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Active contours without edges - Image Processing, IEEE ...

266 IEEE TRANSACTIONS ON Image PROCESSING, VOL. 10, NO. 2, FEBRUARY 2001 Active contours without EdgesTony F. Chan, Member, IEEE,and Luminita A. VeseAbstract In this paper, we propose a new model for Active con-tours to detect objects in a given Image , based on techniques ofcurve evolution, Mumford Shah functional for segmentation andlevel sets. Our model can detect objects whose boundaries are notnecessarily defined by gradient. We minimize an energy which canbe seen as a particular case of the minimal partition problem. Inthe level set formulation, the problem becomes a mean-curvatureflow -like evolving the Active contour , which will stop on the de-sired boundary. However, the stopping term does not depend onthe gradient of the Image , as in the classical Active contour models,but is instead related to a particular segmentation of the Image . Wewill give a numerical algorithm using finite differences.

In the classical snakes and active contour models (see [9], [3], [13], [4]), an edge-detector is used, depending on the gradient of the image , to stop the evolving curve on the boundary of the desired object. We briefly recall these models next. The snake model [9] is: , where (1) Here, , and are positive parameters. The first two terms

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Transcription of Active contours without edges - Image Processing, IEEE ...

1 266 IEEE TRANSACTIONS ON Image PROCESSING, VOL. 10, NO. 2, FEBRUARY 2001 Active contours without EdgesTony F. Chan, Member, IEEE,and Luminita A. VeseAbstract In this paper, we propose a new model for Active con-tours to detect objects in a given Image , based on techniques ofcurve evolution, Mumford Shah functional for segmentation andlevel sets. Our model can detect objects whose boundaries are notnecessarily defined by gradient. We minimize an energy which canbe seen as a particular case of the minimal partition problem. Inthe level set formulation, the problem becomes a mean-curvatureflow -like evolving the Active contour , which will stop on the de-sired boundary. However, the stopping term does not depend onthe gradient of the Image , as in the classical Active contour models,but is instead related to a particular segmentation of the Image . Wewill give a numerical algorithm using finite differences.

2 Finally, wewill present various experimental results and in particular someexamples for which the classical snakes methods based on the gra-dient are not applicable. Also, the initial curve can be anywhere inthe Image , and interior contours are automatically Terms Active contours , curvature, energy minimization,finite differences, level sets, partial differential equations, INTRODUCTIONTHE BASIC idea in Active contour models or snakes is toevolve a curve, subject to constraints from a given Image , in order to detect objects in that Image . For instance, startingwith a curve around the object to be detected, the curve movestoward its interior normal and has to stop on the boundary of a bounded open subset of, withits a given Image , andbe aparameterized the classical snakes and Active contour models (see [9], [3],[13], [4]), an edge-detector is used, depending on the gradientof the Image , to stop the evolving curve on the boundary ofthe desired object.

3 We briefly recall these models snake model [9] is:, where(1)Here,,andare positive parameters. The first two termscontrol the smoothness of the contour (the internal energy),while the third term attracts the contour toward the object inManuscript received June 17, 1999; revised September 27, 2000. This workwas supported in part by ONR under Contract N00014-96-1-0277 and NSF Con-tract DMS-9626755. The associate editor coordinating the review of this man-uscript and approving it for publication was Prof. Robert J. authors are with the Mathematics Department, University of Cali-fornia, Los Angeles, CA 90095-1555 USA (e-mail: Item Identifier S 1057-7149(01) Image (the external energy). Observe that, by minimizingthe energy (1), we are trying to locate the curve at the pointsof maxima, acting as an edge-detector, while keeping asmoothness in the curve (object boundary).)

4 A general edge-detector can be defined by a positive and de-creasing function, depending on the gradient of the Image ,such thatFor instancewhere, a smoother version of, is the convo-lution of the imagewith the Gaussian. The functionis positive inhomogeneous regions, and zero at the problems of curve evolution, the level set method and inparticular the motion by mean curvature of Osher and Sethian[19] have been used extensively, because it allows for cusps,corners, and automatic topological changes. Moreover, the dis-cretization of the problem is made on a fixed rectangular curveis represented implicitly via a Lipschitz function,by, and the evolution of the curve isgiven by the zero-level curve at timeof the the curvein normal direction with speedamountsto solve the differential equation [19]where the setdefines the initial particular case is the motion by mean curvature, whendivis the curvature of the level-curve ofpassing through.

5 The equation becomesdivA geometric Active contour model based on the mean curva-ture motion is given by the following evolution equation [3]:divinin(2)whereedge-function with;is constant;initial level set 7149/01$ 2001 IEEECHAN AND VESE: Active contours without EDGES267 Its zero level curve moves in the normal direction with speedand therefore stops on the desiredboundary, wherevanishes. The constantis a correction termchosen so that the quantitydivremains always positive. This constant may be interpreted as aforce pushing the curve toward the object, when the curvaturebecomes null or negative. Also,is a constraint on the areainside the curve, increasing the propagation other Active contour models based on level sets wereproposed in [13], again using the Image gradient to stop thecurve. The first one isinwhereis a constant, andandare the maximum andminimum values of the magnitude of the Image gradient.

6 Again, the speed of the evolving curve becomes zero on thepoints with highest gradients, and therefore the curve stops onthe desired boundary, defined by strong gradients. The secondmodel [13] is similar to the geometric model [3], related works are [14] and [15].The geodesic model [4] is(3)This is a problem of geodesic computation in a Riemannianspace, according to a metric induced by the Image . Solvingthe minimization problem (3) consists in finding the path ofminimal new length in that metric. A minimizerwill be ob-tained whenvanishes, , when the curveis on the boundary of the object. The geodesic Active contourmodel (3) from [4] also has a level set formulationdivinin(4)Because all these classical snakes and Active contour modelsrely on the edge-function, depending on the Image gradient, to stop the curve evolution, these models can detect onlyobjects with edges defined by gradient.

7 In practice, the dis-crete gradients are bounded and then the stopping functionis never zero on the edges , and the curve may pass throughthe boundary, especially for the models in [3], [13] [15]. If theimageis very noisy, then the isotropic smoothing Gaussianhas to be strong, which will smooth the edges too. In this paper,we propose a different Active contour model , without a stoppingedge-function, a model which is not based on the gradientof the imagefor the stopping process. The stopping term isbased on Mumford Shah segmentation techniques [18]. In thisway, we obtain a model which can detect contours both with orwithout gradient, for instance objects with very smooth bound-aries or even with discontinuous boundaries (for a discussion ondifferent types of contours , we refer the reader to [8]). In addi-tion, our model has a level set formulation, interior contours areautomatically detected, and the initial curve can be anywhere inthe outline of the paper is as follows.

8 In the next section weintroduce our model as an energy minimization and discuss therelationship with the Mumford Shah functional for segmenta-tion. Also, we formulate the model in terms of level set functionsand compute the associated Euler Lagrange equations. In Sec-tion III we present an iterative algorithm for solving the problemand its discretization. In Section IV we validate our model byvarious numerical results on synthetic and real images, showingthe advantages of our model described before, and we end thepaper by a brief concluding related works are [29], [10], [26], and [24] on activecontours and segmentation, [28] and [11] on shape reconstruc-tion from unorganized points, and finally the recent works [20]and [21], where a probability based geodesic Active regionmodel combined with classical gradient based Active contourtechniques is DESCRIPTION OF THEMODELLet us define the evolving curvein, as the boundary of anopen subsetof( , and).

9 In what follows,insi dedenotes the region, andoutsidedenotes method is the minimization of an energy based-segmen-tation. Let us first explain the basic idea of the model in a simplecase. Assume that the imageis formed by two regions of ap-proximatively piecewise-constant intensities, of distinct valuesand. Assume further that the object to be detected is repre-sented by the region with the value. Let denote its boundaryby. Then we haveinside the object [orinside()],andoutside the object [oroutside()]. Now let usconsider the following fitting term:whereis any other variable curve, and the constants,,depending on, are the averages ofinsideand respec-tively outside. In this simple case, it is obvious that, theboundary of the object, is the minimizer of the fitting termThis can be seen easily. For instance, if the curveis outsidethe object, thenand. If the curveisinside the object, thenbut. If the curveis both inside and outside the object, thenand.

10 Finally, the fitting energy is minimized if, , if the curveis on the boundary of the object. These basicremarks are illustrated in Fig. our Active contour model we will minimize the above fit-ting term and we will add some regularizing terms, like the268 IEEE TRANSACTIONS ON Image PROCESSING, VOL. 10, NO. 2, FEBRUARY 2001 Fig. all possible cases in the position of the curve. The fitting termis minimized only in the case when the curve is on the boundary of the of the curve, and (or) the area of the region , we introduce the energy functional, de-fined byLengthAreain sidewhere,,are fixed parameters. In almostall our numerical calculations (see further), we , we consider the minimization problem:Remark 1:In our model , the term Lengthcould bere-written in a more general way asLength, we consider the case of an arbitrary dimension( ,), thencan have the following values:forall,or.


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