampling each image according to the determined according to the determined transformation parameters.transformation parameters.
• registrationregistration - i.e. - i.e. determining the 6 determining the 6 parameters that describe the parameters that describe the rigid body transformation rigid body transformation between each image and a between each image and a reference image.reference image.
• transformationtransformation - i.e. re- - i.e. re-sampling each image sampling each image according to the determined according to the determined transformation parameters.transformation parameters.
The Steps in Motion The Steps in Motion CorrectionCorrection
The Steps in Motion The Steps in Motion CorrectionCorrection
RegistrationRegistrationRegistrationRegistration
• Determine the Determine the rigid body transformationrigid body transformation that minimises the sum of squared that minimises the sum of squared difference between images.difference between images.
• Rigid body transformation is defined by:Rigid body transformation is defined by:– 3 3 translationstranslations - in X, Y & Z directions. - in X, Y & Z directions.
– 3 3 rotationsrotations - about X, Y & Z axes. - about X, Y & Z axes.
• Operations can be represented as Operations can be represented as affineaffine transformation matrixes: transformation matrixes: xx11 = m = m1,11,1xx00 + m + m1,21,2yy00 + m + m1,31,3zz00 + m + m1,41,4
yy11 = m = m2,12,1xx00 + m + m2,22,2yy00 + m + m2,32,3zz00 + m + m2,42,4
zz11 = m = m3,13,1xx00 + m + m3,23,2yy00 + m + m3,33,3zz00 + m + m3,43,4
• Determine the Determine the rigid body transformationrigid body transformation that minimises the sum of squared that minimises the sum of squared difference between images.difference between images.
• Rigid body transformation is defined by:Rigid body transformation is defined by:– 3 3 translationstranslations - in X, Y & Z directions. - in X, Y & Z directions.
– 3 3 rotationsrotations - about X, Y & Z axes. - about X, Y & Z axes.
• Operations can be represented as Operations can be represented as affineaffine transformation matrixes: transformation matrixes: xx11 = m = m1,11,1xx00 + m + m1,21,2yy00 + m + m1,31,3zz00 + m + m1,41,4
yy11 = m = m2,12,1xx00 + m + m2,22,2yy00 + m + m2,32,3zz00 + m + m2,42,4
zz11 = m = m3,13,1xx00 + m + m3,23,2yy00 + m + m3,33,3zz00 + m + m3,43,4
1 0 0 Xtrans
0 1 0 Ytrans
0 0 1 Ztrans
0 0 0 1
1 0 0 0
0 cos() sin() 0
0 sin() cos() 0
0 0 0 1
cos() 0 sin() 0
0 1 0 0
sin() 0 cos() 0
0 0 0 1
cos() sin() 0 0
sin() cos() 0 0
0 0 1 0
0 0 0 1
Translations Pitch Roll Yaw
Rigid body transformations parameterised by:
Residual Errors from fMRIResidual Errors from fMRI• Gaps between slices can cause aliasing artefacts Gaps between slices can cause aliasing artefacts
• Re-sampling can introduce errorsRe-sampling can introduce errors– especially tri-linear interpolationespecially tri-linear interpolation
• Ghosts (and other artefacts) in the imagesGhosts (and other artefacts) in the images– do not move according to the same rigid body rules as do not move according to the same rigid body rules as
the subjectthe subject
• Slices are not acquired simultaneouslySlices are not acquired simultaneously– rapid movements not accounted for by rigid body modelrapid movements not accounted for by rigid body model
• fMRI images are distortedfMRI images are distorted– rigid body model does not model these types of rigid body model does not model these types of
distortiondistortion
• Spin excitation history effectsSpin excitation history effects– variations in residual magnetisationvariations in residual magnetisation
Functions of the estimated motion parameters can be used as Functions of the estimated motion parameters can be used as confounds in subsequent analyses.confounds in subsequent analyses.
Residual Errors from fMRIResidual Errors from fMRI• Gaps between slices can cause aliasing artefacts Gaps between slices can cause aliasing artefacts
• Re-sampling can introduce errorsRe-sampling can introduce errors– especially tri-linear interpolationespecially tri-linear interpolation
• Ghosts (and other artefacts) in the imagesGhosts (and other artefacts) in the images– do not move according to the same rigid body rules as do not move according to the same rigid body rules as
the subjectthe subject
• Slices are not acquired simultaneouslySlices are not acquired simultaneously– rapid movements not accounted for by rigid body modelrapid movements not accounted for by rigid body model
• fMRI images are distortedfMRI images are distorted– rigid body model does not model these types of rigid body model does not model these types of
distortiondistortion
• Spin excitation history effectsSpin excitation history effects– variations in residual magnetisationvariations in residual magnetisation
Functions of the estimated motion parameters can be used as Functions of the estimated motion parameters can be used as confounds in subsequent analyses.confounds in subsequent analyses.
Residual Errors from PETResidual Errors from PET• Incorrect attenuation correction because transmission Incorrect attenuation correction because transmission
scan no longer aligned with emission scans.scan no longer aligned with emission scans.
Residual Errors from PETResidual Errors from PET• Incorrect attenuation correction because transmission Incorrect attenuation correction because transmission
scan no longer aligned with emission scans.scan no longer aligned with emission scans.
TransformationTransformationTransformationTransformation
d1 d2
d3
d4
v1
v4
v2
v3
One if the simplest re-sampling methods is tri-linear interpolation.
Other methods include nearest neighbour re-sampling, and various forms of sinc interpolation using different numbers of neighbouring voxels.
Between Modality Co-registrationBetween Modality Co-registrationBetween Modality Co-registrationBetween Modality Co-registration
• Not based on simply minimising mean Not based on simply minimising mean squared difference between images.squared difference between images.
• A three step approach is used instead.A three step approach is used instead.
1) Simultaneous affine registrations 1) Simultaneous affine registrations between each image and template images between each image and template images of same modality.of same modality.
2) Partitioning of images into grey and 2) Partitioning of images into grey and white matter.white matter.
3) Final simultaneous registration of image 3) Final simultaneous registration of image partitions.partitions.
• Not based on simply minimising mean Not based on simply minimising mean squared difference between images.squared difference between images.
• A three step approach is used instead.A three step approach is used instead.
1) Simultaneous affine registrations 1) Simultaneous affine registrations between each image and template images between each image and template images of same modality.of same modality.
2) Partitioning of images into grey and 2) Partitioning of images into grey and white matter.white matter.
3) Final simultaneous registration of image 3) Final simultaneous registration of image partitions.partitions.
Rigid registration between high resolution structural images and echo planer functional images is a problem. Results are only approximate because of spatial distortions of EPI data.
Third Step - Third Step - Registration of Registration of
Partitions.Partitions.
Third Step - Third Step - Registration of Registration of
Partitions.Partitions.
•Grey and white matter partitions are registered using a rigid body transformation. •Simultaneously minimise sum of squared difference.
First Step - Affine Registrations.First Step - Affine Registrations.First Step - Affine Registrations.First Step - Affine Registrations.
• Requires template images of same modalities.Requires template images of same modalities.
• Both images are registered - using 12 parameter affine transformations - to Both images are registered - using 12 parameter affine transformations - to their corresponding templates by minimising the mean squared difference.their corresponding templates by minimising the mean squared difference.
• Only the rigid-body transformation parameters differ between the two Only the rigid-body transformation parameters differ between the two registrations.registrations.
• This gives:This gives:
– rigid body mapping between the images.rigid body mapping between the images.
– affine mappings between the images and the templates.affine mappings between the images and the templates.
• Requires template images of same modalities.Requires template images of same modalities.
• Both images are registered - using 12 parameter affine transformations - to Both images are registered - using 12 parameter affine transformations - to their corresponding templates by minimising the mean squared difference.their corresponding templates by minimising the mean squared difference.
• Only the rigid-body transformation parameters differ between the two Only the rigid-body transformation parameters differ between the two registrations.registrations.
• This gives:This gives:
– rigid body mapping between the images.rigid body mapping between the images.
– affine mappings between the images and the templates.affine mappings between the images and the templates.
Second Step - Segmentation.Second Step - Segmentation.Second Step - Segmentation.Second Step - Segmentation.
• ‘‘Mixture Model’ cluster analysis Mixture Model’ cluster analysis to classify MR image (or images) to classify MR image (or images) as GM, WM & CSF.as GM, WM & CSF.
• Additional information is Additional information is obtained from obtained from a priori a priori probability images, which are probability images, which are overlaid using previously overlaid using previously determined affine determined affine transformations.transformations.
• ‘‘Mixture Model’ cluster analysis Mixture Model’ cluster analysis to classify MR image (or images) to classify MR image (or images) as GM, WM & CSF.as GM, WM & CSF.
• Additional information is Additional information is obtained from obtained from a priori a priori probability images, which are probability images, which are overlaid using previously overlaid using previously determined affine determined affine transformations.transformations.
Between Modality Coregistration using Mutual InformationBetween Modality Coregistration using Mutual InformationBetween Modality Coregistration using Mutual InformationBetween Modality Coregistration using Mutual Information
PET T1 weightedMRI
An alternative betweenmodality registration methodavailable within SPM99maximises MutualInformation in the 2Dhistogram.
For histograms normalised to integrate to unity, the Mutual Information is defined by:
ij hij log hij
k hik l hlj
Spatial normalisationSpatial normalisationSpatial normalisationSpatial normalisation
• Inter-subject averagingInter-subject averaging– extrapolate findings to the population as a wholeextrapolate findings to the population as a whole
– increase activation signal above that obtained from single subjectincrease activation signal above that obtained from single subject
– increase number of possible degrees of freedom allowed in statistical modelincrease number of possible degrees of freedom allowed in statistical model
• Enable reporting of activations as co-ordinates within a known standard Enable reporting of activations as co-ordinates within a known standard spacespace– e.g. the space described by e.g. the space described by Talairach & TournouxTalairach & Tournoux
• Inter-subject averagingInter-subject averaging– extrapolate findings to the population as a wholeextrapolate findings to the population as a whole
– increase activation signal above that obtained from single subjectincrease activation signal above that obtained from single subject
– increase number of possible degrees of freedom allowed in statistical modelincrease number of possible degrees of freedom allowed in statistical model
• Enable reporting of activations as co-ordinates within a known standard Enable reporting of activations as co-ordinates within a known standard spacespace– e.g. the space described by e.g. the space described by Talairach & TournouxTalairach & Tournoux
• Warp the images such that functionally homologous regions from the Warp the images such that functionally homologous regions from the different subjects are as close together as possibledifferent subjects are as close together as possible– Problems:Problems:
• no exact match between structure and functionno exact match between structure and function
• different brains are organised differentlydifferent brains are organised differently
• computational problems (local minima, not enough information in the images, computational problems (local minima, not enough information in the images, computationally expensive)computationally expensive)
• Compromise by correcting for gross differences followed by smoothing of Compromise by correcting for gross differences followed by smoothing of normalised imagesnormalised images
• Warp the images such that functionally homologous regions from the Warp the images such that functionally homologous regions from the different subjects are as close together as possibledifferent subjects are as close together as possible– Problems:Problems:
• no exact match between structure and functionno exact match between structure and function
• different brains are organised differentlydifferent brains are organised differently
• computational problems (local minima, not enough information in the images, computational problems (local minima, not enough information in the images, computationally expensive)computationally expensive)
• Compromise by correcting for gross differences followed by smoothing of Compromise by correcting for gross differences followed by smoothing of normalised imagesnormalised images
Spatial NormalisationSpatial NormalisationSpatial NormalisationSpatial Normalisation
Spatial Normalisation
Original image
Templateimage
Spatially normalised
Determine the spatial transformation that minimises the sum of squared difference between an image and a linear combination of one or more templates.
Begins with an affine registration to match the size and position of the image.
Followed by a global non-linear warping to match the overall brain shape.
Uses a Bayesian framework to simultaneously maximise the smoothness of the warps.
Deformation field
Six affine registered images.Six affine registered images.Six affine registered images.Six affine registered images. Six basis function registered imagesSix basis function registered imagesSix basis function registered imagesSix basis function registered images
Affine versus affine and non-linear spatial normalisation
EPI
T2 T1 Transm
PD PET
305T1
PD T2 SS
Template Images “Canonical” images
A wider range of different contrasts can be normalised by registering to a linear combination of template 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.
• Bayes rule states: p(q|e) p(e|q) p(q) – p(q|e) is the a posteriori probability of parameters q given errors e.
– p(e|q) is the likelihood of observing errors e given parameters q.
– p(q) is the a priori probability of parameters q.
• Maximum a posteriori (MAP) estimate maximises p(q|e).
• Maximising p(q|e) is equivalent to minimising the Gibbs potential of the posterior distribution (H(q|e), where H(q|e) -log p(q|e)).
• The posterior potential is the sum of the likelihood and prior potentials:
H(q|e) = H(e|q) + H(q) + c– The likelihood potential (H(e|q) -log p(e|q)) is based upon the sum of
squared difference between the images.
– The prior potential (H(q) -log p(q)) penalises unlikely deformations.
• Bayes rule states: p(q|e) p(e|q) p(q) – p(q|e) is the a posteriori probability of parameters q given errors e.
– p(e|q) is the likelihood of observing errors e given parameters q.
– p(q) is the a priori probability of parameters q.
• Maximum a posteriori (MAP) estimate maximises p(q|e).
• Maximising p(q|e) is equivalent to minimising the Gibbs potential of the posterior distribution (H(q|e), where H(q|e) -log p(q|e)).
• The posterior potential is the sum of the likelihood and prior potentials:
H(q|e) = H(e|q) + H(q) + c– The likelihood potential (H(e|q) -log p(e|q)) is based upon the sum of
squared difference between the images.
– The prior potential (H(q) -log p(q)) penalises unlikely deformations.
Bayesian Formulation
Spatial Normalisation - affineSpatial Normalisation - affine
• The first part of spatial normalisation is a The first part of spatial normalisation is a 12 parameter Affine Transformation12 parameter Affine Transformation– 3 translations3 translations
– 3 rotations3 rotations
– 3 zooms3 zooms
– 3 shears3 shears
Empirically generated priors
1000
0100
00)cos()sin(
00)sin()cos(
1000
0)cos(0)sin(
0010
0)sin(0)cos(
1000
0)cos()sin(0
0)sin()cos(0
0001
1000
Z100
Y010
X001
trans
trans
trans
1000
0100
0YZ10
0XZXY1
1000
0Z00
00Y0
000X
shear
shearshear
zoom
zoom
zoom
Find the parameters that minimise the sum of squared difference between the image and template(s) - and also the square of the number of standard deviations away from the expected parameter values.
Spatial Normalisation - Non-linearSpatial Normalisation - Non-linearSpatial Normalisation - Non-linearSpatial Normalisation - Non-linear
• Deformations consist of a linear combination of smooth basis images.• These are the lowest frequency basis images of a 3-D discrete cosine transform (DCT).• Can be generated rapidly from a separable form.
• Algorithm simultaneously minimises– Sum of squared difference between
template and object image .
– Squared distance between the parameters and their known expectation (pTC0
-1 p).
• pTC0-1 p describes the membrane energy
of the deformations..
• Algorithm simultaneously minimises– Sum of squared difference between
template and object image .
– Squared distance between the parameters and their known expectation (pTC0
-1 p).
• pTC0-1 p describes the membrane energy
of the deformations..2
2
1
2
1
=energy membrane
i j k ki
ji
x
u
Templateimage
Templateimage
Affine Registration.(2 = 472.1)
Affine Registration.(2 = 472.1)
Non-linearregistration
withoutregularisation.(2 = 287.3)
Non-linearregistration
withoutregularisation.(2 = 287.3)
Non-linearregistration
usingregularisation.(2 = 302.7)
Non-linearregistration
usingregularisation.(2 = 302.7)
Without the Bayesian formulation, the non-linear spatial normalisation can introduce unnecessary warping into the spatially normalised images.
Segmentation.Segmentation.Segmentation.Segmentation.
• ‘‘Mixture ModelMixture Model’ cluster analysis to classify ’ cluster analysis to classify MR image (or images) as GM, WM & CSF.MR image (or images) as GM, WM & CSF.
• Additional information is obtained from Additional information is obtained from prior probability imagesprior probability images, which are overlaid., which are overlaid.
• Assumes that each MRI voxel is one of a Assumes that each MRI voxel is one of a number of distinct tissue types (clusters).number of distinct tissue types (clusters).
• Each cluster has a (multivariate) normal Each cluster has a (multivariate) normal distribution.distribution.
• ‘‘Mixture ModelMixture Model’ cluster analysis to classify ’ cluster analysis to classify MR image (or images) as GM, WM & CSF.MR image (or images) as GM, WM & CSF.
• Additional information is obtained from Additional information is obtained from prior probability imagesprior probability images, which are overlaid., which are overlaid.
• Assumes that each MRI voxel is one of a Assumes that each MRI voxel is one of a number of distinct tissue types (clusters).number of distinct tissue types (clusters).
• Each cluster has a (multivariate) normal Each cluster has a (multivariate) normal distribution.distribution.
.
• A smooth intensity A smooth intensity modulating function can modulating function can be modelled by a linear be modelled by a linear combination of DCT combination of DCT basis functions.basis functions.
• A smooth intensity A smooth intensity modulating function can modulating function can be modelled by a linear be modelled by a linear combination of DCT combination of DCT basis functions.basis functions.
.The segmented images contain a little non-brain tissue, which can be automatically The segmented images contain a little non-brain tissue, which can be automatically
removed using morphological operations (erosion followed by conditional dilation).removed using morphological operations (erosion followed by conditional dilation).The segmented images contain a little non-brain tissue, which can be automatically The segmented images contain a little non-brain tissue, which can be automatically
removed using morphological operations (erosion followed by conditional dilation).removed using morphological operations (erosion followed by conditional dilation).
More than one image can be used toproduce amulti-spectralclassification.
Morphometric MeasuresMorphometric MeasuresMorphometric MeasuresMorphometric Measures
• Voxel-by-voxelVoxel-by-voxel– where are the differences between where are the differences between
the populations?the populations?
– produce an SPM of regional produce an SPM of regional differencesdifferences
• Univariate - e.g., Voxel-Univariate - e.g., Voxel-Based MorphometryBased Morphometry
• Multivariate - e.g., Tensor-Multivariate - e.g., Tensor-Based MorphometryBased Morphometry
• Volume basedVolume based– is there a difference between the is there a difference between the
populations?populations?
• Multivariate - e.g., Multivariate - e.g., Deformation-Based Deformation-Based MorphometryMorphometry
• Voxel-by-voxelVoxel-by-voxel– where are the differences between where are the differences between
the populations?the populations?
– produce an SPM of regional produce an SPM of regional differencesdifferences
• Univariate - e.g., Voxel-Univariate - e.g., Voxel-Based MorphometryBased Morphometry
• Multivariate - e.g., Tensor-Multivariate - e.g., Tensor-Based MorphometryBased Morphometry
• Volume basedVolume based– is there a difference between the is there a difference between the
populations?populations?
• Multivariate - e.g., Multivariate - e.g., Deformation-Based Deformation-Based MorphometryMorphometry
MANCOVA & CCAMANCOVA & CCAMANCOVA & CCAMANCOVA & CCA
Originalimage
Spatiallynormalised
Partitionedgrey matter
Smoothed
Preparation of images for each subjectPreparation of images for each subjectPreparation of images for each subjectPreparation of images for each subject
Voxel-Based MorphometryVoxel-Based MorphometryVoxel-Based MorphometryVoxel-Based Morphometry
A voxel by voxel statistical analysis is used to detect regional differences in the amount of grey matter between populations.
Deformation-based Morphometrylooks at absolute displacements.
Tensor-based Morphometry looksat local shapes
Morphometric approaches based on deformation fields
Deformation-based morphometry
Deformationfields ...
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.
Remove positional and size information - leave shape
Sex Differences using Deformation-based Morphometry
Sex Differences using Deformation-based Morphometry
Non-linear warps pertaining to sex differences characterised by canonical variates analysis (above), and mean differences (below, mapping from an average female to male brain). In the transverse and coronal sections, the left side of the brain is on the left side of the figure.
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.
TemplateTemplateWarpedOriginal
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.
Relative volumes
Strain tensor
Tensor-based morphometry
High dimensional warpingHigh dimensional warpingHigh dimensional warpingHigh dimensional warping
Millions of parameters are needed for more precise image registration….. Takes a very long time
Relative volumes of brain structures can be computed from the determinants of the deformation fields
Data From the Dementia Research Group, London, UK.
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
