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Level set methods for imaging and application to MRI segmentation Acknowledgements 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 Problem Setting

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Level set methods for imaging and application to MRI segmentation


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acknowledgements
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
problem setting
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
mathematical issues
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
object based segmentation
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
active contours snakes
MRI SegmentationActive Contours - Snakes
  • Image-driven force related to gradient of the image (local gray-value difference)
  • Regularization force is (mean) curvature
active contours snakes7
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
statistical models
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
curve surface representation
MRI SegmentationCurve / Surface Representation
  • Level Set Methods yield boundary representation with appropriate curvatures and subpixel resolution
level set methods
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
level set methods11
MRI SegmentationLevel Set Methods
  • Change of front translated to change of function
level set methods12
MRI SegmentationLevel Set Methods
  • Implicit representation of dynamic shapes
  • with time-dependent level set function
level set methods13
MRI SegmentationLevel Set Methods
  • Evolution of the shape corresponds to evolution of the level set function (and vice versa)
  • Movie by
  • J.Sethian
level set methods14
MRI SegmentationLevel Set Methods
  • Topology change is automatic
  • Movie by
  • J.Sethian
geometric motion
MRI SegmentationGeometric Motion
  • Start for simplicity with the evolution of a curve
  • Evolution in a velocity field , each point evolves via ODE
geometric motion16
MRI SegmentationGeometric Motion
  • Use any parametric representation
  • Due to definition of the level set function
  • Consequently
geometric motion17
MRI SegmentationGeometric Motion
  • By the chain rule
  • Insert ODE for moving points:
geometric motion18
MRI SegmentationGeometric Motion
  • For level set function being a solution of
  • each level set of f is moving with velocity V
geometric motion19
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)
geometric motion20
MRI SegmentationGeometric Motion
  • Normal can be computed from level set function:
  • By the chain rule
geometric motion21
MRI SegmentationGeometric Motion
  • Note that is a tangential direction
  • Hence, is a normal direction, unit normal is given by
geometric motion22
MRI SegmentationGeometric Motion
  • Evolution becomes nonlinear Hamilton-Jacobi equation:
  • „Level set equation“
geometric motion23
MRI SegmentationGeometric Motion
  • Evolution could be anisotropic, i.e. normal velocity depends on the orientation
  • with one-homogeneous extension H , yields Hamilton-Jacobi equation
geometric motion24
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
examples
MRI SegmentationExamples
  • Eikonal equation
  • Positive velocity field yield monotone advancement of fronts
  • Arrival time
  • Solves
examples27
MRI SegmentationExamples
  • Mean curvature flow
  • Classical example of higher-order geometric motion
  • Normal velocity equal to curvature of curve (or mean curvature of surface)
optimal geometries
MRI SegmentationOptimal Geometries
  • Classical problem for optimal geometry:
  • Plateau Problem (Minimal Surface Problem)
  • Minimize area of surface between fixed boundary curves.
optimal geometries30
MRI SegmentationOptimal Geometries
  • Minimal surface (L.T.Cheng, PhD 2002)
optimal geometries31
MRI SegmentationOptimal Geometries
  • Wulff-Shapes: Pb[111] in Cu[111]
  • Surnev et al, J.Vacuum Sci. Tech. A, 1998
mumford shah
MRI SegmentationMumford-Shah
  • Free discontinuity problems:
  • find the set of discontinuity from a noisy observation of a function.
  • Mumford-Shah functional
mumford shah33
MRI SegmentationMumford-Shah
  • Image decomposition
improved model
MRI SegmentationImproved Model
  • Decomposition in
  • 3 parts: smooth, oscillating, edges
object based mumford shah
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
level set formulation
MRI SegmentationLevel Set Formulation
  • Level set function y and
  • Heaviside function H (= 0 negative, = 1 positive)
regularization
MRI SegmentationRegularization
  • For skull segmentation (smooth) regularization based on length minimization is perfect
  • For brain structure (sulci) similar issues as for active contours
adaptive bias parametrization

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MRI Segmentation

Adaptive Bias / Parametrization

Bias of one functional often too strong

Better: use a family of functionals parametrized by

Example: adaptive anisotropy

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

adaptive anisotropy48

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MRI Segmentation

Adaptive Anisotropy

Bias 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

adaptive anisotropy49
MRI SegmentationAdaptive Anisotropy

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

Possible regularization functional

adaptive anisotropy50
MRI SegmentationAdaptive Anisotropy

Improves angles, still loses contrast

adaptive anisotropy51
MRI SegmentationAdaptive Anisotropy

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

adaptive anisotropy52
MRI SegmentationAdaptive Anisotropy

Cartoon reconstruction and orientational classification of aerial images

Berkels, mb, Droske, Nemitz, Rumpf 06

adaptive anisotropy53
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)