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Level set methods for imaging and application to MRI segmentation

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## Level set methods for imaging and application to MRI segmentation

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### Level set methods for imaging and application to MRI segmentation

MRI SegmentationAcknowledgements

- Based on results by Denis Neiter (Ecole Polytechnique) during internship 2007, partly using results by Simon Huffeteau (Ecole Polytechnique), internship 2006
- Using Carsten Wolters‘ MR Data

MRI SegmentationProblem Setting

- Given MR Image(s), find (in an automated way):
- the borders between different head compartments (segmentation)
- an appropriate map of the normal directions, in particular of the brain surface (classification)- a representation useful for further finite element modelling

MRI SegmentationMathematical Issues

- Segmentation needs
- to discriminate noise and textures (small scale structures)
- to incorporate prior knowledge
- - to be flexible with respect to complicated shapes (or even topology)
- First two issues treated via regularization, third via level set methods

MRI SegmentationObject-Based Segmentation

- Classical object based segmentation computes curves (2D) or surfaces (3D) marking the object boundary (contour)
- Traditional approach: start with curve and let it evolve towards the contour by some criteria
- Velocity of evolving curve determined by two counteracting parts: image-driven part and regularization

MRI SegmentationActive Contours - Snakes

- Image-driven force related to gradient of the image (local gray-value difference)
- Regularization force is (mean) curvature

MRI SegmentationActive Contours - Snakes

- Popular, but various shortcomings:
- needs preprocessing of the image (noise removal, intensity map so that edges are in valleys)
- local minima
- issues with narrow structures: big trouble in brain images

MRI SegmentationStatistical Models

- K-Means / C-Means:
- Based on optimization, find a 0-1 function (0 in pixels outside, 1 inside)
- Optimization goal consists of same parts: image-driven and regularization
- No useful boundary representation

MRI SegmentationCurve / Surface Representation

- Level Set Methods yield boundary representation with appropriate curvatures and subpixel resolution

MRI SegmentationLevel Set Methods

- Osher & Sethian, JCP 1987, Sethian, Cambridge Univ. Press 1999, Osher & Fedkiw, Springer, 2002
- Basic idea: implicit shape representation
- with continuous level-set function

MRI SegmentationLevel Set Methods

- Implicit representation of dynamic shapes
- with time-dependent level set function

MRI SegmentationLevel Set Methods

- Evolution of the shape corresponds to evolution of the level set function (and vice versa)
- Movie by
- J.Sethian

MRI SegmentationGeometric Motion

- Start for simplicity with the evolution of a curve
- Evolution in a velocity field , each point evolves via ODE

MRI SegmentationGeometric Motion

- Use any parametric representation
- Due to definition of the level set function
- Consequently

MRI SegmentationGeometric Motion

- For level set function being a solution of
- each level set of f is moving with velocity V

MRI SegmentationGeometric Motion

- In most cases, the full velocity field V is unknown, only normal velocity component v known
- Tangential component of the velocity field is not important anyway, it does not change the motion (only change of parametrization)

MRI SegmentationGeometric Motion

- Note that is a tangential direction
- Hence, is a normal direction, unit normal is given by

MRI SegmentationGeometric Motion

- Evolution becomes nonlinear Hamilton-Jacobi equation:
- „Level set equation“

MRI SegmentationGeometric Motion

- Evolution could be anisotropic, i.e. normal velocity depends on the orientation
- with one-homogeneous extension H , yields Hamilton-Jacobi equation

MRI SegmentationGeometric Motion

- Evolution could be of higher order, e.g. normal velocity depends on the mean curvature
- Level set equation becomes fully nonlinear second-order parabolic PDE

MRI SegmentationExamples

- Eikonal equation
- Positive velocity field yield monotone advancement of fronts
- Arrival time
- Solves

MRI SegmentationExamples

- Mean curvature flow
- Classical example of higher-order geometric motion
- Normal velocity equal to curvature of curve (or mean curvature of surface)

MRI SegmentationOptimal Geometries

- Classical problem for optimal geometry:
- Plateau Problem (Minimal Surface Problem)
- Minimize area of surface between fixed boundary curves.

MRI SegmentationOptimal Geometries

- Wulff-Shapes: Pb[111] in Cu[111]
- Surnev et al, J.Vacuum Sci. Tech. A, 1998

MRI SegmentationMumford-Shah

- Free discontinuity problems:
- find the set of discontinuity from a noisy observation of a function.
- Mumford-Shah functional

MRI SegmentationObject-based Mumford-Shah

- Chan-Vese:
- Approximate smooth component by its mean value inside and outside object
- Curve / Surface can be evolved via simple criterion, in each time step mean-value inside and outside need to be computed
- Via convex relaxation techniques convergence to global minimum can be ensured

MRI SegmentationLevel Set Formulation

- Level set function y and
- Heaviside function H (= 0 negative, = 1 positive)

MRI SegmentationRegularization

- For skull segmentation (smooth) regularization based on length minimization is perfect
- For brain structure (sulci) similar issues as for active contours

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MRI Segmentation

Adaptive Bias / ParametrizationBias of one functional often too strong

Better: use a family of functionals parametrized by

Example: adaptive anisotropy

MRI SegmentationAdaptive Anisotropy

In aerial images the typical anisotropy is rectangular, houses have 90° angles

But not all of them have the same orientation

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MRI Segmentation

Adaptive AnisotropyBias for edges with 90° angles from functional of the form

Ra is rotation matrix for angle a to capture the orientation

Since orientation is not constant over the image, a has to vary and to be found adaptively by minimization

MRI SegmentationAdaptive Anisotropy

To avoid microstructure, variation of a has to be regularized, too

Possible regularization functional

MRI SegmentationAdaptive Anisotropy

Contrast correction by iterative refinement Angle parameter provides classification of orientations in the image

MRI SegmentationAdaptive Anisotropy

Cartoon reconstruction and orientational classification of aerial images

Berkels, mb, Droske, Nemitz, Rumpf 06

MRI SegmentationAdaptive Anisotropy

Analogous problem in segmentation of MRI brain images for EEG/MEG

Adapt anisotropy (locally like sharp ellipse) to find sulci accurately and provide classification of normals (for dipole fitting, source reconstruction)

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