Martin Burger Institut für Numerische und Angewandte Mathematik CeNoS Level set methods for imaging and application to MRI segmentation

Martin Burger 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

Martin Burger MRI Segmentation Problem 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

Martin Burger MRI Segmentation Mathematical 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

Martin Burger MRI Segmentation Object-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

Martin Burger MRI Segmentation Active Contours - Snakes Image-driven force related to gradient of the image (local gray-value difference) Regularization force is (mean) curvature

Martin Burger MRI Segmentation Active 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

Martin Burger MRI Segmentation Statistical 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

Martin Burger MRI Segmentation Curve / Surface Representation Level Set Methods yield boundary representation with appropriate curvatures and subpixel resolution

Martin Burger MRI Segmentation Level 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

Martin Burger MRI Segmentation Level Set Methods Change of front translated to change of function

Martin Burger MRI Segmentation Level Set Methods Implicit representation of dynamic shapes with time-dependent level set function

Martin Burger MRI Segmentation Level Set Methods Evolution of the shape corresponds to evolution of the level set function (and vice versa) Movie by J.Sethian

Martin Burger MRI Segmentation Level Set Methods Topology change is automatic Movie by J.Sethian

Martin Burger MRI Segmentation Geometric Motion Start for simplicity with the evolution of a curve Evolution in a velocity field, each point evolves via ODE

Martin Burger MRI Segmentation Geometric Motion Use any parametric representation Due to definition of the level set function Consequently

Martin Burger MRI Segmentation Geometric Motion By the chain rule Insert ODE for moving points:

Martin Burger MRI Segmentation Geometric Motion For level set function being a solution of each level set of  is moving with velocity V

Martin Burger MRI Segmentation Geometric 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)

Martin Burger MRI Segmentation Geometric Motion Normal can be computed from level set function: By the chain rule

Martin Burger MRI Segmentation Geometric Motion Note that is a tangential direction Hence, is a normal direction, unit normal is given by

Martin Burger MRI Segmentation Geometric Motion Evolution becomes nonlinear Hamilton-Jacobi equation: „Level set equation“

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

Martin Burger MRI Segmentation Geometric 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

Martin Burger MRI Segmentation Examples Eikonal equation Positive velocity field yield monotone advancement of fronts Arrival time Solves

Martin Burger MRI Segmentation Example: Eikonal Equation

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

Martin Burger MRI Segmentation Mean Curvature Flow

Martin Burger MRI Segmentation Optimal Geometries Classical problem for optimal geometry: Plateau Problem (Minimal Surface Problem) Minimize area of surface between fixed boundary curves.

Martin Burger MRI Segmentation Optimal Geometries Minimal surface (L.T.Cheng, PhD 2002)

Martin Burger MRI Segmentation Optimal Geometries Wulff-Shapes: Pb[111] in Cu[111] Surnev et al, J.Vacuum Sci. Tech. A, 1998

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

Martin Burger MRI Segmentation Mumford-Shah Image decomposition

Martin Burger MRI Segmentation Mumford-Shah Limitations

Martin Burger MRI Segmentation Improved Model Decomposition in 3 parts: smooth, oscillating, edges

Martin Burger MRI Segmentation Object-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

Martin Burger MRI Segmentation Level Set Formulation Level set function  and Heaviside function H (= 0 negative, = 1 positive)

Martin Burger MRI Segmentation Reduced Problem: fixed mean value

Martin Burger MRI Segmentation Image Segmentation Noisy Image

Martin Burger MRI Segmentation Image Segmentation Noise level 10%,  =10 3

Martin Burger MRI Segmentation Regularization For skull segmentation (smooth) regularization based on length minimization is perfect For brain structure (sulci) similar issues as for active contours

Martin Burger MRI Segmentation MR Results

Martin Burger MRI Segmentation MR Results

Martin Burger MRI Segmentation Skull Segmentation from MR-PD

Martin Burger MRI Segmentation Head Segmentation

Martin Burger MRI Segmentation Bias of one functional often too strong Better: use a family of functionals parametrized by Example: adaptive anisotropy Adaptive Bias / Parametrization J ( u;® ) ® 2 A

Martin Burger MRI Segmentation In aerial images the typical anisotropy is rectangular, houses have 90° angles But not all of them have the same orientation Adaptive Anisotropy

Martin Burger MRI Segmentation Bias for edges with 90° angles from functional of the form R  is rotation matrix for angle  to capture the orientation Since orientation is not constant over the image,  has to vary and to be found adaptively by minimization Adaptive Anisotropy J ( u;® ) = Z (j v 1 j + j v 2 j) d x ; v = R ® r u

Martin Burger MRI Segmentation To avoid microstructure, variation of  has to be regularized, too Possible regularization functional Adaptive Anisotropy

Martin Burger MRI Segmentation Improves angles, still loses contrast Adaptive Anisotropy

Martin Burger MRI Segmentation Contrast correction by iterative refinement Angle parameter provides classification of orientations in the image Adaptive Anisotropy

Martin Burger MRI Segmentation Cartoon reconstruction and orientational classification of aerial images Berkels, mb, Droske, Nemitz, Rumpf 06 Adaptive Anisotropy

Martin Burger MRI Segmentation 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) Adaptive Anisotropy

Martin Burger MRI Segmentation Fixed Anisotropy, 45° orientation

Martin Burger MRI Segmentation Adaptive Anisotropy

Martin Burger MRI Segmentation Adaptive Anisotropy

Martin Burger MRI Segmentation Adaptive Anisotropy, 3D