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Minimal paths are regarded as a powerful and efficient tool for boundary detection and image segmentation due to its global optimality and the well-established numerical solutions such as fast marching method. In this paper, we introduce a flexible interactive image segmentation model based on the Eikonal partial differential equation (PDE) framework in conjunction with region-based homogeneity enhancement. A key ingredient in the introduced model is the construction of local geodesic metrics, which are capable of integrating anisotropic and asymmetric edge features, implicit region-based homogeneity features and/or curvature regularization. The incorporation of the region-based homogeneity features into the metrics considered relies on an implicit representation of these features, which is one of the contributions of this work. Moreover, we also introduce a way to build simple closed contours as the concatenation of two disjoint open curves. Experimental results prove that the proposed model indeed outperforms state-of-the-art minimal paths-based image segmentation approaches. [ABSTRACT FROM AUTHOR]

The Voronoi diagram-based dual-front scheme is known as a powerful and efficient technique for addressing the image segmentation and domain partitioning problems. In the basic formulation of existing dual-front approaches, the evolving contour can be considered as the interfaces of adjacent Voronoi regions. Among these dual-front models, a crucial ingredient is regarded as the geodesic metrics by which the geodesic distances and the corresponding Voronoi diagram can be estimated. In this paper, we introduce a new dual-front model based on asymmetric quadratic metrics. These metrics considered are built by the integration of the image features and a vector field derived from the evolving contour. The use of the asymmetry enhancement can reduce the risk for the segmentation contours being stuck at false positions, especially when the initial curves are far away from the target boundaries or the images have complicated intensity distributions. Moreover, the proposed dual-front model can be applied for image segmentation in conjunction with various region-based homogeneity terms. The numerical experiments on both synthetic and real images show that the proposed dual-front model indeed achieves encouraging results. [ABSTRACT FROM AUTHOR]

The images used in various practices are often disturbed by noise, such as Gaussian noise, speckled noise, and salt and pepper noise. Images with noise are one of the challenges for segmentation, since the noise may cause inaccurate segmented results. To cope with the effect of noise on images during segmentation, a novel active contour model is proposed in this paper. The newly proposed model consists of fitting term, regularization term and penalty term. The fitting term is designed using a Gaussian kernel function and fractional order differentiation with an adaptively defined fractional order, which applies different orders to different pixels. The regularization term is applied to maintain the smoothness of curves. In order to ensure stable evolution of curves, a penalty term is added into the proposed model. Comparison experiments are conducted to show the effectiveness and efficiency of the proposed model. [ABSTRACT FROM AUTHOR]

Hybrid active contour models with the combination of region and edge information have attracted great interests in image segmentation. To the best of our knowledge, however, the theoretical foundation of these hybrid models with level set evolution is insufficient and limited. More specifically, the weighting factors of their energy terms are difficult to select and are often empirically determined without definite theoretical basis. This problem is particularly prominent in the case of multi-object segmentation when more level set functions must be computed simultaneously. To cope with these challenges, this paper proposes a new level set approach for constructing hybrid active contour models with reliable energy weights, where the weights of region and edge terms can be constrained by the optimization condition deduced from the proposed method. It can be regarded as a general approach since many existing region-based models can be easily used to construct new hybrid models using their equivalent two-phase formulations. Some representative as well as state-of-the-art models are taken as examples to demonstrate the generality of our method. The respective comparative studies validate that under the guidance of the optimization condition, segmentation accuracy, robustness, and computational efficiency can be improved compared with the original models which are used to construct the new hybrid ones. [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]

In this paper, we propose a novel contour-seed pairs learning-based framework for robust and automated cell/nucleus segmentation. Automated granular object segmentation in microscopy images has significant clinical importance for pathology grading of the cell carcinoma and gene expression. The focus of the past literature is dominated by either segmenting a certain type of cells/nuclei or simply splitting the clustered objects without contours inference of them. Our method addresses these issues by formulating the detection and segmentation tasks in terms of a unified regression problem, where a cascade sparse regression chain model is trained and then applied to return object locations and entire boundaries of clustered objects. In particular, we first learn a set of online convolutional features in each layer. Then, in the proposed cascade sparse regression chain, with the input from the learned features, we iteratively update the locations and clustered object boundaries until convergence. In this way, the boundary evidences of each individual object can be easily delineated and be further fed to a complete contour inference procedure optimized by the minimum description length principle. For any probe image, our method enables to analyze free-lying and overlapping cells with complex shapes. Experimental results show that the proposed method is very generic and performs well on contour inferences of various cell/nucleus types. Compared with the current segmentation techniques, our approach achieves state-of-the-art performances on four challenging datasets, i.e., the kidney renal cell carcinoma histopathology dataset, Drosophila Kc167 cellular dataset, differential interference contrast red blood cell dataset, and cervical cytology dataset. [ABSTRACT FROM AUTHOR]

Snakes, or active contours, have been widely used in image processing applications. Typical roadblocks to consistent performance include limited capture range, noise sensitivity, and poor convergence to concavities. This paper proposes a new external force for active contours, called vector field convolution (VFC), to address these problems. VFC is calculated by convolving the edge map generated from the image with the user-defined vector field kernel. We propose two structures for the magnitude function of the vector field kernel, and we provide an analytical method to estimate the parameter of the magnitude function. Mixed VFC is introduced to alleviate the possible leakage problem caused by choosing inappropriate parameters. We also demonstrate that the standard external force and the gradient vector flow (GVF) external force are special cases of VFC in certain scenarios. Examples and comparisons with GVF are presented in this paper to show the advantages of this innovation, including superior noise robustness, reduced computational cost, and the flexibility of tailoring the force field. [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]

Level set methods are a popular way to solve the image segmentation problem. The solution contour is found by solving an optimization problem where a cost functional is minimized. Gradient descent methods are often used to solve this optimization problem since they are very easy to implement and applicable to general nonconvex functionals. They are, however, sensitive to local minima and often display slow convergence. Traditionally, cost functionals have been modified to avoid these problems. In this paper, we instead propose using two modified gradient descent methods, one using a momentum term and one based on resilient propagation. These methods are commonly used in the machine learning community. In a series of 2-D/3-D-experiments using real and synthetic data with ground truth, the modifications are shown to reduce the sensitivity for local optima and to increase the convergence rate. The parameter sensitivity is also investigated. The proposed methods are very simple modifications of the basic method, and are directly compatible with any type of level set implementation. Downloadable reference code with examples is available online. [ABSTRACT FROM PUBLISHER]

This paper presents a novel reaction-diffusion (RD) method for implicit active contours that is completely free of the costly reinitialization procedure in level set evolution (LSE). A diffusion term is introduced into LSE, resulting in an RD-LSE equation, from which a piecewise constant solution can be derived. In order to obtain a stable numerical solution from the RD-based LSE, we propose a two-step splitting method to iteratively solve the RD-LSE equation, where we first iterate the LSE equation, then solve the diffusion equation. The second step regularizes the level set function obtained in the first step to ensure stability, and thus the complex and costly reinitialization procedure is completely eliminated from LSE. By successfully applying diffusion to LSE, the RD-LSE model is stable by means of the simple finite difference method, which is very easy to implement. The proposed RD method can be generalized to solve the LSE for both variational level set method and partial differential equation-based level set method. The RD-LSE method shows very good performance on boundary antileakage. The extensive and promising experimental results on synthetic and real images validate the effectiveness of the proposed RD-LSE approach. [ABSTRACT FROM PUBLISHER]

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]

The state-of-the-art interactive image segmentation algorithms are sensitive to the user inputs and often unable to produce an accurate boundary with a small amount of user interaction. They frequently rely on laborious user editing to refine the segmentation boundary. In this paper, we propose a robust and accurate interactive method based on the recently developed continuous-domain convex active contour model. The proposed method exhibits many desirable properties of an effective interactive image segmentation algorithm, including robustness to user inputs and different initializations, the ability to produce a smooth and accurate boundary contour, and the ability to handle topology changes. Experimental results on a benchmark data set show that the proposed tool is highly effective and outperforms the state-of-the-art interactive image segmentation algorithms. [ABSTRACT FROM AUTHOR]

Active contour models (ACMs) integrated with various kinds of external force fields to pull the contours to the exact boundaries have shown their powerful abilities in object segmentation. However, local minimum problems still exist within these models, particularly the vector field's “equilibrium issues.” Different from traditional ACMs, within this paper, the task of object segmentation is achieved in a novel manner by the Poincaré map method in a defined vector field in view of dynamical systems. An interpolated swirling and attracting flow (ISAF) vector field is first generated for the observed image. Then, the states on the limit cycles of the ISAF are located by the convergence of Newton–Raphson sequences on the given Poincaré sections. Meanwhile, the periods of limit cycles are determined. Consequently, the objects' boundaries are represented by integral equations with the corresponding converged states and periods. Experiments and comparisons with some traditional external force field methods are done to exhibit the superiority of the proposed method in cases of complex concave boundary segmentation, multiple-object segmentation, and initialization flexibility. In addition, it is more computationally efficient than traditional ACMs by solving the problem in some lower dimensional subspace without using level-set methods. [ABSTRACT FROM AUTHOR]

A new formulation of active contours based on explicit functions has been recently suggested. This novel framework allows real-time 3-D segmentation since it reduces the dimensionality of the segmentation problem. In this paper, we propose a B-spline formulation of this approach, which further improves the computational efficiency of the algorithm. We also show that this framework allows evolving the active contour using local region-based terms, thereby overcoming the limitations of the original method while preserving computational speed. The feasibility of real-time 3-D segmentation is demonstrated using simulated and medical data such as liver computer tomography and cardiac ultrasound images. [ABSTRACT FROM PUBLISHER]

In this paper, an iterative narrow-band-based graph cuts (INBBGC) method is proposed to optimize the geodesic active contours with region forces (GACWRF) model for interactive object segmentation. Based on cut metric on graphs proposed by Boykov and Kolmogorov, an NBBGC method is devised to compute the local minimization of GAC. An extension to an iterative manner, namely, INBBGC, is developed for less sensitivity to the initial curve. The INBBGC method is similar to graph-cuts-based active contour (GCBAC) presented by Xu , and their differences have been analyzed and discussed. We then integrate the region force into GAC. An improved INBBGC (IINBBGC) method is proposed to optimize the GACWRF model, thus can effectively deal with the concave region and complicated real-world images segmentation. Two region force models such as mean and probability models are studied. Therefore, the GCBAC method can be regarded as the special case of our proposed IINBBGC method without region force. Our proposed algorithm has been also analyzed to be similar to the Grabcut method when the Gaussian mixture model region force is adopted, and the band region is extended to the whole image. Thus, our proposed IINBBGC method can be regarded as narrow-band-based Grabcut method or GCBAC with region force method. We apply our proposed IINBBGC algorithm on synthetic and real-world images to emphasize its performance, compared with other segmentation methods, such as GCBAC and Grabcut methods. [ABSTRACT FROM PUBLISHER]

In this paper, we propose an approach for tracking an object of interest based on 3-D range data. We employ particle filtering and active contours to simultaneously estimate the global motion of the object and its local deformations. The proposed algorithm takes advantage of range information to deal with the challenging (but common) situation in which the tracked object disappears from the image domain entirely and reappears later. To cope with this problem, a method based on principle component analysis (PCA) of shape information is proposed. In the proposed method, if the target disappears out of frame, shape similarity energy is used to detect target candidates that match a template shape learned online from previously observed frames. Thus, we require no a priori knowledge of the target's shape. Experimental results show the practical applicability and robustness of the proposed algorithm in realistic tracking scenarios. [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]

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]

Centroidal Voronoi tessellations (CVTs) are special Voronoi tessellations whose generators are also the centers of mass (centroids) of the Voronoi regions with respect to a given density function and CVT-based methodologies have been proven to be very useful in many diverse applications in science and engineering. In the context of image processing and its simplest form, CVT-based algorithms reduce to the well-known k-means clustering and are easy to implement. In this paper, we develop edge-weighted centroidal Voronoi tessellation (EWCVT) model for image segmentation and propose some efficient algorithms for its construction. Our EWCVT model can overcome some deficiencies possessed by the basic CVT model; in particular, the new model appropriately combines the image intensity information together with the length of cluster boundaries, and can handle very sophisticated situations. We demonstrate through extensive examples the efficiency, effectiveness, robustness, and flexibility of the proposed method [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]

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]

Active models have been widely used in image processing applications. A crucial stage that affects the ultimate active model performance is initialization. This paper proposes a novel automatic initialization approach for parametric active models in both 2-D and 3-D. The PIG initialization method exploits a novel technique that essentially estimates the external energy field from the external force field and determines the most likely initial segmentation. Examples and comparisons with two state-of-the-art automatic initialization methods are presented to illustrate the advantages of this innovation, including the ability to choose the number of active models deployed, rapid convergence, accommodation of broken edges, superior noise robustness, and segmentation accuracy. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of image segmentation by means of active contours, whose evolution is driven by the gradient flow derived from an energy functional that is based on the Bhattacharyya distance. In particular, given the values of a photo- metric variable (or of a set thereof), which is to be used for classifying the image pixels, the active contours are designed to converge to the shape that results in maximal discrepancy between the empirical distributions of the photometric variable inside and outside of the contours. The above discrepancy is measured by means of the Bhattacharyya distance that proves to be an extremely useful tool for solving the problem at hand. The proposed methodology can be viewed as a generalization of the segmentation methods, in which active contours maximize the difference between a finite number of empirical moments of the ‘inside’ and ‘outside’ distributions. Furthermore, it is shown that the proposed methodology is very versatile and flexible in the sense that it allows one to easily accommodate a diversity of the image features based on which the segmentation should be performed. As an additional contribution, a method for automatically adjusting the smoothness properties of the empirical distributions is proposed. Such a procedure is crucial in situations when the number of data samples (supporting a certain segmentation class) varies considerably in the course of the evolution of the active contour. In this case, the smoothness properties of the empirical distributions have to be properly adjusted to avoid either over-or underestimation artifacts. Finally, a number of relevant segmentation results are demonstrated and some further research directions are discussed. [ABSTRACT FROM AUTHOR]

The partial differential equation driving level-set evolution in segmentation is usually solved using finite differences schemes. In this paper, we propose an alternative scheme based on radial basis functions (RBF5) collocation. This approach provides a continuous representation of both the implicit function and its zero level set. We show that compactly supported RBFs (CSRBFs) are particularly well suited to collocation in the framework of segmentation. In addition, CSRBFs allow us to reduce the computation cost using a kd-tree-based strategy for neighborhood representation. Moreover, we show that the usual reinitialization step of the level set may be avoided by simply constraining the l1-norm of the CSRBF parameters. As a consequence, the final solution is topologically more flexible, and may develop new contours (i.e., new zero-level components), which are difficult to obtain using reinitialization. The behavior of this approach is evaluated from numerical simulations and from medical data of various kinds, such as 3-D CT bone images and echocardiographic ultrasound images. [ABSTRACT FROM AUTHOR]

Tracking an object in a sequence of images can fail due to partial occlusion or clutter. Robustness to occlusion can be increased by tracking the object as a set of "parts" such that not all of these are occluded at the same time. However, successful implementation of this idea hinges upon finding a suitable set of parts. In this paper we propose a novel segmentation, specifically designed to improve robustness against occlusion in the context of tracking. The main result shows that tracking the parts resulting from this segmentation outperforms both tracking parts obtained through traditional segmentations, and tracking the entire target. Additional results include a statistical analysis of the correlation between features of a part and tracking error, and identifying a cost function that exhibits a high degree of correlation with the tracking error. [ABSTRACT FROM AUTHOR]