图像处理水平集方法介绍1课件

1、1Active Contours, Level Sets, and Image SegmentationChunming LiInstitute of Imaging Science Vanderbilt UniversityURL: /licm E-mail: 2OutlineCurve evolution and level set methods:Curve Evolution: from snake to general curve evolutionLevel Set Methods:

2、basic concepts and methods Numerical issues:Difference scheme: upwind scheme, ENO interpolationReinitialization, velocity extension Introduction to calculus of variations and applications:Segmentation: variational level set, shape prior Denoising: diffusion, TV, 32D Level SetClick to play the movieC

3、lick to play the movie43D Level SetClick to play the movieClick to play the movie5More Applications of Level SetsFluid DynamicsComputer Graphics: Surface Rendering, AnimationMaterial ScienceOthers6Advantages of Active Contours and Level SetsNice representation of object boundary:Smooth and closed, g

4、ood for shape analysis and recognition and other applications.Sub-pixel accuracy.Can incorporate various information such as shape prior and motion.Mature mathematical tools can be used: calculus of variations, PDE, differential geometry 7 8Image Segmentation: Classic MethodsThresholdingEdge detecti

5、onAn image of blood vessel 9Parametric Active Contours (Kass et al 1987)For a contour, define energy:10Evolution of Active ContoursSteepest descent:Internal force: Regularizes the contour.External force: Drives the contour towards the desired object boundary.In the classic snake model, the external

6、force field is: 11External Force12Gradient Vector Flow (Xu and Prince, 1998)Find a vector field v(x,y) that minimize the energy functional:Solve two decoupled PDEs:13An Example| /ffGradient vector flow14Advantages of GVF Snake Significantly increase capture range Be able to detect concave object bou

7、ndary 15 16Differential Geometry of Curves A planar curve C is a function , with Tangent vector Normal vector Arc length between and17Curvature Curvature describes how fast the curve changes its direction. Use arc length as the parameter. The unit vector defines the direction of the curve Take deriv

8、ative again: The tangent vector is a unit vector Note that18Curvature (Contd) For general parameterization the curvature can be expressed as: Both and are orthogonal to is parallel to 19Dynamic Curves A dynamic curve is a time dependent curve The tangent vector and normal vector forms a basis of . T

9、he motion of the curve is governed by a curve evolution equation:20Geometric Curve Evolution Theorem: Let be an intrinsic quantity. If evolves according to Then, there exists another parameterization of such that is solution ofwhere at the same point General geometric curve evolution: 21Constant Spe

10、ed Motion and Mean Curvature Motion Constant Speed Motion (Area decreasing/increasing) Mean curvature motion (Length shortening flow)Click to play the movieClick to play the movie22 Constant Speed Motion (Area decreasing flow) Mean curvature motion (Length shortening flow)23Why MCM Shortens Curve Le

11、ngth? Mean curvature motion is the steepest descentflow (or gradient flow) that minimizes arc length of the contour:Click to play the movie24Geodesic Active Contour (Caselles et al, 1997) Minimize a weighted length of C where How to find the geodesic? Euclidean metric: This is a Riemannian metric. S

12、olve the gradient flow equation:25Geodesic Active Contour How does the evolution stop the contour at the edges?External force The external force has two opposite directions on the two sides of the object boundary. The curvature term regularizes the curve: shorten and smooth. Problem: The contour can

13、not progress into concavity. Add a balloon/pressure force:26ExampleClick to play the movie27Difficulties of Parameterized Curve Evolution Re-parameterization during evolution: very difficult for 3D surface Cannot handle topological changes28Solutions McInerney and Terzopoulos, 1995 Ray and Acton, 20

14、03 Li, Liu, and Fox, 2005Some approaches within parametric active contour framework:More natural and better approaches: Level set methods29 30Level Set Representation of Curveszero levelzero level31Advantages of Level Set Methods All computation is carried out on a fixed grid, no need for re-paramet

15、erization. Topological changes are handled naturally. More Also have some disadvantages, will be discussed later. 32Level Set FunctionscontourLevel set functions with the same zero level set33Signed Distance FunctionContour CSigned distance functionCCdistCCCdist inside is x if)x,(x0 outside is x if)

16、x,()x(:by defined isfunction distance signedA 34Useful Calculus Facts in Level Set Formulation If a curve is a level set of a functionthen, the normal vector and the curvature can be computed from the embedding function The surface (or line) integral of a quantity p(x) over the zero level surface (o

17、r contour) of The volume (or area) integral of p(x) over 35From Curve Evolution to Level Set Evolution Curve evolution where F is the speed function, N is normal vector to the curve C Embed the dynamic curve as the zero level set of a time dependent function , i.e. Take derivative with respect to time Level set evolution:36Special Cases Mean curvature flow Constant Speed Motion Geodesic active contours37Demo of GACClick to play the movie38Unstable Evolution and Need for ReinitializationDegraded level set function, 50 iterations, time step=0.1Zero level set of degraded level s