Polar Snakes: a fast and robust parametric active contour model

Transcription

1 Poar Snakes: a fast and robust parametric active contour mode Christophe Coewet To cite this version: Christophe Coewet. Poar Snakes: a fast and robust parametric active contour mode. IEEE Int. Conf. on Image Processing, Nov 2009, Le Caire, Egypt. pp , <inria > HAL Id: inria Submitted on 6 Oct 20 HAL is a muti-discipinary open access archive for the deposit and dissemination of scientific research documents, whether they are pubished or not. The documents may come from teaching and research institutions in France or abroad, or from pubic or private research centers. L archive ouverte puridiscipinaire HAL, est destinée au dépôt et à a diffusion de documents scientifiques de niveau recherche, pubiés ou non, émanant des étabissements d enseignement et de recherche français ou étrangers, des aboratoires pubics ou privés.

2 POLAR SNAKES: A FAST AND ROBUST PARAMETRIC ACTIVE CONTOUR MODEL Christophe Coewet INRIA Rennes - Bretagne Atantique, IRISA, Lagadic team, Rennes, France. ABSTRACT We present in this paper a way to perform a fast and robust image segmentation and to track a contour aong a sequence of images. Our approach is based on a dynamic deformabe mode. More precisey, we revisit the physics basedmode proposed in [] to show the benefit of using a poar description to mode the contour, in particuar to cope with the we-known initiaization probem. Indeed, we show that this way to proceed eads to diagona and constant matrices in the equations of the snake evoution yieding therefore to a faster agorithm. Experimenta resuts on image segmentation and contour tracking vaidate the efficiency of this new formuation. Index Terms Active contour mode, poar description, segmentation, contour tracking.. INTRODUCTION Since their first introduction in [2], active contours or snakes have proven to be very powerfu in various contexts, incuding image segmentation [3], edge detection [4, 5] and target tracking [6, 7]. An active contour is a deformabe curve defined in the image domain that evoves according to interna forces coming from the curve itsef and externa forces generated from the image (most often gradient-based or regionbased information. Numerous approaches are reported in the iterature. Nevertheess, they can be cassified into two main categories: geometric snakes [4, 8] where the curves is described from a eve set representation, or parametric snakes where the curve is either described as a discrete coection of points [2, 9] or by an appropriate parametric function such as superquadrics [] or B-spines [3,6] for exampe. In this paper we are interested in the atter case, that is in parametric active contours. Indeed, we wi consider here robotics issues such as 3D reconstruction by a moving camera or robot contro by visua servoing [0]. In those cases, a high acquisition rate is required and thus the image processing duration has to be as near as possibe to the video rate. Currenty, it is we known that geometric approaches are too much time consuming and consequenty, they cannot be used in this context. In addition, compared to traditiona methods based on points, parametric snakes based on parametric functions are ess time consuming. Such works are reported for exampe in [, 5 7]. However, the major drawback of parametric active contours is that they are very sensitive to the initia position of the contour. Indeed, it has to be chosen cose to the true contour to ensure its convergence to the correct position. Numerous approaches have been proposed to cope with this probem. A pressure force acts as a force that can infate or defate the contour according to the norma at each point beonging to it []. Another approach is to introduce an externa force based on a distance map between a point beonging to the active contour and the cosest boundary point [2]. Mutiresoution methods have aso been proposed to cope with this probem [5,3]. A new externa force, caed GVF, computed as a diffusion of the image gradient map has been presented in [9] to ensure a convergence even from an initia contour far from the true one. However, computing such forces is very time consuming. The main contribution of this paper is to show the benefit of using a poar description as the parametric curve to describe the contour. In addition, we introduce an externa force derived from an energy term based on the area inside the contour. It is very cosed in mind to the pressure forces []. However, since the poar description is based on orthogona functions, the evoution equations (incuding the energy term of this kind of active contour are very simpe and, consequenty, make the computations faster. This paper is structured as foow. First, we revisit in Section 2 the evoution equations of a physics-based deformabe mode as proposed in [] for a generic famiy functions. Next, this resut is used in Section 3 to show that a poar description highy simpifies these equations. Section 4 points out that it becomes very simpe to introduce an area-based energy term. The experimenta resuts are presented in Section EVOLUTION EQUATIONS OF THE CONTOUR First, et us consider the generic famiy curve depending on a n-dimensiona parameter vector q and defined in the image pane by =n (u : x(u = x c + q Φ (u ( =0 x c being a point inside the contour, u [0, 2π] and Φ ( = 0 n some 2d vector functions. Note that writes as a

3 inear form with respect to q. To derive the evoution equation of, we search the evoution of q as proposed in []. More precisey, this curve is supposed to be embedded in a viscous medium and supposed to be undergone to interna forces f i and to externa forces f e. This probem can be set according to the formaism of Lagrangian mechanics when considering q as the generaized coordinates of the system. The Euer-Lagrange equations give for each component q i of q: ( L d dt q i L = Q i (2 where Q i describes the generaized force associated to q i : Q i = f (u (u du. (3 f are the dissipative forces, L the Lagrangian, it is given by L = T U with T the kinetic energy and U the potentia energy. T writes as T = 2 µ C t 2 du (4 with C t = / t and µ the mass of each point supposed to constitute. Most often, U is described from the Tikhonov reguarization terms: U = 2 k C u 2 du + 2 k 2 C uu 2 du (5 where C u = Cq/ u, C uu = Cq/ u. The scaar k wi tend to imit the extension of whie k 2 wi tend to imit its curvature. According to (, we have C t = q Φ C u = q Φ (with Φ = Φ / u C uu = q Φ (with Φ = 2 Φ / 2 u. Therefore, it becomes straightforward to evauate the contribution of U in (2: we have U/ q i = 0 and U ( = k q Φ Φ i + k 2 q Φ Φ i du. (7 Simiary, we have T/ = 0 and eading to C t 2 = 2 q i ( d T = µ ( dt q i (6 q Φ Φ i (8 q Φ Φ i du. (9 Thus, from (7 and (9, the eft part of (2 can be evauated. Concerning, the right part, we first consider a force due to a viscous friction f vis = γc t, where γ is the viscosity of the medium, and, second, a force f im due to the image. In addition, we suppose that f im derives from a potentia energy term E im : f im = E im where E im denotes the spatia gradient of E im in the image. In this paper, we wi use E im = I. Concerning the contribution of f vis, we have: f vis (u du = γ ( Φi q Φ du. (0 Finay, from (7, (9 and (0, (2 can be written under a matrix form: (u M q + C q + Kq = Q(q = E im du ( q where M = [M mn ], C = [C mn ] and K = [K mn ] with M mn = µ Φ mφ n du C q C mn = γ Φ mφ n du K mn = k Φ C m Φ ndu + k 2 Φ m Φ ndu. q (2 To sove the differentia system (, it is more convenient to set the mass µ to zero as proposed in []. Moreover, we use an expicit Euer s method, eading to q k+ = q k + tc ( Q(q Kq (3 where t is the samping rate. In practice, its vaue must be sma since E im is ony vaid in a sma domain around q k. Lets us point out that the computation of matrices C and K can be time consuming. Indeed, they are obtained from an integration aong. Besides, the terms to integrate may be themseves compex to compute. That is why we propose in the next section a way to decrease the computation time of (3. It is based on a judicious choice for Φ. 3. USING A POLAR DESCRIPTION A way to simpify (2 is to use orthogona functions such that their first and second derivative are aso orthogona in the sense of the inner product defined by the integras in (2. Note that it cannot be achieved using traditiona orthogona poynomias. Conversey, it can be done using a poar description of the contour : ( cos u (u : x(u = x c + ρ q (u (4 sinu and by performing a Fourier expansion of the radius ρ q (u: k=h ρ q (u = q 0 + q k cos ku + q k+h sin ku (5 k=

4 where h is the number of harmonics. Thus, we have q = (q 0,,q 2h and consequenty: Φ = ( cos u cos u sinu cos u for 0 h (6 and Φ = ( cos u sin( hu sin u sin( hu for h < 2h. (7 In that case, it is easy to show that the matrices invoved in (2 becomes diagona and constant. Indeed, we simpy have Fig.. Robustness in the case of an outside initiaization for objects of the everyday ife. C = γ diag (2π, π,, π (8 K = k diag (2π, π( + 2,,π( + h 2, π( + 2,, π( + h 2 (9 K 2 =k 2 diag (2π, π( ,,π( + 6h 2 + h 4, π( ,,π( + 6h 2 + h 4 with K and K 2 defined such that K = K + K USING AN AREA-BASED ENERGY TERM (20 We are interested in this section to add an externa force to the active contour mode to hep the curve to reach the true contour even if its initiaization is far from it. However, adding such a force is time consuming since its contribution to (3 requires the computation of an integra of a compex term. It is the case for exampe with the pressure forces proposed in [] where the norma has to be computed at each point. However, as pointed out in [2], this force can aso be interpreted as the gradient of an area energy term E area defined by E area = k 3 S q where S q denotes the area of the inside region deimited by the curve and k 3 a scaar. If k 3 is a positive vaue, the area wi decrease unti the contour is reached. Thus, an initia estimation outside the contour is required. Conversey, if we want to consider an initiaization inside the contour, a negative vaue has to be chosen. Thanks to the poar description, this surface is very simpe to compute from the we-known Green-Riemann s formua. Indeed, we have S q = π ( q k qk 2 (2 that aso eads to a very simpe way to compute E area / invoved in (2. Indeed, we ony have to introduce the foowing diagona and constant matrix: K 3 = k 3 diag (2π, π,, π (22 that we have to add to the matrices K and K 2. Fig. 2. Robustness in the case of an inside initiaization for various objects. 5. EXPERIMENTAL RESULTS The first experiment is devoted to show the robustness of the convergence wrt a bad initiaization of the active contour. The fina goa of this experiment is to aow disabe persons to grasp an object with the hep of a vision system and a robot arm mounted on their whee chair [4]. Therefore, ony one cick has been used. Fig. shows the behavior of the active contour when objects of the everyday ife are considered. The yeow quadriatera is the initia position of the contour, it has been buit from the cick (the green cross and the nearest image border. The curves in bue shows the behavior of the active contour during its convergence; the curve in green depicts the active contour when q no more evoves. As can be seen, using a surface-based energy term is very efficient. For this experiment, the foowing vaues have been used for both objects: h = 20, k = 0, k 2 = 5.0 4, k 3 = 0., γ = 500 (where γ = γ t and k i = k i t. The second experiment (see Fig.2 is aso devoted to the effectiveness of the initiaization process. An inside initiaization has been considered. Here we have chosen : h = 30, k 3 = 0.2 (the other parameters remain unchanged. The eft image shows a segmentation of a red musce of a raw ham. The right one, the segmentation of MRI image of a pork ung. Here again, despite bad initiaizations, the contour converges to the true one. The goa of the ast experiment is to show the capabiity of our approach to achieve rea-time appications (see Fig.3. Indeed, we are interested here to contro by visua servoing [0]

5 Fig. 3. Tracking contour to achieve a visua servoing task. the motion of a 2D utrasound probe hed by a robot. The contro aw is designed so that the current contour (the green one reaches a earned contour (the red one eading to a correct position of the probe. More precisey, the contro aw is based on some 2D images moments (computed from the vector q, see [5] for more detais. Therefore, to achieve such a task the current contour has to be tracked to compute the contro aw. To do that, q k is used to initiaize the contour at time k. Using a poar description aowed to reach the video rate. For this experiment, we have used: h = 6, k = 0, k 2 =.0 2, k 3 = 0 and γ = CONCLUSION We have presented in this paper a physics-based deformabe mode in the case where the contour is described by a poar description. We have shown that this representation eads to a simpe evoution equation of the contour suitabe for rea-time appications as contour tracking for visua servoing issues. In addition, we have aso shown that, with this representation, it is very easy to introduce an additiona area-based energy term that efficienty deas with a contour initiaization far from the contour to reach. Experiments on various objects have vaidated our approach as we to perform image segmentation as for contour tracking. 7. REFERENCES [] D. Terzopouos and D. Metaxas, Dynamic 3d modes with oca and goba deformations : deformabe superquadrics, IEEE Trans. on Pattern Anaysis and Machine Inteigence, vo. 3, pp , May 99. [2] M. Kass, A. Witkin, and D. Terzopouos, Snakes: Active contour modes, Int. Journa of Computer Vision, vo., no. 4, pp , 987. [3] C. Tauber, H. Batatia, G. Morin, and A. Ayache, Robust b-spine snakes for utrasound image segmentation, IEEE Computers in Cardioogy, vo. 3, pp , [4] V. Casees, R. Kimme, and G. Sapiro, Geodesic active contours, Int. Journa of Computer Vision, vo. 22, no., pp. 6 79, 997. [5] P. Brigger, J. Hoeg, and M. Unser, B-spine snakes: a fexibe too for parametric contour detection, IEEE Trans. on Image Processing, vo. 9, no. 9, pp , September [6] B. Basce, P. Bouthemy, N. Deriche, and F. Meyer, Tracking compex primitives in an image sequence, in IAPR Int. Conf. on Pattern Recognition, Jerusaem, October 994, pp [7] F. Precioso, M. Baraud, T. Bu, and M. Unser, Robust rea-time segmentation of images and videos using a smoothing-spine snake-based agorithm, IEEE Trans. on Image Processing, [8] R. Maadi, J.A. Sethian, and B.C. Vemuri, Shape modeing with front propagation: A eve set approach, IEEE Trans. on Pattern Anaysis and Machine Inteigence, vo. 7, no. 2, pp , 995. [9] C. Xu and L. Prince, Snakes, shapes, and gradient vector fow, IEEE Trans. on Image Processing, vo. 7, pp , March 998. [0] F. Chaumette and S. Hutchinson, Visua servo contro, Part I: Basic approaches, IEEE Robotics and Automation Magazine, vo. 3, no. 4, pp , December [] L.D. Cohen, On active contour modes and baoons, CVGIP: Image Understanding, vo. 53, no. 2, pp. 2 28, 99. [2] L.D. Cohen and I. Cohen, Finite-eement methods for active contour modes and baoons for 2-d and 3-d images, IEEE Trans. on Pattern Anaysis and Machine Inteigence, vo. 5, no., pp. 3 47, November 993. [3] B. Leroy, I. Herin, and L.D. Cohen, Muti-resoution agorithms for active contour modes, in Int. Conf. Anaysis and Optimization of Systems, Paris, France, June 996, vo. 4, pp [4] C. Dune, E. Marchand, C. Coewet, and C. Leroux, Active rough shape estimation of unknown objects, in IEEE/RSJ Int. Conf. on Inteigent Robots and Systems, IROS 08, Nice, France, September 2008, pp [5] R. Mebarki, A. Krupa, and C. Coewet, Automatic guidance of an utrasound probe by visua servoing based on b-mode image moments, in Int. Conf. on Medica Image Computing and Computer Assisted Intervention (MICCAI, New York, USA, September 2008.