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International Journal of Computer Vision KL405-03-Caselles February 13, 1997 9:29

International Journal of Computer Vision 22(1), 6179 (1997)c 1997 Kluwer Academic Publishers. Manufactured in The Netherlands.

Geodesic Active Contours

VICENT CASELLESDepartment of Mathematics and Informatics, University of Illes Balears, 07071 Palma de Mallorca, Spain

dmivca0@ps.uib.es

RON KIMMELDepartment of Electrical Engineering, Technion, I.I.T., Haifa 32000, Israel

ron@tx.technion.ac.il

GUILLERMO SAPIROHewlett-Packard Labs, 1501 Page Mill Road, Palo Alto, CA 94304

guille@hpl.hp.com

Received October 17, 1994; Revised February 13, 1995; Accepted July 5, 1995

Abstract. A novel scheme for the detection of object boundaries is presented. The technique is based on activecontours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturallysplit and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries.The proposed approach is based on the relation between active contours and the computation of geodesics or minimaldistance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the imagecontent. This geodesic approach for object segmentation allows to connect classical snakes based on energyminimization and geometric active contours based on the theory of curve evolution. Previous models of geometricactive contours are improved, allowing stable boundary detection when their gradients suffer from large variations,including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution arepresented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimentalresults of applying the scheme to real images including objects with holes and medical data imagery demonstrateits power. The results may be extended to 3D object segmentation as well.

Keywords: dynamic contours, variational problems, differential geometry, Riemannian geometry, geodesics,curve evolution, topology free boundary detection

1. Introduction

Since original work by Kass et al. (1988), extensiveresearch was done on snakes or active contour mo-dels for boundary detection. The classical approach isbased on deforming an initial contour C0 towards theboundary of the object to be detected. The deformationis obtained by trying to minimize a functional designedso that its (local) minimum is obtained at the boundaryof the object. These active contours are examples of

the general technique of matching deformable modelsto image data by means of energy minimization (Blakeand Zisserman, 1987; Terzopoulos et al., 1988). Theenergy functional is basically composed of two com-ponents, one controls the smoothness of the curve andanother attracts the curve towards the boundary. Thisenergy model is not capable of handling changes inthe topology of the evolving contour when direct im-plementations are performed. Therefore, the topologyof the final curve will be as the one of C0 (the initial

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International Journal of Computer Vision KL405-03-Caselles February 13, 1997 9:29

62 Caselles, Kimmel and Sapiro

curve), unless special procedures, many times heuris-tic, are implemented for detecting possible splittingand merging (Leitner and Cinquin, 1991; McInerneyand Terzopoulos, 1995; Szeliski et al., 1993). This is aproblem when an un-known number of objects must besimultaneously detected. This approach is also non-intrinsic, since the energy depends on the parametriza-tion of the curve and is not directly related to the objectsgeometry. As we show in this paper, a kind of re-interpretation of this model solves these problems.See for example (Malladi et al., 1995) for commentson other advantages and disadvantages of energy ap-proaches of deforming contours, as well as an extendedliterature on snakes.

Recently, novel geometric models of active contourswere simultaneously proposed by Caselles et al. (1993)and by Malladi et al. (1994, 1995, ). These modelsare based on the theory of curve evolution and geo-metric flows, which has received a large amount ofattention from the image analysis community in recentyears (Faugeras, 1993; Kimia et al., ; Kimmel andBruckstein, 1995; Kimmel et al., 1995, ; Kimmeland Sapiro, 1995; Niessen et al., 1993; Romeny, 1994;Sapiro et al., 1993, Sapiro and Tannenbaum, 1993a, b,1994a, b, 1995). In these active contours models, thecurve is propagating (deforming) by means of a ve-locity that contains two terms as well, one related tothe regularity of the curve and the other shrinks or ex-pands it towards the boundary. The model is givenby a geometric flow (PDE), based on mean curvaturemotion. This model is motivated by a curve evolutionapproach and not an energy minimization one. It allowsautomatic changes in the topology when implementedusing the level-sets based numerical algorithm (Osherand Sethian, 1988). Thereby, several objects can bedetected simultaneously without previous knowledgeof their exact number in the scene and without usingspecial tracking procedures.

In this paper a particular case of the classical en-ergy snakes model is proved to be equivalent to find-ing a geodesic curve in a Riemannian space with ametric derived from the image content. This meansthat in a certain framework, boundary detection canbe considered equivalent to finding a curve of mini-mal weighted length. This interpretation gives a newapproach for boundary detection via active contours,based on geodesic or local minimal distance computa-tions. Then, assuming that this geodesic active contouris represented as the zero level-set of a 3D function, thegeodesic curve computation is reduced to a geometricflow that is similar to the one obtained in the curve

evolution approaches mentioned above. However, thisgeodesic flow includes a new component in the curvevelocity, based on image information, that improvesthose models.1 The new velocity component allowsus to accurately track boundaries with high variation intheir gradient, including small gaps, a task that was dif-ficult to accomplish with the previous curve evolutionmodels. We also show that the solution to the geodesicflow exists in the viscosity framework, and is uniqueand stable. Consistency of the model is presented aswell, showing that the geodesic curve converges to theright solution in the case of ideal objects. A number ofexamples of real images, showing the above properties,are presented.

The approach here presented has the following mainproperties: 1) Describes the connection between en-ergy and curve evolution approaches of active contours.2) Presents active contours for object detection as ageodesic computation approach. 3) Improves existingcurve evolution models as a result of the geodesic for-mulation. 4) Allows simultaneous detection of interiorand exterior boundaries in several objects without spe-cial contour tracking procedures. 5) Holds formal ex-istence, uniqueness, stability, and consistency results.6) Does not require special stopping conditions.

In Section 2 we present the main result of the paper,the geodesic active contours. This section is dividedin four parts: First, classical energy based snakes arerevisited and commented. A particular case which willbe helpful later is derived and justified. Then, the rela-tion between this energy snakes and geodesic curves isshown and the basics of the proposed active contoursapproach for boundary detection is presented. In thethird part, the level-sets technique for curve evolutionis described, and the geodesic curve flow is incorpo-rated to this framework. The last part, 2.4, presentsfurther interpretation of the geodesic active contoursapproach from the boundary detection perspective andshows its relation to previous deformable models basedon curve evolution. After the model description, the-oretical results concerning the proposed geodesic floware given in Section 3. Experimental results with theproposed approach are given in Section 4 followed byconcluding remarks in Section 5.

2. Geodesic Active Contours

In this section we discuss the connection betweenenergy based active contours (snakes) and the com-putation of geodesics or minimal distance curves in a

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Riemannian space derived from the image. From thisgeodesic model for object detection, we derive a novelgeometric partial differential equation for active con-tours that improves previous curve evolution models.

2.1. Energy Based Active Contours

Let us briefly describe the classical energy basedsnakes. Let C(q): [0, 1] R2 be a parametrized pla-nar curve and let I : [0, a] [0, b] R+ be a givenimage in which we want to detect the objects bound-aries. The classical snakes approach (Kass et al., 1988)associates the curve C with an energy given by

E(C) = 1

0|C (q)|2 dq+

10|C (q)|2 dq

1

0| I (C(q))| dq, (1)

where , , and are real positive constants. The firsttwo terms control the smoothness of the contours to bedetected (internal energy),2 while the third term is re-sponsible for attracting the contour towards the objectin the image (external energy). Solving the problem ofsnakes amounts to finding, for a given set of constants, , and , the curve C that minimizes E . Note thatwhen considering more than one object in the image,for instance for an initial prediction of C surroundingall of them, it is not possible to detect all the objects.Special topology-handling procedures must be added.Actually, the solution without those special proced

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