Image Segmentation Using the Chan-Vese Algorithm

Level set based Image SegmentationHang XiaoJan12, 2013OutlinePart1 : Level set methodPart2 : Image segmentation

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 The level set method (LSM) is a numerical technique for tracking interfaces and shapesIt is easy to handle the topology change of the shapethere are no computational issues when this contour splits from, say, a single circle into two distinct closed curves

3Front propagating problems

Ocean wavesBurning flames

Image contourApplication in physics, chemistry, fluid mechanics, image processing, materials sciences, fabrication of microelectronic components, computer vision, control theory, seismology, computer-aided-design4Interface PropagationInsideOutsideOutsideF = F(L, G, I)F(L,G,I) : speed functionL : local properties, e.g. curvature and normal directionG: Global properties, e.g. heat I: Independent properties, independ of the shape of the front, e.g. underlying fluid velocityArrival time functionAssume F > 0, hence the front moves outwardT(x,y) : the arrival time, characterize the position of the expanding frontDistance = speed * time1D :

2D:

: the initial location of the interfaceOne way to characterize the position of this expanding front is to compute the arrival time T(x,y) of the front6Time-depend level set function

Time-depend level set functionF is inconstant to t

The stationary level set functionF is constant to t

Phi(X, t) = 07Level set function properties

By the chain rule,Front formula:

InsideOutsideOutsideFHamilton-Jacobi equation

X(t) is image point, with certain speedAssume x(t) is the point on the propagating frontEach point moves with certain speed F8Level set method work flow

Initialize level set function t=0Compute speed FCompute level set function in next state t+1 Compute final zero level set Reinitialize level set functiont++

e.g.Arrival time

9Example simple curve evolution

Simple curveComplex curves

Curvature Level setSpeedProperty: Any simple curve will become a ball in the force of curvature

Level set 10Example simple curve evolution

Level set 11Example 3D surface3D curvature

Gaussian curvatureMean curvature

Mean curvatureGaussian curvature

Level set 12Other 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 firePart2 : Image segmentationImage segmentation modelsExplicit methodImplicit method

InsideOutsideOutside

Interface frontEvolution equation

Energy minimization

The Chan-Vese Segmentation Model

Assume the image is piecewise constant

Energy : Sum of difference

Add more regularizing termsThe Chan-Vese Segmentation Model

Energy Minimization Energy function

Substituting into the Euler-Lagrange equation, and applying Greens identity and Greens theorem

we will add some regularizing terms, like the length of the curve , and (or) the area of the region inside

17Effect 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.

Chan-Vese Examples

Parameters: =1, =0, 1 = 2 =1, iterations = 41 Chan-Vese Examples

Change in topologyParameters: =1, =0, 1 = 1, 2 =2, iterations = 45 This is hard in geodesic active contour20Chan-Vese ExamplesBlurred and Noised ImageParameters: =1, =0, 1 = 1, 2 =2, iterations = 45

This is hard in geodesic active contour21Chan-Vese ExamplesChromatic Resemblance Parameters: =10^8, =0, 1 = 2 = 1, iterations = 40

This is hard in geodesic active contour22Chan-Vese ExamplesRestriction on Curve Evolution Direction Parameters: =1, =0, 1 = 2 = 1, iterations = 40

This is hard in geodesic active contour23Chan-Vese ExamplesRestriction on Curve Evolution Direction Parameters: =1, =10^6, 1 = 2 = 1, iterations = 40

This is hard in geodesic active contour24

Detection of lines and curves not necessarily closed

Example of image for which the averages inside and outside are the sameDetection of a simulated minefield, with contour without gradient.

E1: false negative errorE2: false positive errorConclusion: 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 dtWe start with a trivial example; a smooth, synthetically generated image consisting of only two distinct graylevels. The correct segmentation is immediately obvious in this case, and could be obtained easily by a thresholding algorithm. This image is segmented in one iteration with the default parameters of our algorithm.

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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

Top: The noisy imageBottom : the thresholding segmentation resultmax(E1, E2) = 0.5625, min(E1, E2) = 0.1222Evaluation 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.

Overlap errors vs. length penalty . The best segmentation is obtained for 94.

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.

Chan-Vese VS. Mumford-ShahChan-Vese function 2001Mumford-Shah function 1992

MS model CV model

we will add some regularizing terms, like the length of the curve , and (or) the area of the region inside The Mumford-Shah functional is a function which is used to establish an optimality criterion for segmenting an image into sub-regions.

31The Chan-Vese Segmentation Model

we will add some regularizing terms, like the length of the curve , and (or) the area of the region inside

32The Chan-Vese Segmentation Model

Energy Minimization Energy function

Substituting into the Euler-Lagrange equation, and applying Greens identity and Greens theorem

How to add area constant constraint?

we will add some regularizing terms, like the length of the curve , and (or) the area of the region inside

Still we can add other image properties like the gradient to the model.

33Generalizing to Vector-Valued Images

Texture Segmentation

Generate sparse texture features by nonlinear diffusion filtering Brox, Weickert 04, 06 Texture Segmentation

2. Vector-Image based level set methods

Brox, Weickert 04, 06 Texture Segmentation

efficient coarse-to-fine scheme Brox, Weickert 04, 06 Coupling Multiple Active Surfaces (Dufour 2004)

Needs N level set functions for N objects segmentationMulti-Phase Level Set methods

Need log2N Level set functions for segmenting N objects

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

Multi-Phase Level Set methods

Brox, Weickert 04, 06

One Level Set without Coupling Based Multiple object segmentation

Pros. And Cons. For level set methodAdvantageCapture RangeEffect of Local NoiseNo 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. volumeHandling Corners Resolution Changes Jasjit S. et. al. 2002More likely to find the global minima then snake model 43Pros. And Cons. For level set methodDisadvantageInitial Placement of the Contour Hard to stopExpensive running timeJasjit S. et. al. 2002Thanks!!!