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In electrical impedance tomography, we aim to solve the conductivity within a target body through electrical measurements made on the surface of the target. This inverse conductivity problem is severely ill-posed, especially in real applications with only partial boundary data available. Thus regularization has to be introduced. Conventionally regularization promoting smooth features is used, however, the Mumford--Shah regularizer familiar for image segmentation is more appropriate for targets consisting of several distinct objects or materials. It is, however, numerically challenging. We show theoretically through $\Gamma$-convergence that a modification of the Ambrosio--Tortorelli approximation of the Mumford--Shah regularizer is applicable to electrical impedance tomography, in particular the complete electrode model of boundary measurements. With numerical and experimental studies, we confirm that this functional works in practice and produces higher quality results than typical regularizations employed in electrical impedance tomography when the conductivity of the target consists of distinct smoothly-varying regions., Comment: 28 pages, 7 figures

In this work, we propose a p-harmonic Mumford–Shah (MS) functional with adaptive variable exponent 1 ≤ p(x) ≤ 2 according to image gray feature, which provides a model for image segmentation and smoothing. The paper analyzes the physical characteristics of the related p-harmonic equation in local coordinates and explains that diffusion behavior of p-harmonic is superior to that of anisotropic diffusion and isotropic diffusion in essence. Thus the proposed model is more suitable for segmentation and smoothing of noisy images with intensity inhomogeneities while simultaneously preserving edges than the piecewise smooth MS (PSMS) model. Then effective numerical scheme is constructed to handle its computation using level set method. The model is finally applied on a wide variety of image segmentation and smoothing. All these results show that the proposed model is effective.

In this paper we introduce a new phase field approximation of the Mumford-Shah functional similar to the well-known one from Ambrosio and Tortorelli. However, in our setting the phase field is allowed to be a function of bounded variation, instead of an $H^1$-function. In the context of image segmentation, we also show how this new approximation can be used for numerical computations, which contains a total variation minimization of the phase field variable, as it appears in many problems of image processing. A comparison to the classical Ambrosio-Tortorelli approximation, where the phase field is an $H^1$-function, shows that the new model leads to sharper phase fields., 32 pages, 4 figures, 1 table, 39 references

Saliency detection aims to highlight the most relevant objects in an image. Methods using conventional models struggle whenever salient objects are pictured on top of a cluttered background while deep neural nets suffer from excess complexity and slow evaluation speeds. In this paper, we propose a simplified convolutional neural network which combines local and global information through a multi-resolution 4×5 grid structure. Instead of enforcing spacial coherence with a CRF or superpixels as is usually the case, we implemented a loss function inspired by the Mumford-Shah functional which penalizes errors on the boundary. We trained our model on the MSRA-B dataset, and tested it on six different saliency benchmark datasets. Results show that our method is on par with the state-of-the-art while reducing computation time by a factor of 18 to 100 times, enabling near real-time, high performance saliency detection.

The Mumford-Shah (MS) functional is one of the most influential variational model in image segmentation, restoration, and cartooning. Difficult to solve, the Ambrosio-Tortorelli (AT) functional is of particular interest, because minimizers of AT can be shown to converge to a minimizer of MS. This paper takes an interest in a new method for numerically solving the AT model [11]. This method formulates the AT functional in a discrete calculus setting, and by this way is able to capture the set of discontinuities as a one-dimensional set. It is also shown that this model is competitive with total variation restoration methods. We present here the discrete AT models in details, and compare its merit with recent convex relaxations of AT and MS functionals. We also examine the potential of this model for inpainting, and describe its implementation in the DGtal library, an open-source project.

The Ambrosio-Tortorelli functional is a phase-field approximation of the Mumford-Shah functional that has been widely used for image segmentation. The approximation has the advantages of being easy to implement, maintaining the segmentation ability, and $\Gamma$-converging to the Mumford-Shah functional. However, it has been observed in actual computation that the segmentation ability of the Ambrosio-Tortorelli functional varies significantly with different values of the parameter and it even fails to $\Gamma$-converge to the original functional for some cases. In this paper we present an asymptotic analysis on the gradient flow equation of the Ambrosio-Tortorelli functional and show that the functional can have different segmentation behavior for small but finite values of the regularization parameter and eventually loses its segmentation ability as the parameter goes to zero when the input image is treated as a continuous function. This is consistent with the existing observation as well as the numerical examples presented in this work. A selection strategy for the regularization parameter and a scaling procedure for the solution are devised based on the analysis. Numerical results show that they lead to good segmentation of the Ambrosio-Tortorelli functional for real images., Comment: 22 pages

We propose to incorporate a weighted difference of anisotropic and isotropic total variation (TV) norms into a relaxed formulation of the two phase Mumford-Shah (MS) model for image segmentation. We show results exceeding those obtained by the MS model when using the standard TV norm to regularize partition boundaries. In particular, examples illustrating the qualitative differences between the proposed model and the standard MS one are shown. A fast numerical method is introduced to minimize the proposed model utilizing the difference-of-convex algorithm (DCA) and the primal dual hybrid gradient (PDHG) method.

Hierarchies, such as the tree of shapes, are popular representations for image simplification and segmentation thanks to their multiscale structures. Selecting meaningful level lines (boundaries of shapes) yields to simplify image while preserving intact salient structures. Many image simplification and segmentation methods are driven by the optimization of an energy functional, for instance the celebrated Mumford-Shah functional. In this paper, we propose an efficient approach to hierarchical image simplification and segmentation based on the minimization of the piecewise-constant Mumford-Shah functional. This method conforms to the current trend that consists in producing hierarchical results rather than a unique partition. Contrary to classical approaches which compute optimal hierarchical segmentations from an input hierarchy of segmentations, we rely on the tree of shapes, a unique and well-defined representation equivalent to the image. Simply put, we compute for each level line of the image an attribute function that characterizes its persistence under the energy minimization. Then we stack the level lines from meaningless ones to salient ones through a saliency map based on extinction values defined on the tree-based shape space. Qualitative illustrations and quantitative evaluation on Weizmann segmentation evaluation database demonstrate the state-of-the-art performance of our method., Pattern Recognition Letters, Elsevier, 2016

Front propagation models represent an important category of image segmentation techniques in the current literature. These models are normally formulated in a continuous level sets framework and optimized using gradient descent methods. Such formulations result in very slow algorithms that get easily stuck in local solutions and are highly sensitive to initialization. In this paper, we reformulate one of the most influential front propagation models, the Chan-Vese model, in the discrete domain. The graph representability and submodularity of the discrete energy function is established and then max-flow/min-cut approach is applied to perform the optimization of the discrete energy function. Our results show that this formulation is much more robust than the level sets formulation. Our approach is not sensitive to initialization and provides much faster solutions than level sets. The results also depict that our segmentation approach is robust to topology changes, noise and ill-defined edges, i.e., it preserves all the advantages associated with level sets methods.

Image inpainting is an artistic procedure to recover a damaged painting or picture. We propose a novel approach for image inpainting by using the Mumford-Shah (MS) model and the level set method to estimate image structure of the damaged regions. This approach has been successfully used in image segmentation problem. Compared to some other inpainting methods, the MS model approach detects and preserves edges in the inpainting areas. We propose a fast and efficient algorithm that achieves both inpainting and segmentation. In previous works on the MS model, only one or two level set functions are used to segment an image. While this approach works well on simple cases, detailed edges cannot be detected in complicated image structures. Although multi-level set functions can be used to segment an image into many regions, the traditional approach causes extensive computations and the solutions depend on the location of initial curves. Our proposed approach utilizes faster hierarchical level set method and guarantees convergence independent of initial conditions. Because we detect both the main structure and the detailed edges, our approach preserves edges in the inpainting area. Also, exemplar-based approach for filling textured regions is employed. Experimental results demonstrate the advantage of our method.

Variational models have been studied for image segmentation application since the Mumford-Shah functional was introduced in the late 1980s. In this paper, we focus on multiphase segmentation with a new regularization term that yields an unsupervised segmentation model. We propose a functional that automatically chooses a favorable number of phases as it segments the image. The primary driving force of the segmentation is the intensity fitting term while a phase scale measure complements the regularization term. We propose a fast, yet simple, brute-force numerical algorithm and present experimental results showing the robustness and stability of the proposed model.

In this letter, we propose a shape-based active contours method for segmentation, based on a piecewise-constant approximation of the Mumford-Shah (M-S) functional. The Chan-Vese (C-V) formalism in a level set framework is used to formulate our method; however no sign distance function (SDF) is employed in the method proposed here. This method has the topology-free segmentation associated with the C-V algorithm and adds faster convergence, less memory requirement and fast re-initialization. These properties make the algorithm very attractive for video segmentation.

We propose a constrained version of Mumford and Shah's (1989) segmentation model with an information-theoretic point of view in order to devise a systematic procedure to segment brain magnetic resonance imaging (MRI) data for parametric -Map and -weighted images, in both 2-D and 3D settings. Incorporation of a tuning weight in particular adds a probabilistic flavor to our segmentation method, and makes the 3-tissue segmentation possible. Moreover, we proposed a novel method to jointly segment the -Map and calibrate RF Inhomogeneity (JSRIC). This method assumes theaveragevalue of white matter is the same across transverse slices in the central brain region, and JSRIC is able to rectify the flip angles to generate calibrated -Maps. In order to generate an accurate -Map, the determination of optimal flip-angles and the registration of flip-angle images are examined. Our JSRIC method is validated on two human subjects in the 2D -Map modality and our segmentation method is validated by two public databases, BrainWeb and IBSR, of -weighted modality in the 3D setting.

We address the problem of estimating the shape and appearance of a scene made of smooth Lambertian surfaces with piecewise smooth albedo. We allow the scene to have self-occlusions and multiple connected components. This class of surfaces is often used as an approximation of scenes populated by man-made objects. We assume we are given a number of images taken from different vantage points. Mathematically this problem can be posed as an extension of Mumford and Shah's approach to static image segmentation to the segmentation of a function defined on a deforming surface. We propose an iterative procedure to minimize a global cost functional that combines geometric priors on both the shape of the scene and the boundary between smooth albedo regions. We carry out the numerical implementation in the level set framework.

A novel shape prior segmentation model is proposed that utilizes the cliques invariant signature along with a polygonal piecewise constant implementation of the Mumford-Shah Functional. The model will be shown to be useful in the context of difficult segmentation problems including and not limited to segmenting objects amidst clutter, or recognizing objects that contain components with largely varying non-uniform image intensities. In addition, the model will also be shown to be effective for image disocclusion. Lastly, the proposed model can accomplish all the aforementioned tasks both efficiently and with near automation.

We propose a new shape-based modeling technique for applications in imaging problems. Given a collection of shape priors (a shape dictionary), we define our problem as choosing the right dictionary elements and geometrically composing them through basic set operations to characterize desired regions in an image. This is a combinatorial problem solving which requires an exhaustive search among a large number of possibilities. We propose a convex relaxation to the problem to make it computationally tractable. We take some major steps towards the analysis of the proposed convex program and characterizing its minimizers. Applications vary from shape-based characterization, object tracking, optical character recognition, and shape recovery in occlusion, to other disciplines such as the geometric packing problem.

Image segmentation is one of the first steps in any process concerning digital image analysis and its accuracy will go on to determine the quality of this analysis. A classic model used in image segmentation is the Mumford-Shah functional, which includes both the information to pertaining the region and the length of its borders. In this work, by using the concept of loss in Bregman Information a functional is defined which is a generalization of the Mumford-Shah functional, once it is obtained from the proposed function by means of the Squared Euclidean distance as a measure of similarity. The algorithm is constructed by using a fusion criterion, which minimizes the loss in Bregman Information. It is shown that the proposed hierarchical segmentation method generalizes the algorithm which minimizes the piecewise constant Mumford-Shah functional. The results obtained through use of the Generalized I-Divergence, Itakura-Saito and Squared Euclidean distance, show that the algorithm attained a good performance.

In this paper, we propose a new variational model to segment an object belonging to a given shape space using the active contour method, a geometric shape prior and the Mumford-Shah functional. The core of our model is an energy functional composed by three complementary terms. The first one is based on a shape model which constrains the active contour to get a shape of interest. The second term detects object boundaries from image gradients. And the third term drives globally the shape prior and the active contour towards a homogeneous intensity region. The segmentation of the object of interest is given by the minimum of our energy functional. This minimum is computed with the calculus of variations and the gradient descent method that provide a system of evolution equations solved with the well-known level set method. We also prove the existence of this minimum in the space of functions with bounded variation. Applications of the proposed model are presented on synthetic and medical images.

We propose an efficient algorithm for minimizing the piecewise constant Mumford-Shah functional of image segmentation. It is based on the threshold dynamics of Merriman, Bence, and Osher for evolving an interface by its mean curvature. We show that a very fast minimization can be achieved by alternating the solution of a linear parabolic partial differential equation and simple thresholding.

International audience; We consider the Chan and Vese multiphase segmentationbased on the partition of an image minimizing an energyinvolving a region-based data fidelity term and a regularization term. The common implementation of the continuousoptimization of this segmentation framework, with multiplelevel set functions, raises some numerical issues which leadto poor performance of the method when handling more thantwo phases. We propose a general formulation of the multi-phase model, and a permutation method, incorporated in thelevel-set based implementation of the multi-phase approachto handle in an original way the so-called hidden phase problem.

This paper studies the gradient flow of a regularized Mumford-Shah functional proposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with L 2 ×L ∞ initial data possesses a global weak solution, and it has a unique global in time strong solution, which has at most finite number of point singularities in the space-time, when the initial data are in H 1 × H 1 ∩ L ∞ . A family of fully discrete approximation schemes using low order finite elements is proposed for the gradient flow. Convergence of a subsequence (resp. the whole sequence) of the numerical solutions to a weak solution (resp. the strong solution) of the gradient flow is established as the mesh sizes tend to zero, and optimal and suboptimal order error estimates, which depend on 1 and 1 ke only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation k = o(h 1 2 ). Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.

We prove that a suitable rescaling of biased Perona–Malik energies, defined in the discrete setting, Γ-converges to an anisotropic version of the Mumford–Shah functional. Numerical results are discussed.

We cast the problem of multiframe stereo reconstruction of a smooth surface as the global region segmentation of a collection of images of the scene. Dually, the problem of segmenting multiple calibrated images of an object becomes that of estimating the solid shape that gives rise to such images. We assume that the radiance of the scene results in piecewise homogeneous image statistics. This simplifying assumption covers Lambertian scenes with constant albedo as well as fine homogeneous textures, which are known challenges to stereo algorithms based on local correspondence. We pose the segmentation problem within a variational framework, and use fast level set methods to find the optimal solution numerically. Our algorithm does not work in the presence of strong photometric features, where traditional reconstruction algorithms do. It enjoys significant robustness to noise under the assumptiong it is designed for.

We cast the problem of shape reconstruction of a scene as the global region segmentation of a collection of calibrated images. We assume that the scene is composed of a number of smooth surfaces and a background, both of which support smooth Lambertian radiance functions. We formulate the problem in a variational framework, where the solution (both the shape and radiance of the scene) is a minimizer of a global cost functional which combines a geometric prior on shape, a smoothness prior on radiance and a data fitness score. We estimate the shape and radiance via an alternating minimization: The radiance is computed as the solutions of partial differential equations defined on the surface and the background. The shape is estimated using a gradient descent flow, which is implemented using the level set method. Our algorithm works for scenes with smooth radiances as well as fine homogeneous textures, which are known challenges to traditional stereo algorithms based on local correspondence.

We present a modification of the Mumford-Shah functional and its cartoon limit which facilitates the incorporation of a statistical prior on the shape of the segmenting contour. By minimizing a single energy functional, we obtain a segmentation process which maximizes both the grey value homogeneity in the separated regions and the similarity of the contour with respect to a set of training shapes. We propose a closed-form, parameter-free solution for incorporating invariance with respect to similarity transformations in the variational framework. We show segmentation results on artificial and real-world images with and without prior shape information. In the cases of noise, occlusion or strongly cluttered background the shape prior significantly improves segmentation. Finally we compare our results to those obtained by a level set implementation of geodesic active contours.

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 overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies, including triple junctionss 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.

In this work, we first address the problem of simultaneous image segmentation and smoothing by approaching the Mumford-Shah paradigm from a curve evolution perspective. In particular, we let a set of deformable contours define the boundaries between regions in an image where we model the data via piecewise smooth functions and employ a gradient flow to evolve these contours. Each gradient step involves solving an optimal estimation problem for the data within each region, connecting curve evolution and the Mumford-Shah functional with the theory of boundary-value stochastic processes. The resulting active contour model offers a tractable implementation of the original Mumford-Shah model (i.e., without resorting to elliptic approximations which have traditionally been favored for greater ease in implementation) to simultaneously segment and smoothly reconstruct the data within a given image in a coupled manner. Various implementations of this algorithm are introduced to increase its speed of convergence. We also outline a hierarchical implementation of this algorithm to handle important image features such as triple points and other multiple junctions. Next, by generalizing the data fidelity term of the original Mumford-Shah functional to incorporate a spatially varying penalty, we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing. This more general model leads us to a novel PDE-based approach for simultaneous image magnification, segmentation, and smoothing, thereby extending the traditional applications of the Mumford-Shah functional which only considers simultaneous segmentation and smoothing.

International audience; Many methods relying on the morphological notion of shapes, (i.e., connected components of level sets) have been proved to be very useful for pattern analysis and recognition. Selecting meaningful level lines (boundaries of level sets) yields to simplify images while preserving salient structures. Many image simplification and/or segmentation methods are driven by the optimization of an energy functional, for instance the Mumford-Shah functional. In this article, we propose an efficient shape-based morphological filtering that very quickly compute to a locally (subordinated to the tree of shapes) optimal solution of the piecewise-constant Mumford- Shah functional. Experimental results demonstrate the efficiency, usefulness, and robustness of our method, when applied to image simplification, pre-segmentation, and detection of affine regions with viewpoint changes.

Image segmentation is intended to group perceptually similar pixels into 2D regions, and the corresponding border is gained at the same time. Video segmentation generalizes this concept to the grouping of pixels into spatio-temporal regions that exhibit coherence in both appearance and motion, but this generalization pose the complexity of spatio-temporal grouping, and in order to overcome this complexity, the existing video segmentation methods have extended the image segmentation methods to 3D domain. In these volumetric approaches, it is not known a priori, which regions to track, what frames contain those regions, or the time-direction for tracking (forward or backward). In this paper we present an efficient and scalable method for spatio-temporal segmentation obtained by minimizing a 2D+time extension of the simplified Mumford-Shah functional. The 2D+time extension permits to write the Mumford-Shah functional as an a multiscale energy, which is minimized on a hierarchy of video domain. The construction of this hierarchy based on the 2D-shapes of video images, and Scale-invariant feature transform (SIFT).

A typical task of image segmentation is to partition a given image into regions of homogeneous property. In this paper we focus on the problem of further detecting scales of discontinuities of the image. The approach uses a recently developed iterative numerical algorithm for minimizing the Mumford-Shah functional which is based on topological derivatives. For the scale selection we use a squared norm of the gradient at edge points. During the iteration progress, the square norm, as a function varied with iteration numbers, provides information about different scales of the discontinuity sets. For realistic image data, the graph of the norm function is regularized by using total variation minimization to provide stable separation. We present the details of the algorithm and document various numerical experiments.