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Shape complexity is a hard-to-quantify quality, mainly due to its relative nature. Biased by Euclidean thinking, circles are commonly considered as the simplest. However, their constructions as digital images are only approximations to the ideal form. Consequently, complexity orders computed in reference to circle are unstable. Unlike circles which lose their circleness in digital images, squares retain their qualities. Hence, we consider squares (hypercubes in $\mathbb Z^{n}$) to be the simplest shapes relative to which complexity orders are constructed. Using the connection between $L^\infty $ norm and squares we effectively encode squareness-adapted simplification through which we obtain multi-scale complexity measure, where scale determines the level of interest to the boundary. The emergent scale above which the effect of a boundary feature (appendage) disappears is related to the ratio of the contacting width of the appendage to that of the main body. We discuss what zero complexity implies in terms of information repetition and constructibility and what kind of shapes in addition to squares have zero complexity. [ABSTRACT FROM AUTHOR]

In this paper, we propose an image segmentation model that incorporates convexity shape priori using level set representations. In the past decade, several discrete and continuous methods have been developed to solve this problem. Our method comes from the observation that the signed distance function of a convex region must be a convex function. Based on this observation, we transfer the complicated geometrical convexity shape priori into some simple constraints on the signed distance function. We propose a simple algorithm to keep these constraints exactly. The proposed method could be easily applied to level set based segmentation models, such as the well-known Chan-Vese mode and the active contour models. By setting some good initial curves, the proposed method can easily segment convex objects from images with complicated background. We demonstrate the performance of the proposed methods on both synthetic images and real images, as well as the comparison to some state-of-the-art methods. [ABSTRACT FROM AUTHOR]

The active contour models with level set evolution have been visited with a vast number of methods for image segmentation. They can be mainly classified into region-based and edge-based models, and it has been validated that the hybrid variants combining both region and edge information can improve the segmentation performance. However, to the best of our knowledge, the theoretical foundation of collaboration mechanism between the region and the edge information is limited. Specifically, most existing hybrid models are just combining all the energy terms together, resulting in great challenges of choosing an appropriate weight coefficient for each term and accommodating different modalities of imaging. To overcome these difficulties, this paper proposes a region and edge synergetic level set framework named RESLS. It provides an approach to construct new hybrid level set models using a normalized intensity indicator function that allows the region information easily embedding into the edge-based model. In this case, the energy weights of region and edge terms can be constrained by the global optimization condition deduced from the framework. Some representative as well as state-of-the-art models are taken as examples to demonstrate the generality of our method. The experiments validate that under the guidance of the optimization condition, the weighting parameter of each term can be reliably chosen. Meanwhile, the segmentation accuracy, robustness, and computational efficiency of RESLS can be improved compared with its component models. [ABSTRACT FROM AUTHOR]

Level set methods are widely used for image segmentation because of their convenient shape representation for numerical computations and capability to handle topological changes. However, in spite of the numerous works in the literature, the use of level set methods in image segmentation still has several drawbacks. These shortcomings include formation of irregularities of the signed distance function, sensitivity to initialization, lack of locality, and expensive computational cost, which increases dramatically as the number of objects to be simultaneously segmented grows. In this paper, we propose a novel parametric level set method called disjunctive normal level set (DNLS), and apply it to both two-phase (single object) and multiphase (multiobject) image segmentations. DNLS is a differentiable model formed by the union of polytopes, which themselves are created by intersections of half-spaces. We formulate the segmentation algorithm in a Bayesian framework and use a variational approach to minimize the energy with respect to the parameters of the model. The proposed DNLS can be considered as an open framework that allows the use of different appearance models and shape priors. Compared with the conventional level sets available in the literature, the proposed DNLS has the following major advantages: it requires significantly less computational time and memory, it naturally keeps the level set function regular during the evolution, it is more suitable for multiphase and local region-based image segmentations, and it is less sensitive to noise and initialization. The experimental results show the potential of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents a fast and robust level set method for image segmentation. To enhance the robustness against noise, we embed a Markov random field (MRF) energy function to the conventional level set energy function. This MRF energy function builds the correlation of a pixel with its neighbors and encourages them to fall into the same region. To obtain a fast implementation of the MRF embedded level set model, we explore algebraic multigrid (AMG) and sparse field method (SFM) to increase the time step and decrease the computation domain, respectively. Both AMG and SFM can be conducted in a parallel fashion, which facilitates the processing of our method for big image databases. By comparing the proposed fast and robust level set method with the standard level set method and its popular variants on noisy synthetic images, synthetic aperture radar (SAR) images, medical images, and natural images, we comprehensively demonstrate the new method is robust against various kinds of noises. In particular, the new level set method can segment an image of size $500 \times 500$ within 3 s on MATLAB R2010b installed in a computer with 3.30-GHz CPU and 4-GB memory. [ABSTRACT FROM AUTHOR]

We address the two inherently related problems of segmentation and interpolation of 3D and 4D sparse data and propose a new method to integrate these stages in a level set framework. The interpolation process uses segmentation information rather than pixel intensities for increased robustness and accuracy. The method supports any spatial configurations of sets of 2D slices having arbitrary positions and orientations. We achieve this by introducing a new level set scheme based on the interpolation of the level set function by radial basis functions. The proposed method is validated quantitatively and/or subjectively on artificial data and MRI and CT scans and is compared against the traditional sequential approach, which interpolates the images first, using a state-of-the-art image interpolation method, and then segments the interpolated volume in 3D or 4D. In our experiments, the proposed framework yielded similar segmentation results to the sequential approach but provided a more robust and accurate interpolation. In particular, the interpolation was more satisfactory in cases of large gaps, due to the method taking into account the global shape of the object, and it recovered better topologies at the extremities of the shapes where the objects disappear from the image slices. As a result, the complete integrated framework provided more satisfactory shape reconstructions than the sequential approach. [ABSTRACT FROM PUBLISHER]

A new variational level set approach is proposed for lung nodule segmentation in lung CT scans. A general lung nodule shape model is proposed using implicit spaces as a signed distance function. The shape model is fused with the image intensity statistical information in a variational segmentation framework. The nodule shape model is mapped to the image domain by a global transformation that includes inhomogeneous scales, rotation, and translation parameters. A matching criteria between the shape model and the image implicit representations is employed to handle the alignment process. Transformation parameters evolve through gradient descent optimization to handle the shape alignment process and hence mark the boundaries of the nodule “head.” The embedding process takes into consideration the image intensity as well as prior shape information. A nonparametric density estimation approach is employed to handle the statistical intensity representation of the nodule and background regions. The proposed technique does not depend on nodule type or location. Exhaustive experimental and validation results are demonstrated on 742 nodules obtained from four different CT lung databases, illustrating the robustness of the approach. [ABSTRACT FROM AUTHOR]

Due to its intrinsic nature which allows to easily handle complex shapes and topological changes, the level set method (LSM) has been widely used in image segmentation. Nevertheless, LSM is computationally expensive, which limits its applications in real-time systems. For this purpose, we propose a new level set algorithm, which uses simultaneously edge, region, and 2D histogram information in order to efficiently segment objects of interest in a given scene. The computational complexity of the proposed LSM is greatly reduced by using the highly parallelizable lattice Boltzmann method (LBM) with a body force to solve the level set equation (LSE). The body force is the link with image data and is defined from the proposed LSE. The proposed LSM is then implemented using an NVIDIA graphics processing units to fully take advantage of the LBM local nature. The new algorithm is effective, robust against noise, independent to the initial contour, fast, and highly parallelizable. The edge and region information enable to detect objects with and without edges, and the 2D histogram information enable the effectiveness of the method in a noisy environment. Experimental results on synthetic and real images demonstrate subjectively and objectively the performance of the proposed method. [ABSTRACT FROM PUBLISHER]

We propose a compressive sensing algorithm that exploits geometric properties of images to recover images of high quality from few measurements. The image reconstruction is done by iterating the two following steps: 1) estimation of normal vectors of the image level curves, and 2) reconstruction of an image fitting the normal vectors, the compressed sensing measurements, and the sparsity constraint. The proposed technique can naturally extend to nonlocal operators and graphs to exploit the repetitive nature of textured images to recover fine detail structures. In both cases, the problem is reduced to a series of convex minimization problems that can be efficiently solved with a combination of variable splitting and augmented Lagrangian methods, leading to fast and easy-to-code algorithms. Extended experiments show a clear improvement over related state-of-the-art algorithms in the quality of the reconstructed images and the robustness of the proposed method to noise, different kind of images, and reduced measurements. [ABSTRACT FROM PUBLISHER]

In this paper, a novel variational model based on prior shapes for simultaneous object classification and segmentation is proposed. Given a set of training shapes of multiple object classes, a sparse linear combination of training shapes in a low-dimensional representation is used to regularize the target shape in variational image segmentation. By minimizing the proposed variational functional, the model is able to automatically select the reference shapes that best represent the object by sparse recovery and accurately segment the image, taking into account both the image information and the shape priors. For some applications under an appropriate size of training set, the proposed model allows artificial enlargement of the training set by including a certain number of transformed shapes for transformation invariance, and then the model remains jointly convex and can handle the case of overlapping or multiple objects presented in an image within a small range. Numerical experiments show promising results and the potential of the method for object classification and segmentation. [ABSTRACT FROM AUTHOR]

The level set method is a popular technique for tracking moving interfaces in several disciplines, including computer vision and fluid dynamics. However, despite its high flexibility, the original level set method is limited by two important numerical issues. First, the level set method does not implicitly preserve the level set function as a distance function, which is necessary to estimate accurately geometric features, s.a. the curvature or the contour normal. Second, the level set algorithm is slow because the time step is limited by the standard Courant–Friedrichs–Lewy (CFL) condition, which is also essential to the numerical stability of the iterative scheme. Recent advances with graph cut methods and continuous convex relaxation methods provide powerful alternatives to the level set method for image processing problems because they are fast, accurate, and guaranteed to find the global minimizer independently to the initialization. These recent techniques use binary functions to represent the contour rather than distance functions, which are usually considered for the level set method. However, the binary function cannot provide the distance information, which can be essential for some applications, s.a. the surface reconstruction problem from scattered points and the cortex segmentation problem in medical imaging. In this paper, we propose a fast algorithm to preserve distance functions in level set methods. Our algorithm is inspired by recent efficient \ell^1 optimization techniques, which will provide an efficient and easy to implement algorithm. It is interesting to note that our algorithm is not limited by the CFL condition and it naturally preserves the level set function as a distance function during the evolution, which avoids the classical re-distancing problem in level set methods. We apply the proposed algorithm to carry out image segmentation, where our methods prove to be 5–6 times faster than standard distance preserving level set techniques. We also present two applications where preserving a distance function is essential. Nonetheless, our method stays generic and can be applied to any level set methods that require the distance information. [ABSTRACT FROM AUTHOR]

Existing computer simulations of aircraft infrared signature (IRS) do not account for dispersion induced by uncertainty on input data, such as aircraft aspect angles and meteorological conditions. As a result, they are of little use to estimate the detection performance of IR optronic systems; in this case, the scenario encompasses a lot of possible situations that must be indeed addressed, but cannot be singly simulated. In this paper, we focus on low-resolution infrared sensors and we propose a methodological approach for predicting simulated IRS dispersion of poorly known aircraft and performing aircraft detection on the resulting set of low-resolution infrared images. It is based on a sensitivity analysis, which identifies inputs that have negligible influence on the computed IRS and can be set at a constant value, on a quasi-Monte Carlo survey of the code output dispersion, and on a new detection test taking advantage of level sets estimation. This method is illustrated in a typical scenario, i.e., a daylight air-to-ground full-frontal attack by a generic combat aircraft flying at low altitude, over a database of 90 000 simulated aircraft images. Assuming a white noise or a fractional Brownian background model, detection performances are very promising. [ABSTRACT FROM AUTHOR]

Dual-energy computerized tomography has gained great interest because of its ability to characterize the chemical composition of a material rather than simply providing relative attenuation images as in conventional tomography. The purpose of this paper is to introduce a novel polychromatic dual-energy processing algorithm, with an emphasis on detection and characterization of piecewise constant objects embedded in an unknown cluttered background. Physical properties of the objects, particularly the Compton scattering and photoelectric absorption coefficients, are assumed to be known with some level of uncertainty. Our approach is based on a level-set representation of the characteristic function of the object and encompasses a number of regularization techniques for addressing both the prior information we have concerning the physical properties of the object and the fundamental physics-based limitations associated with our ability to jointly recover the Compton scattering and photoelectric absorption properties of the scene. In the absence of an object with appropriate physical properties, our approach returns a null characteristic function and, thus, can be viewed as simultaneously solving the detection and characterization problems. Unlike the vast majority of methods that define the level-set function nonparametrically, i.e., as a dense set of pixel values, we define our level set parametrically via radial basis functions and employ a Gauss–Newton-type algorithm for cost minimization. Numerical results show that the algorithm successfully detects objects of interest, finds their shape and location, and gives an adequate reconstruction of the background. [ABSTRACT FROM PUBLISHER]

We present an algorithm to compute the connected components of upper or lower level sets from level lines. The running time is O (n + \vert \tau \vert), where n is the size of the image and \vert \tau \vert is the number of level lines. [ABSTRACT FROM AUTHOR]

Intensity inhomogeneity often occurs in real-world images, which presents a considerable challenge in image segmentation. The most widely used image segmentation algorithms are region-based and typically rely on the homogeneity of the image intensities in the regions of interest, which often fail to provide accurate segmentation results due to the intensity inhomogeneity. This paper proposes a novel region-based method for image segmentation, which is able to deal with intensity inhomogeneities in the segmentation. First, based on the model of images with intensity inhomogeneities, we derive a local intensity clustering property of the image intensities, and define a local clustering criterion function for the image intensities in a neighborhood of each point. This local clustering criterion function is then integrated with respect to the neighborhood center to give a global criterion of image segmentation. In a level set formulation, this criterion defines an energy in terms of the level set functions that represent a partition of the image domain and a bias field that accounts for the intensity inhomogeneity of the image. Therefore, by minimizing this energy, our method is able to simultaneously segment the image and estimate the bias field, and the estimated bias field can be used for intensity inhomogeneity correction (or bias correction). Our method has been validated on synthetic images and real images of various modalities, with desirable performance in the presence of intensity inhomogeneities. Experiments show that our method is more robust to initialization, faster and more accurate than the well-known piecewise smooth model. As an application, our method has been used for segmentation and bias correction of magnetic resonance (MR) images with promising results. [ABSTRACT FROM AUTHOR]

We propose a variant of the Mumford–Shah model for the segmentation of a pair of overlapping objects with additive intensity value. Unlike standard segmentation models, it does not only determine distinct objects in the image, but also recover the possibly multiple membership of the pixels. To accomplish this, some a priori knowledge about the smoothness of the object boundary is integrated into the model. Additivity is imposed through a soft constraint which allows the user to control the degree of additivity and is more robust than the hard constraint. We also show analytically that the additivity parameter can be chosen to achieve some stability conditions. To solve the optimization problem involving geometric quantities efficiently, we apply a multiphase level set method. Segmentation results on synthetic and real images validate the good performance of our model, and demonstrate the model's applicability to images with multiple channels and multiple objects. [ABSTRACT FROM AUTHOR]

We propose a new approach to front propagation algorithms based on a topological variant of well-composedness which contrasts with previous methods based on simple point detection. This provides for a theoretical justification, based on the digital Jordan separation theorem, for digitally “gluing” evolved well-composed objects separated by well-composed curves or surfaces. Additionally, our framework can be extended to more relaxed topologically constrained algorithms based on multisimple points. For both methods this framework has the additional benefit of obviating the requirement for both a user-specified connectivity and a topologically-consistent marching cubes/squares algorithm in meshing the resulting segmentation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel 3-D deformable model that is based upon a geometrically induced external force field which can be conveniently generalized to arbitrary dimensions. This external force field is based upon hypothesized interactions between the relative geometries of the deformable model and the object boundary characterized by image gradient. The evolution of the deformable model is solved using the level set method so that topological changes are handled automatically. The relative geometrical configurations between the deformable model and the object boundaries contribute to a dynamic vector force field that changes accordingly as the deformable model evolves. The geometrically induced dynamic interaction force has been shown to greatly improve the deformable model performance in acquiring complex geometries and highly concave boundaries, and it gives the deformable model a high invariancy in initialization configurations. The voxel interactions across the whole image domain provide a global view of the object boundary representation, giving the external force a long attraction range. The bidirectionality of the external force field allows the new deformable model to deal with arbitrary cross-boundary initializations, and facilitates the handling of weak edges and broken boundaries. In addition, we show that by enhancing the geometrical interaction field with a nonlocal edge-preserving algorithm, the new deformable model can effectively overcome image noise. We provide a comparative study on the segmentation of various geometries with different topologies from both synthetic and real images, and show that the proposed method achieves significant improvements against existing image gradient techniques. [ABSTRACT FROM PUBLISHER]

This paper investigates variational region-level criterion for supervised and unsupervised texture-based image segmentation. The focus is given to the demonstration of the effectiveness and robustness of this region-based formulation compared to most common variational approaches. The main contributions of this global criterion are twofold. First, the proposed methods circumvent a major problem related to classical texture based segmentation approaches. Existing methods, even if they use different and various texture features, are mainly stated as the optimization of a criterion evaluating punctual pixel likelihoods or similarity measure computed within a local neighborhood. These approaches require sufficient dissimilarity between the considered texture features. An additional limitation is the choice of the neighborhood size and shape. These two parameters and especially the neighborhood size significantly influence the classification performances: the neighborhood must be large enough to capture texture structures and small enough to guarantee segmentation accuracy. These parameters are often set experimentally. These limitations are mitigated with the proposed variational methods stated at the region-level. It resorts to an energy criterion defined on image where regions are characterized by nonparametric distributions of their responses to a set of filters. In the supervised case, the segmentation algorithm consists in the minimization of a similarity measure between region-level statistics and texture prototypes and a boundary based functional that imposes smoothness and regularity on region boundaries. In the unsupervised case, the data-driven term involves the maximization of the dissimilarity between regions. The proposed similarity measure is generic and permits optimally fusing various types of texture features. It is defined as a weighted sum of Kullback–Leibler divergences between feature distributions. The optimization of the proposed variational criteria is carried out using a level-set formulation. The effectiveness and the robustness of this formulation at region-level, compared to classical active contour methods, are evaluated for various Brodatz and natural images. [ABSTRACT FROM AUTHOR]

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 level set toward desired locations. The distance regularization term is defined with a potential function such that the derived level set evolution has a unique forward-and-backward (FAB) diffusion effect, which is able to maintain a desired shape of the level set function, particularly a signed distance profile near the zero level set. This yields a new type of level set evolution called distance regularized level set evolution (DRLSE). The distance regularization effect eliminates the need for reinitialization and thereby avoids its induced numerical errors. In contrast to complicated implementations of conventional level set formulations, a simpler and more efficient finite difference scheme can be used to implement the DRLSE formulation. DRLSE also allows the use of more general and efficient initialization of the level set function. In its numerical implementation, relatively large time steps can be used in the finite difference scheme to reduce the number of iterations, while ensuring sufficient numerical accuracy. To demonstrate the effectiveness of the DRLSE formulation, we apply it to an edge-based active contour model for image segmentation, and provide a simple narrowband implementation to greatly reduce computational cost. [ABSTRACT FROM AUTHOR]

This paper presents an extension of our recently introduced MAC model to deal with the initialization dependency problem that commonly appears in edge-based approaches. Its dynamic force field, unique bidirectionality, and constrained diffusion-based level set evolution provide great freedom in contour initialization and show significant improvements in initialization independency compared to other edge-based techniques. It handle more sophisticated topological changes than splitting and merging. It provides new potentials for edge-based active contour methods, particularly when detecting and localizing objects with unknown location, geometry, and topology. [ABSTRACT FROM AUTHOR]

In the field of image segmentation, most level-set-based active-contour approaches take advantage of a discrete representation of the associated implicit function. We present in this paper a different formulation where the implicit function is modeled as a continuous parametric function expressed on a B-spline basis. Starting from the active-contour energy functional, we show that this formulation allows us to compute the solution as a restriction of the variational problem on the space spanned by the B-splines. As a consequence, the minimization of the functional is directly obtained in terms of the B-spline coefficients. We also show that each step of this minimization may be expressed through a convolution operation. Because the B-spline functions are separable, this convolution may in turn be performed as a sequence of simple 1-D convolutions, which yields an efficient algorithm. As a further consequence, each step of the level-set evolution may be interpreted as a filtering operation with a B-spline kernel. Such filtering induces an intrinsic smoothing in the algorithm, which can be controlled explicitly via the degree and the scale of the chosen B-spline kernel. We illustrate the behavior of this approach on simulated as well as experimental images from various fields. [ABSTRACT FROM AUTHOR]

In current level set image segmentation methods, the number of regions is assumed to known beforehand. As a result, it remains constant during the optimization of the objective functional. How to allow it to vary is an important question which has been generally avoided. This study investigates a region merging prior related to regions area to allow the number of regions to vary automatically during curve evolution, thereby optimizing the objective functional implicitly with respect to the number of regions. We give a statistical interpretation to the coefficient of this prior to balance its effect systematically against the other functional terms. We demonstrate the validity and efficiency of the method by testing on real images of intensity, color, and motion. [ABSTRACT FROM AUTHOR]

The Mumford-Shah model is one of the most successful image segmentation models. However, existing algorithms for the model are often very sensitive to the choice of the initial guess. To make use of the model effectively, it is essential to develop an algorithm which can compute a global or near global optimal solution efficiently. While gradient descent based methods are well-known to find a local minimum only, even many stochastic methods do not provide a practical solution to this problem either. In this paper, we consider the computation of a global minimum of the multiphase piecewise constant Mumford-Shah model. We propose a hybrid approach which combines gradient based and stochastic optimization methods to resolve the problem of sensitivity to the initial guess. At the heart of our algorithm is a well-designed basin hopping scheme which uses global updates to escape from local traps in a way that is much more effective than standard stochastic methods. In our experiments, a very high-quality solution is obtained within a few stochastic hops whereas the solutions obtained with simulated annealing are incomparable even after thousands of steps. We also propose a multiresolution approach to reduce the computational cost and enhance the search for a global minimum. Furthermore, we derived a simple but useful theoretical result relating solutions at different spatial resolutions. [ABSTRACT FROM AUTHOR]

Edge-based and region-based active contours are frequently used in image segmentation. While edges characterize small neighborhoods of pixels, region descriptors characterize entire image regions that may have overlapping probability densities. In this paper, we propose to characterize image regions locally by defining Local Region Descriptors (LRDs). These are essentially feature statistics from pixels located within windows centered on the evolving contour, and they may reduce the overlap between distributions. LRDs are used to define general-form energies based on level sets. In general, a particular energy is associated with an active contour by means of the logarithm of the probability density of features conditioned on the region. In order to reduce the number of local minima of such energies, we introduce two novel functions for constructing the energy functional which are both based on the assumption that local densities are approximately Gaussian. The first uses a similarity measure between features of pixels that involves confidence intervals. The second employs a local Markov Random Field (MRF) model. By minimizing the associated energies, we obtain active contours that can segment objects that have largely overlapping global probability densities. Our experiments show that the proposed method can accurately segment natural large images in very short time when using a fast level-set implementation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a natural framework that allows any region-based segmentation energy to be re-formulated 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 region-based active contour energy and instill in it the benefits of localization. We describe this framework and demonstrate the localization of three well-known 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 in-depth 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. [ABSTRACT FROM AUTHOR]

The implicit framework of the level-set method has several advantages when tracking propagating fronts. Indeed, the evolving contour is embedded in a higher dimensional level-set function and its evolution can be phrased in terms of a Eulerian formulation. The ability of this intrinsic method to handle topological changes (merging and breaking) makes it useful in a wide range of applications (fluid mechanics, computer vision) and particularly in image segmentation, the main subject of this paper. Nevertheless, in some applications, this topological flexibility turns out to be undesirable: for instance, when the shape to be detected has a known topology, or when the resulting shape must be homeomorphic to the initial one. The necessity of designing topology-preserving processes arises in medical imaging, for example, in the human cortex reconstruction. It is known that the human cortex has a spherical topology so throughout the reconstruction process this topological feature must be preserved. Therefore, we propose in this paper a segmentation model based on an implicit level-set formulation and on the geodesic active contours, in which a topological constraint is enforced. [ABSTRACT FROM AUTHOR]

Geometric active contour models are very popular partial differential equation-based tools in image analysis and computer vision. We present a new multigrid algorithm for the fast evolution of level-set-based geometric active contours and compare it with other established numerical schemes. We overcome the main bottleneck associated with most numerical implementations of geometric active contours, namely the need for very small time steps to avoid instability, by employing a very stable fully 2-D implicit-explicit time integration numerical scheme. The proposed scheme is more accurate and has improved rotational invariance properties compared with alternative split schemes, particularly when big time steps are utilized. We then apply properly designed multigrid methods to efficiently solve the occurring sparse linear system. The combined algorithm allows for the rapid evolution of the contour and convergence to its final configuration after very few iterations. Image segmentation experiments demonstrate the efficiency and accuracy of the method. [ABSTRACT FROM AUTHOR]

The level sets of a map are the sets of points with level above a given threshold. The connected components of the level sets, thanks to the inclusion relation, can be organized in a tree structure, that is called the component tree. This tree, under several variations, has been used in numerous applications. Various algorithms have been proposed in the literature for computing the component tree. The fastest ones (considering the worst-case complexity) have been proven to run in O(n ln(n)). In this paper, we propose a simple to implement quasi-linear algorithm for computing the component tree on symmetric graphs, based on Tarjan's union-find procedure. We also propose an algorithm that computes the n most significant lobes of a map. [ABSTRACT FROM AUTHOR]

The popularity of level sets for segmentation is mainly based on the sound and convenient treatment of regions and their boundaries. Unfortunately, this convenience is so far not known from level set methods when applied to images with more than two regions. This communication introduces a comparatively simple way how to extend active contours to multiple regions keeping the familiar quality of the two-phase case. We further suggest a strategy to determine the optimum number of regions as well as initializations for the contours. [ABSTRACT FROM AUTHOR]

We address the issue of image sequence analysis jointly in space and time. While typical approaches to such an analysis consider two image frames at a time, we propose to perform this analysis jointly over multiple frames. We concentrate on spatiotemporal segmentation of image sequences and on analysis of occlusion effects therein. The segmentation process is three-dimensional (3-D); we search for a volume carved out by each moving object in the image sequence domain, or "object tunnel:' a new space-time concept. We pose the problem in variational framework by using only motion information (no intensity edges). The resulting formulation can be viewed as volume competition, a 3-D generalization of region competition. We parameterize the unknown surface to be estimated, but rather than using an active-surface approach, we embed it into a higher dimensional function and apply the level-set methodology. We first develop simple models for the detection of moving objects over static back- ground; no motion models are needed. Then, in order to improve segmentation accuracy, we incorporate motion models for objects and background. We further extend the method by including explicit models for occluded and newly exposed areas that lead to "occlusion volumes:' another new space-time concept Since, in this case, multiple volumes are sought, we apply a multiphase variant of the level-set method. We present various experimental results for synthetic and natural image sequences. [ABSTRACT FROM AUTHOR]

We propose a cell detection and tracking solution using image-level sets computed via threshold decomposition. In contrast to existing methods where manual initialization is required to track individual cells, the proposed approach can automatically identify and track multiple cells by exploiting the shape and intensity characteristics of the cells. The capture of the cell boundary is considered as an evolution of a closed curve that maximizes image gradient along the curve enclosing a homogeneous region. An energy functional dependent upon the gradient magnitude along the cell boundary, the region homogeneity within the cell boundary and the spatial overlap of the detected cells is minimized using a variational approach. For tracking between frames, this energy functional is modified considering the spatial and shape consistency of a cell as it moves in the video sequence. The integrated energy functional complements shape-based segmentation with a spatial consistency based tracking technique. We demonstrate that an acceptable, expedient solution of the energy functional is possible through a search of the image-level lines: boundaries of connected components within the level sets obtained by threshold decomposition. The level set analysis can also capture multiple cells in a single frame rather than iteratively computing a single active contour for each individual cell. Results of cell detection using the energy functional approach and the level set approach are presented along with the associated processing time. Results of successful tracking of rolling leukocytes from a number of digital video sequences are reported and compared with the results from a correlation tracking scheme. [ABSTRACT FROM AUTHOR]

A new approach to tracking using active contours is presented. The class of objects to be tracked is assumed to be characterized by a probability distribution over some variable, such as intensity, color, or texture. The goal of the algorithm is to find the region within the current image, such that the sample distribution of the interior of the region most closely matches the model distribution. Two separate criteria for matching distributions are examined, and the curve evolution equations are derived in each case. The flows are shown to perform well in experiments. [ABSTRACT FROM AUTHOR]

Segmentation of motion in an image sequence is one of the most challenging problems in image processing, while at the same time one that finds numerous applications. To date, a wealth of approaches to motion segmentation have been proposed. Many of them suffer from the local nature of the models used. Global models, such as those based on Markov random fields, perform, in general, better. In this paper, we propose a new approach to motion segmentation that is based on a global model. The novelty of the approach is twofold. First, inspired by recent work of other researchers we formulate the problem as that of region competition, but we solve it using the level set methodology. The key features of a level set representation, as compared to active contours, often used in this context, are its ability to handle variations in the topology of the segmentation and its numerical stability. The second novelty of the paper is the formulation in which, unlike in many other motion segmentation algorithms, we do not use intensity boundaries as an accessory; the segmentation is purely based on motion. This permits accurate estimation of motion boundaries of an object even when its intensity boundaries are hardly visible. Since occasionally intensity boundaries may prove beneficial, we extend the formulation to account for the coincidence of motion and intensity boundaries. In addition, we generalize the approach to multiple motions. We discuss possible discretizations of the evolution (PDE) equations and we give details of an initialization scheme so that the results could be duplicated. We show numerous experimental results for various formulations on natural images with either synthetic or natural motion. [ABSTRACT FROM AUTHOR]