Image Registration John Ashburner Smooth Realign Normalise Segment With slides b
Gary, Steve, Meteorologist has reference to this Academic Journal, PHwiki organized this Journal Image Registration John Ashburner Smooth Realign Normalise Segment 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 Preliminaries Smooth Rigid-Body in addition to Affine Trans as long as mations Optimisation in addition to Objective Functions Trans as long as mations in addition to Interpolation Intra-Subject Registration Inter-Subject Registration
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Smooth Be as long as e convolution Convolved with a circle Convolved with a Gaussian Smoothing is done by convolution. Each voxel after smoothing effectively becomes the result of applying a weighted region of interest (ROI). Image Registration Registration – i.e. Optimise the parameters that describe a spatial trans as long as mation between the source in addition to reference (template) images Trans as long as mation – i.e. Re-sample according to the determined trans as long as mation parameters 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 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- 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 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)
Trans as long as mation 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.
Contents Preliminaries Intra-Subject Registration Realign Mean-squared difference objective function Residual artifacts in addition to distortion correction Coregister Inter-Subject Registration 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 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 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
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 Preliminaries Intra-Subject Registration Realign Coregister Mutual In as long as mation objective function Inter-Subject 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. 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)
Contents Preliminaries Intra-Subject Registration Inter-Subject Registration Normalise Affine Registration Nonlinear Registration Regularisation Segment 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 – Procedure Minimise mean squared difference from template image(s) Non-linear registration Affine registration
Spatial Normalisation – Templates EPI T2 T1 Transm PD PET 305 T1 PD T2 SS Template Images Canonical images A wider range of contrasts can be registered 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 as long as normalising lesioned brains. T1 PD PET Spatial Normalisation – Affine The first part is a 12 parameter affine trans as long as m 3 translations 3 rotations 3 zooms 3 shears Fits overall shape in addition to size Algorithm simultaneously minimises Mean-squared difference between template in addition to source image Squared distance between parameters in addition to their expected values (regularisation) 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)
References Friston et al. Spatial registration in addition to normalisation of images. Human Brain Mapping 3:165-189 (1995). Collignon et al. Automated multi-modality image registration based on in as long as mation theory. IPMI95 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 de as long as mations in EPI time series. Neuroimage 13:903-919 (2001). Ashburner & Friston. Unified Segmentation. NeuroImage in press (2005).
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