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1 Active Contours, Level Sets, and Image Segmentation Chunming Li Institute of Imaging Science Vanderbilt University URL:

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2 Outline 1.Curve evolution and level set methods: Curve Evolution: from snake to general curve evolution Level Set Methods: basic concepts and methods 2.Numerical issues: Difference scheme: upwind scheme, ENO interpolation… Reinitialization, velocity extension … 3.Introduction to calculus of variations and applications: Segmentation: variational level set, shape prior … Denoising: diffusion, TV, …

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3 2D Level Set Click to play the movie

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4 3D Level Set Click to play the movie

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5 More Applications of Level Sets 1.Fluid Dynamics 2.Computer Graphics: Surface Rendering, Animation… 3.Material Science 4.Others…

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6 Advantages of Active Contours and Level Sets 1.Nice representation of object boundary: Smooth and closed, good for shape analysis and recognition and other applications. 2.Sub-pixel accuracy. 3.Can incorporate various information such as shape prior and motion. 4.Mature mathematical tools can be used: calculus of variations, PDE, differential geometry …

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7 Snake Models

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8 Image Segmentation: Classic Methods Thresholding Edge detection An image of blood vessel

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9 Parametric Active Contours (Kass et al 1987) For a contour, define energy:

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10 Evolution of Active Contours Steepest descent: Internal force: Regularizes the contour. External force: Drives the contour towards the desired object boundary. In the classic snake model, the external force field is:

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11 External Force

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12 Gradient Vector Flow (Xu and Prince, 1998) Find a vector field v(x,y) that minimize the energy functional: Solve two decoupled PDEs:

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13 An Example Gradient vector flow

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14 Advantages of GVF Snake Significantly increase capture range Be able to detect concave object boundary

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15 General Curve Evolution

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16 Differential Geometry of Curves A planar curve C is a function, with Tangent vector Normal vector Arc length between and

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17 Curvature 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 derivative again: The tangent vector is a unit vector Note that

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18 Curvature (Cont ’ d) For general parameterization the curvature can be expressed as: Both and are orthogonal to is parallel to

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19 Dynamic Curves A dynamic curve is a time dependent curve The tangent vector and normal vector forms a basis of. The motion of the curve is governed by a curve evolution equation:

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20 Geometric Curve Evolution Theorem: Let be an intrinsic quantity. If evolves according to Then, there exists another parameterization of such that is solution of where at the same point General geometric curve evolution:

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21 Constant Speed Motion and Mean Curvature Motion Constant Speed Motion (Area decreasing/increasing) Mean curvature motion (Length shortening flow) Click to play the movie

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22 Constant Speed Motion (Area decreasing flow) Mean curvature motion (Length shortening flow)

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23 Why MCM Shortens Curve Length? Mean curvature motion is the steepest descent flow (or gradient flow) that minimizes arc length of the contour: Click to play the movie

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24 Geodesic 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. Solve the gradient flow equation:

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25 Geodesic 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 cannot progress into concavity. Add a balloon/pressure force:

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26 Example Click to play the movie

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27 Difficulties of Parameterized Curve Evolution Re-parameterization during evolution: very difficult for 3D surface Cannot handle topological changes

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28 Solutions McInerney and Terzopoulos, 1995 Ray and Acton, 2003 Li, Liu, and Fox, 2005 Some approaches within parametric active contour framework: More natural and better approaches: Level set methods

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29 Level Set Methods

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30 Level Set Representation of Curves zero level

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31 Advantages of Level Set Methods All computation is carried out on a fixed grid, no need for re-parameterization. Topological changes are handled naturally. More … Also have some disadvantages, will be discussed later.

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32 Level Set Functions contour Level set functions with the same zero level set

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33 Signed Distance Function Contour C Signed distance function

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34 Useful Calculus Facts in Level Set Formulation If a curve is a level set of a function then, 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 (or contour) of The volume (or area) integral of p(x) over

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35 From 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:

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36 Special Cases Mean curvature flow Constant Speed Motion Geodesic active contours

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37 Demo of GAC Click to play the movie

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38 Unstable Evolution and Need for Reinitialization Degraded level set function, 50 iterations, time step=0.1 Zero level set of degraded level set function

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39 Procedure of Standard Level Set Methods Reinitialization Initialization Evolve level set function converge? N Y stop

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40 More Details Difference Scheme: finite difference, upwind scheme. ENO interpolation Velocity Extension Narrow Band Implementation More… Next time

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41 Questions

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42 Thank you

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