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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]

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]

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]

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]

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]

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]