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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

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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.

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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.

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Within-subject registration

Motion correction Coregistration

Registration of the fMRI scans (across time)

With-in subject registrations– Assumption: same individual, so there should be a good linear solution.

Registration of fMRI scans with high resolution image.

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Rigid Body Transforms

Translation Rotation

By measuring and correcting for translations and rotations, we can adjust for an object’s movement in an image.

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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.

YawPitch

Roll

ZX

Y

Translation Rotation

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Motion Correction

Motion correction aligns all in time series.

Translations and rotations only– ‘rigid body registration’– Assumes brain size

and shape identical across images.

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Motion Correction Cost Function

=

ResliceTarget

When aligned, Difference squared ~ 0

2

ResliceTarget

=

2 When unaligned, Difference squared > 0

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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 Image 2 Difference Difference²

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Local Minima

Search algorithm is iterative:

1. move the image a little bit.

2. Test cost function

3. 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

Valu

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f C

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Local Minimum

Global Minimum

Translation in X

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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.

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Affine Transforms (aka linear, geometric)

Translation Rotation Zoom Shear

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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.

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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

T1 image Coregistered FLAIR

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Between-subject: Normalization

Allows inference about general population

Subject 2

Subject 1

TemplateAverage activation

Normalization

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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.

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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.

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Coordinates - normalization

Different people’s brains look different ‘Normalizing’ adjusts overall size and orientation

Raw Images Normalized Images

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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

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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.

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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

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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.

PC

AC

Y-Y+

Z+

Z-

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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).

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Affine Transforms

Co-linear points remain co-linear after any affine transform.

Transform influences entire image.

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Spatial Processing

Non-linear transforms can match features that could not aligned with affine transforms.

SPM uses basis functions.

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Nonlinear functions and normalization

Linear Only

http://imaging.mrc-cbu.cam.ac.uk/imaging/SpmMiniCourse

Linear + NonlinearScans from 6 people

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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

www.fil.ion.ucl.ac.uk/spm/course/www.pubmed.com/19195496/

Affine template

DARTELtemplate

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Regularization

Regularization penalizes bending energyWhat is the best way to show graph points with

a smooth line? Heavy regularization is a poor fit, heavy regularization causes local distortion

Medium RegularizationHeavy Regularization Little Regularization

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Regularization

Regularization is a parameter that you can adjust that influences non-linear normalization

http://www.fmri.ox.ac.uk/fsl/fnirt/

Medium Regularization Little Regularization

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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.

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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.

www.loni.ucla.edu/~thompson/

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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)

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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.

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Interpolation

Each lower image rotated 12º.

Left looks jagged, right looks smooth.

Different reslicing interpolation.

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1D Interpolation

How do we estimate values that occur between discrete samples?

Three popular methods:1. Nearest neighbor (box)

2. Linear (tent)

3. Spline/Sinc

Weather analogy: if it was 25º at 9am, and 31º at 12am, what would you estimate the temperature was at 10am?

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Interpolation

How do we estimate values that occur between discrete samples?

Three popular methods:1. Nearest neighbor (box)

2. Linear (tent)

3. Spline/Sinc

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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

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Linear Interpolation – High Frequency Loss

Original

90o

180o

360o10o

20o

•Linear interpolation loses high frequencies

•Multiple successive resampling will lead to blurry image

•Solution: Minimize number of times the data is resliced.

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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.

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2D Sinc Function 1D Sinc Function

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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

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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

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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.

2= 1

2=2

2=3

2=4f(x)

Dispersion Differs

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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

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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

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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

Gau

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nM

edia

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Gaussian Noise Gaussian Smooth

Spike Noise

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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.

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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

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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.

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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.

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Vectors

1, 0

Vectors– Vectors have a direction and a length

The 2D vector [1,0] points East and has a length of 1The 2D vector [0,1] points North and has a length of 1The 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]

0, 1

1, 2

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2D Rotation matrix

1, 0

0, 1

2D rotation matrix as two vectors (horizontal and vertical)– Can be described by four numbers

1, 0

0, 2.7,-.7

.7, .7

Original Zoom Rotation

1, 0

1, 1

Shear

-1, 0

0, 1

Flip

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2D Transformation matrix

1, 0, 0

0, 1, 0

0, 0, 1

Transformation matrix is a rotation matrix plus translation values (encodes x,y origin of vectors)– 9 values, final row is always 0 0 1 (must have as many

rows as columns).Original

1, 0, 1

0, 1, 0

0, 0, 1

2, 0, .2

0, 1, .2

0, 0, 1

Translated+ZoomTranslated

All affine transforms can be combined into a single matrix

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3D Matrices are 4x4

We can generate 4x4 matrices that will allow us to work with 3D images.

A 4x4 matrix, but the last row always ‘0 0 0 1’fx = (x*i)+(y*j)+(z*k)+l

fy = (x*m)+(y*n)+(z*o)+p

fz = (x*q)+(y*r)+(z*s)+t

i j km n o

lp

q r s t

0 0 0 1

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Matrices and 3D space

3D matrices work just like 2D matrices– The identity matrix still has 1’s along the diagonal– Translations are the values in the final column (constants)– Zooms are done by scaling values of a row.– Shears are values added to the relevant orthogonal value– Rotations use sine/cosine in dimensions of plane.– Our twelve numbers can store all of the possible rotations,

shears, translations and scaling. Simply multiply previous matrix with our transform (order crucial).