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Level set based Image Segmentation Hang Xiao Jan12, 2013 Slide 2 Outline Part1 : Level set method Part2 : Image segmentation Slide 3 Part1 : Level set method In mathematics, a level set of a real-valued function f of n variables is a set of the form STANLEY OSHER 1988 Slide 4 Front propagating problems Ocean waves Burning flames Image contour Slide 5 Interface Propagation Inside Outside F = F(L, G, I) F(L,G,I) : speed function L : local properties, e.g. curvature and normal direction G: Global properties, e.g. heat I: Independent properties, independ of the shape of the front, e.g. underlying fluid velocity Slide 6 Arrival time function Assume F > 0, hence the front moves outward T(x,y) : the arrival time, characterize the position of the expanding front Distance = speed * time 1D : 2D: : the initial location of the interface Slide 7 Time-depend level set function F is inconstant to t The stationary level set function F is constant to t Slide 8 Level set function properties By the chain rule, Front formula: Inside Outside F Hamilton-Jacobi equation Slide 9 Level set method work flow Initialize level set function t=0 Compute speed F Compute level set function in next state t+1 Compute final zero level set Reinitialize level set function t++ e.g. Arrival time Slide 10 Example simple curve evolution Simple curve Complex curves Curvature Level set Speed Property: Any simple curve will become a ball in the force of curvature Slide 11 Example simple curve evolution Slide 12 Example 3D surface 3D curvature Gaussian curvature Mean curvature Gaussian curvature Slide 13 Other examples The particle level set method was used to represent the interface separating water from air as water is being poured into a glass Level set method is used to simulate a ball catching on fire Slide 14 Part2 : Image segmentation Slide 15 Image segmentation models Explicit method Implicit method Inside Outside Interface front Evolution equation Energy minimization Slide 16 The Chan-Vese Segmentation Model Assume the image is piecewise constant Energy : Sum of difference Add more regularizing terms Slide 17 The Chan-Vese Segmentation Model Energy Minimization Energy function Substituting into the Euler-Lagrange equation, and applying Greens identity and Greens theorem Slide 18 Effect of Parameters controls the importance of the length of C in the minimization. if 1, then a few large closed curves will retain in the steady state, compared to many small ones. This may be useful in ignoring noise, or grouping objects of similar characteristics controls the the importance of area inside C. If 1, like the geometric active contour model, it forces C to move strictly inward. Also, the speed at which C evolves inward increases. The relative balance between 1 and 2 determines which side, inside or outside, has higher importance in minimizing the regional variance. This is useful in segmenting blurred images: for example, if one wishes to completely enclose the blurred object, 2 > 1 will ensure this. Slide 19 Chan-Vese Examples Parameters: =1, =0, 1 = 2 =1, iterations = 41 Slide 20 Chan-Vese Examples Change in topology Parameters: =1, =0, 1 = 1, 2 =2, iterations = 45 Slide 21 Chan-Vese Examples Blurred and Noised Image Parameters: =1, =0, 1 = 1, 2 =2, iterations = 45 Slide 22 Chan-Vese Examples Chromatic Resemblance Parameters: =10^8, =0, 1 = 2 = 1, iterations = 40 Slide 23 Chan-Vese Examples Restriction on Curve Evolution Direction Parameters: =1, =0, 1 = 2 = 1, iterations = 40 Slide 24 Chan-Vese Examples Restriction on Curve Evolution Direction Parameters: =1, =10^6, 1 = 2 = 1, iterations = 40 Slide 25 Detection of lines and curves not necessarily closed Example of image for which the averages inside and outside are the same Detection of a simulated minefield, with contour without gradient. Slide 26 E1: false negative error E2: false positive error Conclusion: The image can be segmented well in one iteration for timesteps greater than about 0.001. Time steps (dt) evalution Overlap error after one iteration for small or large dt Slide 27 Conclusion: 1). When the timestep is small the segmentation converges. 2). However, if we take the timestep too large, the results diverge after a few iterations even though the segmentation is initially good. Overlap error vs. number of iterations for a small timestep Overlap error vs. number of iterations for a large timestep Iteration number evalution Slide 28 Top: The noisy image Bottom : the thresholding segmentation result max(E1, E2) = 0.5625, min(E1, E2) = 0.1222 Evaluation on noisy image: 1) Use the true segmentation from the noiseless image (ground truth) for forming the overlap error 2) Perform five iterations of the algorithm for each fixed value of c1, c2 3) Vary one parameter at a time and plot the overlap errors against that parameter. Slide 29 Overlap errors vs. length penalty . The best segmentation is obtained for 94. Slide 30 Overlap error vs. 1. Changing 1 from 1 does not improve the segmentation, since our foreground and background regions both have uniform intensity before noise is added. The results for varying 2 are similar. Slide 31 Chan-Vese VS. Mumford-Shah Chan-Vese function 2001 Mumford-Shah function 1992 MS model CV model Slide 32 The Chan-Vese Segmentation Model Slide 33 Energy Minimization Energy function Substituting into the Euler-Lagrange equation, and applying Greens identity and Greens theorem How to add area constant constraint? Slide 34 Generalizing to Vector-Valued Images Slide 35 Texture Segmentation 1.Generate sparse texture features by nonlinear diffusion filtering Brox, Weickert 04, 06 Slide 36 Texture Segmentation 2. Vector-Image based level set methods Brox, Weickert 04, 06 Slide 37 Texture Segmentation efficient coarse-to-fine scheme Brox, Weickert 04, 06 Slide 38 Coupling Multiple Active Surfaces (Dufour 2004) Needs N level set functions for N objects segmentation Slide 39 Multi-Phase Level Set methods Need log 2 N Level set functions for segmenting N objects Slide 40 Multi-Phase Level Set methods Results on a synthetic image, with a triple junction, using the 4-phase piecewise constant model with 2 level set functions. We also show the zero level sets of 1 and 2 Slide 41 Multi-Phase Level Set methods Brox, Weickert 04, 06 Slide 42 One Level Set without Coupling Based Multiple object segmentation Slide 43 Pros. And Cons. For level set method Advantage Capture Range Effect of Local Noise No Need of Elasticity Coefficients Suitability for Medical Image Segmentation Normal Computation Integration of Regional Statistics Flexible Topology Extension to high dimension Incorporation of Regularizing Terms e.g. volume Handling Corners Resolution Changes Jasjit S. et. al. 2002 Slide 44 Pros. And Cons. For level set method Disadvantage Initial Placement of the Contour Hard to stop Expensive running time Jasjit S. et. al. 2002 Slide 45 Thanks!!!