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The Ambrosio–Tortorelli approximation scheme with weighted underlying metric is investigated. It is shown that it Γ-converges to a Mumford–Shah image segmentation functional depending on the weight ω ⁢ d ⁢ x , where ω is a special function of bounded variation, and on its values at the jumps. [ABSTRACT FROM AUTHOR]

This note addresses the question of convergence of critical points of the Ambrosio–Tortorelli functional in the one-dimensional case under pure Dirichlet boundary conditions. An asymptotic analysis argument shows the convergence to two possible limits points: either a globally affine function or a step function with a single jump at the middle point of the space interval, which are both critical points of the one-dimensional Mumford–Shah functional under a Dirichlet boundary condition. As a byproduct, non minimizing critical points of the Ambrosio–Tortorelli functional satisfying the energy convergence assumption as in (Babadjian, Millot and Rodiac (2022)) are proved to exist. [ABSTRACT FROM AUTHOR]

Motivated by the success in image processing, the Mumford–Shah functional has attracted extensive attentions in geometry processing. Existing methods, mainly focusing on discretizations on the triangulated mesh, either over-smooth sharp features or are sensitive to noises or outliers. In this paper, we first introduce a nonsmooth nonconvex Mumford–Shah model for a feature-preserving filtering of face normal field to ameliorate the staircasing artifacts that appear in the original Mumford–Shah total variation (MSTV) and develop an alternating minimization scheme based on alternating direction method of multipliers to realize the proposed model. After restoring the face normal field, vertex updating is then employed by incorporating the oriented normal constraints and discontinuities to achieve a detail-preserving reconstruction of mesh geometry. Extensive experimental results demonstrate the effectiveness of the above shape optimization routine for various geometry processing applications such as mesh denoising, mesh inpainting and mesh segmentation. [Display omitted] • We present a feature-preserving normal filter using nonsmooth nonconvex Mumford–Shah regularization. • We propose a method for vertex updating by incorporating both the oriented normal constraints and the discontinuity function. • We demonstrate the superiority of our approach visually and numerically. [ABSTRACT FROM AUTHOR]

We give an alternative proof of the regularity, up to the loose end, of minimizers, resp. critical points of the Mumford-Shah functional when they are sufficiently close to the cracktip, resp. they consist of a single arc terminating at an interior point. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Recent state-of-the-art image segmentation algorithms are mostly based on deep neural networks, thanks to their high performance and fast computation time. However, these methods are usually trained in a supervised manner, which requires large number of high quality ground-truth segmentation masks. On the other hand, classical image segmentation approaches such as level-set methods are formulated in a self-supervised manner by minimizing energy functions such as Mumford-Shah functional, so they are still useful to help generate segmentation masks without labels. Unfortunately, these algorithms are usually computationally expensive and often have limitation in semantic segmentation. In this paper, we propose a novel loss function based on Mumford-Shah functional that can be used in deep-learning based image segmentation without or with small labeled data. This loss function is based on the observation that the softmax layer of deep neural networks has striking similarity to the characteristic function in the Mumford-Shah functional. We show that the new loss function enables semi-supervised and unsupervised segmentation. In addition, our loss function can also be used as a regularized function to enhance supervised semantic segmentation algorithms. Experimental results on multiple datasets demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We propose a new Γ-convergent discrete approximation of the Mumford–Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford–Shah functional in any dimension. [ABSTRACT FROM AUTHOR]

In this paper we exhibit a family of stationary solutions of the Mumford–Shah functional in R 3 , arbitrary close to a crack-front. Unlike other examples, known in the literature, those are topologically non-minimizing in the sense of Bonnet [4] . We also give a local version in a finite cylinder and prove an energy estimate for minimizers. Numerical illustrations indicate the stationary solutions are unlikely minimizers and show how the dependence on axial variable impacts the geometry of the discontinuity set. A self-contained proof of the stationarity of the cracktip function for the Mumford–Shah problem in 2D is presented. [ABSTRACT FROM AUTHOR]

This paper is the first part of an ongoing project aimed at providing a local minimality criterion, based on a second variation approach, for the triple point configurations of the Mumford-Shah functional. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper it is shown that any regular critical point of the Mumford–Shah functional, with positive definite second variation, is an isolated local minimizer with respect to competitors which are sufficiently close in the L 1 -topology. A global minimality result in small tubular neighborhoods of the discontinuity set is also established. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a new unconditionally stable hybrid numerical method for minimizing the piecewise constant Mumford–Shah functional of image segmentation. The model is based on the Allen–Cahn equation and an operator splitting technique is used to solve the model numerically. We split the governing equation into two linear equations and one nonlinear equation. One of the linear equations and the nonlinear equation are solved analytically due to the availability of closed-form solutions. The other linear equation is discretized using an implicit scheme and the resulting discrete system of equations is solved by a fast numerical algorithm such as a multigrid method. We prove the unconditional stability of the proposed scheme. Since we incorporate closed-form solutions and an unconditionally stable scheme in the solution algorithm, our proposed scheme is accurate and robust. Various numerical results on real and synthetic images with noises are presented to demonstrate the efficiency, robustness, and accuracy of the proposed method. [Copyright &y& Elsevier]

We study the properties of absolute minimal and equilibrium states of generalized Mumford—Shah functionals, with applications to models of quasistatic brittle fracture propagation. The main results, theorems 7.3, 8.4 and 9.1, concern a priori inequalities between energy release rate and energy concentration for 3D cracks with complex shapes, seen as outer measures living on the crack edge. [ABSTRACT FROM PUBLISHER]

Abstract: The aim of this paper is to prove the existence of extension operators for SBV functions from periodically perforated domains. This result will be the fundamental tool to prove the compactness in a noncoercive homogenization problem. [Copyright &y& Elsevier]

Abstract: 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. [ABSTRACT FROM AUTHOR]

The goal of this work is to identify a density function of a physical body from a given X-ray data. The mathematical relation between parameter and data is described by the Radon transform. We propose a piecewise smooth Mumford-Shah model for the simultaneous inversion and segmentation of the tomography data. In our approach the functional variable is eliminated by solving a classical variational problem for each fixed geometry. The solution is then inserted in the Mumford-Shah cost functional leading to a geometrical optimization problem for the singularity set. The resulting shape optimization problem is solved using shape sensitivity calculus and propagation of shape variables in the level-set form. The optimality system for the fixed geometry has the form of a coupled system of integro-differential equations on variable and irregular domains. A new finite difference method-based approach for the solution of the optimality system is presented. Here a standard five-point stencil is used on regular points of an underlying uniform grid and modifications of the standard stencil are made at points close to the boundary. The optimality system is solved iteratively. Numerical experiments are presented. [ABSTRACT FROM AUTHOR]

Critical points of a variant of the Ambrosio-Tortorelli functional, for which non-zero Dirichlet boundary conditions replace the fidelity term, are investigated. They are shown to converge to particular critical points of the corresponding variant of the Mumford-Shah functional; those exhibit many symmetries. That Dirichlet variant is the natural functional when addressing a problem of brittle fracture in an elastic material. [ABSTRACT FROM AUTHOR]

Abstract: We consider a conducting body which presents some (unknown) perfectly insulating defects, such as cracks or cavities, for instance. We aim to reconstruct the defects by performing measurements of current and voltage type on a (known and accessible) part of the boundary of the conductor. A crucial step in this reconstruction is the determination of the electrostatic potential inside the conductor, by the electrostatic boundary measurements performed. Since the defects are unknown, we state such a determination problem as a free-discontinuity problem for the electrostatic potential in the framework of special functions of bounded variation. We provide a characterisation of the looked for electrostatic potential and we approximate it with the minimum points of a sequence of functionals, which take also in account the error in the measurements. These functionals are related to the so-called Mumford–Shah functional, which acts as a regularizing term and allows us to prove existence of minimizers and Γ-convergence properties. [Copyright &y& Elsevier]

Abstract: A level-set based approach for the determination of a piecewise constant density function from data of its Radon transform is presented. Simultaneously, a segmentation of the reconstructed density is obtained. The segmenting contour and the corresponding density are found as minimizers of a Mumford–Shah like functional over the set of admissible contours and – for a fixed contour – over the space of piecewise constant densities which may be discontinuous across the contour. Shape sensitivity analysis is used to find a descent direction for the cost functional which leads to an update formula for the contour in the level-set framework. The descent direction can be chosen with respect to different metrics. The use of an L 2-type and an H 1-type metric is proposed and the corresponding steepest descent flow equations are derived. A heuristic approach for the insertion of additional components of the density is presented. The method is tested for several data sets including synthetic as well as real-world data. It is shown that the method works especially well for large data noise (∼10% noise). The choice of the H 1-metric for the determination of the descent direction is found to have positive effect on the number of level-set steps necessary for finding the optimal contours and densities. [Copyright &y& Elsevier]

We show that limits of Mumford-Shah minimizers in product domains Ω = Ω′× (0, t), t small, are Mumford-Shah minimizers in one less dimension. The main ingredient of the proof is a symmetry argument from Dal Maso, Morel, and Solimini. [ABSTRACT FROM AUTHOR]

Some applications in remote sensing require estimating a field containing a discontinuity whose exact location is a priori unknown. Such fields of interest include sea surface temperature in oceanography and soil moisture in hydrology. For the former, oceanic fronts form a temperature discontinuity, while in the latter sharp changes exist across the interface between soil types. To complicate the estimation process, remotely sensed measurements often exhibit regions of missing observations due to occlusions such as cloud cover. Similarly, water surface and ground-based sensors usually provide only an incomplete set of measurements. Traditional methods of interpolation and smoothing for estimating the fields from such potentially sparse measurements often blur across the discontinuities in the field. [ABSTRACT FROM AUTHOR]

Recently, Chan and Vese developed an active contour model for image segmentation and smoothing by using piecewise constant and smooth representation of an image. Tsai et al. also independently developed a segmentation and smoothing method similar to the Chan and Vese piecewise smooth approach. These models are active contours based on the Mumford-Shah variational approach and the level-set method. In this paper, we develop a new hierarchical method which has many advantages compared to the Chan and Vese multiphase active contour models. First, unlike previous works, the curve evolution partial differential equations (PDEs) for different level-set functions are decoupled. Each curve evolution PDE is the equation of motion of just one level-set function, and different level-set equations of motion are solved in a hierarchy. This decoupling of the motion equations of the level-set functions speeds up the segmentation process significantly. Second, because of the coupling of the curve evolution equations associated with different level-set functions, the initialization of the level sets in Chan and Vese's method is difficult to handle. In fact, different initial conditions may produce completely different results. The hierarchical method proposed in this paper can avoid the problem due to the choice of initial conditions. Third, in this paper, we use the diffusion equation for denoising. This method, therefore, can deal with very noisy images. In general, our method is fast, flexible, not sensitive to the choice of initial conditions, and produces very good results. [ABSTRACT FROM AUTHOR]

In recent years, diffusion tensor imaging (DTI) has become a popular in vivo diagnostic imaging technique in Radiological sciences. In order for this imaging technique to be more effective, proper image analysis techniques suited for analyzing these high dimensional data need to be developed. In this paper, we present a novel definition of tensor "distance" grounded in concepts from information theory and incorporate it in the segmentation of DTI. In a DTI, the symmetric positive definite (SPD) diffusion tensor at each voxel can be interpreted as the covariance matrix of a local Gaussian distribution. Thus, a natural measure of dissimilarity between SPD tensors would be the Kullback-Leibler (KL) divergence or its relative. We propose the square root of the i-divergence (symmetrized KL) between two Gaussian distributions corresponding to the diffusion tensors being compared and this leads to a novel closed form expression for the "distance" as well as the mean value of a DTI. Unlike the traditional Frobenius norm-based tensor distance, our "distance" is affine invariant, a desirable property in segmentation and many other applications. We then incorporate this new tensor "distance" in a region based active contour model for DTI segmentation. Synthetic and real data experiments are shown to depict the performance of the proposed model. [ABSTRACT FROM AUTHOR]

We study by means of $\Gamma$-convergence the asymptotics of the rescaled Mumford-Shah functional when $\varepsilon \to 0$ and prove the existence of a $\Gamma$-limit. The limit functional is easy to handle and can be used as a simple approximation to the original Mumford-Shah functional. Moreover, its minimizers can be interpreted as a sort of asymptotic probability distribution of the sets $\Gamma$. Some examples illustrate the use of this method in image segmentation. [ABSTRACT FROM AUTHOR]

We cast the problem of multiframe stereo reconstruction of a smooth shape 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 assumptions it is designed for. [ABSTRACT FROM AUTHOR]

In this paper we give a partial answer to a conjecture of De Giorgi, namely we prove that in dimension two the regular part of the discontinuity set of a local minimizer of the homogeneous Mumford–Shah functional is analytic with the exception of at most a countable number of isolated points. [Copyright &y& Elsevier]

We present a generative approach to model-based motion segmentation by incorporating a statistical shape prior into a novel variational segmentation method. The shape prior statistically encodes a training set of object outlines presented in advance during a training phase.In a region competition manner the proposed variational approach maximizes the homogeneity of the motion vector field estimated on a set of regions, thus evolving the separating discontinuity set. Due to the shape prior, this discontinuity set is not only sensitive to motion boundaries but also favors shapes according to the statistical shape knowledge.In numerical examples we verify several properties of the proposed approach: for objects which cannot be easily discriminated from the background by their appearance, the desired motion segmentation is obtained, although the corresponding segmentation based on image intensities fails. The region-based formulation facilitates convergence of the contour from its initialization over fairly large distances, and the estimated flow field is progressively improved during the gradient descent minimization. Due to the shape prior, partial occlusions of the moving object by ‘unfamiliar’ objects are ignored, and the evolution of the motion boundary is effectively restricted to the subspace of familiar shapes. [Copyright &y& Elsevier]

In this paper, we introduce a Mumford–Shah framework to restore the face normal field on the triangulated surface. To effectively discretize Γ -convergence approximation of the Mumford–Shah model, we first define an edge function space and its associated differential operators. They are helpful for directly diffusing the discontinuity function over mesh edges instead of computing the approximated discontinuity function via pointwise diffusion in existing discretizations. Then, by using the operators in the proposed function space, two Mumford–Shah-based denoising methods are presented, which can produce denoised results with neat geometric features and locate geometric discontinuities exactly. Our Mumford–Shah framework overcomes the limitations of existing techniques that blur the discontinuity function, be less able to preserve geometric features, be sensitive to surface sampling, and require a postprocessing to form feature curves from located discontinuity vertices. Intensive experimental results on a variety of surfaces show the superiority of our denoising methods qualitatively and quantitatively. • Two coupled function spaces and associated operators are given out over meshes, which can describe the edge function space and its operators for directly diffusing the function over edges. • Two Mumford–Shah-based models are formulated in the proposed function spaces, which are more able to produce high quality denoised results with neat features and at the same time locate discontinuities accurately. • Two efficient algorithms based on alternating minimization are presented to solve the proposed Mumford–Shah regularizations. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of correspondence establishment in binocular stereo vision. We suggest a novel spatially continuous approach for stereo matching based on the variational framework. The proposed method suggests a unique regularization term based on Mumford-Shah functional for discontinuity preserving, combined with a new energy functional for occlusion handling. The evaluation process is based on concurrent minimization of two coupled energy functionals, one for domain segmentation (occluded versus visible) and the other for disparity evaluation. In addition to a dense disparity map, our method also provides an estimation for the half-occlusion domain and a discontinuity function allocating the disparity/depth boundaries. Two new constraints are introduced improving the revealed discontinuity map. The experimental tests include a wide range of real data sets from the Middlebury stereo database. The results demonstrate the capability of our method in calculating an accurate disparity function with sharp discontinuities and occlusion map recovery. Significant improvements are shown compared to a recently published variational stereo approach. A comparison on the Middlebury stereo benchmark with subpixel accuracies shows that our method is currently among the top-ranked stereo matching algorithms. [ABSTRACT FROM PUBLISHER]