THE SUBJECT of this study is motion-based tracking, the

Transcription

1 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 11, NOVEMBER Spatiotemporal Motion Boundary Detection and Motion Boundary Velocity Estimation for Tracking Moving Objects With a Moving Camera: A Level Sets PDEs Approach With Concurrent Camera Motion Compensation Rosario Feghali and Amar Mitiche Abstract The purpose of this study is to investigate a method of tracking moving objects with a moving camera. This method estimates simultaneously the motion induced by camera movement. The problem is formulated as a Bayesian motion-based partitioning problem in the spatiotemporal domain of the image sequence. An energy functional is derived from the Bayesian formulation. The Euler Lagrange descent equations determine simultaneously an estimate of the image motion field induced by camera motion and an estimate of the spatiotemporal motion boundary surface. The Euler Lagrange equation corresponding to the surface is expressed as a level-set partial differential equation for topology independence and numerically stable implementation. The method can be initialized simply and can track multiple objects with nonsimultaneous motions. Velocities on motion boundaries can be estimated from geometrical properties of the motion boundary. Several examples of experimental verification are given using synthetic and real-image sequences. Index Terms Level-sets PDEs, motion detection, optical flow, spatiotemporal analysis, tracking. I. INTRODUCTION THE SUBJECT of this study is motion-based tracking, the process of following image objects in their movement through an image sequence on the basis of motion information alone. Although color, image statistics of the moving objects, or contrast at motion boundaries can drive or assist tracking, motion-based tracking relies on motion information alone. When there is a single moving object, the goal of tracking is simply to locate the object in every image of the sequence. The process is not a mere classification of pixels, one without some form of spatial coherence, because the totality of the image of the moving object or the boundary of this image, and only this image or its boundary, are to be located. When several objects are in movement, motion-based tracking has often been defined (in [13], [28], for instance), albeit implicitly, as follows: Given an aggregate of moving regions (or motion boundaries) in the Manuscript received December 8, 2002; revised January 14, This work was supported by NSERC under Grant OGP The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Philippe Salembier. The authors are with the INRS-EMT, Montréal, QC H5A 1K6, Canada ( rosario.feghali@crc.ca; mitiche@inrs-emt. uquebec.ca). Digital Object Identifier /TIP first image of a sequence, identify the aggregate in every subsequent image. This definition does not include correspondence between moving objects from one frame of the sequence to the next. In motion-based tracking, this correspondence can be sought under assumptions. For instance, in [19], the image of the objects to track and a model of their motion are given at the onset of tracking and the problem is stated as finding the match between successive images that best conforms to the motions models. In [22], the problem is stated as motion-based segmentation of the sequence spatiotemporal domain under the assumption that the number of moving objects and the objects motion models are given. These are determined by clustering separately from the segmentation process. In [22], the moving objects contours are detected in the spatiotemporal domain, and motion is computed along these contours. Motion along the moving contours is a legitimate tracking information although it cannot resolve correspondence when occlusions occur without the intervention of some external process. The motion-based method we investigate in this study follows the characterization of tracking in [19], [22]. It does not resolve occlusions, but provides information that can be used for that purpose. Before describing this method, we review the literature of related work. Detection and tracking is one of the most challenging problems in computer vision. It plays a significant role in numerous useful applications [7], [23], [29], [32] such as robotics, meteorology, communications, biomedicine, and image sequence database retrieval. In many such applications, cameras are mounted on mobiles, such as airplanes, satellites, and motor vehicles. Moving cameras are also used in active vision systems. Active cameras, rather than passively observing their environment, play an active role in the interpretation of the observed scene. Examples of active vision applications include visual servoing and autonomous robot navigation. A moving camera poses new challenges. When a camera moves, it generates motion over the entire image positional array. In this situation, the problem cannot be solved directly by simple motion detection as in the case of a static camera. Methods of detection and tracking with a moving camera fall in one of two categories. In one category, methods assume that camera motion is given as an input or that the background scene has distinctive image properties. This leads to constraints that are valid only in background regions. Moving objects regions violate these /04$ IEEE

2 1474 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 11, NOVEMBER 2004 constraints and can be thus detected [24], [33]. Methods in the other category [7], [20] assume that background motion is represented by a parametric model, often derived from perspective projection models [9]. The parameters of image motion induced by camera movement are estimated before tracking and used to obtain a camera-motion compensated image sequence on which tracking is performed. Methods within this category perform tracking generally in three consecutive steps. 1) Estimation of background motion parameters: This first step extracts background feature points in each image of the sequence, followed by temporal matching of these feature points. Temporal correspondences yield a system of equations solved using various numerical methods (e.g., the Gauss Newton method, the Levenberg Marquardt method, or the least median of squares method [1]). 2) Detection of moving objects based on residual motion, where residual motion is the image motion after camera motion compensation. Detection methods include Bayesian classification [32], Markov random field models [25], and clustering techniques. 3) Tracking: This is a time integration step at the object level. Tracking is performed frame by frame and correspondence is achieved using contour matching algorithms [10], [29] or region matching algorithms, an example of which is the use of the mean shift iterations [6]. A major problem with such sequential multistep methods is that errors propagate from one step to the other without a mechanism for correction. The operations in different steps are not integrated by feedback and are, therefore, prone to cumulative errors from one step to another. Tracking proper can be contour based or region based. Formal statement of contour tracking uses some form of active contour evolution [5], [16], [17]. With region tracking, region trajectories are traced based on consistency with motion measurements or properties of the regions intensity patterns and geometry [2], [11]. Quite recently, both contour and region tracking have been formulated using level-sets partial differential equations (PDEs) [4], [13], [18], [19], [26], [27]. The level-sets formalism is momentous because it accounts for topology changes during contour evolution and it can be implemented by stable numerical methods. Current methods of tracking formulated via level-sets track moving image objects frame by frame. They require that these objects be segmented beforehand by some external process [13], [18], [19], [27]. Some schemes assume that a good estimate of image motion is available at each instant of time, to be used as data [4], [18], or that the background has known properties to be used for identification [13]. Under other assumptions, the background intensity pattern contrasts strongly with the pattern of moving objects, and interframe intensity difference statistics invariant in time can be computed [27]. Current PDE-based methods also share the following shortcomings. Except for [19], they are not valid when there is camera motion, and only to those objects that are identified as moving at the start of tracking can be tracked; no other object can be tracked that comes into motion after tracking of objects is started. This study is along the vein of our previous one on tracking by explicit processing in the spatiotemporal domain [21], [22]. There was no reference in that study to the image motion induced by camera movement and, therefore, no estimation of this motion. As in [21], [22], we start by stating the problem as a Bayesian motion-based partitioning problem in the spatiotemporal domain. Representing background motion by a parametric model, the approach simultaneously estimates the model parameters while evolving a spatiotemporal surface so that as the end of its evolution it enfolds the volume generated by moving objects and thus partitions the spatiotemporal domain into the background region on the one hand and the foreground region of the moving objects on the other hand. Surface evolution is implemented via level-sets partial differential equations to afford topology free and numerically stable solutions. Furthermore, motion velocities can be estimated along motion boundaries from geometrical properties of the spatiotemporal surface. The approach has the following characteristics: a) it allows camera movement, b) it allows several objects that have nonsimultaneous motions, c) it does not require prior estimation of camera motion, and d) it is implemented via level-set PDEs to allow topology free processing and numerically stable computation. Finally, e) it allows explicit recovery of motion along motion boundaries. The remainder of this paper is organized as follows. Section II gives the formulation of the proposed tracking approach. Section III presents discussions and limitations of the approach. Section IV gives experimental results and Section V contains a conclusion. II. FORMULATION Let be an image sequence defined over into, where is the time interval of the image sequence, and an open subset of. Let be a closed surface in, the region enclosed by, its complement in, and the partition. We assume that background motion can be fully characterized by a set of parameters defined by a vector. With being a motion measurement, the MAP estimate of(, )is is ignored because it is independent of and. is the observation data term, and is the a priori term. Assuming conditional independence of the motion measurement for yields The problem is then equivalent to minimizing the following functional: (1) (2)

3 FEGHALI AND MITICHE: SPATIOTEMPORAL MOTION BOUNDARY DETECTION AND MOTION BOUNDARY VELOCITY ESTIMATION 1475 The first two terms on the right of (2) will be defined by the observation model. The last term will be defined by the prior model. A. Observation Model Assuming small range motion so that motion is of small extent between consecutive instants of observation, let be the normal component of optical velocity, given by for for and being the spatial gradient and the temporal derivative of, respectively. We selected as a measurement of motion activity because of its intuitive significance as a cue for motion and its successful use in other studies [7]. Define (3) of this motion, respectively. For the purpose of motion detection and tracking, this model is sufficient to account for camera tilt, pan, and fronto-parallel translation, particularly when objects are relatively distant. In other circumstances, such as movement of camera in depth, an affine model may be more appropriate. Because and are global parameters ( and describe the motion induced at every point of the image domain by camera movement), we will consider,, and to be independent variables. 1 Assuming that parameters and are constant during the observation period, and because The minimization of with respect to gives (4) where is the normal component of the image motion due to camera motion. is a function of, the parameters of the image motion due to camera motion. In a noiseless image sequence, is zero in the background, while in the regions of moving objects, it denotes motion activity due to moving objects intrinsic motions. We choose the following observation model: for for (5) where and are positive real constants and is the proportional to symbol. This choice will favor partitions where points in have while points in have. Therefore, we will be seeking a partition where is high both in and. In other words, this model biases the estimate toward surfaces that partition the spatiotemporal image into regions of contrasting motion activity. High residual motion activity occurs in the region enclosed by the surface and low residual motion activity in the complement of this region. B. Prior Model We use the standard regularization term, independent of This prior has a smoothing effect on the evolving surface by favoring surface estimates that have small area. (6) Similar equations can be written for an affine model. However, we would have a more complex set of equations in such a case. For the minimization with respect to, we start by rewriting (7) as The first term of (9) is independent of and does not intervene in the minimization with respect to and can be discarded. The second term is a volume integral that we transform into a surface integral by applying Gauss divergence theorem. The functional derivative of with respect to is then given by (the derivation, and related ones, can be found in [3], [5], [15], [22], and [30]) (8) (9) (10) where is the outward unit normal to and is its mean curvature. The descent equations of minimization are given by C. Euler Lagrange Equations Maximizing the a posteriori probability equivalent to minimizing the following energy functional is (11) (7) Background motion parameters represented by intervene in the first two terms of (7). The third term depends only on geometrical properties of the spatiotemporal surface. Following here, we assume that motion is translational. Therefore,, where and are the horizontal and vertical components 1 An alternative would be to use the Horn-and-Shunk equation and estimate a and b by least squares over the complement of S. In such a case, the derivation of the Euler Lagrange equations should take into account the dependency of these parameters on S, which may result in additional related terms in the velocity of evolution of S.

4 1476 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 11, NOVEMBER 2004 Fig. 1. At the end of its evolution, the spatiotemporal surface enfolds the volume generated by moving objects. This evolution is simultaneously coupled with the background motion parameters estimation. D. Level-Sets Representation Execution of the descent equation with respect to by explicit representation of as a set of points cannot implement changes in the topology of. To obtain a topology-free evolution of this surface, and a numerically stable scheme, we adopt a level-set representation [31]. Surface is represented implicitly as as the zero level of a one-parameter family of functions, indexed by algorithmic time (12) where,, and are the spatiotemporal variables and. Taking the derivative of (12) with respect to

5 FEGHALI AND MITICHE: SPATIOTEMPORAL MOTION BOUNDARY DETECTION AND MOTION BOUNDARY VELOCITY ESTIMATION 1477 Fig. 2. Tracking results for rotating fish sequence. One frame interval, upper left to lower right. yields the level-set partial differential equation that dictates the evolution of (13) which can be rewritten equivalently (14)

6 1478 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 11, NOVEMBER 2004 Fig. 3. Velocities along motion boundaries of rotating fish sequence. One frame interval, upper left to lower right. Using (10), and defining to be negative inside and positive outside ( on, bydefinition) so that is oriented as, i.e., we get such that Let the normal to the surface be expressed as (15) Assuming rank, the two nonzero eigenvalues of this matrix give the principal curvatures [8]. Mean curvature is expressed as the negative of the mean of the two principal curvatures, which is half of trace of (16)

7 FEGHALI AND MITICHE: SPATIOTEMPORAL MOTION BOUNDARY DETECTION AND MOTION BOUNDARY VELOCITY ESTIMATION 1479 Fig. 4. Curvature coefficient choice. A high curvature coefficient yields a smooth oval shape surrounding moving objects. A lower curvature coefficient yields better contour location: =10in the upper part of the figure and =5in the lower part. Fig. 5. Convergence of (a, b) corresponding to step 2) in the given algorithm, in the case of the rotating fish sequence., the mean curvature is ex- In terms of the level-set function pressed as According to (10) and (15), speed evolves normal to itself at a (17) (18) By construction, can be recovered at any instant of observation as the zero-level surface of function regardless of its topology. Let the initial position of be a surface that subsumes the volume generated by the moving objects. With the proper choice

8 1480 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 11, NOVEMBER 2004 Fig. 6. Evolution of the spatiotemporal surface for the Person-Person sequence. of the constant coefficients, we will have the following behavior of. While is in the background, we will have and : will move inward, remain smooth because of the curvature term, and the speed of evolution will vary little because of the constant term. Whenever and wherever it reaches the boundary of a moving object,. The term acts to prevent the surface from penetrating into the region of motion activity past the motion boundary, while has the same spatiotemporal smoothing effect. The algorithm is summarized as follows. 1) Initialize and.

9 FEGHALI AND MITICHE: SPATIOTEMPORAL MOTION BOUNDARY DETECTION AND MOTION BOUNDARY VELOCITY ESTIMATION 1481 Fig. 7. Tracking results for Person-Person sequence. One frame interval, upper left to lower right. 2) Perform one iteration of the first two descent equations for and in (11). 3) Evolve using one iteration of the level-set PDE descent (15). 4) Return to step 2) until convergence. The initialization is done by setting to (0, 0). is set to a parallelepipedic shape enfolding all the spatiotemporal volume of the image sequence. Note that step 2) can be iterated serveral times or until convergence. The algorithm terminates when the surface has not evolved during a number of iterations. E. Choices of Parameters We provide a guideline on appropriate relative choices of parameters. The normal residual motion is approximately zero in the background case of a negative curvature, the choice of the following constraint has to comply to (19) The other case of a positive curvature helps the surface to move inwards and does not need to be taken into consideration. In the foreground of a moving object region, is given by To drive the surface toward the moving object boundaries we need an outward velocity (, ). In the case of a positive mean curvature, the choice of has to comply to the following constraint (20) In order to drive the surface inwards, in the direction of motion boundaries, we need a negative velocity (, ). In the In the case of a negative curvature (21)

10 1482 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 11, NOVEMBER 2004 Fig. 8. Velocities along motion boundaries of Person-Person sequence. One frame interval, upper left to lower right. which is most of the time true, assuming that in the foreground, residual motion is sufficiently high, i.e.,. (19), (20), and (21) determine a range of acceptable values for. Large values lead to smoother motion boundary surfaces in the spatiotemporal domain compromising location accuracy, whereas small values yield less regular motion boundary surfaces with a better location accuracy. Choices of and reflect the balance between the inward vs. outward velocity. Experimental verifications show that and is a good combination for all the sequences tested. F. Estimation of Velocities Along Motion Boundaries Within this scheme, motion boundaries at time are obtained by intersecting the spatiotemporal surface by the plane. Velocities at motion boundaries can be recovered from the spatiotemporal surface as follows. Let be a motion boundary point trajectory in spatiotemporal space. The tangent vector to is, where is the velocity at. If we assume no occlusion and that the trajectory of a motion boundary point is located on the spatiotemporal surface, then the velocity vector along this trajectory is tangent to the surface and, therefore, is orthogonal to the surface normal, yielding the following geometric constraint on velocity (22) If, then (22) determines the component of velocity in the direction of. This constraint is valid at any point where the spatiotemporal surface is regular. Note that (22) is similar to the Horn and Schunck gradient equation.

11 FEGHALI AND MITICHE: SPATIOTEMPORAL MOTION BOUNDARY DETECTION AND MOTION BOUNDARY VELOCITY ESTIMATION 1483 Fig. 9. Spatiotemporal surface evolution in the case of the Walker sequence. However, the Horn and Schunck gradient equation is theoretically not valid at motion boundaries. To estimate velocity from the component given by (22), we add a regularization constraint where velocities are considered

12 1484 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 11, NOVEMBER 2004 Fig. 10. Tracking results for the Walker sequence. One frame interval, upper left to lower right.

13 FEGHALI AND MITICHE: SPATIOTEMPORAL MOTION BOUNDARY DETECTION AND MOTION BOUNDARY VELOCITY ESTIMATION 1485 Fig. 11. Velocities along motion boundaries of Walker sequence. One frame interval, upper left to lower right.

14 1486 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 11, NOVEMBER 2004 Fig. 12. Spatiotemporal surface evolution in the case of the Pedestrian Car sequence. to vary smoothly along motion boundaries. We proceed with the Hildreth iterations [12]. These iterations minimize where is the contour along which velocities are estimated. (23) III. EXPERIMENTAL VERIFICATION We validate the method on synthetic and real-image sequences. For every sequence tested, we first show the spatiotemporal surface at several stages during its evolution. Second, we provide tracking results obtained by taking tem-

15 FEGHALI AND MITICHE: SPATIOTEMPORAL MOTION BOUNDARY DETECTION AND MOTION BOUNDARY VELOCITY ESTIMATION 1487 Fig. 13. Tracking results for the Pedestrian Car sequence. One frame interval, upper left to lower right. poral cuts across the spatiotemporal surface. Finally, we show velocities estimated along motion boundaries. A. Rotating Fish Sequence The Rotating Fish sequence is a synthetically generated sequence where a fish is rotating against a background in translational motion. Fig. 1 shows the evolution of the spatiotemporal surface. The surface is simply initialized to a parallelepipedic shape. Fig. 2 shows the tracking results. Fig. 3 shows velocities along motion boundaries for the Rotating Fish sequence at different instants of time. Fig. 4 shows the effect of the curvature coefficient. Throughout its evolution, we simultaneously refine the estimation of the camera-induced motion parameters (see Fig. 5). At the end of its evolution, the surface enfolds the volume generated by the rotating fish. We note that locations of motion boundaries are accurately delineated specifically at acute corners of the fish. The surface velocity used to evolve the spatiotemporal surface is expressed as (24) Velocities estimates along motion boundaries are computed from geometrical properties of the spatiotemporal surface. We note that the field correctly denote the rotation movement of the fish.

16 1488 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 11, NOVEMBER 2004 Fig. 14. Velocities along motion boundaries of Pedestrian Car sequence. One frame interval, upper left to lower right. B. Person-Person Sequence The Person-Person sequence is an example of a sequence with multiple objects having nonsimultaneous motions. Note that the camera is static in this case. Fig. 6 shows the evolution of the spatiotemporal surface, initialized to to a parallelepipedic shape that enfolds all the volume of the image sequence. At the end of its evolution, the surface enfolds the volume generated by the two moving persons. Fig. 7 shows the tracking results. Initially, the person on the right moves while the person on the left is stationary. Both motion are detected as soon as they occur, although they do not start at the same time. The velocity used to evolve the spatiotemporal surface is expressed as (25)

17 FEGHALI AND MITICHE: SPATIOTEMPORAL MOTION BOUNDARY DETECTION AND MOTION BOUNDARY VELOCITY ESTIMATION 1489 Fig. 8 shows velocities along motion boundaries for the Person- Person sequence at different instants of time. We note that the motion field is consistent with the movement of the two persons. C. Walker Sequence The Walker sequence is a natural sequence where a man is walking on a street. A moving camera results in an apparent translational background motion. Figs. 9 and 10 show the spatiotemporal surface evolution and tracking results. The surface velocity in use is (26) Because tracking is solely based on a motion activity measurement, motion boundaries do not include portions of the pedestrian that are static during the person s movement. For instance, the bottom parts of the pedestrian s legs exhibit relatively low motion at certain instants in the sequence and are excluded. Velocities along motion boundaries are presented in Fig. 11 and are consistent with the pedestrian s overall motion trajectory. D. Pedestrian-Car Sequence Fig. 12 shows the spatiotemporal surface evolution in the case of the Pedestrian Car sequence. Fig. 13 presents tracking results for Pedestrian-Car sequence. The image sequence is that of a scene where a car drives by a walking pedestrian on a background with a translational motion. This example illustrates how a motion boundary splits when the images of two moving objects (the pedestrian and the car in this case) separate. Velocities along motion boundaries are given in Fig. 14. The pedestrian and the car have two contrasted motions. The regularization term used to compute an estimate of the velocities yields a smooth field that is biased to an average overall motion. Once the two objects separate, velocities correctly denote each the individual motions. IV. CONCLUSION We proposed an new approach to detect motion boundaries in the spatiotemporal domain and estimate velocity along these boundaries, to serve the purpose of tracking with a moving camera. Advantages of this approach include the detection of nonsimultaneous motions and simple initialization. Execution via the level-set partial differential equations enables topology free execution and numerically stable implementation. In contrast with current tracking methods, the estimation of camera-induced motion parameters is not performed as a separate step. Here, we exploit the existing coupling between both problems by simultaneously performing motion boundaries detection and camera-induced motion parameters estimation through a greedy scheme [34]. A method to estimate velocities along motion boundaries is proposed that is based on geometrical properties of the spatiotemporal surface. Experimental results show the validity of the proposed tracking approach and its potential. REFERENCES [1] S. Araki, T. Matsuoka, N. Yokoya, and H. Takemura, Real-time tracking of multiple moving object contours in a moving camera image sequence, IEICE Trans. Inform. Syst., vol. E83D, no. 7, pp , July [2] S. Ayer, P. Schroeter, and J. Bigun, Segmentation of moving objects by robust motion parameter estimation over multiple frames, in Proc. Eur. Conf. Computer Vision, vol. II, Stockholm, Sweden, 1994, pp [3] G. Aubert, M. Barlaud, O. Faugeras, and S. Jehan-Besson, Image segmentation using active contours: Calculus of variations or shape gradients?, Projet CReATIVe, I3S, Rapport de recherche, I3S/RR FR, May [4] M. Bertalmio, G. Sapiro, and G. Randall, Morphing active contours, IEEE Trans. Pattern Anal. Machine Intell., vol. 22, pp , July [5] V. Caselles, R. Kimmel, and G. Sapiro, Geodesic active contours, Int. J. Comput. Vis., vol. 22, no. 1, pp , [6] D. Comaniciu and V. Ramesh, Mean shift and optimal prediction for efficient object tracking, in Proc. Int. Conf. Image Processing, vol. 3, Vancouver, BC, Canada, Sept , 2000, pp [7] I. Cohen and G. Medioni, Detecting and tracking moving objects for video surveillance, in IEEE Proc. Computer Vision and Pattern Recognition, Fort Collins, CO, June 23 25, [8] M. P. Do Carmo, Differential Geometry of Curves and Surfaces. Englewood Cliffs, NJ: Prentice-Hall, [9] O. Faugeras, Three-Dimensional Computer Vision: A Geometric Viewpoint. MA: MIT Press, [10] J. H. Han and J. S. Park, Contour matching using epipolar geometry, IEEE Trans. Pattern Anal. Machine Intell., vol. 22, pp , Apr [11] F. Heitz and P. Bouthemy, Multimodal estimation of discontinuous optical flow using Markov random fields, IEEE Trans. Pattern Anal. Machine Intell., vol. 15, pp , Dec [12] E. Hildreth, Computation underlying the measurement of visual motion, Artif. Intell., vol. 23, no. 3, pp , [13] S. Jehan-Besson, M. Barlaud, and G. Aubert, Detection and tracking of moving objects using a new level set based method, in Proc. Int. Conf. Pattern Recognition, Barcelona, Spain, Sept [14], A 3-step algorithm using region-based active contours for video objects detection, EURASIP J. Appl. Signal Process., vol. 6, pp , [15], DREAM2S: deformable regions driven by an eulerian accurate minimization medthod, Int. J. Comput. Vis., vol. 1, no. 53, pp , [16] M. Kass, A. Witkin, and D. Terzopoulos, Snakes: Active contour models, in Proc. 1st Int.Conf. Computer Vision, 1987, pp [17] F. Leymarie and M. Levine, Tracking deformable objects in the plane using an active contour model, IEEE Trans. Pattern Anal. Machine Intell., vol. 15, pp , June [18] A. R. Mansouri, A. Olivier, and J. Konrad, Topology-independent region tracking with level sets, in Proc. Int. Conf. Image Processing, vol. 3, Vancouver, BC, Canada, Sept , 2000, pp [19] A. Mansouri, Region tracking via level set PDEs without motion computation, IEEE Trans. Pattern Anal. Machine Intell., vol. 24, pp , July [20] R. Mech and M. Wollborn, A noise robust method for 2D shape estimation of moving objects in video sequences considering a moving camera, Signal Processing, vol. 66, no. 2, pp , April [21] A. Mitiche, R. Feghali, and A. Mansouri, Tracking moving objects as spatiotemporal boundary detection, in Southwest Symposium on Image Analysis and Interpretation, Santa Fe, NM, 2002, pp [22], Motion tracking as spatiotemporal boundary detection, J. Robot. Auotmat. Syst., vol. 43, no. 1, pp , [23] M. Maziere and F. Chassaing, Segmentation and tracking of video objects: suited to content-based video indexing, interactive television and production systems, in ICIP, vol. 2, 2000, pp [24] R. C. Nelson, Qualitative detection of motion by a moving observer, Int. J. Comput. Vis., vol. 7, no. 1, pp , [25] J. M. Odobez and P. Bouthemy, Detection of multiple moving objects using multiscale MRF with camera motion compensation, in Proc. Int. Conf. Image Process., Austin, TX, Nov

18 1490 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 11, NOVEMBER 2004 [26] S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on the Hamilton-Jacobi formulation, J. Comput. Phys., vol. 79, pp , [27] N. Paragios and R. Deriche, Geodesic active contours and level sets for the detection and tracking of moving objects, IEEE Trans. Pattern Anal. Machine Intell., vol. 22, no. 3, pp , Mar [28], Geodesic active regions: a new paradigm to deal with frame partition problems in computer vision, J. Vis. Commun. Image Representation, vol. 13, pp , [29] V. Philomin, R. Duraiswami, and L. Davis, Pedestrian tracking from a moving vehicle, in Proc. IEEE Intelligent Vehicles Symp., Dearborn, MI, Oct. 3 5, 2000, pp [30] S. Kischenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi, Conformal curvature flows: From phase transitions to active vision, Archive Rational Mech. Anal., vol. 134, pp , [31] J. Sethian, Level Set Methods. Cambridge, U.K.: Cambridge Univ. Press, [32] A. Strehl and J. K. Aggarwal, MODEEP: a motion-based object detection and pose estimation method for airborne FLIR sequences, Mach. Vis. Applicat., vol. 11, no. 6, pp , Apr [33] W. B. Thompson and T. G. Pong, Detecting moving objects, Int. J. Comput. Vis., vol. 4, pp , [34] S. C. Zhu, T. S. Lee, and A. L. Yuille, Region competition: Unifying snakes, region growing, and Bayes/MDL for multiband image segmentation, IEEE Trans. Pattern Anal. Machine Intell., vol. 18, pp , Sept Rosario Feghali received the B.E. degree in electrical engineering from the École Supérieure des Ingénieurs de Beyrouth in 1998 and the Ph.D. degree from the Institut National de Recherche Scientifique, Montreal, QC, Canada. He is currently a Research Scientist with the Communications Research Centre, an agency of Industry Canada. His research interests are in the fields of computer vision and video compression. Amar Mitiche received the Licence És Sciences in mathematics from the University of Algiers, Algiers, Algeria, and the Ph.D. degree in computer science from the University of Texas, Austin. He is currently a Professor in the Department of telecommunications, Institut National de Recherche Scientifique, Montreal, QC, Canada. His research interests include computer vision, motion analysis in monocular and stereoscopic image sequences (detection, estimation, segmentation, and tracking) with a focus on methods based on level-set PDEs, and written text recognition with a focus on neural networks methods.