Transcript 01_Image_Registration

Image
Registration
John Ashburner
Pre-processing Overview
fMRI time-series
Anatomical MRI
Template
Statistics or
whatever
Smoothed
Estimate
Spatial Norm
Motion Correct
Smooth
Coregister
 m11

 m21

 m31


 0
m12
m22
m13
m23
m32
0
m33
0
Spatially
normalised
m14 

m24 
m34 
1 
Deformation
Contents
* Preliminaries
*
*
*
*
Smooth
Rigid-Body and Affine Transformations
Optimisation and Objective Functions
Transformations and Interpolation
* Intra-Subject Registration
* Inter-Subject Registration
Smooth
Smoothing is done by convolution.
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
Image Registration
• Registration - i.e. Optimise the parameters
that describe a spatial transformation between
the source and reference (template) images
• Transformation - i.e. Re-sample according to
the determined transformation parameters
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
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
* The order of the operations matters
1


0

0

0
0
1
0
0
0
1
0
0
Xtrans   1
Ytrans   0

Ztrans   0
1
Translations



0
0
cosΦ
 sinΦ
0
sinΦ
cosΦ
0
0
Pitch
about x axis
0   cosΘ
0   0

0    sinΘ
0
1
0
sinΘ
0
cosΘ
1 
0
0



0
Roll
about y axis
0   cosΩ
0    sinΩ

0  0

1 


0
sinΩ
cosΩ
0
0
0
1
0 
0 

0
0
0
1 
Yaw
about z axis

Voxel-to-world Transforms
* Affine transform associated with each image
* Maps from voxels (x=1..nx, y=1..ny, z=1..nz) to some world coordinate system. e.g.,
* Scanner co-ordinates - images from DICOM toolbox
* T&T/MNI coordinates - spatially normalised
* Registering image B (source) to image A (target) will
update B’s voxel-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
Left- and Right-handed Coordinate
Systems
* NIfTI format files are stored in either a left- or right-handed system
* Indicated in the header
* Talairach & Tournoux uses a right-handed system
* Mapping between them sometimes requires a flip
* Affine transform has a negative determinant
Optimisation
* Image registration is done by 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
Objective
function
Most probable solution
(global optimum)
Local optimum
Local optimum
Value of parameter
Objective Functions
* 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)
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
B-spline Interpolation
A continuous function is represented by
a linear combination of basis functions
B-splines are piecewise polynomials
2D B-spline basis functions
of degrees 0, 1, 2 and 3
Nearest neighbour and
trilinear interpolation are
the same as B-spline
interpolation with degrees
0 and 1.
Contents
* Preliminaries
* Intra-Subject Registration
* Realign
* Mean-squared difference objective function
* Residual artifacts and distortion correction
* Coregister
* Inter-Subject Registration
Mean-squared Difference
* Minimising mean-squared difference works for intramodal registration (realignment)
* Simple relationship between intensities in one
image, versus those in the other
* Assumes normally distributed differences
Residual Errors from aligned fMRI
* Re-sampling can introduce interpolation errors
* especially tri-linear interpolation
* Gaps between slices can cause aliasing artefacts
* 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 modelled
as confounds in subsequent analyses
Movement by Distortion Interaction of
fMRI
*Subject disrupts B0 field, rendering it
inhomogeneous
* distortions in phase-encode direction
*Subject moves during EPI time series
*Distortions vary with subject orientation
*shape varies
Movement by distortion interaction
Correcting for distortion changes using
Unwarp
Estimate reference from
mean of all scans.
Estimate new distortion
fields for each image:
Estimate
movement
parameters.
• estimate rate of change
of field with respect to
the current estimate of
movement parameters
in pitch and roll.

Unwarp time
series.
+
B0 
B0 
Andersson et al, 2001
Contents
* Preliminaries
* Intra-Subject Registration
* Realign
* Coregister
* Mutual Information objective function
* Inter-Subject Registration
Inter-modal registration
• Match images from same
subject but different
modalities:
–anatomical localisation of
single subject activations
–achieve more precise
spatial normalisation of
functional image using
anatomical image.
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)
Contents
* Preliminaries
* Intra-Subject Registration
* Inter-Subject Registration
* Normalise
* Affine Registration
* Nonlinear Registration
* Regularisation
* Segment
* DARTEL
Spatial Normalisation - Procedure
* Minimise mean squared difference from template
image(s)
Affine registration
Non-linear registration
T2
T1
Transm
T1
305
EPI
PD
PET
PD
T2
Template Images
SS
“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.
Spatial Normalisation - Templates
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)
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
Spatial Normalisation - Overfitting
Without
regularisation,
the non-linear
Target
spatial
image
normalisation
can introduce
unnecessary
warps.
Non-linear
registration
using
regularisation.
(2 = 302.7)
Affine
registration.
(2 = 472.1)
Non-linear
registration
without
regularisation.
(2 = 287.3)
Contents
* Preliminaries
* Intra-Subject Registration
* Inter-Subject Registration
* Normalise
* Segment
* Gaussian mixture model
* Intensity non-uniformity correction
* Deformed tissue probability maps
* DARTEL
Segmentation
* Segmentation in SPM5 also estimates a spatial
transformation that can be used for spatially
normalising images.
* It uses a generative model, which involves:
* Mixture of Gaussians (MOG)
* Bias Correction Component
* Warping (Non-linear Registration) Component
Mixture of Gaussians (MOG)
* Classification is based on a Mixture of Gaussians model
(MOG), which represents the intensity probability density by a
number of Gaussian distributions.
Frequency
Image Intensity
Belonging Probabilities
Belonging
probabilities
are assigned
by normalising
to one.
Non-Gaussian Intensity Distributions
* Multiple Gaussians per tissue class allow non-Gaussian
intensity distributions to be modelled.
* E.g. accounting for partial volume effects
Modelling a Bias Field
* A bias field is modelled as a linear combination
of basis functions.
Corrupted image
Bias Field
Corrected image
Tissue Probability Maps
* Tissue probability maps (TPMs) are used instead of
the proportion of voxels in each Gaussian as the
prior.
ICBM Tissue Probabilistic Atlases. These tissue probability maps are
kindly provided by the International Consortium for Brain Mapping, John C.
Mazziotta and Arthur W. Toga.
Deforming the Tissue Probability Maps
* Tissue probability
images are deformed
so that they can be
overlaid on top of the
image to segment.
Optimisation
* The “best” parameters are those that minimise this
objective function.
* Optimisation involves finding them.
* Begin with starting estimates, and repeatedly change
them so that the objective function decreases each time.
Steepest Descent
Start
Optimum
Alternate between
optimising different groups
of parameters
Spatially
normalised
BrainWeb
phantoms (T1,
T2 and PD)
Tissue
probability
maps of GM
and WM
Cocosco, Kollokian, Kwan & Evans. “BrainWeb: Online Interface to a 3D MRI Simulated Brain Database”. NeuroImage 5(4):S425 (1997)
Contents
* Preliminaries
* Intra-Subject Registration
* Inter-Subject Registration
* Normalise
* Segment
* DARTEL
*
*
*
*
Flow field parameterisation
Objective function
Template creation
Examples
A one-to-one mapping
* Many models simply add a smooth displacement to an identity transform
* One-to-one mapping not enforced
* Inverses approximately obtained by subtracting the displacement
* Not a real inverse
Small deformation approximation
DARTEL
* Parameterising the deformation
* φ(0)(x) = x
1
* φ(1)(x) = ∫ u(φ(t)(x))dt
* u is a flow field to be estimated
t=0
* Scaling and squaring is used to
generate deformations.
Scaling and squaring example
Forward and backward
transforms
Registration objective function
*
Simultaneously minimize the sum of:
*
Likelihood component
*
*
*
Drives the matching of the images.
Multinomial assumption
Prior component
*
*
A measure of deformation roughness
Regularises the registration.
* ½uTHu
*
A balance between the two terms.
Likelihood Model
* Current DARTEL model is multinomial for matching
tissue class images.
* Template represents probability of obtaining different
tissues at each point.
log p(t|μ,ϕ) = ΣjΣk tjk log(μk(ϕj))
t
μ
ϕ
– individual GM, WM and background
– template GM, WM and background
– deformation
Prior Model
Simultaneous registration of GM to GM and
WM to WM
Subject 1
Grey matter
White matter
Grey matter
White matter
Grey matter
White matter
Grey matter
Template
Grey matter
White matter
White matter
Subject 2
Subject 4
Subject 3
Template
Initial
Average
Iteratively generated
from 471 subjects
Began with rigidly
aligned tissue
probability maps
Used an inverse
consistent
formulation
After a few
iterations
Final
template
Grey matter
average of 452
subjects – affine
Grey matter
average of 471
subjects
Initial
GM images
Warped
GM images
VBM Pre-processing
* Use Segment button for
characterising intensity
distributions of tissue classes.
* DARTEL import, to generate
rigidly aligned grey and white
matter maps.
* DARTEL warping to generate
“modulated” warped grey
matter.
* Smooth.
* Statistics.
References
* Friston et al. Spatial registration and normalisation of images.
Human Brain Mapping 3:165-189 (1995).
* Collignon et al. Automated multi-modality image registration based on
information theory. IPMI’95 pp 263-274 (1995).
* Ashburner et al. Incorporating prior knowledge into image registration.
NeuroImage 6:344-352 (1997).
* Ashburner & Friston. Nonlinear spatial normalisation using basis functions.
Human Brain Mapping 7:254-266 (1999).
* Thévenaz et al. Interpolation revisited.
IEEE Trans. Med. Imaging 19:739-758 (2000).
* Andersson et al. Modeling geometric deformations in EPI time series.
Neuroimage 13:903-919 (2001).
* Ashburner & Friston. Unified Segmentation.
NeuroImage 26:839-851 (2005).
* Ashburner. A Fast Diffeomorphic Image Registration Algorithm. NeuroImage
38:95-113 (2007).