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

Mathematical Methods for the Segmentation of Medical ImagesMathematical Methods for the

Segmentation of Medical Images

Carsten WoltersCarsten WoltersCarsten WoltersCarsten Wolters

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

Lecture, December 10, 2013Lecture, December 10, 2013

StructureStructure

• Fuzzy segmentation techniques

• Levelset segmentation techniques


• Levelset segmentation techniques

Unsupervised clusteringUnsupervised clusteringUnsupervised clusteringUnsupervised clustering

• A clustering method is called A clustering method is called unsupervisedunsupervised, if it automatically , if it automatically

determines the intensity ranges for a user-given number of determines the intensity ranges for a user-given number of

clustersclusters

[G. Lohmann, Volumetric Image Analysis, John Wiley & Sons, 1998][C.Wolters, Vorlesungsskriptum, 2013]

• ClusteringClustering methods classify voxels by simple intensity range methods classify voxels by simple intensity range

partitioningpartitioning

• The The ISODATAISODATA algorithm is a representative of unsupervised algorithm is a representative of unsupervised

clustering, minimizing the following objective function:clustering, minimizing the following objective function:

Unsupervised clustering: The ISODATA algorithmUnsupervised clustering: The ISODATA algorithmUnsupervised clustering: The ISODATA algorithmUnsupervised clustering: The ISODATA algorithm

[G. Lohmann, Volumetric Image Analysis, John Wiley & Sons, 1998][C.Wolters, Vorlesungsskriptum, 2013]

Fuzzy C-means segmentationFuzzy C-means segmentationFuzzy C-means segmentationFuzzy C-means segmentation

• FCM minimizes the following objective function:FCM minimizes the following objective function:

• Fuzzy C-Means (FCM) Fuzzy C-Means (FCM) segmentation algorithms do not force segmentation algorithms do not force

a voxel to belong to exclusively one class, but assign a a voxel to belong to exclusively one class, but assign a

membership value withmembership value with

[J.K. Udupa & S. Samarasekera, Graph.Models and Image Processing, 58, 1999][C.Wolters, Vorlesungsskriptum, 2013]

• FCM thus takes partial volume averaging effects into account.FCM thus takes partial volume averaging effects into account.

Adaptive Fuzzy C-means segmentationAdaptive Fuzzy C-means segmentationAdaptive Fuzzy C-means segmentationAdaptive Fuzzy C-means segmentation

• Inhomogeneities are well modeled by the product of the original image with a Inhomogeneities are well modeled by the product of the original image with a

smoothly varying multiplier fieldsmoothly varying multiplier field (Dawant et al., (Dawant et al., IEEE Trans.Med.Imag.IEEE Trans.Med.Imag., 1993, 1993))

• The following factors cause intensity inhomogeneities in MRI:The following factors cause intensity inhomogeneities in MRI:

– Radio frequency excitation field inhomogeneityRadio frequency excitation field inhomogeneity (McVeigh et al., (McVeigh et al., Med.Phys., 1986)Med.Phys., 1986)

– eddy current driven field gradientseddy current driven field gradients (Simmons et al., (Simmons et al., Magn.Reson.Med., 1994)Magn.Reson.Med., 1994)

– RF penetration and standing wave effectsRF penetration and standing wave effects (Bottomley & Andrew, (Bottomley & Andrew, Phys.Med.Biol., 1978)Phys.Med.Biol., 1978)

[D.L. Pham & J.L. Prince, Pattern Recognition Letters, 20, 1999][C.Wolters, Vorlesungsskriptum, 2013]

• Functional of the Functional of the adaptive fuzzy C-means (AFCM)adaptive fuzzy C-means (AFCM) algorithm: algorithm:

AFCM: Derivation of the algorithm AFCM: Derivation of the algorithm (centroids c)(centroids c)

AFCM: Derivation of the algorithm AFCM: Derivation of the algorithm (centroids c)(centroids c)

AFCM: Derivation of the algorithm AFCM: Derivation of the algorithm (membership function u)(membership function u)

AFCM: Derivation of the algorithm AFCM: Derivation of the algorithm (membership function u)(membership function u)

AFCM: Derivation of the algorithm AFCM: Derivation of the algorithm (multiplier m)(multiplier m)

AFCM: Derivation of the algorithm AFCM: Derivation of the algorithm (multiplier m)(multiplier m)

[D.L. Pham & J.L. Prince, Pattern Recognition Letters, 20, 1999]

AFCM: The algorithm AFCM: The algorithm AFCM: The algorithm AFCM: The algorithm [D.L. Pham & J.L. Prince, Pattern Recognition Letters, 20, 1999]

[C.Wolters, Vorlesungsskriptum, 2013]

Example (PD-MRI): Results of ISODATA and AFCMExample (PD-MRI): Results of ISODATA and AFCMExample (PD-MRI): Results of ISODATA and AFCMExample (PD-MRI): Results of ISODATA and AFCM[C.Wolters, Vorlesungsskriptum, 2013]

Segmentation result ISODATA (C=2)





Segmentation result AFCM (C=2)



Corrected Images

(AFCM)

Corrected Images

(AFCM)






Example (T1-MRI): Results of ISODATA and AFCMExample (T1-MRI): Results of ISODATA and AFCMExample (T1-MRI): Results of ISODATA and AFCMExample (T1-MRI): Results of ISODATA and AFCM[C.Wolters, Vorlesungsskriptum, 2013]









StructureStructure


• Geometric deformable model reconstructions


• Geometric deformable model reconstructions

Geometric deformable modelsGeometric deformable modelsGeometric deformable modelsGeometric deformable models

• TheThe level set technique level set technique represents the boundary contour represents the boundary contour

of a domain implicitly as the zero of a domain implicitly as the zero

level of alevel of a level set function level set function : :

[Han, Xu and Prince, IEEE Trans.Pattern Anal.Mach.Intell., 25, 2003][Han, Pham, Tosun, Rettmann, Xu & Prince, NeuroImage, 23, 2004]

[M.Burger, Talk at IBB, May 2008]

• Geometric deformable modelsGeometric deformable models are based on the theory of are based on the theory of

front evolutionfront evolution and are implemented using the and are implemented using the level setlevel set

numerical method numerical method (Osher & Sethian, 1988; Sethian, 1999)(Osher & Sethian, 1988; Sethian, 1999)

• In a velocity field , each point evolves via the ODEIn a velocity field , each point evolves via the ODE


• By the chain rule:By the chain rule:



• For any parametric representationFor any parametric representation

and due to the definition of the level set function and due to the definition of the level set function

we findwe find

• We thus obtain:We thus obtain:


• By the chain rule:By the chain rule:



• For any parametric representationFor any parametric representation

and due to the definition of the level set function and due to the definition of the level set function

we additionally findwe additionally find

• Because is a tangential direction, has to be a radial Because is a tangential direction, has to be a radial

direction:direction:




• This finally leads toThis finally leads to

Example: Topology-preserving Geometric Deformable Example: Topology-preserving Geometric Deformable Model (TGDM)Model (TGDM)


[Han, Xu and Prince, IEEE Trans.Pattern Anal.Mach.Intell., 25, 2003]

• For the reconstruction of the gray matter/white matter boundary of the For the reconstruction of the gray matter/white matter boundary of the

human brain, a combination of an expansion/contraction speed human brain, a combination of an expansion/contraction speed RR and the and the

mean curvature mean curvature κκ was proposed for the normal velocity: was proposed for the normal velocity:

• TGDM: “Simple point” constraint that prevents topology-changeTGDM: “Simple point” constraint that prevents topology-change

• Choice of expansion/contraction speed:Choice of expansion/contraction speed:

Original contour Pass over simple point Split at non-simple point

Geometric deformable models: The TGDM algorithmGeometric deformable models: The TGDM algorithmGeometric deformable models: The TGDM algorithmGeometric deformable models: The TGDM algorithm




[Han, Pham, Tosun, Rettmann, Xu & Prince, NeuroImage, 23, 2004]

Inner cortical surface for R=1 and =-0.02

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

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