Spatial Preprocessing

Ged Ridgway

University College London

Thanks to:

John Ashburner, Klaas Enno Stephan, and the FIL Methods Group.Meike Grol, Chloe Hutton, Jesper Andersson, Floris de Lange,Miranda van Turennout, Lennart Verhagen, …

Terminology of fMRI

Structural (T1) images: - high resolution- to distinguish different types of tissue

Functional (T2*) images:- lower spatial resolution- to relate changes in BOLD signal to an experimental manipulation

Time series: A large number of images that are acquired in temporal order at a specific rate

tConditi

on AConditi

on B

subjects

sessions

runs

single run

volume

slices

Terminology of fMRI

TR = repetition timetime required to scan one volume

voxel

Slice thicknesse.g., 3 mm

Scan Volume:Field of View

(FOV),e.g. 192 mm

Axial slices

3 mm

3 mm

3 mm

Voxel Size(volumetric pixel)

Matrix Sizee.g., 64 x 64

In-plane resolution192 mm / 64

= 3 mm

Terminology of fMRI

fMRI Time-series Movie

SpatialNormalisation

fMRI time-series

Smoothing

Anatomical Reference

Statistical Parametric Map

Parameter Estimates

General Linear Model

Design matrix

Overview of SPM Analysis

MotionCorrection

Preprocessing in SPM

• Realignment – With non-linear unwarping for EPI fMRI

• Slice-time correction• Coregistration• Normalisation• Segmentation• Smoothing

The many guises of affine registration

• Manual reorientation• Rigid intra-modal realignment

– Motion correction of fMRI time-series

– Within-subject longitudinal registration of serial sMRI

• Rigid inter-modal coregistration– Aligning structural and (mean) functional images

– Multimodal structural registration, e.g. T1-T2

• Affine inter-subject registration– First stage of non-linear spatial normalisation

– Approximate alignment of tissue probability maps

Other types of registration in SPM

• Non-linear spatial normalisation– Registering different subjects to a standard template

• Unified segmentation and normalisation– Warping standard-space tissue probability maps to a

particular subject (can normalise using the inverse)

• High-dimensional warping– Modelling small longitudinal deformations (e.g. AD)

• DARTEL– Smooth large-deformation warps using flows– Normalisation to group’s average shape template

Image “headers” contain information that lets us map from voxel indices to “world” coordinates in mm

Modifying this mapping lets us reorient (and realign or coregister) the image(s)

Manual reorientation

Manual reorientation

(Bi)linear

Manual reorientation

Interpolation

(Bi)linearNearest Neighbour Sinc

Interpolation

• Applying the transformation parameters, and re-sampling the data onto the same grid of voxels as the target image (usually)– AKA reslicing, regridding, transformation, and writing (as in

normalise - write)• Nearest neighbour gives the new voxel the value of

the closest corresponding voxel in the source• Linear interpolation uses information from all

immediate neighbours (2 in 1D, 4 in 2D, 8 in 3D)• NN and linear interp. correspond to zeroth and first

order B-spline interpolation, higher orders use more information in the hope of improving results– (Sinc interpolation is an alternative to B-spline)

Manual reorientation – Reslicing

Reoriented

(1x1x3 mm voxel size)

Resliced

(to 2 mm cubic)

Quantifying image alignment

• Registration intuitively relies on the concept of aligning images to increase their similarity– This needs to be mathematically formalised– We need practical way(s) of measuring similarity

• Using interpolation we can find the intensity at equivalent voxels– (equivalent according to the current transformation

parameter estimates)

Voxel similarity measures

• Mean-squared difference• Correlation coefficient• Joint histogram measures

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Pairs of voxel intensities

Intra-modal similarity measures

• Mean squared error (minimise)– AKA sum-squared error, RMS error, etc.

(monotonically related). AKA “least squares”– Assumes simple relationship between intensities– Optimal (only) if differences are i.i.d. Gaussian– Okay for fMRI realignment or sMRI-sMRI coreg

• Correlation-coefficient (maximise)– Slightly more general, e.g. T1-T1 inter-scanner– Invariant under affine transformation of intensities– AKA Normalised Cross-Correlation, Zero-NCC

Automatic image registration

• Quantifying the quality of the alignment with a measure of image similarity allows computational estimation of transformation parameters

• This is the basis of both realignment and coregistration in SPM– Allowing more complex geometric transformations or

warps leads to more flexible spatial normalisation

• Let’s look at an example of realignment next– We will return to coregistration later...

Option of second pass, re-registering to mean from first pass, improves results if first image is not perfectly representative of all others

How much motion is significant?

S(x) ~ 70-90%

Edge:

Inside:S(x) ~ 10-20%

0.05 pixel shift

(187 m)

0.25 pixel shift

(~1 mm)

3 - 5% S

3 - 5% S

Slide from Kent Kiehl’s 2005 (USA) Course Workshop

Motion in fMRI

Minimising movements is one of the most

important factors for ensuring good data quality

• Even small head movements can be a major problem:– Movement artefacts add up to the residual variance

and reduce sensitivity.– Data may get completely lost if sudden movements

occur during a single volume– Movements may be correlated with the task

performed

fMRI motion prevention

1. Constrain the volunteer’s head

2. Instruct him/her explicitly to remain as calm as possible, not to talk between sessions, and swallow as little as possible

3. Do not scan for too long – everyone will move after while!

fMRI motion correction

• Each volume in the time-series is realigned– Using rigid-body registration

• Assumes that brain shape doesn’t change– Least squares cost function

• Realigned images must be resliced for analysis– Not necessary if they will be normalised anyway

• An example of severe motion…

Even after realignment, as much as 90% of the variance can be accounted for by effects of movementThis can be caused by e.g.:

1. Movement between and within slice acquisition

2. Interpolation artefacts due to resampling

3. Non-linear distortions and drop-out due to inhomogeneity of the magnetic field

Incorporate movement parametersas confounds in the statistical model

Residual errors in realigned fMRI

Movement-by-distortion interaction

• Subject disrupts B0 field, rendering it inhomogeneous– distortions occur along

phase-encode direction• Subject moves during EPI

time series– Distortions vary with

subject orientation – Shape varies

Co-registration example

• Rigid intra-modality inter-subject registration• Purely to illustrate concepts• Coreg more commonly used to register a

structural image to a mean functional one– Allows more accurate anatomical localisation– Assists spatial normalisation (see later)– Examples in SPM8 Manual, Chapters 28, 29

ReferenceAKA templateAKA fixed image(single_subj_T1)

SourceAKA template (!)AKA moving image(auditory/sM00223)

x = 2.52 20.86 28.79 -0.20 -0.02 -0.01

M = spm_matrix(x)M = 0.999 -0.011 -0.016 2.522 0.008 0.981 -0.195 20.857 0.018 0.195 0.981 28.790 0 0 0 1.000det(M)ans = 1.000

spm_defaults; global defaultsopts = defaults.coreg.estimateopts = cost_fun: 'nmi' sep: [4 2] tol: [1x12 double] fwhm: [7 7]opts.cost_fun = 'ncc'; % intra-modal

x = spm_coreg(ref, src, opts) % …

iteration 1...3.72 0 0 0 0 0 | -0.507313.72 15.95 0 0 0 0 | -0.558933.72 15.95 19.61 0 0 0 | -0.628153.72 15.95 19.61 -0.1837 0 0 | -0.64343.72 15.95 19.61 -0.1837 -0.00151 0 | -0.643413.72 15.95 19.61 -0.1837 -0.00151 -0.00063 | -0.64341iteration 2...

...

iteration 8...2.447 20.97 28.73 -0.1905 -0.003633 -0.01026 | -0.714862.447 20.97 28.73 -0.1905 -0.003633 -0.01026 | -0.714862.447 20.97 28.74 -0.1905 -0.003633 -0.01026 | -0.714872.447 20.97 28.74 -0.1905 -0.003633 -0.01026 | -0.714872.447 20.97 28.74 -0.1905 -0.003633 -0.01026 | -0.714872.447 20.97 28.74 -0.1905 -0.003633 -0.01031 | -0.71487

Svx2Rvx = inv(ref.mat)*inv(M)*src.mat

Inter-modal similarity measures

• Often derived from joint and marginal entropies– Entropies via probabilities from histograms

• Mutual Information (maximise)– MI= ab P(a,b) log2 [P(a,b)/( P(a) P(b) )]

– Related to entropy: MI = H(a) + H(b) - H(a,b)

• H(a) = -a P(a) log2P(a)

• H(a,b) = -a P(a,b) log2P(a,b)

• Normalised MI = (H(a) + H(b)) / H(a,b)

Joint and marginal histograms

Joint histogram sharpness correlates with image alignmentMutual information and related measures attempt to quantify this

Initially registered T1 and T2 templatesAfter deliberate misregistration(10mm relative x-translation)

Joint histogram based registration

SpatialNormalisation

fMRI time-series

Smoothing

Anatomical Reference

Statistical Parametric Map

Parameter Estimates

General Linear Model

Design matrix

Overview of SPM Analysis

MotionCorrection

Spatial Normalisation

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

• Make results from different studies comparable by aligning them to standard space– e.g. The T&T convention, using the MNI template

Standard spaces

The MNI template follows the convention of T&T, but doesn’t match the particular brain

Recommended reading: http://imaging.mrc-cbu.cam.ac.uk/imaging/MniTalairach

The Talairach Atlas The MNI/ICBM AVG152 Template

Coordinate system sense

• Analyze™ files are stored in a left-handed system• Talairach space has the opposite (right-handed) sense• Mapping between them requires a reflection or “flip”

– Affine transform with a negative determinant

x

y

z

xy

z

z

x

y

Rotated example

Compromise by correcting gross differences followed by smoothing of normalised images

However…

1. No exact match between structure and function

2. Different brains – different structural topology

3. Computational problems (local minima, etc.)

Spatial normalisation – Issues

We want to warp the images to match functionally homologous regions from different subjects

Spatial Normalisation – Procedure

• Start with a 12 DF affineregistration– 3 translations, 3 rotations

3 zooms, 3 shears– Fits overall shape and size

• Refine the registration withnon-linear deformations

• Algorithm simultaneously minimises– Mean-squared difference (Gaussian likelihood)– Squared distance between parameters and their

expected values (regularisation with Gaussian prior)

Spatial Normalisation – Warping

Deformations are modelled with a linear combination of non-linear basis functions

Spatial Normalisation – DCT basis

The lowest frequencies of a 3D discrete cosine transform (DCT) provide a smooth basis

plot(spm_dctmtx(50, 5))

spm_dctmtx(5,5)ans = 0.447 0.602 0.512 0.372 0.195 0.447 0.372 -0.195 -0.602 -0.512 0.447 0.000 -0.633 -0.000 0.633 0.447 -0.372 -0.195 0.602 -0.512 0.447 -0.601 0.512 -0.372 0.195

% Note, pinv(x)=x’, projection P=x*x’ P{n} = x(:,1:n)*x(:,1:n)’ P{N} == eye(N) P{n<N} projects to smoother approx.

EPI

T2 T1 Transm

PD PET

Spatial normalisation can be weighted so that non-brain voxels do not influence the result.

A wider range of contrasts can be registered to a linear combination of template images.

T1 PD

PET

Spatial Normalisation– Templates and masks

More specific weighting masks can be used to improve normalisation of lesioned brains.

Spatial Normalisation – Results

Non-linear registrationAffine registration

Optimisation

• Find the “best” parameters according to an “objective function” (minimised or maximised)

• Objective functions can often be related to a probabilistic model, e.g. Maximum Likelihood

Value of parameter

Objective function

Global optimum(most probable)

Local optimumLocal optimum

Optimisation – poor local optima

Optimisation – regularisation

• The “best” parameters according to the objective function may not be realistic

• In addition to similarity, regularisation terms or constraints are often needed to ensure a reasonable solution is found– Also helps avoid poor local optima– These can be considered as priors in a Bayesian

framework, e.g. converting ML to MAP:• log(posterior) = log(likelihood) + log(prior) + c

Templateimage

Affine registration.(2 = 472.1)

Non-linearregistration

withoutregularisation.(2 = 287.3)

Non-linearregistration

usingregularisation.(2 = 302.7)

Without regularisation, the non-linear normalisation can introduce unnecessary deformation

Spatial Normalisation – Overfitting

Normalising fMRI time-series

• Two main options for fMRI normalisation:

1. Structural normalisation (unified segmentation)– Rigidly coregister structural MRI to mean functional– Normalise structural and apply warp (write) to fMRI

2. EPI normalisation– Normalise mean functional to EPI template– Apply warp to fMRI time-series– Optional coregistration and norm-write of structural

for visualisation purposes

Estimate normalization parameters (T1 -> T1 template)

Apply the normalization parameters to the T1 image

And apply the normalization parameters to the (coregistered) functional images

T1 image

EPI time series

Normalising structural to T1 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

Normalising time series to EPI template

Custom EPItemplate

Original EPIimage

Custom EPItemplate (note different coords)

NormalisedEPI image

Smoothing

• Why would we deliberately blur the data?– Suppress noise and effects due to differences in

anatomy by averaging over neighbouring voxels– Achieve better spatial overlap, enhanced sensitivity,

and greater validity of statistical assumptions– Reduce the effective number of multiple comparisons

when correcting results using Random Field Theory

• How is it implemented?– Convolution with a 3D Gaussian kernel, of specified

Full-width at half-maximum (FWHM) in mm

Example ofGaussian smoothing in one-dimension

A 2D Gaussian Kernel

The Gaussian kernel is separable we can smooth 2D data with two 1D convolutions.

Generalisation to 3D is simple and efficient

Before convolution Convolved with a circle Gaussian convolution

Each voxel after smoothing effectively represents a weighted average over its local region of interest (ROI)

Smoothing – a link to ROI analysis

Preprocessing in SPM

• Realignment – With non-linear unwarping for EPI fMRI

• Slice-time correction• Coregistration• Normalisation• Segmentation• Smoothing

SPM8’s unified tissue segmentation and spatial normalisation procedure will be covered in the VBM lecture