Signal Processing for Functional Brain Imaging:A few topics revisited & preprocessing pipeline

Dimitri Van De VilleMedical Image Processing Lab, EPFL/UniGEdimitri.vandeville@epfl.ch

May 21, 2015

OverviewA few old friends revisited

Pattern recognition: the kernel trickNetwork theory: modularity indexConfirmatory versus exploratory analysis

The fMRI preprocessing pipelineRealignmentCo-registrationNormalization

Examples of exam questions

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Linear classification (by hyperplane) can be limiting

Project into (even) higher-dimensional feature space V where linear

separation corresponds to non-linear separation in original space

Kernel trick

3

'

Linear classification (by hyperplane) can be limiting

Project into (even) higher-dimensional feature space V where linearseparation corresponds to non-linear separation in original space

However, this new feature space V is implicit because in practice we use gen-eralized kernel functions:

K(x1,x2) = h'(x1),'(x2)iV

where h·, ·iV should be a proper inner product, that is, K(·, ·) should satisfyMercer’s condition:

NX

i=1

NX

j=1

K(xi,xj)cicj � 0

for all finite sequences of points (x1, . . . ,xN ) and real-valued coefficients(c1, . . . , cN ). This also means that the kernel matrix [K(xi,xj)]i,j shouldbe positive semi-definite (i.e., all eigenvalues� 0)

Kernel trick

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Commonly used kernel functions K(x1,x2) = h'(x1),'(x2)i:Polynomial (homogeneous): K(x1,x2) = (x

T1 x2)

p

Polynomial (inhomogeneous): K(x1,x2) = (x

T1 x2 + 1)

p

Gaussian radial basis kernel: K(x1,x2) = exp(�� kx1 � x2k2), � > 0

Example: polynomial kernel K(x1,x2) = (xT1 x2)2

For D = 2, developing the kernel function leads to (binomial theorem)

K(x1,x2) = (x1,1x2,1 + x1,2x2,2)2

= (x1,1x2,1)2+ 2x1,1x2,1x1,2x2,2 + (x1,2x2,2)

2

=

D[x

21,1

p2x1,1x1,2 x

21,2], [x

22,1

p2x2,1x2,2 x

22,2]

E

From which we can identify ' : (x1, x2) 7! (x

21,p2x1x2, x

22)

For general D, this kernel maps to

�D+12

�=

D(D+1)2 dimensions (multinomial

theorem)

Kernel trick

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Based on concept of assortative network: network in which signifi-

cant fraction of edges run between vertices of same class

Total number of edges between vertices same class

1

2

X

n,m

An,m�sn,sm

Expected number of edges between all pairs of vertices of same type

1

2

X

n,m

knkm2M

�sn,sm

Modularity Q is the fraction (=divided by M ) of the difference between these

two quantities

Q =1

2M

X

n,m

✓An,m � knkm

2M

◆�sn,sm < 1

Can also be solved with spectral clustering algorithm

Network modularity

6[Newman, 2003, 2006]

Types of data analysis

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Confirmatory Exploratory

Does the model fit the data well? Is there anything interesting in the data?

Problem Data

Model Analysis

Results

Results depend on the model Can lead to unexpected results

Problem Data

Analysis Model

Results

• GLM• rt-FMRI

• unsupervised: ICA• supervised: classification

inter-subject

intra-subject

Pre-processing pipelineraw fMRI data

slice-timingcorrection

mean realigned

data out

smoothing

Gaussiankernel

normalization

templatebrain

normalization segmentation atlasing

!

brainatlas

tissue probabilitymaps

extraction (regression,regional averaging,...)

co-registration

structuralMRI data

co-registeredstructural

underlay

realignment& unwarp

field map

Realignment3D rigid body transformation

3 translation parameters3 rotation angles

Expressed in homogeneous coordinates:

note that the order of the operations matters!

Artefacts might “move”differently

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0

BB@

1 0 0 x

trans

0 1 0 y

trans

0 0 1 z

trans

0 0 0 1

1

CCA

| {z }translations

⇥

0

BB@

1 0 0 0

0 cos(�) sin(�) 0

0 � sin(�) cos(�) 0

0 0 0 1

1

CCA

| {z }pitch (around x-axis)

⇥

0

BB@

cos(✓) 0 sin(✓) 0

0 1 0 0

� sin(✓) 0 cos(✓) 0

0 0 0 1

1

CCA

| {z }roll (around y-axis)

⇥

0

BB@

cos(') sin(') 0 0

� sin(') cos(') 0 0

0 0 1 0

0 0 0 1

1

CCA

| {z }yaw (around z-axis)

RealignmentObjective function

Mean-squared difference (minimize)

Reference imageFirst, last, middle, mean (iteratively)

Applying transformation to image will result in ‘off-voxel’ positions

Resampling requiresinterpolation stepImage is represented incontinuous-domain by linearcombination of basis functionsand then sampled at newpositions

Gaps between slices can leadto aliasing artefacts

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2D B-splinebasis functions(degree 0 to 3)

Realignment (example)

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before realignment

joint histogram joint histogram

after realignmentreference image

source image source image

gray

scal

e va

lues

grayscale values

P(g1,g2)

Realignment (example)Applied to realignseries of fMRI volumesAlthough volumesare realigned,motion can stillleave residualsignal changes!Motion parameterscan be includedas regressors ofno interest in GLM

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Realignment: framewise displacement

13[Power et al, NeuroImage, 2010]

FDi = |�xi|+ |�yi|+ |�zi|+ |��i|+ |�✓i|+ |�'i|�xi = xi�1 � xi

Between-modality registrationDifferent contrasts!

Mean-squared differences does not make sense

Objective functionMutual information (maximize)

Co-registration

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Co-registration (example)

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before realignment

joint histogram joint histogram

after realignmentreference image(mean realigned)

source image(structural)

source image(structural)

P(g1,g2)

NormalizeInter-subject averaging

Increase sensitivity with more subjectsExtrapolate findings to the population as a whole

Standard coordinate systemTalairach & Tournoux space

aim to define standardized coordinates for neurosurgery,based on single post-mortem dissection of a human brainJ. Talairach, G. Szikla "Atlas of stereotactic concepts to the surgery of epilepsy", 1967 (second edition in 1988 with Tournoux)

MNI spaceMNI305: based on 305 normal MRI scans [Collins, Evans,...]ICBM152: based on 152 normal MRI scans, matched to MNI305Colin27: “canonical” image, based on 27 scans of same subject, matched to MNI305

3D rigid body transformation is too limited to match different brains

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Jean Talairach(1911-2007)

MNI space

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MNI305 ICBM152

Colin27

[http://www.bic.mni.mcgill.ca/ServicesAtlases/HomePage]

NormalizeAffine transform

3 translations3 rotations3 zooms3 shears

Fits overall shape and sizeof brainsObjective function (minimize)

Mean-squared difference between template and imageWeighting with mask of intracranial voxelsMask can be extended to exclude lesionedbrain regions

Squared distance between parameters and ‘expected’ values (regularization)

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NormalizeNon-linear transform

Linear combination ofsmooth functions(e.g., first 3D DCT basis functions)

Objective functioncombines minimum-squared differencewith regularizationagain

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NormalizeAffine versus non-linear transform

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affine non-linear

template

Normalize (example)

21

originaltemplate

non-linearwith regularization

(MSE=302.7)

non-linearwithout regularization(MSE=287.3)

affine(MSE=472.1)

Example questions

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Example question 1

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Example question 2

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Example question 3

25

Example question 3

26

Example question 4

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Example question 6

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Test-run: question 7

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Example question 1

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Example question 2

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When the threshold parameter is exceeded, the null hypothesis (=contrast

can be explained by noise) is rejected. This control specificity; i.e., the

probability that the wrong decision was taken (=false positive).

Controlling specificity (false positives) only relates to falsely declaring

voxels to be active. However, the neurosurgeon wants to make sure that

truely active voxels are detected, which relates to sensitivity (~false

negatives)

Example question 3

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1 -1

The F-test can be used for that, either with the contrast [1 -1] or [-1 1].

Example question 3

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XT XT

(XTX)-1XT

The square matrix (XTX) needs to be invertible. Since it is basically a

correlation matrix of the regressors, that means that the regressors need

to be linearly independent.

Example question 4

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With a given probability p, we can rewire the existing

connections; i.e., we take a random edge, and replace one node of

the edge with a randomly selected other node. By increasing the

probability, we get from a regular to a small-world graph, and

then into a completely random one.

Example question 6

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The large number of dimensions !

FMRI has many voxels to be considered (10’000 to 100’000) and the

number of instances to learn from is limited (10 to 1’000 at most).

Therefore, it’s a high-dimensional learning problem (curse of

dimensionality), which is difficult and prone to overfitting.

Test-run: question 7

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The mapping stimuli periodically cycle through positions in visual space.

As a result, voxels in primary visual cortices have cyclical responses that

peak at different times, depending on when the stimulus passes through

the voxels' receptive field.

In other words, all signals will have the same magnitide reponses, but

different phases, and the phase encodes the location to which they

respond.

Good luck!

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Wednesday July 1, 8h15-11h15