Conference Paper

Fuzzy Region Competition: A Convex Two-Phase Segmentation Framework

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Abstract

We describe a novel framework for two-phase image segmentation, namely the Fuzzy Region Competition. The functional involved in several existing models related to the idea of Region Competition is extended by the introduction of a fuzzy membership function. The new problem is convex and the set of its global solutions turns out to be stable under thresholding, operation that also provides solutions to the corresponding classical formulations. The advantages are then shown in the piecewise-constant case. Finally, motivated by medical applications such as angiography, we derive a fast algorithm for segmenting images into two non-overlapping smooth regions. Compared to existing piecewise-smooth approaches, this last model has the unique advantage of featuring closed-form solutions for the approximation functions in each region based on normalized convolutions. Results are shown on synthetic 2D images and real 3D volumes.

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... Global minimization solution is important to design robust image segmentation models (and other image processing models) because a global solution is independent of a good initial position of the contour unlike e.g. standard active contour models [29,8,30]. Although [10,6] define a global minimization approach for the Chan-Vese's model [11], the geodesic active contour model [8,30] and the Mumford-Shah's model [50,39], this global minimization approach can naturally be applied to any active contour model as shown in [38]. Traditionally active contour segmentation models are solved using the level set method [43] which depends on slow minimization process and needs regular re-distancing the level set function. ...
... We present the continuous global minimization approach for unsupervised image segmentation introduced in [10, 6] (see also [38]). The variational segmentation model presented in this section will be extended to a non-local framework in the rest of the paper. ...
... Finally, we would like to point out the continuous global minimization approach [10,6] can also be used in other image processing problems. Other image segmentation models based on the proposed global optimization approach [10,6] have been proposed in the literature, see [34,38,40,26,49]. Besides, Kolev, Klodt, Brox, Esedoglu and Cremers in [31] and Zach, Pock and Bischof in [52] used a global optimization framework to define global 3D surface reconstruction, independent of the initial condition. ...
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⋆ Abstract. New image denoising models based on non-local image information have been recently introduced in the literature. These so-called "non-local" denoising models provide excellent results because these models can denoise smooth regions or/and textured regions simultaneously, unlike standard denoising models. Stan- dard variational models s.a. Total Variation-based models are defined to work in a small local neighborhood, which is enough to denoise smooth regions. However, textures are not local in nature and requires semi-local/non-local information to be denoised efficiently. Several papers have introduced non-local filters and non-local variational models for image denoising. Yet, few studies have been done to develop unsupervised image segmentation models based on non-local information. This will be the goal of this paper. We define and study three unsupervised non-local seg- mentation models. These models will be based on the continuous global minimiza- tion approach for image segmentation recently introduced in (10, 6). The energy of (10, 6) is a first order energy composed of the weighted Total Variation norm and a linear term. The first proposed non-local segmentation model will extend the Total Variation regularization term of (10, 6) to the non-local Total Variation energy. We will see that the non-local energy can segment fine and small structures better than the standard Total Variation energy. The second model will extend the data-based term of (10, 6) to a non-local term using the Chan-Vese model. The proposed non-local Chan-Vese model will overcome the main limitation of the orig- inal model, that does not work with local intensity inhomogeneities. Finally, the third model will also extend the data-based term of (10, 6) to a non-local term using the Mumford-Shah energy. The original Mumford-Shah energy is designed to work for piecewise smooth images only. We suggest to extend it to textures, defining a non-local Mumford-Shah model that works with real-world images. Numerical minimization schemes presented in this paper are based on continuous and discrete (graph cut) approaches. Experimental results will illustrate the improvements pro- vided by the three proposed non-local unsupervised segmentation models.
... In addition, the level set-based methods are computationally inefficient. To address the issues mentioned before, a method presented in [20] provides a solution for the two-region segmentation using fuzzy membership functions. This method is then extended by Li et al. [21] to be applied to the multiregion segmentation of images corrupted without any noises or with weak noises. ...
... This method is then extended by Li et al. [21] to be applied to the multiregion segmentation of images corrupted without any noises or with weak noises. In [20] and [21], two constraints with respect to the family of fuzzy membership functions guarantee the convexity of the models. These two constraints are strictly met in each single iteration process in two-region segmentation, whereas we found that they probably cannot be satisfied in the multiregion segmentation of images containing strong noise. ...
... Consequently, the segmented regions {R 1 , R 2 , · · · , R N } are attained by (20) and (31). ...
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This article proposes an unsupervised multiregion segmentation method for fully polarimetric synthetic aperture radar (polSAR) images based on the improved fuzzy active contour model. Different from most of the active contour models that are based on the utilization of only statistical information, the proposed method makes better use of information from polarimetric data. In addition to the statistical information, an edge detector modified from the ratio of exponentially weighted averages (ROEWA) operator, a sliding window algorithm for the total received power, and a ratio operator with respect to scattering mechanisms are integrated to the proposed active contour model. We then present a layer-based fuzzy active contour framework to solve our model. The general fuzzy active contour framework is computationally much more efficient compared with the level set-based framework; however, it cannot be applied to the multiregion segmentation of SAR images due to its low robustness to strong noise. The proposed approach includes the advantages of the general fuzzy active contour framework and has good robustness. Using two fully polSAR images demonstrates that the proposed method can achieve higher efficiency and a better segmentation performance in comparison with the commonly used active contour methods.
... There have been many soft segmentation methods [17][18][19][20][21][22]. Mory and Ardon extended the original region competition model [8] to a fuzzy region competition method [19,20]. ...
... There have been many soft segmentation methods [17][18][19][20][21][22]. Mory and Ardon extended the original region competition model [8] to a fuzzy region competition method [19,20]. The technique generalizes some existing supervised and unsupervised region-based models. ...
... (1) Initialization: (i) Initialize and ; (ii) Initialize ; (iii) Update and with (19). ...
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Soft segmentation is more flexible than hard segmentation. But the membership functions are usually sensitive to noise. In this paper, we propose a multiphase soft segmentation model for nearly piecewise constant images based on stochastic principle, where pixel intensities are modeled as random variables with mixed Gaussian distribution. The novelty of this paper lies in three aspects. First, unlike some existing models where the mean of each phase is modeled as a constant and the variances for different phases are assumed to be the same, the mean for each phase in the Gaussian distribution in this paper is modeled as a product of a constant and a bias field, and different phases are assumed to have different variances, which makes the model more flexible. Second, we develop a bidirection projected primal dual hybrid gradient (PDHG) algorithm for iterations of membership functions. Third, we also develop a novel algorithm for explicitly computing the projection from to simplex for any dimension using dual theory, which is more efficient in both coding and implementation than existing projection methods.
... Mory and Ardon [19] used the concept of fuzzy region competition to unify and extend it to model texture segmentation problems. Here, each pixel is assigned a probability, instead of a precise membership integer, of belonging to a particular region. ...
... Mathematically, each phase may be characterized by a distinct constant as defined in [5] and in two phases with N = 2, one may simply try to find a piecewise constant solution u taking two constant values in {0, 1} (or two other constants {α 1 , α 2 } for membership identity). However it is often advantageous to look for a solution u that takes values in [0, 1] in a fuzzy membership framework [19]. ...
... A general form of two-phase segmentation problem can be represented [19] as ...
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Image segmentation is a fundamental task for many computer vision and image processing applications. There exist many useful and reliable models for two-phase segmentation. However, the multiphase segmentation is a more challenging problem than two phase segmentation, mainly due to strong dependence on initialization of solutions. In this paper we propose a reliable hierarchical algorithm for multiphase texture image segmentation by making full use of two-phase texture models in a fuzzy membership framework. Application of the new algorithm to the synthetic and real medical imaging data demonstrate more satisfactory results than existing algorithms.
... More recent, some new developments, such as the convex techniques [7][8] and some fast algorithms [9][10], have improved the segmentation in both efficiency and accuracy for optical images. Since the successes of those models are founded on the assumption that the images are corrupted by some additive Gaussian noises, they are not directly suitable for non-Gaussian noise problems. ...
... Mory et al. [8] introduced to use the fuzzy membership function to represent the region and minimize the twophase fuzzy region competition energy. Following the idea of [8], we introduce the fuzzy membership function [0,1] j u ∈ to replace the level set function j φ , and relax the multiphase segmentation model (2) into a soft form ...
... Mory et al. [8] introduced to use the fuzzy membership function to represent the region and minimize the twophase fuzzy region competition energy. Following the idea of [8], we introduce the fuzzy membership function [0,1] j u ∈ to replace the level set function j φ , and relax the multiphase segmentation model (2) into a soft form ...
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In this paper, we present a novel variational framework for multiphase synthetic aperture radar (SAR) image segmentation based on the fuzzy region competition method. A new energy functional is proposed to integrate the Gamma model and the edge detector based on the ratio of exponentially weighted averages (ROEWA) operator within the optimization process. To solve the optimization problem efficiently, the functional is firstly modified to be convex and differentiable by using the fuzzy membership functions. And then the constrained optimization problem is converted to an unconstrained one by using the variable splitting techniques and the augmented Lagrangian method (ALM). Finally the energy is minimized with an alternative iterative minimization algorithm. The effectiveness of our proposed algorithm is validated by experiments on both synthetic and real SAR images.
... A more recent extension of the aforementioned twophase approaches is proposed in [21], where the authors develop a multiphase image segmentation method based on fuzzy region competition [25,35]. They start from the general N -phase region competition functional, introduce fuzzy membership functions and replace the error term by the data fidelity of the Mumford-Shah and Chan-Vese model, respectively. ...
... The adjoint of the derivative of F 2 (u i ) is expressed in the same manner as in (25). We have ...
Preprint
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In this paper, we propose a variational image segmentation framework for multichannel multiphase image segmentation based on the Chan-Vese active contour model. The core of our method lies in finding a variable u encoding the segmentation, by minimizing a multichannel energy functional that combines the information of multiple images. We create a decomposition of the input, either by multichannel filtering, or simply by using plain natural RGB, or medical images, which already consist of several channels. Subsequently we minimize the proposed functional for each of the channels simultaneously. Our model meets the necessary assumptions such that it can be solved efficiently by optimization techniques like the Chambolle-Pock method. We prove that the proposed energy functional has global minimizers, and show its stability and convergence with respect to noisy inputs. Experimental results show that the proposed method performs well in single- and multichannel segmentation tasks, and can be employed to the segmentation of various types of images, such as natural and texture images as well as medical images.
... A variational model integrating statistical information, scattering mechanism, and total received power is then developed. Considering the low computational efficiency attained by using level sets, we employed fuzzy membership functions and dual projection method to solve our model [10][11][12]. ...
... and then project The reader is referred to [10][11][12] for more details and proof. ...
... In the recent years, the segmentation problem has been studied extensively [3,5,6,9,14,16,17,[19][20][21][22][23][24]26,28,[30][31][32][33][34][35][36][37][38][39][40] using different methods. ...
... Along with the Mumford-Shah model [25], other variational methods have been successfully used for image segmentation such as region competition [23,24,40], geodesic active contour [8], geodesic active region [27]. In variational formulation, image segmentation is achieved by solving an energy minimization problem, which includes some useful information such as prior shape and constraints on the regularity of object boundaries. ...
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In this paper, a new stochastic variational PDE model is developed, using instead of hard segmentation soft segmentation. In this way, each pixel is allowed to belong to each image pattern with some probability. Our work proposes a functional with variable exponent, which provides a more accurate model for image segmentation and denoising. The diffusion resulting from the proposed model is a combination between TV-based and isotropic smoothing. The modeling procedure, computational implementation and results are explored in detail and numerical examples of real and synthetic images are presented.
... Dans [9] les auteurs ont montré que cette fonctionnelle admettait un minimum égal presque partout à une fonction indicatrice. En procédant de la même manière que dans [9] et [10], l'utilisation d'un terme de compétition de régions r plus évolué peut être envisagée. ...
... Dans [9] les auteurs ont montré que cette fonctionnelle admettait un minimum égal presque partout à une fonction indicatrice. En procédant de la même manière que dans [9] et [10], l'utilisation d'un terme de compétition de régions r plus évolué peut être envisagée. Pour minimiser l'équation (4) par rapport à u sous la contrainte u ∈ BV [0,1] (Ω), on résout le problème suivant qui a le même ensemble de minimiseurs : ...
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... There have been many soft segmentation methods. Mory and Ardon extended the original region competition model [4] to a fuzzy region competition method [18] [19]. The technique generalizes some existing supervised and unsupervised region-based model. ...
... By Γ -convergence theory, the term converges to the length of the boundary in the sense of Γ -convergence [37] [38] as  goes to zero. Now, in combination of (15), (16), (18) and (19), the final proposed segmentation model is the minimization of: ...
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In this paper, we propose a new soft multi-phase segmentation model where it is assumed that the pixel intensities are distributed as a Gaussian mixture. The model is formulated as a minimization problem through the use of the maximum likelihood estimator and phase-transition theory. The mixture coefficients, which are estimated using a spatially varying mean and variance procedure, are used for image segmentation. The experimental results indicate the effectiveness of the method.
... The Fuzzy Region Competition framework, developed by Mory and Ardon (2007), avoids these drawbacks, because its functional is convex related to the fuzzy membership function. So, optimal global solutions with a weak sensitivity to initial conditions can be obtained. ...
... With the success of the Region Competition algorithm (Zhu and Yuille 1996), several variational region-based methods that use its ideas have been developed. Mory and Ardon (2007) proposed the two-phase Fuzzy Region Competition framework, which has a fuzzy approach and aims to segment an image I over the domain ⊂ R m in two regions based on its intensity distributions. The Fuzzy Region Competition minimization problem, in a general formulation, is given by: ...
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This paper presents a new multiphase image segmentation model based on Fuzzy Region Competition. The proposed model approximates image regions by probability density functions and uses a supervised approach in the segmentation process. The strategy of the proposed model is to perform a two-phase Fuzzy Region Competition model iteratively for image segmentation. The hard partition is obtained in each step from a determined fuzzy membership functions consequently, the segmentation process is soft, while the final result is hard, due to the simplicity of avoiding non-overlapping and vacuum regions. Finally, several experiments on multiphase images are presented to demonstrate the efficiency and robustness of the proposed model when dealing with noisy, texturized and natural images.
... However, one object recovery is often interfered with the intensity distribution of background and the intensity inhomogeneity of image [24,25]. Some global-to-local strategies [24][25][26][27][28][29][30][31] are used to overcome such problems. Active contours can also be classified into the conservative and the nonconservative according to whether the driving force for snake evolution is the gradient flow of an energy functional or not. ...
... Remark 1 When subregions of object and background have a similar feature distribution, some global-to-local strategies [27][28][29][30][31] can be used to overcome the problem mentioned above, but such treatment may lead to new problems. For example, one can use the local data of A − , whose domain is near the active curve, to estimate p − , in order to remove the interference of background regions. ...
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... The L 1 norm was also used for segmentation purposes in Li et al. (2016) and Jung (2017). Following the idea of Mory and Ardon (2007) and Li et al. (2016) we relax the functional by using fuzzy membership functions, which assume that each point can be in several regions simultaneosly with a certain probability. As a generalization of the notion of characteristic functions, fuzzy membership functions satisfy the following two constrains: ...
... But the set BV (Ω; {0, 1}) is not convex and, moreover, the Euler-Lagrange equation for non-continuous functions leads to difficulties in numerical implementations. Thus, following the idea proposed by Mory and Ardon (2007) and Li et al. (2016) we relax the characteristic functions to be fuzzy membership functions, which are associated to the notion of fuzzy sets 1 , firstly introduced by Zadeh (1965). Fuzzy membership functions belong to the set ...
Thesis
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... The above mentioned explicit active contour and implicit level set methods both assume that each pixel belongs to a unique region. Different from the two methods, a representative method using a fuzzy membership function considers that each pixel can simultaneously belong to several regions with probability in [0, 1], see [2,[18][19][20]. This type of method has distinct advantages, such as the convex energy functional that guarantees the convergence and stability of the solution, and larger feasible set to find better segmentation results. ...
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... In the future, we will consider applying the weighted anisotropic-isotropic penalty to other types of segmentation models, such as piecewise-smooth models [27,23], the Potts models [46,49,52], and fuzzy segmentation models [39,30,31]. Since the two-stage methods are generally faster than the CV methods, we plan to develop a weighted anisotropic-isotropic variant as a faster alternative. ...
Preprint
In a class of piecewise-constant image segmentation models, we incorporate a weighted difference of anisotropic and isotropic total variation (TV) to regularize the partition boundaries in an image. To deal with the weighted anisotropic-isotropic TV, we apply the difference-of-convex algorithm (DCA), where the subproblems can be minimized by the primal-dual hybrid gradient method (PDHG). As a result, we are able to design an alternating minimization algorithm to solve the proposed image segmentation models. The models and algorithms are further extended to segment color images and to perform multiphase segmentation. In the numerical experiments, we compare our proposed models with the Chan-Vese models that use either anisotropic or isotropic TV and the two-stage segmentation methods (denoising and then thresholding) on various images. The results demonstrate the effectiveness and robustness of incorporating weighted anisotropic-isotropic TV in image segmentation.
... Discontinuous functions are used in piecewise constant level set methods [15,16,17] to show distinct phases. Continuous functions should be used in methods of fuzzy membership function [18,19]. To show the probability of belonging to some specific region these functions ranging from 0 to 1. ...
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In medical imaging and many other areas, selective image segmentation plays a key role. In this paper, we introduced a unique and novel convex selective segmentation model which contains two stages. The first stage is to achieve a regular approximation associated with the Mumford-shah model to the mark region in the given image. The approximation yields a greater value for the mark region and smaller values for others. In the second stage, we make use of this approximation and implement a thresholding technique to retrieve the object of interest. The approximation can be achieved by the alternating direction method. Experimental results on medical and noisy images are given to testify to the importance of the proposed method. The comparisons show that the proposed method works better than other existing methods.
... Both the parametric/explicit active contour method and the level set method assume that each pixel belongs to a unique region. An alternative way to represent various regions is to use fuzzy membership functions [6,14,25,34], which describe the levels of possible membership. Fuzzy membership functions assume that each pixel can be in several regions simultaneously with probability in [0, 1]. ...
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In this paper, we propose a variational multiphase image segmentation model based on fuzzy membership functions and L1-norm fidelity. Then we apply the alternating direction method of multipliers to solve an equivalent problem. All the subproblems can be solved efficiently. Specifically, we propose a fast method to calculate the fuzzy median. Experimental results and comparisons show that the L1-norm based method is more robust to outliers such as impulse noise and keeps better contrast than its L2-norm counterpart. Theoretically, we prove the existence of the minimizer and analyze the convergence of the algorithm.
... It should be mentioned that the proposed framework is potentially applicable on similar methods which are capable of solving two-phase segmentation problems, as is the case with the model of Mory et al. [64]. ...
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A principled method for active contour (AC) parameterization remains a challenging issue in segmentation research, with a potential impact on the quality, objectivity, and robustness of the segmentation results. This paper introduces a novel framework for automated adjustment of region-based AC regularization and data fidelity parameters. Motivated by an isomorphism between the weighting factors of AC energy terms and the eigenvalues of structure tensors, we encode local geometry information by mining the orientation coherence in edge regions. In this light, the AC is repelled from regions of randomly oriented edges and guided toward structured edge regions. Experiments are performed on four state-of-the-art AC models, which are automatically adjusted and applied on benchmark datasets of natural, textured and biomedical images and two image restoration models. The experimental results demonstrate that the obtained segmentation quality is comparable to the one obtained by empirical parameter adjustment, without the cumbersome and time-consuming process of trial and error.
... This is the well-known Rudin-Osher-Fatemi model [41], which can be solved by many different methods such as Chambolle's fast duality projection algorithm [2,7], semismooth Newton's method [27], and the multilevel optimization method [5]. In this paper, we employ the fast duality projection algorithm of Chambolle due to its efficiency, effectiveness, ease of implementation [20], and high numerical stability [26]. Then the solution is given by ...
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Inverse lithography technology formulates the photomask synthesis as an inverse mathematical problem. To solve this, we propose a variational functional and develop a robust computational algorithm, where the proposed functional takes into account the process variations and incorporates several regularization terms that can control the mask complexity. We establish the existence of the minimizer of the functional, and in order to optimize it effectively, we adopt an alternating minimization procedure with Chambolle's fast duality projection algorithm. Experimental results show that our proposed algorithm is effective in synthesizing high quality photomasks as compared with existing methods.
... Consequently, the initialization has to be closed to the desired solution. In [Mory & Ardon, 2007], a relaxed version of (2.20) is shown to be convex, which eludes the dependency on the initialization. ...
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Within the wide field of medical imaging research, image segmentation is one of the earliest but still open topics. This thesis focuses on model-based segmentation methods, which achieve a good trade-off between genericity and ability to carry prior information on the target organ. Our goal is to build an efficient segmentation framework that is able to leverage all kinds of external information, i.e. annotated databases via statistical learning, other images from the patient via co-segmentation and user input via live interactions. This work is based on the implicit template deformation framework, a variational method relying on an implicit representation of shapes. After improving the mathematical formulation of this approach, we show its potential on challenging clinical problems. Then, we introduce different generalizations, all independent but complementary, aimed at enriching both the shape and appearance model exploited. The diversity of the clinical applications addressed shows the genericity and the effectiveness of our contributions.
... A fixed number of iterations (10 in the current version) of the fuzzy region competition algorithm was performed. 6 Step 4: centreline. Using a minimal path algorithm, 7 the centrelines of the main branches of the aorta (main trunk and iliac arteries) are computed based on both lumen and thrombus. ...
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Objective The aim of this study was to evaluate an automatic, deep learning based method (Augmented Radiology for Vascular Aneurysm [ARVA]), to detect and assess maximum aortic diameter, providing cross sectional outer to outer aortic wall measurements. Methods Accurate external aortic wall diameter measurement is performed along the entire aorta, from the ascending aorta to the iliac bifurcations, on both pre- and post-operative contrast enhanced computed tomography angiography (CTA) scans. A training database of 489 CTAs was used to train a pipeline of neural networks for automatic external aortic wall measurements. Another database of 62 CTAs, including controls, aneurysmal aortas, and aortic dissections scanned before and/or after endovascular or open repair, was used for validation. The measurements of maximum external aortic wall diameter made by ARVA were compared with those of seven clinicians on this validation dataset. Results The median absolute difference with respect to expert’s measurements ranged from 1 mm to 2 mm among all annotators, while ARVA reported a median absolute difference of 1.2 mm. Conclusion The performance of the automatic maximum aortic diameter method falls within the interannotator variability, making it a potentially reliable solution for assisting clinical practice.
... The soft segmentation method assumes that each image pixel can be in several regions and the probability in each region is represented by soft membership function valued in [0, 1]. Recently, many two-phase soft segmentation models are proposed [2, 6, 10, 17, 18], in which one soft membership function is used in the functionals such that the functionals are convex with respect to the membership. The convexity ensures that the new methods are not sensitive to initialization and global minima can be found. ...
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In this paper, we propose a variational soft segmentation framework inspired by the level set formulation of multiphase Chan-Vese model. We use soft membership functions valued in [0,1] to replace the Heaviside functions of level sets (or characteristic functions) such that we get a representation of regions by soft membership functions which automatically satisfies the sum to one constraint. We give general formulas for arbitrary N-phase segmentation, in contrast to Chan-Vese’s level set method only 2m -phase are studied. To ensure smoothness on membership functions, both total variation (TV) regularization and H 1 regularization used as two choices for the definition of regularization term. TV regularization has geometric meaning which requires that the segmentation curve length as short as possible, while H 1 regularization has no explicit geometric meaning but is easier to implement with less parameters and has higher tolerance to noise. Fast numerical schemes are designed for both of the regularization methods. By changing the distance function, the proposed segmentation framework can be easily extended to the segmentation of other types of images. Numerical results on cartoon images, piecewise smooth images and texture images demonstrate that our methods are effective in multiphase image segmentation.
... Bresson et al also proposed a fast and easy to implement numerical minimization scheme based on the dual formulation of the TV norm as proposed in [6,2]. Finally, this algorithm has shown it efficiency on medical image as in [22]. ...
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We present an approach for unsupervised segmentation of natural and textural images based on active contour, differential geometry and information theoretical concept. More precisely, we propose a new texture descriptor which intrinsically defines the geometry of textural regions using the shape operator borrowed from differential geometry. Then, we use the popular Kullback-Leibler distance to define an active contour model which distinguishes the background and textural objects of interest represented by the probability density functions of our new texture descriptor. We prove the existence of a solution to the proposed segmentation model. Finally, a fast and easy to implement texture segmentation algorithm is introduced to extract meaningful objects. We present promising synthetic and real-world results and compare our algorithm to other state-of-the-art techniques.
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In this paper we propose a general variational segmentation model for multiphase texture segmentation based on fuzzy region competition principle. An important strength of the proposed framework is that different region terms (e.g. mutual information Kim et al. (2005) [1], local histogram Ni et al. (2009) [2] models for texture-based segmentation, and piecewise constant intensity model Chan and Vese (2001) [3] for intensity-based segmentation) can be included as appropriate to the problem. Constraints of different phases are considered by introducing Lagrangian multipliers into the energy functional, and a fast numerical solution is achieved by employing the fast dual projection algorithm Chambolle (2004) [4]. The proposed model has been applied to synthetic and natural images in order to make comparisons with other competing models in literature. Our results demonstrate superiority in dealing with multiphase texture segmentation problems. To demonstrate its usefulness in biomedical applications we have applied the new model to two retinal image segmentation problems: segmentation of capillary non-perfusion regions in fluorescein angiogram and segmentation of cellular layers of the retina in optical coherence tomography, and evaluated against the gold standard set by experts. The generalized overlap analysis shows good agreement for both applications. As a generic segmentation technique our new model has the potential to be extended for wider applications.
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In the context of mathematical modeling of complex vessel tree structures with deformable models, we present a novel level set formulation to evolve both the vessel surface and its centerline. The implicit function is computed as the convolution of a geometric primitive, representing the centerline, with localized kernels of continuously-varying scales allowing accurate estimation of the vessel width. The centerline itself is derived as the characteristic function of an underlying signed medialness function, to enforce a tubular shape for the segmented object, and evolves under shape and medialness constraints. Given a set of initial medial loci and radii, this representation first allows for simultaneous recovery of the vessels centerlines and radii, thus enabling surface reconstruction. Secondly, due to the topological adaptivity of the level set segmentation setting, it can handle tree-like structures and bifurcations without additional junction detection schemes nor user inputs. We discuss the shape parameters involved, their tuning and their influence on the control of the segmented shapes, and we present some segmentation results on synthetic images, 2D angiographies, 3D rotational angiographies and 3D-CT scans.
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Conference Paper
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Conference Paper
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Conference Paper
In this paper, we propose an unsupervised variational two-phase image segmentation model based on Fuzzy Region Competition. This model uses probability density functions to design image regions and to set a homogeneity criterion for the competition between regions. The key idea of the proposed model is to optimize the probability distribution parameters while the segmentation procedure takes place. The experiments in natural and noisy images showed that the proposed model is robust in relation to noise and presents better segmentation results using texturized images than the unsupervised piecewise constant case of Fuzzy Region Competition method.
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Book
Les contours actifs sont des courbes déformables que l'on vient positionner dans les images pour y capturer des structures d'intérêt : on parle de segmentation d'images. La plupart du temps, cet ajustement est formulé comme l'optimisation d'une fonctionnelle d'énergie, caractérisée par la présence de nombreux minima locaux, correspondant à des solutions peu pertinentes. Dans ce chapitre, nous passons en revue les principaux modèles de contours actifs existants, puis nous décrivons des solutions récemment développées, assurant la détermination de solutions globalement optimales. Il s'agit, d'une part, d'algorithmes de calcul de chemins optimaux et, d'autre part, de techniques de relaxation convexe. Les premiers, adaptés à la recherche de courbes optimales entre deux points, procèdent par propagation d'une distance géodésique et rétro-parcours. Les secondes, applicables à certaines formes de contours actifs orientés région, se positionnent dans un espace convexe, en cherchant une approximation de la fonction caractéristique des régions, et optimisent une fonctionnelle elle aussi convexe.
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This letter proposes a novel fuzzy multiphase image segmentation model. In the model, we introduce wavelet based regularization on the membership functions which are used as indicators of different regions. By using principal component analysis (PCA) features data as descriptors, the proposed model can segment texture and natural images. To efficiently solve the model, we formulate a fast iterative shrinkage algorithm for multiphase image segmentation. Experimental results show that the proposed method achieves better segmentation results compared with some other classical variational segmentation methods.
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In this paper, a novel level set segmentation model integrating the intensity and texture terms is proposed to segment complicated two-phase nature images. Firstly, an intensity term based on the global division algorithm is proposed, which can better capture intensity information of image than the Chan–Vese model (CV). Particularly, the CV model is a special case of the proposed intensity term under a certain condition. Secondly, a texture term based on the adaptive scale local variation degree (ASLVD) algorithm is proposed. The ASLVD algorithm adaptively incorporates the amplitude and frequency components of local intensity variation, thus, it can extract the non-stationary texture feature accurately. Finally, the intensity term and the texture term are jointly incorporated into level set and used to construct effective image segmentation model named as the Intensity-Texture model. Since the intensity term and the texture term are complementary for image segmentation, the Intensity-Texture model has strong ability to accurately segment those complicated two-phase nature images. Experimental results demonstrate the effectiveness of the proposed Intensity-Texture model.
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The authors present a fully automatic method of color theme extraction and transfer for fabric color design. For real-life fabrics, such extraction and transfer is performed through a highly time consuming and knowledge intensive process, known as color theme design. Specifically color and tone style adjustments are part of a generic process of cognition involved in the creation of new fabric designs. The authors explicitly formalize the process of color theme extraction from a set of images as a process of color mood based hierarchical data clustering and optimization. They begin with image sorting within a cognitive theme and color compatibility learning from large datasets. They then propose fully automatic color-texture association and color transfer algorithms which satisfy the criteria used in professional fabric pattern design while ensuring the plausibility of the cognitive theme preserved color transfer from the images to fabric patterns. Lastly, the color transfer process is formulated as a constrained optimization problem that is solved efficiently by total variation minimization. The use of color theme associations can automatically generate new fabric designs that rival complex commercial designs that are otherwise difficult to generate even by experienced designers. The authors’ fully automatic color theme preserving transfer method leads to a new approach to fabric design that significantly save time and cost for both fashion designers and computer artists.
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Segmentation is an important problem in various applications. There exist many effective models designed to locate all features and their boundaries in an image. However such global models are not suitable for automatically detecting a single object among many objects of an image, because nearby objects are often selected as well. Several recent works can provide selective segmentation capability but unfortunately when generalized to three dimensions, they are not yet effective or efficient. This paper presents a selective segmentation model which is inherently suited for efficient implementation. With the added solver by a fast nonlinear multigrid method for the inside domain of a zero level set function, the over methodology leads to an effective and efficient algorithm for 3D selective segmentation. Numerical experiments show that our model can produce efficient results in terms of segmentation quality and reliability for a large class of 3D images.
Thesis
L’essor récent de l’imagerie hybride combinant la Tomographie par Emission de Positons (TEP) à l’Imagerie par Résonance Magnétique (IRM) est une opportunité permettant d’exploiter des images d’un même territoire anatomo-pathologique obtenues simultanément et apportant des informations complémentaires. Cela représente aussi un véritable défi en raison de la différence de nature et de résolution spatiale des données acquises. Cette nouvelle technologie offre notamment des perspectives attrayantes en oncologie, et plus particulièrement en neuro-oncologie grâce au contraste qu’offre l’image IRM entre les tissus mous. Dans ce contexte et dans le cadre du projet PIM (Physique et Ingénierie pour la Médecine) de l’Université Paris-Saclay, l’objectif de cette thèse a été de développer un processus de segmentation multimodale adapté aux images TEP et IRM, comprenant une méthode de détection des volumes tumoraux en TEP et IRM, et une technique de segmentation précise du volume tumoral IRM. Ce processus doit être suffisamment générique pour s’appliquer à diverses pathologies cérébrales, différentes par leur nature même et par l’application clinique considérée. La première partie de la thèse aborde la détection de tumeurs par une approche hiérarchique. Plus précisément, la méthode de détection, réalisée sur les images IRM ou TEP, repose sur la création d’un nouveau critère de contexte spatial permettant de sélectionner les lésions potentielles par filtrage d’une représentation de l’image par max-tree. La deuxième partie de la thèse concerne la segmentation du volume tumoral sur les images IRM par une méthode variationnelle par ensembles de niveaux. La méthode de segmentation développée repose sur la minimisation d’une énergie globalement convexe associée à une partition d’une image RM en régions homogènes guidée par des informations de la TEP. Enfin, une dernière partie étend les méthodes proposées précédemment à l’imagerie multimodale IRM, notamment dans le cadre de suivi longitudinal. Les méthodes développées ont été testées sur plusieurs bases de données, chacune correspondant à une pathologie cérébrale et un radiotraceur TEP distincts. Les données TEP-IRM disponibles comprennent, d’une part, des examens de méningiomes et de gliomes acquis sur des machines séparées, et d’autre part, des examens réalisés sur le scanner hybride du Service Hospitalier Frédéric Joliot d’Orsay dans le cadre de recherches de tumeurs cérébrales. La méthode de détection développée a aussi été adaptée à l’imagerie multimodale IRM pour la recherche de lésions de sclérose en plaques ou le suivi longitudinal. Les résultats obtenus montrent que la méthode développée, reposant sur un socle générique, mais étant aussi modulable à travers le choix de paramètres, peut s’adapter à diverses applications cliniques. Par exemple, la qualité de la segmentation des images issues de la machine combinée a été mesurée par le coefficient de Dice, la distance de Hausdorff (DH) et la distance moyenne (DM), en prenant comme référence une segmentation manuelle de la tumeur validée par un expert médical. Les résultats expérimentaux sur ces données montrent que la méthode détecte les lésions visibles à la fois sur les images TEP et IRM, et que la segmentation contoure correctement la lésion (Dice, DH et DM valant respectivement 0, 85 ± 0, 09, 7, 28 ± 5, 42 mm et 0, 72 ± 0, 36mm).
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In oncological thoracic imaging, computerized tomography (CT) and positron emission tomography (PET) are widely used jointly, for diagnosis or treatment planing. The development of combined scanners enables the acquisition of pairs of CT-PET volumes, allowing their joint exploitation in clinical routine, without the prerequisite for complex registration. One goal of this thesis work was to propose a segmentation method jointly exploiting PET and CT image information. The proposed methodology therefore focuses on a detailed segmentation of the CT images, using PET information to guide the tumor segmentation. The framework of variational segmentation methods is used to design our algorithms and the specific constraints based on PET information. In addition to target structures for radiotherapy (tumors, nodules), organs at risk which need to be preserved from radiations, must be segmented. An additional goal of this thesis is to provide segmentation methods for these organs. The methods rely on strong a priori knowledge on the non-parametric intensity distributions and on the shapes of the different organs. A final goal of the thesis is to propose a methodological framework for the segmentation of tumors in the context of longitudinal follow up of patients with registered images. The proposed segmentation methods were tested on multiple data sets. When manual tracing is available, quantitive comparisons of the segmentations are presented, demonstrating the performance and accuracy of the proposed segmentation framework.
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We present a multi-phase image segmentation method based on the histogram of the Gabor feature space, which consists of a set of Gabor-filter responses with various orientations, scales and frequencies. Our model replaces the error function term in the original fuzzy region competition model with squared 2-Wasserstein distance function, which is a metric to measure the distance of two histograms. The energy functional is minimized by alternative minimization method and the existence of closed-form solutions is guaranteed when the exponent of the fuzzy membership term being 1 or 2. We test our model on both simple synthetic texture images and complex natural images with two or more phases. Experimental results are shown and compared to other recent results.
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In order to help doctor make diagnosis of vascular diseases quickly and accurately, for Computed Tomography images containing head and neck, a novel automatic segmentation algorithm of blood vessel is proposed. First of all, some special points in carotid and vertebral arteries are located automatically and considered as seeds. Then, vessel can be assumed as tubular structure which is formed through rolling spherical model from one end to the other, and during rolling spherical center and radius are adjusted by internal and external energy, so a series of spherical models can be acquired by sampling along vessel. At last, the whole vessel can be extracted by interpolation between adjacent spherical models. Experimental results showed that the vessels can be segmented from background accurately, even if vessel and bone are close to each other, where the gray value of bone is very similar to vessel.
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Existing image segmentation methods usually consider only intensity or texture features, and thus cannot work well on images with both varying intensity and salient texture patterns. This paper aims to solve this problem by exploring both intensity and texture features for better image segmentation. To this end, we propose a novel variational image segmentation model, which uses constant vectors to represent intensity means in different image subregions and structured dictionaries to encode local texture patterns in images. We develop an effective algorithm to implement the proposed variational model by seeking for intensity means and a specific dictionary for the input image and meanwhile computing the representation coefficients of image patches with respect to the learned dictionary. We derive closed-form solutions for all these components, which result in the high efficiency of our proposed method. Extensive evaluation experiments with comparison to existing methods have been done on both synthetic and real-world images. The results demonstrate the superiority of our proposed method in effectively segmenting images with both unsmooth intensity variations and salient texture patterns.
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In this paper, a wavelet transform based fuzzy region competition image segmentation algorithm is proposed. Utilizing the superiority of texture classification based on wavelet transform, this algorithm processes the image with undecimated wavelet transform under the fuzzy region competition algorithm, and then sets up the statistical characteristics of parameters in the wavelet domain first. Based on the parameters described above, the energy function is constructed accordingly. In the stage of solving the equation, a fast iterative algorithm is adopted to accelerate the procedure. The image segmentation algorithm proposed here can decrease the sensitivity of the signal-to-noise ratio, which often occurrs in traditional segmentation methods. Corresponding experiments are carried out. In these experiments, active contours without edges and the fuzzy region competition image segmentation algorithm are adopted as the references to evaluate our algorithm. Experimental results show that our proposed algorithm has promising performance on the segmentation qualities and the segmentation rate than the active contours without edges and the fuzzy region competition image segmentation algorithm.
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Selective image segmentation is an important topic in medical imaging and real applications. In this paper, we propose a weighted variational selective image segmentation model which contains two steps. The first stage is to obtain a smooth approximation related to Mumford-Shah model to the target region in the input image. Using weighted function, the approximation provides a larger value for the target region and smaller values for other regions. In the second stage, we make use of this approximation and perform a thresholding procedure to obtain the object of interest. The approximation can be obtained by the alternating direction method of multipliers and the convergence analysis of the method can be established. Experimental results for medical image selective segmentation are given to demonstrate the usefulness of the proposed method. We also do some comparisons and show that the performance of the proposed method is more competitive than other testing methods.
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SUMMARYA crucial aspect of spectral image analysis is the identification of the materials present in the object or scene being imaged and to quantify their abundance in the mixture. An increasingly useful approach to extracting such underlying structure is to employ image classification and object identification techniques to compressively represent the original data cubes by a set of spatially orthogonal bases and a set of spectral signatures. Owing to the increasing quantity of data usually encountered in hyperspectral data sets, effective data compressive representation is an important consideration, and noise and blur can present data analysis problems. In this paper, we develop image segmentation methods for hyperspectral space object material identification. We also couple the segmentation with a hyperspectral image data denoising/deblurring model and propose this method as an alternative to a tensor factorization methods proposed recently for space object material identification. The model provides the segmentation result and the restored image simultaneously. Numerical results show the effectiveness of our proposed combined model in hyperspectral material identification. Copyright © 2010 John Wiley & Sons, Ltd.
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The active contour/snake model (9, 2, 10) is one of the most well- known segmentation variational models in image processing. However this model suers from the existence of local minima which makes the initial guess critical for getting satisfactory results. In this paper, we propose to solve this problem by nding global minimizers of the active contour model following the original work of Chan, Esedo glu and Nikolova (4). Our approach uses the weighted total variation norm to link the standard active contour segmentation model with the denoising model of Rudin-Osher-Fatemi (15) and the Chan-Vese active contour segmentation models (5, 18) based on the Mumford-Shah functional (12).
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It is shown how false operator responses due to missing or uncertain data can be significantly reduced or eliminated. It is shown how operators having a higher degree of selectivity and higher tolerance against noise can be constructed using simple combinations of appropriately chosen convolutions. The theory is based on linear operations and is general in that it allows for both data and operators to be scalars, vectors or tensors of higher order. Three new methods are represented: normalized convolution, differential convolution and normalized differential convolution. All three methods are examples of the power of the signal/certainty-philosophy, i.e., the separation of both data and operator into a signal part and a certainty part. Missing data are handled simply by setting the certainty to zero. In the case of uncertain data, an estimate of the certainty must accompany the data. Localization or windowing of operators is done using an applicability function, the operator equivalent to certainty, not by changing the actual operator coefficients. Spatially or temporally limited operators are handled by setting the applicability function to zero outside the window
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We propose an algorithm for minimizing the total variation of an image, and provide a proof of convergence. We show applications to image denoising, zooming, and the computation of the mean curvature motion of interfaces.
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Image sharpening in the presence of noise is formulated as a non-convex variational problem. The energy functional incorporates a gradient-dependent potential, a convex fidelity criterion and a high order convex regularizing term. The first term attains local minima at zero and some high gradient magnitude, thus forming a triple well-shaped potential (in the one-dimensional case). The energy minimization flow results in sharpening of the dominant edges, while most noisy fluctuations are filtered out.
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We propose an algorithm for minimizing the total variation of an image, and provide a proof of convergence. We show applications to image denoising, zooming, and the computation of the mean curvature motion of interfaces.
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We propose an algorithm for minimizing the total variation of an image, and provide a proof of convergence. We show applications to image denoising, zooming, and the computation of the mean curvature motion of interfaces.
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We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is also a generalization of an active contour model without edges based 2-phase segmentation, developed by the authors earlier in T. Chan and L. Vese (1999. In Scale-Space'99, M. Nilsen et al. (Eds.), LNCS, vol. 1682, pp. 141–151) and T. Chan and L. Vese (2001. IEEE-IP, 10(2):266–277). The multiphase level set formulation is new and of interest on its own: by construction, it automatically avoids the problems of vacuum and overlap; it needs only log n level set functions for n phases in the piecewise constant case; it can represent boundaries with complex topologies, including triple junctions; in the piecewise smooth case, only two level set functions formally suffice to represent any partition, based on The Four-Color Theorem. Finally, we validate the proposed models by numerical results for signal and image denoising and segmentation, implemented using the Osher and Sethian level set method.
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A coupled level set method for the motion of multiple junctions (of, e.g., solid, liquid, and grain boundaries), which follows the gradient flow for an energy functional consisting of surface tension (proportional to length) and bulk energies (proportional to area), is developed. The approach combines the level set method of S. Osher and J. A. Sethian with a theoretical variational formulation of the motion by F. Reitich and H. M. Soner. The resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions. The gradient projection method leads to a coupled system of perturbed (by curvature terms) Hamilton-Jacobi equations. The coupling is enforced using a single Lagrange multiplier associated with a constraint which essentially prevents (a) regions from overlapping and (b) the development of a vacuum. The numerical implementation is relatively simple and the results agree with (and go beyond) the theory as given in [12]. Other applications of this methodology, including the decomposition of a domain into subregions with minimal interface length, are discussed. Finally, some new techniques and results in level set methodology are presented. (C) 1996 Academic Press, Inc.
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In this paper, we propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford--Shah functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by gradient. We minimize an energy which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. We will give a numerical algorithm using finite differences. Finally, we will present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.
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We propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by the gradient. We minimize an energy which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. We give a numerical algorithm using finite differences. Finally, we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.
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We present a novel statistical and variational approach to image segmentation based on a new algorithm, named region competition. This algorithm is derived by minimizing a generalized Bayes/minimum description length (MDL) criterion using the variational principle. The algorithm is guaranteed to converge to a local minimum and combines aspects of snakes/balloons and region growing. The classic snakes/balloons and region growing algorithms can be directly derived from our approach. We provide theoretical analysis of region competition including accuracy of boundary location, criteria for initial conditions, and the relationship to edge detection using filters. It is straightforward to generalize the algorithm to multiband segmentation and we demonstrate it on gray level images, color images and texture images. The novel color model allows us to eliminate intensity gradients and shadows, thereby obtaining segmentation based on the albedos of objects. It also helps detect highlight regions
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We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is also a generalization of an active contour model without edges based 2-phase segmentation, developed by the authors earlier in (Chan and Vese 1999), (Chan and Vese 2001). The multiphase level set formulation is new and of interest on its own: by construction, it automatically avoids the problems of vacuum and overlap; it needs only log n level set functions for n phases in the piecewise constant case; it can represent boundaries with complex topologies, including triple junctions; in the piecewise smooth case, only two level set functions formally suce to represent any partition, based on The Four-Color Theorem. Finally, we validate the proposed models by numerical results for signal and image denoising and segmentation, implemented using the Osher and Sethian level set method.