Contents

## Spatial Preprocessing John Ashburner john@fil.ion.ucl.ac.uk Overview of SPM Analysis Contents Smoothing Smoothing

Bailey, Katherine, Contributing Editor has reference to this Academic Journal, PHwiki organized this Journal Spatial Preprocessing John Ashburner john@fil.ion.ucl.ac.uk Smoothing Rigid registration Spatial normalisation With slides by Chloe Hutton in addition to Jesper Andersson Overview of SPM Analysis Motion Correction Smoothing Spatial Normalisation General Linear Model Statistical Parametric Map fMRI time-series Parameter Estimates Design matrix Anatomical Reference Contents Smoothing Rigid registration Spatial normalisation

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Smoothing Be as long as e convolution Convolved with a circle Convolved with a Gaussian Each voxel after smoothing effectively becomes the result of applying a weighted region of interest (ROI). 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 Contents Smoothing Rigid registration Rigid-body trans as long as ms Optimisation & objective functions Interpolation Spatial normalisation

Within-subject Registration Assumes there is no shape change, in addition to motion is rigid-body Used by [realign] in addition to [coregister] functions The steps are: Registration – i.e. Optimising the parameters that describe a rigid body trans as long as mation between the source in addition to reference images Trans as long as mation – i.e. Re-sampling according to the determined trans as long as mation Affine Trans as long as ms Rigid-body trans as long as mations 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: Y=Mx 2D Affine Trans as long as ms Translations by tx in addition to 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 in addition to sy x1 = sx x0 y1 = sy y0 Shear x1 = x0 + h y0 y1 = y0

2D Affine Trans as long as ms Translations by tx in addition to 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 in addition to 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 Trans as long as mations A 3D rigid body trans as long as m 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 Voxel-to-world Trans as long as ms Affine trans as long as m associated with each image Maps from voxels (x=1 nx, y=1 ny, z=1 nz) to some world co-ordinate 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 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

Left- in addition to Right-h in addition to ed Coordinate Systems Analyze files are stored in a left-h in addition to ed system Talairach & Tournoux uses a right-h in addition to ed system Mapping between them requires a flip Affine trans as long as m with a negative determinant 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 Value of parameter Objective function Most probable solution (global optimum) Local optimum Local optimum Objective Functions as long as Image Registration Intra-modal Mean squared difference (minimise) Normalised cross correlation (maximise) Entropy of difference (minimise) Inter-modal (or intra-modal) Mutual in as long as mation (maximise) Normalised mutual in as long as mation (maximise) Entropy correlation coefficient (maximise) AIR cost function (minimise)

Mean-squared Difference Minimising mean-squared difference works as long as intra-modal registration (realignment) Simple relationship between intensities in one image, versus those in the other Assumes normally distributed differences Gauss-newton Optimisation Works best as long as least-squares Minimum is estimated by fitting a quadratic at each iteration 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. Inter-modal registration

Mutual In as long as mation Used as long as 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) in addition to H(a,b) = -a P(a,b) log2P(a,b) Image Trans as long as mations Images are re-sampled. An example in 2D: as long as y0=1 ny0 % loop over rows as long as x0=1 nx0 % loop over pixels in row x1 = tx(x0,y0,q) % trans as long as m according to q y1 = ty(x0,y0,q) if 1×1 nx1 & 1y1ny1 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 in addition to y1 are not integers 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 B-splines are piecewise polynomials A continuous function is represented by a linear combination of basis functions 2D B-spline basis functions of degrees 0, 1, 2 in addition to 3 Nearest neighbour in addition to trilinear interpolation are the same as B-spline interpolation with degrees 0 in addition to 1. 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 as long as 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 Residual Errors from aligned fMRI 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 as long as distortion changes using Unwarp Estimate movement parameters. Estimate new distortion fields as long as each image: estimate rate of change of field with respect to the current estimate of movement parameters in pitch in addition to roll. Estimate reference from mean of all scans. Unwarp time series. + Andersson et al, 2001 Contents Smoothing Rigid registration Spatial normalisation Affine registration Nonlinear registration Regularisation

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 St in addition to ard coordinate system e.g., Talairach & Tournoux space 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 in addition to function Different brains are organised differently Computational problems (local minima, not enough in as long as mation in the images, computationally expensive) Compromise by correcting gross differences followed by smoothing of normalised images Very hard to define a one-to-one mapping of cortical folding Use only approximate registration.

Spatial Normalisation – Non-linear Algorithm simultaneously minimises Mean squared difference between template in addition to source image Squared distance between parameters in addition to their known expectation De as long as mations consist of a linear combination of smooth basis functions These are the lowest frequencies of a 3D discrete cosine trans as long as m (DCT) Spatial Normalisation – Overfitting Template image Affine registration. (2 = 472.1) Non-linear registration without regularisation. (2 = 287.3) Non-linear registration using regularisation. (2 = 302.7) Without regularisation, the non-linear spatial normalisation can introduce unnecessary warps. References Friston et al (1995): Spatial registration in addition to normalisation of images. Human Brain Mapping 3:165-189 Collignon et al (1995): Automated multi-modality image registration based on in as long as mation theory. IPMI95 pp 263-274 Andersson et al (2001): Modeling geometric de as long as mations in EPI time series. Neuroimage 13:903-919 Thévenaz et al (2000): Interpolation revisited. IEEE Trans. Med. Imaging 19:739-758. Ashburner et al (1997): Incorporating prior knowledge into image registration. NeuroImage 6:344-352 Ashburner et al (1999): Nonlinear spatial normalisation using basis functions. Human Brain Mapping 7:254-266

## Bailey, Katherine Contributing Editor

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