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Using prior shapes in geometric active contours in a variational framework
 IJCV
, 2002
"... Abstract. In this paper, we report an active contour algorithm that is capable of using prior shapes. The energy functional of the contour is modified so that the energy depends on the image gradient as well as the prior shape. The model provides the segmentation and the transformation that maps the ..."
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Cited by 112 (3 self)
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Abstract. In this paper, we report an active contour algorithm that is capable of using prior shapes. The energy functional of the contour is modified so that the energy depends on the image gradient as well as the prior shape. The model provides the segmentation and the transformation that maps the segmented contour to the prior shape. The active contour is able to find boundaries that are similar in shape to the prior, even when the entire boundary is not visible in the image (i.e., when the boundary has gaps). A level set formulation of the active contour is presented. The existence of the solution to the energy minimization is also established. We also report experimental results of the use of this contour on 2d synthetic images, ultrasound images and fMRI images. Classical active contours cannot be used in many of these images.
Image Segmentation Using Active Contours: Calculus Of Variations Or Shape Gradients?
 SIAM Applied Mathematics
, 2002
"... We consider the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals. We show that one can go from one class to the other by solving Poisson's or Helmholtz's equation with wellchosen boundary conditions. Using this equiva ..."
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Cited by 103 (29 self)
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We consider the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals. We show that one can go from one class to the other by solving Poisson's or Helmholtz's equation with wellchosen boundary conditions. Using this equivalence, we study the case of a large class of region functionals by standard methods of the calculus of variations and derive the corresponding EulerLagrange equations. We revisit this problem using the notion of shape derivative and show that the same equations can be elegantly derived without going through the unnatural step of converting the region integrals into boundary integrals. We also define a larger class of region functionals based on the estimation and comparison to a prototype of the probability density distribution of image features and show how the shape derivative tool allows us to easily compute the corresponding Gateaux derivatives and EulerLagrange equations. We finally apply this new functional to the problem of regions segmentation in sequences of color images. We briefly describe our numerical scheme and show some experimental results.
Dynamical statistical shape priors for level set based tracking
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2006
"... Abstract. In recent years, researchers have proposed to introduce statistical shape knowledge into the level set method in order to cope with insufficient lowlevel information. While these priors were shown to drastically improve the segmentation of images or image sequences, so far the focus has b ..."
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Cited by 101 (8 self)
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Abstract. In recent years, researchers have proposed to introduce statistical shape knowledge into the level set method in order to cope with insufficient lowlevel information. While these priors were shown to drastically improve the segmentation of images or image sequences, so far the focus has been on statistical shape priors that are timeinvariant. Yet, in the context of tracking deformable objects, it is clear that certain silhouettes may become more or less likely over time. In this paper, we tackle the challenge of learning dynamical statistical models for implicitly represented shapes. We show how these can be integrated into a segmentation process in a Bayesian framework for image sequence segmentation. Experiments demonstrate that such shape priors with memory can drastically improve the segmentation of image sequences. 1 Level Set Based Image Segmentation In 1988, Osher and Sethian [16] introduced the level set method 1 as a means to implicitly propagate boundaries C(t) in the image plane Ω ⊂ R 2 by evolving an
A level set algorithm for minimizing the MumfordShah functional in image processing
 IEEE WORKSHOP ON VARIATIONAL AND LEVEL SET METHODS
, 2001
"... We show how the piecewisesmooth MumfordShah segmentation problem [25] can be solved using the level set method of S. Osher and J. Sethian [26]. The obtained algorithm can be simultaneously used to denoise, segment, detectextract edges, and perform active contours. The proposed model is also a gen ..."
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Cited by 93 (11 self)
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We show how the piecewisesmooth MumfordShah segmentation problem [25] can be solved using the level set method of S. Osher and J. Sethian [26]. The obtained algorithm can be simultaneously used to denoise, segment, detectextract edges, and perform active contours. The proposed model is also a generalization of a previous active contour model without edges, proposed by the authors in [12], and of its extension to the case with more than two segments for piecewiseconstant segmentation [11]. Based on the Four Color Theorem, we can assume that in general, at most two level set functions are sufficient to detect and represent distinct objects of distinct intensities, with triple junctions, or Tjunctions.
A nonparametric statistical method for image segmentation using information theory and curve evolution
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2005
"... In this paper, we present a new informationtheoretic approach to image segmentation. We cast the segmentation problem as the maximization of the mutual information between the region labels and the image pixel intensities, subject to a constraint on the total length of the region boundaries. We as ..."
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Cited by 79 (1 self)
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In this paper, we present a new informationtheoretic approach to image segmentation. We cast the segmentation problem as the maximization of the mutual information between the region labels and the image pixel intensities, subject to a constraint on the total length of the region boundaries. We assume that the probability densities associated with the image pixel intensities within each region are completely unknown a priori, and we formulate the problem based on nonparametric density estimates. Due to the nonparametric structure, our method does not require the image regions to have a particular type of probability distribution and does not require the extraction and use of a particular statistic. We solve the informationtheoretic optimization problem by deriving the associated gradient flows and applying curve evolution techniques. We use levelset methods to implement the resulting evolution. The experimental results based on both synthetic and real images demonstrate that the proposed technique can solve a variety of challenging image segmentation problems. Furthermore, our method, which does not require any training, performs as good as methods based on training.
Optimal surface segmentation in volumetric images  a graphtheoretic approach
 IEEE TRANS. PATTERN ANAL. MACHINE INTELL
, 2006
"... Efficient segmentation of globally optimal surfaces representing object boundaries in volumetric data sets is important and challenging in many medical image analysis applications. We have developed an optimal surface detection method capable of simultaneously detecting multiple interacting surfaces ..."
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Cited by 77 (5 self)
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Efficient segmentation of globally optimal surfaces representing object boundaries in volumetric data sets is important and challenging in many medical image analysis applications. We have developed an optimal surface detection method capable of simultaneously detecting multiple interacting surfaces, in which the optimality is controlled by the cost functions designed for individual surfaces and by several geometric constraints defining the surface smoothness and interrelations. The method solves the surface segmentation problem by transforming it into computing a minimum st cut in a derived arcweighted directed graph. The proposed algorithm has a loworder polynomial time complexity and is computationally efficient. It has been extensively validated on more than 300 computersynthetic volumetric images, 72 CTscanned data sets of differentsized plexiglas tubes, and tens of medical images spanning various imaging modalities. In all cases, the approach yielded highly accurate results. Our approach can be readily extended to higherdimensional image segmentation.
Spatially adaptive techniques for level set methods and incompressible flow
 Comput. Fluids
"... Since the seminal work of [92] on coupling the level set method of [69] to the equations for twophase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes ..."
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Cited by 74 (15 self)
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Since the seminal work of [92] on coupling the level set method of [69] to the equations for twophase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interestingly, this work also demonstrated the largest weakness of the level set method, i.e. mass or information loss characteristic of most Eulerian capturing techniques. In fact, [92] introduced a partial differential equation for battling this weakness, without which their work would not have been possible. In this paper, we discuss both historical and most recent works focused on improving the computational accuracy of the level set method focusing in part on applications related to incompressible flow due to both its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as HamiltonJacobi WENO [46], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [27] and the coupled level set volume of fluid method [91], and adaptive gridding techniques such as the octree approach to free surface flows proposed in [56].
Localizing regionbased active contours
 IEEE TRANS. ON IMAGE PROCESSING
, 2008
"... In this paper, we propose a natural framework that allows any regionbased segmentation energy to be reformulated in a local way. We consider local rather than global image statistics and evolve a contour based on local information. Localized contours are capable of segmenting objects with heterog ..."
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Cited by 73 (3 self)
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In this paper, we propose a natural framework that allows any regionbased segmentation energy to be reformulated in a local way. We consider local rather than global image statistics and evolve a contour based on local information. Localized contours are capable of segmenting objects with heterogeneous feature profiles that would be difficult to capture correctly using a standard global method. The presented technique is versatile enough to be used with any global regionbased active contour energy and instill in it the benefits of localization. We describe this framework and demonstrate the localization of three wellknown energies in order to illustrate how our framework can be applied to any energy. We then compare each localized energy to its global counterpart to show the improvements that can be achieved. Next, an indepth study of the behaviors of these energies in response to the degree of localization is given. Finally, we show results on challenging images to illustrate the robust and accurate segmentations that are possible with this new class of active contour models.
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.
Distance Regularized Level Set Evolution and Its Application to Image Segmentation
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2010
"... Level set methods have been widely used in image processing and computer vision. In conventional level set formulations, the level set function typically develops irregularities during its evolution, which may cause numerical errors and eventually destroy the stability of the evolution. Therefore, ..."
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Cited by 72 (1 self)
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Level set methods have been widely used in image processing and computer vision. In conventional level set formulations, the level set function typically develops irregularities during its evolution, which may cause numerical errors and eventually destroy the stability of the evolution. Therefore, a numerical remedy, called reinitialization, is typically applied to periodically replace the degraded level set function with a signed distance function. However, the practice of reinitialization not only raises serious problems as when and how it should be performed, but also affects numerical accuracy in an undesirable way. This paper proposes a new variational level set formulation in which the regularity of the level set function is intrinsically maintained during the level set evolution. The level set evolution is derived as the gradient flow that minimizes an energy functional with a distance regularization term and an external energy that drives the motion of the zero