Math in image processing

Math in image processingMath in image processing


Nyquist theoremNyquist theorem


Discrete Fourier TransformationDiscrete Fourier Transformation

Math in image processingMath in image processingImage enhancement: scalingImage enhancement: scaling

Math in image processingMath in image processingImage enhancement: histogram equalizationImage enhancement: histogram equalization

cumulative histogramcumulative histogram improved imageimproved image

Math in image processingMath in image processingImage enhancement: filtering (low- or high-pass)Image enhancement: filtering (low- or high-pass)

Reducing the amplitudes ofReducing the amplitudes oflow-freq peak we can avoid somelow-freq peak we can avoid someof the artefacts.of the artefacts.

Math in image processing: Math in image processing: segmentationsegmentation

An idea is to find the clusters orAn idea is to find the clusters orsubsets of the image which can besubsets of the image which can beconsidered (or its characteristics) considered (or its characteristics) as homogeneous.as homogeneous.

skullskull

CSF (cerebrospinal fluid)CSF (cerebrospinal fluid)

White matter or Grey matterWhite matter or Grey matter


First and simple way to do itFirst and simple way to do itmanually (frequently is applied, for manually (frequently is applied, for example, in the case of tumour example, in the case of tumour segmentation).segmentation).


ThresholdingThresholding


Edge-based segmentationEdge-based segmentation

GradientGradient

DirectionDirection

MR

AM

RA

Ed

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Ed

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ed

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Low

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FSL: BETFSL: BET

Fslview allows one to visualise the results in three projectionsFslview allows one to visualise the results in three projections


FSL: BETFSL: BET

Artefacts from bad parametrisationArtefacts from bad parametrisation


FSL: BETFSL: BET

Extracted/rendered brainExtracted/rendered brain Extracted/rendered skullExtracted/rendered skull


FSL: FASTFSL: FAST

Rendered T1 raw dataRendered T1 raw data

WMWM

GMGM

CSFCSF


FreeSurferFreeSurfer

How to segment properly the GMHow to segment properly the GM


FreeSurferFreeSurfer

Convert it into computer modelConvert it into computer model

Math in image processing: registrationMath in image processing: registration

Cerebral cortex can be segmented in specialCerebral cortex can be segmented in specialregions (Broadmann, 1909).regions (Broadmann, 1909).

The question: how can we compare the sameThe question: how can we compare the sameregions for different subjects?regions for different subjects?

Math in image processing: registrationMath in image processing: registrationIdea is to find a transformation T whichIdea is to find a transformation T whichallows one to align two images. One image is fixed, when another is moving.allows one to align two images. One image is fixed, when another is moving.

where S is a similarity, P is a penaltywhere S is a similarity, P is a penalty

Different variants of the similarity functions:Different variants of the similarity functions:

Sum of squared differencesSum of squared differences

Mutual informationMutual information


What kind of geometrical transformation we can use?What kind of geometrical transformation we can use?

Rotation (rigid body transformation)Rotation (rigid body transformation)



Rotation Rotation Translation (rigid body transformation)Translation (rigid body transformation)



Rotation Rotation Translation Translation

Scaling (nonrigid)Scaling (nonrigid)


Simple 2D caseSimple 2D case

The same case but generalized to 3DThe same case but generalized to 3D

Math in image processing: registrationMath in image processing: registrationAffine transformation:Affine transformation: wiki page examplewiki page example

Math in image processing: registrationMath in image processing: registrationNon-linear transformationsNon-linear transformations

Math in image processing: registrationMath in image processing: registrationNon-linear transformations: often before it we do affine transformationNon-linear transformations: often before it we do affine transformation

Warp functionsWarp functions

This transformation is reversible!This transformation is reversible!

Math in image processing: fittingMath in image processing: fitting

XX

YY

We need to fit the measured dataWe need to fit the measured datato model functionto model function


XX

YY

We need to fit the measured dataWe need to fit the measured datato model functionto model functionIf function is linear it's more or If function is linear it's more or less easy to do: y = ax + bless easy to do: y = ax + b


XX

YY

We need to fit the measured dataWe need to fit the measured datato model functionto model functionIf function is linear it's more or If function is linear it's more or less easy to do: y = ax + bless easy to do: y = ax + b

OutlierOutlier

Math in image processing: fittingMath in image processing: fittingRobust estimators in regression methodsRobust estimators in regression methods

ProblemsProblems

1.1. Install FSL (http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/) and try different Install FSL (http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/) and try different utilities from the package.utilities from the package.

2.2. Install SPM (http://www.fil.ion.ucl.ac.uk/spm/) and try different Install SPM (http://www.fil.ion.ucl.ac.uk/spm/) and try different utilities from the package.utilities from the package.

3.3. Install FreeSurfer (http://freesurfer.net/) and try different Install FreeSurfer (http://freesurfer.net/) and try different utilities form the package.utilities form the package.

4.4. Install ITK-SNAP, try to segment the images from the given examples.Install ITK-SNAP, try to segment the images from the given examples.5.5. Extract the brain using BET utility with minimal artefactsExtract the brain using BET utility with minimal artefacts6.6. Extract WM, GM, and CSF tissues using FAST with minimal artefactsExtract WM, GM, and CSF tissues using FAST with minimal artefacts7.7. Are the unit transformation in registration procedure commutative?Are the unit transformation in registration procedure commutative?8.8. Perform a coregistration of 4D volumes of diffusion dataset Perform a coregistration of 4D volumes of diffusion dataset

LiteratureLiterature

Bankman, Handbook of medical imaging. Processing and analysisBankman, Handbook of medical imaging. Processing and analysisSmith, Digital signal processingSmith, Digital signal processingSonka and Fitzpattrick, Handbook of medical imaging, vol.2. Medical imageSonka and Fitzpattrick, Handbook of medical imaging, vol.2. Medical imageprocessing and analysisprocessing and analysis

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