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53
Dense estimation of fluid flows
 IEEE Trans. Pattern Anal. Machine Intell
, 2002
"... AbstractÐIn this paper, we address the problem of estimating and analyzing the motion of fluids in image sequences. Due to the great deal of spatial and temporal distortions that intensity patterns exhibit in images of fluids, the standard techniques from Computer Vision, originally designed for qua ..."
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Cited by 128 (41 self)
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AbstractÐIn this paper, we address the problem of estimating and analyzing the motion of fluids in image sequences. Due to the great deal of spatial and temporal distortions that intensity patterns exhibit in images of fluids, the standard techniques from Computer Vision, originally designed for quasirigid motions with stable salient features, are not well adapted in this context. We thus investigate a dedicated minimizationbased motion estimator. The cost function to be minimized includes a novel data term relyingon an integrated version of the continuity equation of fluid mechanics, which is compatible with large displacements. This term is associated with an original secondorder divcurl regularization which prevents the washing out of the salient vorticity and divergence structures. The performance of the resulting fluid flow estimator is demonstrated on meteorological satellite images. In addition, we show how the sequences of dense motion fields we estimate can be reliably used to reconstruct trajectories and to extract the regions of high vorticity and divergence. Index TermsÐFluid motion, continuity equation, divcurl regularization, nonconvex minimization, trajectories, vorticity, and divergence concentration. 1
Motion competition: a variational approach to piecewise parametric motion segmentation
 Int. J. Comput. Vision
, 2005
"... Abstract. We present a novel variational approach for segmenting the image plane into a set of regions of parametric motion on the basis of two consecutive frames from an image sequence. Our model is based on a conditional probability for the spatiotemporal image gradient, given a particular veloci ..."
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Cited by 73 (10 self)
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Abstract. We present a novel variational approach for segmenting the image plane into a set of regions of parametric motion on the basis of two consecutive frames from an image sequence. Our model is based on a conditional probability for the spatiotemporal image gradient, given a particular velocity model, and on a geometric prior on the estimated motion field favoring motion boundaries of minimal length. Exploiting the Bayesian framework, we derive a cost functional which depends on parametric motion models for each of a set of regions and on the boundary separating these regions. The resulting functional can be interpreted as an extension of the MumfordShah functional from intensity segmentation to motion segmentation. In contrast to most alternative approaches, the problems of segmentation and motion estimation are jointly solved by continuous minimization of a single functional. Minimizing this functional with respect to its dynamic variables results in an eigenvalue problem for the motion parameters and in a gradient descent evolution for the motion discontinuity set. We propose two different representations of this motion boundary: an explicit splinebased implementation which can be applied to the motionbased tracking of a single moving object, and an implicit multiphase level set implementation which allows for the segmentation of an arbitrary number of multiply connected moving objects. Numerical results both for simulated ground truth experiments and for realworld sequences demonstrate the capacity of our approach to segment objects based exclusively on their relative motion.
Shape Recovery Algorithms Using Level Sets in 2D/3D Medical Imagery: A StateoftheArt Review
, 2001
"... The class of geometric deformable models, socalled level sets, has brought tremendous impact to medical imagery due to its capability to preserve topology and fast shape recovery. In an effort to facilitate a clear and full understanding of these powerful stateoftheart applied mathematical tools ..."
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Cited by 52 (2 self)
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The class of geometric deformable models, socalled level sets, has brought tremendous impact to medical imagery due to its capability to preserve topology and fast shape recovery. In an effort to facilitate a clear and full understanding of these powerful stateoftheart applied mathematical tools, this paper is an attempt to explore these geometric methods, their implementations and integration of regularization terms to improve the robustness of these topologically independent propagating curves/surfaces. This paper first presents the origination of the level sets, followed by the taxonomy tree of level sets. We then derive the fundamental equation of curve/surface evolution and zerolevel curves/surfaces. The paper then focuses on the first core class of level sets, the socalled level sets "without regularizers". The next section is devoted on a second kind, socalled level sets "with regularizers". In this class, we present four kinds of systems on the design of the regularizers. Next, the paper presents a third kind of level sets, socalled the "bubblebased" techniques. An entire section is dedicated to optimization and quantification techniques for shape recovery when used with the level sets. Finally, the paper concludes with 22 general merits and four demerits on level sets and the future of level sets in medical image segmentation. We present the applications of level sets to complex shapes likethehuman cortex acquired via MRI for neurological image analysis.
Analysis of HalfQuadratic Minimization Methods for Signal and Image Recovery
, 2003
"... Abstract. We address the minimization of regularized convex cost functions which are customarily used for edgepreserving restoration and reconstruction of signals and images. In order to accelerate computation, the multiplicative and the additive halfquadratic reformulation of the original cost ..."
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Cited by 51 (8 self)
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Abstract. We address the minimization of regularized convex cost functions which are customarily used for edgepreserving restoration and reconstruction of signals and images. In order to accelerate computation, the multiplicative and the additive halfquadratic reformulation of the original costfunction have been pioneered in Geman and Reynolds [IEEE Trans. Pattern Anal. Machine Intelligence, 14 (1992), pp. 367–383] and Geman and Yang [IEEE Trans. Image Process., 4 (1995), pp. 932–946]. The alternate minimization of the resultant (augmented) costfunctions has a simple explicit form. The goal of this paper is to provide a systematic analysis of the convergence rate achieved by these methods. For the multiplicative and additive halfquadratic regularizations, we determine their upper bounds for their rootconvergence factors. The bound for the multiplicative form is seen to be always smaller than the bound for the additive form. Experiments show that the number of iterations required for convergence for the multiplicative form is always less than that for the additive form. However, the computational cost of each iteration is much higher for the multiplicative form than for the additive form. The global assessment is that minimization using the additive form of halfquadratic regularization is faster than using the multiplicative form. When the additive form is applicable, it is hence recommended. Extensive experiments demonstrate that in our MATLAB implementation, both methods are substantially faster (in terms of computational times) than the standard MATLAB Optimization Toolbox routines used in our comparison study.
A Variational Framework for Simultaneous Motion Estimation and Restoration of MotionBlurred Video
, 2007
"... Figure 1. From two real blurred frames (left), we automatically and simultaneously estimate the motion region, the motion vector, and the image intensity of the foreground (middle). Based on this and the background intensity we reconstruct the two frames (right). ..."
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Cited by 36 (1 self)
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Figure 1. From two real blurred frames (left), we automatically and simultaneously estimate the motion region, the motion vector, and the image intensity of the foreground (middle). Based on this and the background intensity we reconstruct the two frames (right).
Video Object Segmentation Using Eulerian RegionBased Active Contours
 in International Conference on Computer Vision
, 2001
"... We address the problem of moving object segmentation using active contours. As far as segmentation of moving objects is concerned, regionbased terms must be incorporated in the evolution equation of the active contour, in addition to classical boundarybased terms. In this paper, we propose a gener ..."
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Cited by 34 (19 self)
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We address the problem of moving object segmentation using active contours. As far as segmentation of moving objects is concerned, regionbased terms must be incorporated in the evolution equation of the active contour, in addition to classical boundarybased terms. In this paper, we propose a general framework for regionbased active contours. Novel aspects of the segmentation method include a new Eulerian proof to compute the evolution equation of the active contour from the minimization of a criterion, and the introduction of functions named "descriptors" of the regions. In this proof, the dynamical scheme is directly introduced in the criterion before differentiation. With such a method, the case of descriptors depending on the evolution of the curve, i.e. depending upon features globally attached to the region, can readily be taken into account. The variation of these descriptors upon the evolution of the curve induces additional terms in the evolution equation of the active contour. The proof ensures the fastest decrease of the active contour towards a minimum of the criterion. Inside this theoretical framework, a set of descriptors is evaluated on real sequences for the detection of moving objects. 1.
Variational SpaceTime Motion Segmentation
, 2003
"... We propose a variational method for segmenting image sequences into spatiotemporal domains of homogeneous motion. To this end, we formulate the problem of motion estimation in the framework of Bayesian inference, using a prior which favors domain boundaries of minimal surface area. We derive a cost ..."
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Cited by 33 (5 self)
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We propose a variational method for segmenting image sequences into spatiotemporal domains of homogeneous motion. To this end, we formulate the problem of motion estimation in the framework of Bayesian inference, using a prior which favors domain boundaries of minimal surface area. We derive a cost functional which depends on a surface in spacetime separating a set of motion regions, as well as a set of vectors modeling the motion in each region. We propose a multiphase level set formulation of this functional, in which the surface and the motion regions are represented implicitly by a vectorvalued level set function. Joint minimization of the proposed functional results in an eigenvalue problem for the motion model of each region and in a gradient descent evolution for the separating interface. Numerical results on realworld sequences demonstrate that minimization of a single cost functional generates a segmentation of spacetime into multiple motion regions.
A variational framework for image segmentation combining motion estimation and shape regularization
 IEEE Conf. on Comp. Vis. and Patt. Recog
, 2003
"... Based on a geometric interpretation of the optic flow constraint equation, we propose a conditional probability on the spatiotemporal image gradient. We consistently derive a variational approach for the segmentation of the image domain into regions of homogeneous motion. The proposed energy functi ..."
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Cited by 27 (8 self)
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Based on a geometric interpretation of the optic flow constraint equation, we propose a conditional probability on the spatiotemporal image gradient. We consistently derive a variational approach for the segmentation of the image domain into regions of homogeneous motion. The proposed energy functional extends the MumfordShah functional from gray value segmentation to motion segmentation. It depends on the spatiotemporal image gradient calculated from only two consecutive images of an image sequence. Moreover, it depends on motion vectors for a set of regions and a boundary separating these regions. In contrast to most alternative approaches, the problems of motion estimation and motion segmentation are jointly solved by minimizing a single functional. Numerical evaluation with both explicit and implicit (level set based) representations of the boundary shows the strengths and limitations of our approach. 1. Introduction and Related
Statistical Shape Knowledge in Variational Motion Segmentation
 IMAGE AND VISION COMPUTING
, 2002
"... We present a generative approach to modelbased motion segmentation by incorporating a statistical shape prior into a novel variational segmentation method. The shape prior statistically encodes a training set of object outlines presented in advance during a training phase. In a region ..."
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Cited by 27 (3 self)
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We present a generative approach to modelbased motion segmentation by incorporating a statistical shape prior into a novel variational segmentation method. The shape prior statistically encodes a training set of object outlines presented in advance during a training phase. In a region
An integral solution to surface evolution PDEs via geocuts
 In ECCV
, 2006
"... Abstract. We introduce a new approach to modelling gradient flows of contours and surfaces. While standard variational methods (e.g. level sets) compute local interface motion in a differential fashion by estimating local contour velocity via energy derivatives, we propose to solve surface evolution ..."
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Cited by 25 (7 self)
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Abstract. We introduce a new approach to modelling gradient flows of contours and surfaces. While standard variational methods (e.g. level sets) compute local interface motion in a differential fashion by estimating local contour velocity via energy derivatives, we propose to solve surface evolution PDEs by explicitly estimating integral motion of the whole surface. We formulate an optimization problem directly based on an integral characterization of gradient flow as an infinitesimal move of the (whole) surface giving the largest energy decrease among all moves of equal size. We show that this problem can be efficiently solved using recent advances in algorithms for global hypersurface optimization [4, 2, 11]. In particular, we employ the geocuts method [4] that uses ideas from integral geometry to represent continuous surfaces as cuts on discrete graphs. The resulting interface evolution algorithm is validated on some 2D and 3D examples similar to typical demonstrations of levelset methods. Our method allows for computation of gradient flows for hypersurfaces with respect to a fairly general class of continuous functionals and it is flexible with respect to distance metrics on the space of contours/surfaces. Our preliminary tests for standard L2 distance metric demonstrate numerical stability, topological changes and an absence of any oscillatory motion. Index Terms — front propagation, gradient flows for hypersurfaces, mincut/maxflow algorithms on graphs, geocuts, integral geometry. 2 1