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 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**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**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**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**MRI Segmentation**Active Contours - Snakes • Image-driven force related to gradient of the image (local gray-value difference) • Regularization force is (mean) curvature**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**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**MRI Segmentation**Curve / Surface Representation • Level Set Methods yield boundary representation with appropriate curvatures and subpixel resolution**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**MRI Segmentation**Level Set Methods • Change of front translated to change of function**MRI Segmentation**Level Set Methods • Implicit representation of dynamic shapes • with time-dependent level set function**MRI Segmentation**Level Set Methods • Evolution of the shape corresponds to evolution of the level set function (and vice versa) • Movie by • J.Sethian**MRI Segmentation**Level Set Methods • Topology change is automatic • Movie by • J.Sethian**MRI Segmentation**Geometric Motion • Start for simplicity with the evolution of a curve • Evolution in a velocity field , each point evolves via ODE**MRI Segmentation**Geometric Motion • Use any parametric representation • Due to definition of the level set function • Consequently**MRI Segmentation**Geometric Motion • By the chain rule • Insert ODE for moving points:**MRI Segmentation**Geometric Motion • For level set function being a solution of • each level set of f is moving with velocity V**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)**MRI Segmentation**Geometric Motion • Normal can be computed from level set function: • By the chain rule**MRI Segmentation**Geometric Motion • Note that is a tangential direction • Hence, is a normal direction, unit normal is given by**MRI Segmentation**Geometric Motion • Evolution becomes nonlinear Hamilton-Jacobi equation: • „Level set equation“**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**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**MRI Segmentation**Examples • Eikonal equation • Positive velocity field yield monotone advancement of fronts • Arrival time • Solves**MRI Segmentation**Example: Eikonal Equation**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)**MRI Segmentation**Mean Curvature Flow**MRI Segmentation**Optimal Geometries • Classical problem for optimal geometry: • Plateau Problem (Minimal Surface Problem) • Minimize area of surface between fixed boundary curves.**MRI Segmentation**Optimal Geometries • Minimal surface (L.T.Cheng, PhD 2002)**MRI Segmentation**Optimal Geometries • Wulff-Shapes: Pb[111] in Cu[111] • Surnev et al, J.Vacuum Sci. Tech. A, 1998**MRI Segmentation**Mumford-Shah • Free discontinuity problems: • find the set of discontinuity from a noisy observation of a function. • Mumford-Shah functional**MRI Segmentation**Mumford-Shah • Image decomposition**MRI Segmentation**Mumford-Shah • Limitations**MRI Segmentation**Improved Model • Decomposition in • 3 parts: smooth, oscillating, edges**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**MRI Segmentation**Level Set Formulation • Level set function y and • Heaviside function H (= 0 negative, = 1 positive)**MRI Segmentation**Reduced Problem: fixed mean value**MRI Segmentation**Image Segmentation • Noisy Image**MRI Segmentation**Image Segmentation • Noise level 10%, l=103**MRI Segmentation**Regularization • For skull segmentation (smooth) regularization based on length minimization is perfect • For brain structure (sulci) similar issues as for active contours**MRI Segmentation**MR Results**MRI Segmentation**MR Results**MRI Segmentation**Skull Segmentation from MR-PD**MRI Segmentation**Head Segmentation**(**) J A 2 ® u ; ® MRI Segmentation Adaptive Bias / Parametrization Bias of one functional often too strong Better: use a family of functionals parametrized by Example: adaptive anisotropy**MRI Segmentation**Adaptive Anisotropy In aerial images the typical anisotropy is rectangular, houses have 90° angles But not all of them have the same orientation**Z**( ) ( j j j j ) d J R r + u ; ® v v x v u = = 1 2 ® ; 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**MRI Segmentation**Adaptive Anisotropy To avoid microstructure, variation of a has to be regularized, too Possible regularization functional**MRI Segmentation**Adaptive Anisotropy Improves angles, still loses contrast