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Spatial Processing

Spatial Processing

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Spatial Processing

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  1. Spatial Processing • Chris Rorden • Spatial Registration • Motion correction • Coregistration • Normalization • Interpolation • Spatial Smoothing • Advanced notes: • Spatial distortions of EPI scans • Image intensity distortions • Matrix mathematics

  2. Why spatially register data? • Statistics computed individually for voxels. • Only meaningful if voxel examines same region across images. • Therefore, images must be in spatially registered with each other.

  3. Spatial Registration • We use spatial registration to align images • Motion correction (realignment) adjusts for an individual’s head movements. • Coregistration aligns two images of different modalities from the same individual. • Normalization aligns images from different people.

  4. Within-subject registration • With-in subject registrations • Assumption: same individual, so there should be a good linear solution. Coregistration Motion correction Registration of the fMRI scans (across time) Registration of fMRI scans with high resolution image.

  5. Rigid Body Transforms Translation Rotation • By measuring and correcting for translations and rotations, we can adjust for an object’s movement in an image.

  6. How many parameters? • Each transform can be applied in 3 dimensions. • Therefore, if we correct for both rotation and translation, we will compute 6 parameters. Translation Rotation X Z Pitch Yaw Roll Y

  7. Motion Correction • Motion correction aligns all in time series. • Translations and rotations only • ‘rigid body registration’ • Assumes brain size and shape identical across images.

  8. Motion Correction Cost Function 2 When aligned, Difference squared ~ 0 = Reslice Target 2 When unaligned, Difference squared > 0 = Reslice Target

  9. Motion correction cost function • Motion correction uses least squares to check if images are a good match (aka minimum sum of squares). • Smaller difference2 = better match (‘least squares’). • Iterative: moves image a bit at a time until match is worse. Image 1 Difference Image 2 Difference²

  10. Local Minima • Search algorithm is iterative: • move the image a little bit. • Test cost function • Repeat until cost function does not get better. • Search algorithm can get stuck at local minima: cost function suggests that no matter how the transformation parameters are changed a minimum has been reached Value of Cost Function Local Minimum Global Minimum Translation in X

  11. Coregistration • Coregistration is more complicated than motion correction • Rigid body not enough: • Size differs between images (must rescale: zooms). • fMRI scans often have spatial distortion not seen in other scans (must skew: shears). • Least squares cost function will fail: relative contrast of gray matter, white matter, CSF and air differences between images.

  12. Affine Transforms (aka linear, geometric) Zoom Shear Translation Rotation

  13. Coregistration • Coregistration is used to align images of different modalities from the same individual • Uses ‘mutual information cost function’: Note aligned images have neater histograms. • Uses entropy reduction instead of variance reduction as cost function.

  14. Coregistration • Used within individual, so linear transforms should be sufficient • Typically 12 parameters (translation, rotation, zooms, shear each in 3 dimensions) • Though note that different MRI sequences create different non-linear distortions Coregistered FLAIR T1 image

  15. Between-subject: Normalization • Allows inference about general population Subject 1 Template Subject 2 Average activation Normalization

  16. Why normalize? • Stereotaxic coordinates analogous to longitude • Universal description for anatomical location • Allows other to replicate findings • Allows between-subject analysis: crucial for inference that effects generalize across humanity.

  17. Normalization • Normalization attempts to register scans from different people. • We align each persons brain to a template. • Template often created from multiple people (so it is fairly average in shape, size, etc). • We typically use template that is in the same modality as the image we want to normalize • Therefore, variance cost function. • If different groups use similar templates, they can talk in common coordinates. Popular MNI Templatebased on T1-weighted scans from 152 individuals.

  18. Coordinates - normalization • Different people’s brains look different • ‘Normalizing’ adjusts overall size and orientation Normalized Images Raw Images

  19. SPM uses modality specific template • MNI T1 template, plus custom templates • By default, FSL uses MNI T1 template for all modalities • Requires intra-modal cost functions T1 T2* PET

  20. Coordinates - Earth • For earth (2D surface) we use latitude and longitude • Origin for latitude is equator • Explicit: defined by axis of rotation • Origin for longitude is Greenwich. • Arbitrary: could be Paris • What is crucial is that we we agree on the same origin. • For the brain • left-right side explicit: Interhemispheric Fissure analogous to equator • How about Anterior-Posterior and Superior-Inferior? We need an origin for these coordinates.

  21. Coordinates - Talairach • Anterior Commissure (AC) is the origin for neuroscience. • We measure distance from AC • 57x-67x0 means ‘right posterior middle’. • Three values: left-right, posterior-anterior, ventral-dorsal

  22. Coordinates - Talairach • Axis for axial plane is defined by anterior commissure (AC) and posterior commissure (PC). • Both are small regions that are clear to see on most scans. Y+ Y- PC Z+ Z- AC

  23. Templates • Original Talairach-Tournoux atlas based on histological slices from one 69-year old woman. • Single brain may not be representative • No MRI scans from this woman • Modern templates were at some stage aligned to images from the Montreal Neurological Institute. • MNI space slightly different from T&T atlas (larger in every dimension).

  24. Affine Transforms • Co-linear points remain co-linear after any affine transform. • Transform influences entire image.

  25. Spatial Processing • Non-linear transforms can match features that could not aligned with affine transforms. • SPM uses basis functions.

  26. Nonlinear functions and normalization Linear Only Scans from 6 people Linear + Nonlinear

  27. Recent algorithms • Affine transforms (e.g. FSL’s FLIRT): 12 degrees of freedom • Translation, Rotation, Scaling, Shear *3 dimensions • Nonlinear basis functions (SPM5) thousands of dof • Diffeomorphic algorithms (SPM8’s DARTEL, ANT) millions dof Affine template DARTEL template

  28. Regularization • Regularization penalizes bending energy • What is the best way to show graph points with a smooth line? Heavy regularization is a poor fit, heavy regularization causes local distortion Heavy Regularization Medium Regularization Little Regularization

  29. Regularization • Regularization is a parameter that you can adjust that influences non-linear normalization Medium Regularization Little Regularization

  30. Spatial Processing • Affine Transforms are robust – they influence the entire brain • Note that non-linear functions can have local effects. • This can improve normalization • This can also lead to image distortion. • E.G. In stroke patients, the injured region may not match the intensity of the template… Area of brain injury looks different on scan from stroke patient. ‘Perfect’ alignment with template will still have high cost function in injured region.

  31. Sulcal matching • Normalization conducted on smoothed images. • We are not trying to precisely match sulci (would cause local distortion). • Sulcal matching approximate • old images: DARTEL somewhat better Post-normalization alignment of calcarine sulcus, precentral gyrus, superior temporal gyrus. •

  32. Alternatives • SPM and FSL normalize overall brain shape. • Individual sulci largely ignored. • What are different normalization strategies? • Sulci are crucial for some tasks (Herschl’s gyrus and hearing) • Perhaps less so for others (e.g. Amunts et. al 2004 with Broca’s variability)

  33. Alternatives • SPM/FSL normalization will roughly match orientation and shape of head. • Good if function is localized to proportional part of brain • Poor if function is localized to specific sulci (e.g. early visual area V1 tied to calcarine fissure). • Alternatively, use sulci as cost function (Goebel et al., 2006). • Image below: mean sulcal position for 12 people after standard normalization (left) followed by sucal registration (middle). • Note: This technique improves sulcal alignment, but distorts cortical size.

  34. Interpolation Each lower image rotated 12º. Left looks jagged, right looks smooth. Different reslicing interpolation.

  35. 1D Interpolation  • How do we estimate values that occur between discrete samples? • Three popular methods: • Nearest neighbor (box) • Linear (tent) • Spline/Sinc   Weather analogy: if it was 25º at 9am, and 31º at 12am, what would you estimate the temperature was at 10am?

  36. Interpolation  • How do we estimate values that occur between discrete samples? • Three popular methods: • Nearest neighbor (box) • Linear (tent) • Spline/Sinc  

  37. Linear Interpolation • For neuroimaging we usually use linear interpolation. • Much more accurate than nearest neighbor. • There is some loss of high frequencies. • Since we spatially smooth data after spatial registration, we will lose high frequencies eventually. 1D Linear Interpolation Weighted mean of 2 samples 2D Bilinear Interpolation Weighted mean of 4 samples 3D Trilinear Interpolation Weighted mean of 8 samples

  38. 10o 20o 360o 90o 180o Linear Interpolation – High Frequency Loss Original • Linear interpolation loses high frequencies • Multiple successive resampling will lead to blurry image • Solution: Minimize number of times the data is resliced.

  39. Advanced Interpolation • Sinc interpolation can retain high frequency information. • Computation very time consuming (in theory, infinite extent) • FSL: Windowing options limit extent • SPM: Splines are used for rapid approximation • Not necessary if you will heavily blur your data with a broad smoothing kernel. 2D Sinc Function 1D Sinc Function

  40. Interpolation versus smoothing • Interpolation kernel is always 100% at 0 and 0% at all other integers. • This means that interpolated estimates always cross through control points (observations). • Smoothing kernels blur an observation with its neighbors. Interpolation Smoothing Kernel Observations

  41. Spatial Smoothing • Each voxel is noisy. However, neighbors tend to show similar effect. Smoothing results in a more stable signal. • Smooth also helps statistics: smoothed data tends to be more ‘normal’ – fits our assumptions. Also, allows RFT thresholding (see Statistics lecture). Gaussian Smoothing =

  42. FWHM • Smoothing is a form of convolution: the output intensity based on weighted-influence of neighbors. • The most popular kernel is the gaussian function (a normal distribution). • The ‘full width half maximum’ adjusts the amount of gaussian smoothing. • FWHM is a measure of dispersion (like standard deviation or variance) • Large FWHMs lead to more blurry images. • For fMRI, we typically use a FWHM that is ~x2..x3 our original resolution (e.g. 8mm for 3x3x3mm data). • However, the FWHM tunes the size of region we will be best able to detect. • E.G. If you want to look for a brain region that is around 10mm diameter, use a 10mm FWHM. Dispersion Differs

  43. Smoothing • Spatial smoothing useful for between-subject analyses. • Spatial normalization is only approximate: smoothing minimizes individual sulcal variability. • Smoothing controls for variation in functional localization between people. None 4mm 8mm 12mm

  44. Smoothing Limits Inference • Consider a study that observes ‘increased’ activation for strong versus weak motor movements. • After smoothing we can not distinguish between: • Increased activation of the same population of neurons • Recruitment of more neighboring neurons. • Example: note that after smoothing broad low contrast looks line looks like focused high contrast line. 2D 1D

  45. Smoothing Alternatives • Gaussian smoothing is great for ‘normal’ (Gaussian) noise: lots of small errors, very few outliers. • Gaussian poor for spike noise • Outlier contaminates neighbors • Alternatives if your data has spikes: • Median filters • FSL’s SUSAN Gaussian Noise Gaussian Smooth Gaussian Median Filter Spike Noise

  46. Inhomogeneity artifacts • The head distorts the magentic field. • Shimming attempts to make field level homogeneous. • Even after shimming, there will be varying field strengths. • Specifically, regions with large density changes (sinus/bone of frontal lobe). • This inhomogeneity leads to intensity and spatial distortion.

  47. Spatial unwarping • We can measure field homogeneity. • This can be used to unwarp images (FSL’s B0 unwarping, SPM’s FieldMap). Structural Unwarped EPI Raw EPI

  48. Intensity unwarping • Motion correction creates a spatially stabilized image. • However, head motion also changes image intensity – some regions of the brain will appear brighter/darker. • SPM: EPI unwarping corrects for brightness changes (right) • FSL: You can add motion parameters to statistical model (FEAT stats page). • Problem: We will lose statistical power if head motion is task related, e.g. pitch head every time we press a button • Above: motion related image intensity changes.

  49. Bias correction • Inhomogeneity also leads to variability in image intensity. • Bias correct anatomical scans (e.g. SPM’s segmentation, N3). • Field homogeneity issues more severe with higher field strength. • Parallel Imaging (collecting MRI with multiple coils) can dramatically reduce effects.

  50. Vectors • Vectors • Vectors have a direction and a length • The 2D vector [1,0] points East and has a length of 1 • The 2D vector [0,1] points North and has a length of 1 • The 2D vector [1,2] points North-East and has a length of 2.23 • 2D: two values [x,y], 3D: three values [x,y,z] 1, 2 0, 1 1, 0