Mathematical Methods for the Registration of Medical Images, Part II Carsten Wolters Institut für Biomagnetismus und Biosignalanalyse, Westfälische Wilhelms-Universität

Mathematical Methods for the Registration of Medical Images, Part II

Mathematical Methods for the Registration of Medical Images, Part II

Carsten WoltersCarsten WoltersCarsten WoltersCarsten Wolters

Institut für Biomagnetismus und Biosignalanalyse, Westfälische Wilhelms-Universität MünsterInstitut für Biomagnetismus und Biosignalanalyse, Westfälische Wilhelms-Universität Münster

Lecture, Dec.3, 2013Lecture, Dec.3, 2013

StructureStructure

• Parametric image registration techniques

• Non-parametric image registration techniques

• Non-parametric registration for DTI




StructureStructure

• Parametric image registration techniques (see Vorlesung 26.Nov.2013)



• Parametric image registration techniques (see Vorlesung 26.Nov.2013)



StructureStructure







General framework General framework [Modersitzki, Numerical Methods for Image Registration, Oxford University Press, 2004]

General framework: Gateau differentiabilityGeneral framework: Gateau differentiability

General frameworkGeneral framework[Modersitzki, Numerical Methods for Image Registration, Oxford University Press, 2004]

General frameworkGeneral framework

General framework: Euler-Lagrange General framework: Euler-Lagrange equationsequations

General framework: DiscretizationGeneral framework: Discretization

General framework: The algorithm for General framework: The algorithm for non-parametric registrationnon-parametric registration

StructureStructure

• Non-parametric image registration techniquesNon-parametric image registration techniques

– Elastic registrationElastic registration

– Fluid registrationFluid registration

– Diffusion registrationDiffusion registration

Elastic registrationElastic registration[Modersitzki, Numerical Methods for Image Registration, Oxford University Press, 2004]

Elastic registrationElastic registration

Elastic registrationElastic registrationElastic registrationElastic registration

• The arising PDE is called the Navier-Lamé equation.The arising PDE is called the Navier-Lamé equation.

• The proof of the Theorem shows that implicit boundary The proof of the Theorem shows that implicit boundary conditions are needed. conditions are needed.

• In practice: explicit periodic boundary conditions are often used, In practice: explicit periodic boundary conditions are often used, since they allow the computation of eigenfunctions and since they allow the computation of eigenfunctions and eigenvalues of the Navier-Lame operator and thus the eigenvalues of the Navier-Lame operator and thus the development of an O(N log N) direct FFT-solver of the discrete development of an O(N log N) direct FFT-solver of the discrete system.system.

• Elastic registration needs affine pre-registration since it Elastic registration needs affine pre-registration since it penalizes linear deformations u(x)=Cx+dpenalizes linear deformations u(x)=Cx+d

Elastic registrationElastic registration

A (n1=5, n2=7) for periodic boundary conditions

[Modersitzki, Numerical Mathematics and Scientific Computation, 2004]

Elastic registration: An exampleElastic registration: An example

Result for periodic boundary conditions=0, =500

Reference image

Template image



Elastic registration: An exampleElastic registration: An example

StructureStructure





Fluid registrationFluid registration[Modersitzki, Numerical Methods for Image Registration, Oxford University Press, 2004]

Euler and Lagrange coordinatesEuler and Lagrange coordinates

Fluid registrationFluid registration[Modersitzki, Numerical Methods for Image Registration, Oxford University Press, 2004]

Fluid registrationFluid registration

• For Stokes fluids with very slow motion flow, we get the PDE for For Stokes fluids with very slow motion flow, we get the PDE for

fluid registration:fluid registration:

Fluid registrationFluid registrationFluid registrationFluid registration

• The only difference to the algorithm of elastic registration is the The only difference to the algorithm of elastic registration is the

additional Euler step to compute the deformation from the additional Euler step to compute the deformation from the

velocity following velocity following

• The Euler step can be implemented using a centered finite The Euler step can be implemented using a centered finite difference approximation for and a forward FD approx. difference approximation for and a forward FD approx. forfor

Result for periodic boundary conditions. =0, =5000.


Fluid registration: An exampleFluid registration: An example

StructureStructure





Diffusion registrationDiffusion registration




Reference image

Template image



Diffusion registration: An exampleDiffusion registration: An example

Result for =1, =0.2.


Velocity-based diffusion registration: An Velocity-based diffusion registration: An exampleexample

StructureStructure







EPI susceptibility artifact: An exampleEPI susceptibility artifact: An exampleEPI susceptibility artifact: An exampleEPI susceptibility artifact: An example

OutlineOutline

• IntroductionIntroduction

• MethodMethod

• ResultsResults

• DiscussionDiscussion


• MethodMethod

• ResultsResults


OutlineOutline


• MethodMethod

• ResultsResults



• MethodMethod

• ResultsResults


Introduction: Reason for field distortionsIntroduction: Reason for field distortions• Fast (whole brain images in seconds) acquisition Fast (whole brain images in seconds) acquisition

scheme of echo-planar imaging (EPI) is most frequently scheme of echo-planar imaging (EPI) is most frequently

used for diffusion-weighted imaging (DWI) and used for diffusion-weighted imaging (DWI) and

diffusion-tensor imaging (DTI) diffusion-tensor imaging (DTI) (Stehling et al., 1991) (Stehling et al., 1991) • Low acquisition bandwidth in phase encoding Low acquisition bandwidth in phase encoding

gradient direction causes high sensitivity against small gradient direction causes high sensitivity against small

perturbations of the magnetic field perturbations of the magnetic field (Chang & (Chang &

Fitzpatrick,1992)Fitzpatrick,1992)• Field inhomogeneities are caused by susceptibility Field inhomogeneities are caused by susceptibility

differences and they scale with the Bdifferences and they scale with the B00 field strength field strength

(Jezzard & Clare, 1999)(Jezzard & Clare, 1999)

Introduction: Correction approachesIntroduction: Correction approaches• Field map approachesField map approaches (Jezzard & Balaban, 1995)(Jezzard & Balaban, 1995) suffer suffer

from long acquisition time and subject motion as well as from long acquisition time and subject motion as well as

regularization to overcome problems near tissue edges and regularization to overcome problems near tissue edges and

regions with large inhomogeneities regions with large inhomogeneities (Holland et al., 2010) (Holland et al., 2010) • Point Spread FunctionPoint Spread Function strategy strategy (Robson et al., 1997)(Robson et al., 1997) also also

takes several minutes for a full brain takes several minutes for a full brain (Holland et al., 2010) (Holland et al., 2010) • Direct registrationDirect registration of EPI to anatomical images of EPI to anatomical images (Merhof et (Merhof et

al., 2007; Tao et al., 2009)al., 2007; Tao et al., 2009)• Reversed gradient approachReversed gradient approach (Chang & Fitzpatrick, 1992; (Chang & Fitzpatrick, 1992;

Morgan et al., 2004) Morgan et al., 2004)

Our contribution: Reversed gradient approach with Our contribution: Reversed gradient approach with

diffeomorphic transformation and thus meaningful intensity diffeomorphic transformation and thus meaningful intensity

modulations modulations (Ruthotto et al., 2012)(Ruthotto et al., 2012)

OutlineOutline


• MethodMethod

• ResultsResults



• MethodMethod

• ResultsResults


MethodMethod

• Assuming a known field inhomogeneity and the known Assuming a known field inhomogeneity and the known

direction of spatial mismatch , a forward model that relates the direction of spatial mismatch , a forward model that relates the

unobservable undistorted image to the observation unobservable undistorted image to the observation

was introduced by was introduced by (Chang & Fitzpatrick, 1992)(Chang & Fitzpatrick, 1992): :

• Requirement Requirement (Chang & Fitzpatrick, 1992): (Chang & Fitzpatrick, 1992): Measurement parameters have to Measurement parameters have to

be chosen such that the intensity modulation remains positive, be chosen such that the intensity modulation remains positive,

corresponding to the invertibility of the mapping:corresponding to the invertibility of the mapping:

MethodMethodMethodMethod

Method: Functional to be minimizedMethod: Functional to be minimizedMethod: Functional to be minimizedMethod: Functional to be minimized

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(Holland et al., 2010) (Holland et al., 2010)

(Ruthotto et al., 2012) (Ruthotto et al., 2012)


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Method: Implementation and discretizationMethod: Implementation and discretization

• Follow guidelines of Follow guidelines of (Modersitzki, 2009)(Modersitzki, 2009)• our model is implemented in SPM and as an extension to our model is implemented in SPM and as an extension to

the freely available FAIR toolbox in Matlabthe freely available FAIR toolbox in Matlab (Modersitzki, (Modersitzki,

2009)2009)• Use a discretize-then-optimize approach on a hierarchy of Use a discretize-then-optimize approach on a hierarchy of

levels to improve stability against local minima and give levels to improve stability against local minima and give

speed speed (Modersitzki, 2009)(Modersitzki, 2009)• For computationally expensive routines such as image For computationally expensive routines such as image

interpolation and regularization, parallelized C-code in a interpolation and regularization, parallelized C-code in a

matrix free fashion is used matrix free fashion is used (Modersitzki, 2009)(Modersitzki, 2009)• Memory consumption is kept to a minimum and method Memory consumption is kept to a minimum and method

runs on standard computer runs on standard computer

Method: MR measurementsMethod: MR measurements• DT-MRI measured using Stejskal-Tanner spin-echo EPIDT-MRI measured using Stejskal-Tanner spin-echo EPI• Voxels of 1.875mm x 1.875mm x 3.6mmVoxels of 1.875mm x 1.875mm x 3.6mm• Contrast parameters: TR=9473, TE=95msContrast parameters: TR=9473, TE=95ms• 20 direct. images, equally distrib. on sphere 20 direct. images, equally distrib. on sphere (Jones, 2004)(Jones, 2004) • Bandwidth in phase-encoding direction - selected as Bandwidth in phase-encoding direction - selected as

anterior-posterior- was 9.1 Hz pixelanterior-posterior- was 9.1 Hz pixel-1-1, in frequency-encoding , in frequency-encoding

direction, it was 1675 Hz pixeldirection, it was 1675 Hz pixel-1-1

• With one exception (difference 14%), the difference of mass With one exception (difference 14%), the difference of mass

(i.e. integral over I(i.e. integral over I11-I-I22) was less than 1.5%) was less than 1.5%

• Also only the image pair with “flat” diffusion gradient Also only the image pair with “flat” diffusion gradient

(b=0 s mm(b=0 s mm-2-2) is required for our correction, we acquired two ) is required for our correction, we acquired two

full data sets with reversed gradients to investigate impact of full data sets with reversed gradients to investigate impact of

correction on Fractional Anisotropy (FA)correction on Fractional Anisotropy (FA)

OutlineOutline


• MethodMethod

• ResultsResults



• MethodMethod

• ResultsResults


ResultsResultsResultsResults







OutlineOutline


• TheoryTheory

• ResultsResults



• TheoryTheory

• ResultsResults


DiscussionDiscussion

• Necessary additional amount of measurement time: <1minNecessary additional amount of measurement time: <1min• Compared to Compared to (Holland et al., 2010)(Holland et al., 2010), a key novelty of our, a key novelty of our

reversed gradient approachreversed gradient approach is the additional nonlinear is the additional nonlinear

regularizer that guarantees positive intensity modulations regularizer that guarantees positive intensity modulations

and diffeomorphic geometrical transformations independent and diffeomorphic geometrical transformations independent

of the actual choice of regularization parametersof the actual choice of regularization parameters• Not very sensitive to choice of regularization parameters, Not very sensitive to choice of regularization parameters,

our choice was fine for all 6 of our investigated DTI datasetsour choice was fine for all 6 of our investigated DTI datasets• Effective reduction of distortions and further reduction Effective reduction of distortions and further reduction

possible when combining our approach with parallel imaging possible when combining our approach with parallel imaging • When compared to When compared to (Chang & Fitzpatrick, 1992; Morgan et (Chang & Fitzpatrick, 1992; Morgan et

al., 2004; Weiskopf et al., 2005)al., 2004; Weiskopf et al., 2005), our approach doesn’t need , our approach doesn’t need

any pre-segmentation and edge detectionany pre-segmentation and edge detection

DiscussionDiscussion

• When compared to When compared to (Andersson et al., 2003; Skare & (Andersson et al., 2003; Skare &

Andersson, 2005)Andersson, 2005), computational expenses were lowered , computational expenses were lowered

due to non-parametric transformation, high-end optimization due to non-parametric transformation, high-end optimization

and multi-level techniquesand multi-level techniques• Approach is not limited to EPI, as the underlying physical Approach is not limited to EPI, as the underlying physical

distortion model was first developed for any inhomogeneity-distortion model was first developed for any inhomogeneity-

induced artifacts in MR induced artifacts in MR (Chang & Fitzpatrick, 1992)(Chang & Fitzpatrick, 1992)• The assumption of no signal loss (mass-preservation) is The assumption of no signal loss (mass-preservation) is

well satisfied by spin-echo, but it is only approximately well satisfied by spin-echo, but it is only approximately

fulfilled by gradient echo schemes commonly used in fMRI, fulfilled by gradient echo schemes commonly used in fMRI,

but first fMRI results suggest that our method is also but first fMRI results suggest that our method is also

valuable for fMRIvaluable for fMRI

Thank you for your attention!Thank you for your attention!

Methods group at IBB MünsterMethods group at IBB MünsterMethods group at IBB MünsterMethods group at IBB Münster

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Mathematical Methods for the Registration of Medical Images, Part II Carsten Wolters Institut für Biomagnetismus und Biosignalanalyse, Westfälische Wilhelms-Universität