239 results on '"Mumford-Shah functional"'
Search Results
2. Hyperparameter selection for Discrete Mumford–Shah.
- Author
-
Lucas, Charles-Gérard, Pascal, Barbara, Pustelnik, Nelly, and Abry, Patrice
- Abstract
This work focuses on a parameter-free joint piecewise smooth image denoising and contour detection. Formulated as the minimization of a discrete Mumford–Shah functional and estimated via a theoretically grounded alternating minimization scheme, the bottleneck of such a variational approach lies in the need to fine-tune their hyperparameters, while not having access to ground truth data. To that aim, a Stein-like strategy providing optimal hyperparameters is designed, based on the minimization of an unbiased estimate of the quadratic risk. Efficient and automated minimization of the estimate of the risk crucially relies on an unbiased estimate of the gradient of the risk with respect to hyperparameters. Its practical implementation is performed using a forward differentiation of the alternating scheme minimizing the Mumford–Shah functional, requiring exact differentiation of the proximity operators involved. Intensive numerical experiments are performed on synthetic images with different geometry and noise levels, assessing the accuracy and the robustness of the proposed procedure. The resulting parameter-free piecewise-smooth estimation and contour detection procedure, not requiring prior image processing expertise nor annotated data, can then be applied to real-world images. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
3. A note on the one-dimensional critical points of the Ambrosio–Tortorelli functional.
- Author
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Babadjian, Jean-François, Millot, Vincent, and Rodiac, Rémy
- Subjects
- *
BRITTLE fractures - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
4. A Semi-supervised Deep Learning-Based Approach with Multiphase Active Contour Loss for Left Ventricle Segmentation from CMR Images
- Author
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Trinh, Minh-Nhat, Nguyen, Nhu-Toan, Tran, Thi-Thao, Pham, Van-Truong, Kacprzyk, Janusz, Series Editor, Pal, Nikhil R., Advisory Editor, Bello Perez, Rafael, Advisory Editor, Corchado, Emilio S., Advisory Editor, Hagras, Hani, Advisory Editor, Kóczy, László T., Advisory Editor, Kreinovich, Vladik, Advisory Editor, Lin, Chin-Teng, Advisory Editor, Lu, Jie, Advisory Editor, Melin, Patricia, Advisory Editor, Nedjah, Nadia, Advisory Editor, Nguyen, Ngoc Thanh, Advisory Editor, Wang, Jun, Advisory Editor, Poonia, Ramesh Chandra, editor, Singh, Vijander, editor, Singh Jat, Dharm, editor, Diván, Mario José, editor, and Khan, Mohammed S., editor
- Published
- 2022
- Full Text
- View/download PDF
5. Feature-preserving Mumford–Shah mesh processing via nonsmooth nonconvex regularization.
- Author
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Wang, Chunxue, Liu, Zheng, and Liu, Ligang
- Subjects
- *
STRUCTURAL optimization , *IMAGE processing , *MULTIPLIERS (Mathematical analysis) , *INPAINTING - Abstract
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]
- Published
- 2022
- Full Text
- View/download PDF
6. Soft Image Segmentation: On the Clustering of Irregular, Weighted, Multivariate Marked Networks
- Author
-
Ceré, Raphaël, Bavaud, François, Barbosa, Simone Diniz Junqueira, Series Editor, Filipe, Joaquim, Series Editor, Kotenko, Igor, Series Editor, Sivalingam, Krishna M., Series Editor, Washio, Takashi, Series Editor, Yuan, Junsong, Series Editor, Zhou, Lizhu, Series Editor, Ghosh, Ashish, Series Editor, Ragia, Lemonia, editor, Laurini, Robert, editor, and Rocha, Jorge Gustavo, editor
- Published
- 2019
- Full Text
- View/download PDF
7. Endpoint regularity for 2d Mumford-Shah minimizers: On a theorem of Andersson and Mikayelyan.
- Author
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De Lellis, Camillo, Focardi, Matteo, and Ghinassi, Silvia
- Abstract
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]
- Published
- 2021
- Full Text
- View/download PDF
8. Numerical Implementation of the Ambrosio-Tortorelli Functional Using Discrete Calculus and Application to Image Restoration and Inpainting
- Author
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Foare, Marion, Lachaud, Jacques-Olivier, Talbot, Hugues, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Kerautret, Bertrand, editor, Colom, Miguel, editor, and Monasse, Pascal, editor
- Published
- 2017
- Full Text
- View/download PDF
9. Approximation of the Mumford–Shah functional by phase fields of bounded variation.
- Author
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Belz, Sandro and Bredies, Kristian
- Subjects
- *
MARKOV random fields , *FUNCTIONS of bounded variation , *IMAGE segmentation , *IMAGE processing , *IMAGE denoising - Abstract
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]
- Published
- 2021
- Full Text
- View/download PDF
10. Hyperparameter selection for Discrete Mumford–Shah
- Author
-
Lucas, Charles-Gérard, Pascal, Barbara, Pustelnik, Nelly, Abry, Patrice, Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure de Lyon (ENS de Lyon)-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Laboratoire des Sciences du Numérique de Nantes (LS2N), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-École Centrale de Nantes (Nantes Univ - ECN), Nantes Université (Nantes Univ)-Nantes Université (Nantes Univ)-Nantes université - UFR des Sciences et des Techniques (Nantes univ - UFR ST), Nantes Université - pôle Sciences et technologie, Nantes Université (Nantes Univ)-Nantes Université (Nantes Univ)-Nantes Université - pôle Sciences et technologie, and Nantes Université (Nantes Univ)
- Subjects
non-convex minimization ,contour detection ,Image and Video Processing (eess.IV) ,Electrical Engineering and Systems Science - Image and Video Processing ,Optimization and Control (math.OC) ,Stein Unbiased Risk Estimate ,[INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV] ,Signal Processing ,FOS: Electrical engineering, electronic engineering, information engineering ,FOS: Mathematics ,Mumford-Shah functional ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Electrical and Electronic Engineering ,Mathematics - Optimization and Control - Abstract
International audience; This work focuses on a parameter-free joint piecewise smooth image denoising and contour detection. Formulated as the minimization of a discrete Mumford-Shah functional and estimated via a theoretically grounded alternating minimization scheme, the bottleneck of such a variational approach lies in the need to fine-tune their hyperparameters, while not having access to ground truth data.To that aim, a Stein-like strategy providing optimal hyperparameters is designed, based on the minimization of an unbiased estimate of the quadratic risk.Efficient and automated minimization of the estimate of the risk crucially relies on an unbiased estimate of the gradient of the risk with respect to hyperparameters. Its practical implementation is performed using a forward differentiation of the alternating scheme minimizing the Mumford-Shah functional, requiring exact differentiation of the proximity operators involved. Intensive numerical experiments are performed on synthetic images with different geometry and noise levels, assessing the accuracy and the robustness of the proposed procedure.The resulting parameter-free piecewise-smooth estimation and contour detection procedure, not requiring prior image processing expertise nor annotated data, can then be applied to real-world images.
- Published
- 2022
11. Mumford–Shah Loss Functional for Image Segmentation With Deep Learning.
- Author
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Kim, Boah and Ye, Jong Chul
- Subjects
- *
DEEP learning , *IMAGE segmentation , *ARTIFICIAL neural networks , *SUPERVISED learning , *CHARACTERISTIC functions , *ENERGY function - Abstract
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]
- Published
- 2020
- Full Text
- View/download PDF
12. Region-based segmentation on evolving surfaces with application to 3D reconstruction of shape and piecewise constant radiance
- Author
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Jin, H L, Yezzi, A J, and Soatto, Stefano
- Subjects
variational methods ,Mumford-Shah functional ,image segmentation ,multi-view stereo ,level set methods ,curve evolution on manifolds - Abstract
We consider the problem of estimating the shape and radiance of a scene from a calibrated set of images under the assumption that the scene is Lambertian and its radiance is piecewise constant. We model the radiance segmentation explicitly using smooth curves on the surface that bound regions of constant radiance. We pose the scene reconstruction problem in a variational framework, where the unknowns are the surface, the radiance values and the segmenting curves. We propose an iterative procedure to minimize a global cost functional that combines geometric priors on both the surface and the curves with a data fitness score. We carry out the numerical implementation in the level set framework.
- Published
- 2004
13. Bregman Divergence Applied to Hierarchical Segmentation Problems
- Author
-
Ferreira, Daniela Portes L., Backes, André R., Barcelos, Celia A. Zorzo, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, and Pardo, Alvaro, editor
- Published
- 2015
- Full Text
- View/download PDF
14. Real-Time Minimization of the Piecewise Smooth Mumford-Shah Functional
- Author
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Strekalovskiy, Evgeny, Cremers, Daniel, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Kobsa, Alfred, Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Fleet, David, editor, Pajdla, Tomas, editor, Schiele, Bernt, editor, and Tuytelaars, Tinne, editor
- Published
- 2014
- Full Text
- View/download PDF
15. Discrete stochastic approximations of the Mumford–Shah functional.
- Author
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Ruf, Matthias
- Subjects
- *
STOCHASTIC approximation , *ASYMPTOTIC homogenization , *STOCHASTIC convergence , *LATTICE theory , *FINITE differences - Abstract
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]
- Published
- 2019
- Full Text
- View/download PDF
16. Scale and Edge Detection with Topological Derivatives
- Author
-
Dong, Guozhi, Grasmair, Markus, Kang, Sung Ha, Scherzer, Otmar, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Kuijper, Arjan, editor, Bredies, Kristian, editor, Pock, Thomas, editor, and Bischof, Horst, editor
- Published
- 2013
- Full Text
- View/download PDF
17. The Beltrami-Mumford-Shah Functional
- Author
-
Sochen, Nir, Bar, Leah, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Bruckstein, Alfred M., editor, ter Haar Romeny, Bart M., editor, Bronstein, Alexander M., editor, and Bronstein, Michael M., editor
- Published
- 2012
- Full Text
- View/download PDF
18. Stationarity of the crack-front for the Mumford–Shah problem in 3D.
- Author
-
Lemenant, Antoine and Mikayelyan, Hayk
- Subjects
- *
FRACTURE mechanics , *FINITE element method , *GEOMETRY , *GROUP theory , *NUMERICAL analysis - Abstract
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]
- Published
- 2018
- Full Text
- View/download PDF
19. A SECOND ORDER LOCAL MINIMALITY CRITERION FOR THE TRIPLE JUNCTION SINGULARITY OF THE MUMFORD-SHAH FUNCTIONAL.
- Author
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CRISTOFERI, RICCARDO
- Subjects
- *
PROTOTYPES , *FRACTURE mechanics , *SURFACE energy , *MATHEMATICAL analysis , *MATHEMATICAL functions - Abstract
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]
- Published
- 2018
- Full Text
- View/download PDF
20. Level Set Approaches and Adaptive Asymmetrical SVMs Applied for Nonideal Iris Recognition
- Author
-
Roy, Kaushik, Bhattacharya, Prabir, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Kamel, Mohamed, editor, and Campilho, Aurélio, editor
- Published
- 2009
- Full Text
- View/download PDF
21. Mumford-Shah regularization in electrical impedance tomography with complete electrode model
- Author
-
Aku Seppänen, Tuomo Valkonen, Jyrki Jauhiainen, Department of Mathematics and Statistics, and University of Helsinki
- Subjects
FUNCTIONALS ,Applied Mathematics ,ill-posed inverse problem ,Computer Science Applications ,Theoretical Computer Science ,65K10 (Primary), 35R30, 68U10, 35Q93 (Secondary) ,Mathematics - Analysis of PDEs ,Optimization and Control (math.OC) ,Signal Processing ,FOS: Mathematics ,Mumford-Shah functional ,111 Mathematics ,variational regularisation ,Mathematics - Optimization and Control ,image segmentation ,SET ,Mathematical Physics ,Analysis of PDEs (math.AP) ,electrical impedance tomography ,APPROXIMATION - Abstract
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
- Published
- 2022
22. Segmentations for Piecewise Smooth Pictures in PERMON.
- Author
-
Pecha, Marek and Čermák, Martin
- Subjects
- *
IMAGE segmentation , *PIECEWISE affine systems , *COMPUTER software , *COMPUTATIONAL intelligence , *DIGITAL images - Abstract
In this paper we present segmentation method for piecewise smooth pictures and our implemented software. We describe image segmentation problem and its difficulties in real applications. Since image segmentation is a complicated problem, we focus on the segmentation method only for piecewise smooth pictures based on the Mumford-Shah functional and its connection to spectral methods. We have developed software for the piecewise image segmentation; currently, we focus on decreasing the execution time of massively parallel computations and quality of results. The results conclude the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
23. Hierarchical image simplification and segmentation based on Mumford–Shah-salient level line selection.
- Author
-
Xu, Yongchao, Géraud, Thierry, and Najman, Laurent
- Subjects
- *
IMAGE segmentation , *FEATURE selection , *IMAGE analysis , *IMAGE processing , *PATTERN recognition systems , *COMPUTER science - Abstract
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]
- Published
- 2016
- Full Text
- View/download PDF
24. Hyperparameter selection for the Discrete Mumford-Shah functional
- Author
-
Lucas, Charles-Gérard, Pascal, Barbara, Pustelnik, Nelly, Abry, Patrice, Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon, Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), and Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)
- Subjects
non-convex minimization ,contour detection ,Stein Unbiased Risk Estimate ,[INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV] ,Mumford-Shah functional ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] - Abstract
This work focuses on joint piecewise smooth image reconstruction and contour detection, formulated as the minimization of a discrete Mumford-Shah functional, performed via a theoretically grounded alternating minimization scheme. The bottleneck of such variational approaches lies in the need to finetune their hyperparameters, while not having access to ground truth data. To that aim, a Stein-like strategy providing optimal hyperparameters is designed, based on the minimization of an unbiased estimate of the quadratic risk. Efficient and automated minimization of the estimate of the risk crucially relies on an unbiased estimate of the gradient of the risk with respect to hyperparameters, whose practical implementation is performed thanks to a forward differentiation of the alternating scheme minimizing the Mumford-Shah functional, requiring exact differentiation of the proximity operators involved. Intensive numerical experiments are performed on synthetic images with different geometries and noise levels, assessing the accuracy and the robustness of the proposed procedure. The resulting parameterfree piecewise-smooth reconstruction and contour detection procedure, not requiring prior image processing expertise, is thus amenable to real-world applications.
- Published
- 2021
25. Mumford-Shah and Potts Regularization for Manifold-Valued Data.
- Author
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Weinmann, Andreas, Demaret, Laurent, and Storath, Martin
- Abstract
Mumford-Shah and Potts functionals are powerful variational models for regularization which are widely used in signal and image processing; typical applications are edge-preserving denoising and segmentation. Being both non-smooth and non-convex, they are computationally challenging even for scalar data. For manifold-valued data, the problem becomes even more involved since typical features of vector spaces are not available. In this paper, we propose algorithms for Mumford-Shah and for Potts regularization of manifold-valued signals and images. For the univariate problems, we derive solvers based on dynamic programming combined with (convex) optimization techniques for manifold-valued data. For the class of Cartan-Hadamard manifolds (which includes the data space in diffusion tensor imaging (DTI)), we show that our algorithms compute global minimizers for any starting point. For the multivariate Mumford-Shah and Potts problems (for image regularization), we propose a splitting into suitable subproblems which we can solve exactly using the techniques developed for the corresponding univariate problems. Our method does not require any priori restrictions on the edge set and we do not have to discretize the data space. We apply our method to DTI as well as Q-ball imaging. Using the DTI model, we obtain a segmentation of the corpus callosum on real data. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
26. Convex Cardinal Shape Composition.
- Author
-
Aghasi, Alireza and Romberg, Justin
- Subjects
IMAGING systems ,CONVEX functions ,REAL variables ,GEOMETRY ,COMBINATORICS - Abstract
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. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
27. Discrete stochastic approximations of the Mumford–Shah functional
- Author
-
Matthias Ruf
- Subjects
49M25, 68U10, 49J55, 49J45 ,Approximations of π ,Applied Mathematics ,010102 general mathematics ,Isotropy ,Finite difference ,16. Peace & justice ,01 natural sciences ,Homogenization (chemistry) ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Lattice (order) ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Anisotropy ,Mumford–Shah functional ,Mathematical Physics ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We propose a $\Gamma$-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., Comment: 47 pages, reorganized version
- Published
- 2019
28. Stable regular critical points of the Mumford–Shah functional are local minimizers.
- Author
-
Bonacini, M. and Morini, M.
- Subjects
- *
CRITICAL point theory , *FUNCTIONALS , *TOPOLOGY , *NEIGHBORHOODS , *PROBLEM solving - Abstract
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]
- Published
- 2015
- Full Text
- View/download PDF
29. On the Stability of MAP Estimation with Hierarchical Prior Distributions.
- Author
-
Helin, Tapio and Lassas, Matti
- Subjects
- *
INVERSE problems , *ESTIMATION theory , *LINEAR statistical models , *INVERSE Gaussian distribution , *GAUSSIAN processes - Abstract
The maximum a posteriori (MAP) estimates for linear inverse problems are studied using hierarchical Gaussian models. The stability of this point estimate is considered with respect to different discretizations. We analyze the phenomena which appear when the discretization becomes finer. An edge-preserving Bayesian reconstruction method for signal restoration problems is introduced and studied with arbitrarily fine discretization. Moreover, different noise asymptotics are considered for the inverse problem. We show that the maximum a posteriori and conditional mean estimates converge under different conditions. Finally, we discuss connection of this method to Mumford–Shah functional. This paper reviews results from [6]. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
30. Smoothing of Data Using Mumford-Shah Type Functionals.
- Author
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Mucha, Katharina and Bärwolff, Günter
- Subjects
- *
IMAGE processing , *SCALAR field theory , *LAGRANGE equations , *FINITE volume method , *NONLINEAR evolution equations , *NEWTON-Raphson method - Abstract
As results of 3d sensors using the ”time of flight” technology we get noise information of the shape of objects together with intensities. From the mathematical point of view these information are scalar fields d and I. For filtering and smoothing of images we minimize a Mumford-Shah functional by solving the boundary value problem of the relevant Euler-Lagrange equations. For the numerical solution of the boundary value problem we use a finite volume discretization and we solve the resulting nonlinear equation system on the finite volume grid by Newtons method and the steepest descent method. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
31. Computer-assisted segmentation of brain tumor lesions from multi-sequence Magnetic Resonance Imaging using the Mumford-Shah model.
- Author
-
Zoghbi, Jihan M., Mamede, Marcelo H., and Jackowski, Marcel P.
- Abstract
Segmentation of brain lesions in Magnetic Resonance Imaging (MRI) is a difficult task to be mastered by the specialist. This is due to the presence of noise, partial volume effects and susceptibility artifacts in the images and on the borders of the regions of interest. These problems can interfere with the results when manual segmentation is used. Manual segmentation uses local anatomic information based on the user's background; that implies the necessity of constant human intervention. Deformable model approaches attempt to minimize these drawbacks by outlining the region of interest semi-automatically. These methods have been shown to be effective in the extraction of the lesion boundaries in brain MR images. The proposed method employs the multi-channel version of the Mumford-Shah model via level set methods in order to segment multi-sequence brain magnetic resonance (MR) images: FLAIR (Fluid attenuated inversion recovery), T1 and T2- weighted images. Results showed that segmentation of multi-sequence images using this methodology yielded superior results than using each sequence alone. As a consequence, medical doctors can exploit the segmentation results to follow up their patients' status by controlling the evolution or involution of brain lesions. [ABSTRACT FROM PUBLISHER]
- Published
- 2010
- Full Text
- View/download PDF
32. An unconditionally stable hybrid method for image segmentation.
- Author
-
Li, Yibao and Kim, Junseok
- Subjects
- *
STABILITY theory , *HYBRID systems , *IMAGE segmentation , *NUMERICAL analysis , *MATHEMATICAL constants , *MATHEMATICAL proofs - Abstract
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]
- Published
- 2014
- Full Text
- View/download PDF
33. Endpoint regularity for $2d$ Mumford-Shah minimizers: On a theorem of Andersson and Mikayelyan
- Author
-
Silvia Ghinassi, Camillo De Lellis, and Matteo Focardi
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,1,α ,regularity ,Cracktip ,Endpoint regularity ,Mumford-Shah functional ,Mumford-Shah minimizers ,01 natural sciences ,010101 applied mathematics ,Arc (geometry) ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,0101 mathematics ,Mumford–Shah functional ,Interior point method ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
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., Comment: 27 pages. v3: corrected typos, added proof of (8.1), corrected acknowledgements. To appear in Journal de Math\'ematiques Pures et Appliqu\'ees. For Errata see https://www.math.ias.edu/delellis/sites/math.ias.edu.delellis/files/Errata-crackip.pdf
- Published
- 2020
- Full Text
- View/download PDF
34. Local minimality results for the Mumford-Shah functional via monotonicity
- Author
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Ilaria Fragalà, Alessandro Giacomini, and Dorin Bucur
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,Free discontinuity functionals ,Local minimality ,Monotonicity formulas ,free discontinuity functionals ,Monotonic function ,local minimality ,35A16 ,94A08 ,35J25 ,35Q74 ,49J45 ,monotonicity formulas ,28A75 ,Mumford–Shah functional ,Analysis ,35R35 ,Mathematics - Abstract
Let [math] be a bounded piecewise [math] open set with convex corners, and let ¶ MS ( u ) : = ∫ Ω | ∇ u | 2 d x + α ℋ 1 ( J u ) + β ∫ Ω | u − g | 2 d x ¶ be the Mumford–Shah functional on the space [math] , where [math] and [math] . We prove that the function [math] such that ¶ − Δ u + β u = β g in Ω , ∂ u ∕ ∂ ν = 0 on ∂ Ω ¶ is a local minimizer of [math] with respect to the [math] -topology. This is obtained as an application of interior and boundary monotonicity formulas for a weak notion of quasiminimizers of the Mumford–Shah energy. The local minimality result is then extended to more general free discontinuity problems taking into account also boundary conditions.
- Published
- 2020
35. Image segmentation and selective smoothing based on p-harmonic Mumford–Shah functional
- Author
-
Shuaijie Li and Peng Li
- Subjects
Level set method ,Computer science ,Anisotropic diffusion ,Isotropy ,02 engineering and technology ,Image segmentation ,01 natural sciences ,Atomic and Molecular Physics, and Optics ,Electronic, Optical and Magnetic Materials ,010101 applied mathematics ,Feature (computer vision) ,Computer Science::Computer Vision and Pattern Recognition ,0202 electrical engineering, electronic engineering, information engineering ,Piecewise ,020201 artificial intelligence & image processing ,Segmentation ,0101 mathematics ,Electrical and Electronic Engineering ,Mumford–Shah functional ,Algorithm ,Smoothing - Abstract
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.
- Published
- 2018
36. Existence of minimizers of the Mumford-Shah functional with singular operators and unbounded data.
- Author
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Fornasier, Massimo, March, Riccardo, and Solombrino, Francesco
- Abstract
We consider the regularization of linear inverse problems by means of the minimization of a functional formed by a term of discrepancy to data and a Mumford-Shah functional term. The discrepancy term penalizes the L distance between a datum and a version of the unknown function which is filtered by means of a non-invertible linear operator. Depending on the type of the involved operator, the resulting variational problem has had several applications: image deblurring, or inverse source problems in the case of compact operators, and image inpainting in the case of suitable local operators, as well as the modeling of propagation of fracture. We present counterexamples showing that, despite this regularization, the problem is actually in general ill-posed. We provide, however, existence results of minimizers in a reasonable class of smooth functions out of piecewise Lipschitz discontinuity sets in two dimensions. The compactness arguments we developed to derive the existence results stem from geometrical and regularity properties of domains, interpolation inequalities, and classical compactness arguments in Sobolev spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
37. Image segmentation methods and an application to brain images
- Author
-
Lindner, Marko, Nicolai, Christoph, Lindner, Marko, and Nicolai, Christoph
- Abstract
Die vorliegende Masterarbeit behandelt eine Klasse von Methoden zur sogenannten Bildsegmentierung. Dabei wird ein digitales Bild automatisch in einzelne Objekte zerlegt. Dies könnten zum Beispiel einzelne Personen sowie der Hintergrund auf einem Foto sein. In dieser Arbeit liegen die Anwendungsfelder aber im medizintechnischen Bereich bei Aufnahmen des menschlichen Gehirns. Bei diesen zwei- oder dreidimensionalen Aufnahmen des Gehirns sollen zum Beispiel die Regionen der sogenannten grauen Masse von denen weißer Masse sowie vom Hirnbalken und dem Hintergrund unterschieden werden. Die einzelnen Regionen können dann vermessen und im Laufe der Zeit beobachtet werden. Re- oder degenerative Prozesse im Gehirn können so nicht nur subjektiv durch den Arzt verfolgt, sondern automatisch quantifiziert und aufgezeichnet werden. Einfache Verfahren erstellen eine solche Segmentierung durch sogenanntes Thresholding. Hier werden die einzelnen Bildpunkte ausschließlich anhand ihres Helligkeitswerts einem Objekt zugeordnet. Bei Aufnahmen mit fließenden Übergängen zwischen den Helligkeitsstufen erhält man damit aber extrem viele kleine Segmente an den Übergangszonen. In der Masterarbeit geht es nun um die analytische Herleitung und effiziente Implementierung eines alternativen Ansatzes, die Minimierung des sogenannten Mumford-Shah-Funktionals. Die in der Arbeit vorgestellte Methode unterscheidet verschiedene Bildpunkte auch anhand deren Helligkeit, vermeidet aber gleichzeitig die Entstehung langer Segmentkanten und damit die Unterteilung in zu viele kleine Objekte., In this master thesis, a class of methods for image segmentation is discussed. Image segmentation is the decomposition of an image in separate objects. An example for such a decomposition is the separation of individual persons and the background in a photograph. The focus of this thesis is medical image processing. An exemplary task is the segmentation of two- or three-dimensional images of the human brain, where the different objects could be white matter, grey matter, the corpus callosum and the background. Changes over time in the structure of the brain can then not only be subjectively assessed by a physician, but also be automatically quantified and observed. Basic methods compute such a segmentation by so-called thresholding. Here, image pixels are assigned to a region only by their brightness. For images with soft transitions between the different brightness levels, this yields many small segments along the object borders. The topic of this master thesis is the analytical derivation and efficient implementation of an alternative approach, the minimization of the so-called Mumford-Shah functional. The method discussed in this thesis also considers the brightness of the image pixels, but avoids the subdivision in too many small objects.
- Published
- 2019
38. Edge Detection Filter based on Mumford-Shah Green Function.
- Author
-
Mahmoodi, Sasan
- Subjects
ALGORITHMS ,GREEN'S functions ,BOUNDARY element methods ,IMAGE processing ,STOCHASTIC processes - Abstract
In this paper, we propose an edge detection algorithm based on the Green function associated with the Mumford-Shah segmentation model. This Green function has a singularity at its center. A regularization method is therefore proposed here to obtain an edge detection filter known here as the Bessel filter. This filter is robust in the presence of noise, and its implementation is simple. It is demonstrated here that this filter is scale invariant. A mathematical argument is also provided to prove that the gradient magnitude of the convolved image with this filter has local maxima in discontinuities of the original image. The Bessel filter enjoys better overall performance (the product of the detection performance and localization indices) in Canny-like criteria than the state-of-the-art filters in the literature. Quantitative and qualitative evaluations of the edge detection algorithms investigated in this paper on synthetic and real world benchmark images confirm the theoretical results presented here, indicating the scale invariant property of the Bessel filter. The numerical complexity of the algorithm proposed here is as low as any convolution-based edge detection algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
39. A Mumford-Shah-Like Method for Limited Data Tomography with an Application to Electron Tomography.
- Author
-
Klann, Esther
- Subjects
INVERSE problems ,TOMOGRAPHY ,ALGORITHMS ,MATHEMATICAL functions ,PIECEWISE linear approximation - Abstract
In this article the Mumford-Shah-like method of [R. Ramlau and W. Ring, J. Comput. Phys., 221 (2007), pp. 539-557] for complete tomographic data is generalized and applied to limited angle and region of interest tomography data. With the Mumford-Shah-like method, one reconstructs a piecewise constant function and simultaneously a segmentation from its (complete) Radon transform data. For limited data, the ability of the Mumford-Shah-like method to find a segmentation, and by that the singularity set of a function, is exploited. The method is applied to generated data from a torso phantom. The results demonstrate the performance of the method in reconstructing the singularity set, the density distribution itself for limited angle data, and also some quantitative information about the density distribution for region of interest data. As a second example limited angle region of interest tomography is considered as a simplified model for electron tomography (ET). For this problem we combine Lambda tomography and the Mumford-Shah-like method. The combined method is applied to simulated ET data. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
40. A Priori Inequalities between Energy Release Rate and Energy Concentration for 3D Quasistatic Brittle Fracture Propagation.
- Author
-
Buliga, Marius
- Subjects
- *
BRITTLENESS , *FRACTURE mechanics , *DENSITY functionals , *A priori , *MATHEMATICAL inequalities , *EQUILIBRIUM - Abstract
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]
- Published
- 2011
- Full Text
- View/download PDF
41. An extension theorem in SBV and an application to the homogenization of the Mumford–Shah functional in perforated domains
- Author
-
Cagnetti, F. and Scardia, L.
- Subjects
- *
ASYMPTOTIC homogenization , *INTEGRAL representations , *STOCHASTIC convergence , *MATHEMATICAL functions , *MATHEMATICAL analysis , *NUMERICAL analysis - Abstract
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]
- Published
- 2011
- Full Text
- View/download PDF
42. Image segmentation and inpainting using hierarchical level set and texture mapping
- Author
-
Du, Xiaojun, Cho, Dongwook, and Bui, Tien D.
- Subjects
- *
IMAGE processing , *INPAINTING , *TEXTURE mapping , *ESTIMATION theory , *ALGORITHMS , *STOCHASTIC convergence , *EXPERIMENTAL design , *NUMERICAL analysis - Abstract
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]
- Published
- 2011
- Full Text
- View/download PDF
43. Numerical treatment of the Mumford-Shah model for the inversion and segmentation of X-ray tomography data.
- Author
-
Hoetzl, Elena and Ring, Wolfgang
- Subjects
- *
TOMOGRAPHY , *X-rays , *INVERSE problems , *FINITE differences , *LEVEL set methods - Abstract
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]
- Published
- 2010
- Full Text
- View/download PDF
44. Shape-Based Active Contours for Fast Video Segmentation.
- Author
-
Mahmoodi, Sasan
- Subjects
ALGORITHMS ,EQUATIONS ,SIGNAL processing ,SIGNAL theory ,INFORMATION measurement - Abstract
n 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. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
45. CRITICAL POINTS OF AMBROSIO-TORTORELLI CONVERGE TO CRITICAL POINTS OF MUMFORD-SHAH IN THE ONE-DIMENSIONAL DIRICHLET CASE.
- Author
-
Francfort, Gilles A., Le, Nam Q., and Serfaty, Sylvia
- Subjects
- *
DIRICHLET principle , *CONJUGATE gradient methods , *NUMERICAL solutions to equations , *ELASTICITY , *MATHEMATICAL analysis - Abstract
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]
- Published
- 2009
- Full Text
- View/download PDF
46. A variational approach to the reconstruction of cracks by boundary measurements
- Author
-
Rondi, Luca
- Subjects
- *
FUNCTIONS of bounded variation , *REAL variables , *FUNCTIONALS , *ELECTRICAL conductors - Abstract
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]
- Published
- 2007
- Full Text
- View/download PDF
47. A Mumford–Shah level-set approach for the inversion and segmentation of X-ray tomography data
- Author
-
Ramlau, Ronny and Ring, Wolfgang
- Subjects
- *
LEVEL set methods , *RADON transforms , *DENSITY functionals , *GEOMETRIC tomography - Abstract
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]
- Published
- 2007
- Full Text
- View/download PDF
48. Mumford-Shah minimizers on thin plates.
- Author
-
David, Guy
- Subjects
MATHEMATICAL symmetry ,SURFACE plates ,MATHEMATICS ,STOCHASTIC convergence ,FRACTIONAL calculus ,GEOMETRIC surfaces - Abstract
Copyright of Calculus of Variations & Partial Differential Equations is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
- Published
- 2006
- Full Text
- View/download PDF
49. Variational Approaches on Discontinuity Localization and Field Estimation in Sea Surface Temperature and Soil Moisture.
- Author
-
Sun, Walter, Çetin, Müjdat, Thacker, W. Carlisle, Chin, T. Mike, and Wilisky, Alan S.
- Subjects
- *
SOIL moisture , *TEMPERATURE , *AQUATIC sciences , *REMOTE sensing , *A priori , *MULTIVARIATE analysis - Abstract
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]
- Published
- 2006
- Full Text
- View/download PDF
50. DTT Segmentation Using an Information Theoretic Tensor Dissimilarity Measure.
- Author
-
Zhizhou Wang and Vemuri, Baba C.
- Subjects
- *
MEDICAL imaging systems , *GAUSSIAN distribution , *DISTRIBUTION (Probability theory) , *DIAGNOSTIC imaging , *IMAGE analysis , *MEDICAL equipment - Abstract
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]
- Published
- 2005
- Full Text
- View/download PDF
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