### Citations

2827 |
Sobolev spaces
- Adams
- 1975
(Show Context)
Citation Context ... k ,φ1 k ,φ2 )} k is a bounded sequence of real numbers and, consequently, {( k 1 zk ,z2 k ,φ1 k ,φ2 )} k k∈N is uniformly bounded in (BV)2 × (H 1 (�)) 2 . Thus, the Sobolev compact embedding theorem =-=[2]-=- and the compact embedding of BV into L1 [16, chapter 5] guarantee the existence of a subsequence, denoted again by {( z1 k ,z2 k ,φ1 k ,φ2 )} k , and also the existence k∈N of (z1,z2,φ1,φ2) ∈ (L1(�))... |

1469 |
Fronts propagation with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations
- Osher, Sethian
- 1988
(Show Context)
Citation Context ...us solutions for inverse problems have existed for a long time. Total variation regularization and other techniques have been studied so far [1, 7, 9, 21, 52]. Recently, level set-type of methods, cf =-=[35]-=-, have become popular due to their superior performance for many real-life inverse problems related to discontinuous functions, see [3, 4, 8, 10–13, 17, 19, 20, 24–26, 29–32, 34, 36, 37, 39, 41–44, 46... |

1286 | Measure theory and fine properties of functions - Evans, Gariepy - 1992 |

97 | A level-set approach for inverse problems involving obstacles, - Santosa - 1996 |

61 | Hanke M and Neubauer A 1996 Regularization of Inverse Problems (Dordrecht - Engl |

55 | A survey on level set methods for inverse problems and optimal design, - Burger, Osher - 2005 |

41 | An augmented Lagrangian method for identifying discontinuous parameters in elliptic systems
- Chen, Zou
- 1999
(Show Context)
Citation Context ... piecewise constant. The interests in detecting discontinuous solutions for inverse problems have existed for a long time. Total variation regularization and other techniques have been studied so far =-=[1, 7, 9, 21, 52]-=-. Recently, level set-type of methods, cf [35], have become popular due to their superior performance for many real-life inverse problems related to discontinuous functions, see [3, 4, 8, 10–13, 17, 1... |

30 | A survey on multiple level set methods with applications for identifying piecewise constant functions
- Tai, Chan
(Show Context)
Citation Context ...surable subsets �j ⊂ �,j = 1,...,N and constants {c1,c2,...,cN} such that |�| = ∑ j |�j | and u(x) = cj a.e. in �j . In order to represent the unknown function u, we use a multiple level-set approach =-=[8, 45, 51]-=-. This is done as follows: first we introduce the H 1 functions {φj } p j=1 , where p ∈ N is the smallest integer satisfying 2p � N (for simplicity of the presentation we shall assume in this paper th... |

29 |
Identification of discontinuous parameters in flow equations.
- Gutman
- 1990
(Show Context)
Citation Context ... piecewise constant. The interests in detecting discontinuous solutions for inverse problems have existed for a long time. Total variation regularization and other techniques have been studied so far =-=[1, 7, 9, 21, 52]-=-. Recently, level set-type of methods, cf [35], have become popular due to their superior performance for many real-life inverse problems related to discontinuous functions, see [3, 4, 8, 10–13, 17, 1... |

29 |
S and Fatemi E 1992 Nonlinear total variation based noise removal algorithms
- Rudin, Osher
(Show Context)
Citation Context ...lving (1) combined with a ROF total variation regularization. The Rudin–Osher–Fatemi (ROF) model as well as its several variations are quite well-known PDEs-based methods in image analysis (cf, e.g., =-=[5, 38]-=-). It is worth noting that ROF-type models in image analysis involve the minimization of functionals defined in BV(�) and, in this context, well posedness (the existence of minimizers) is well known [... |

20 |
Iterative methods for the reconstruction of an inverse potential problem, Inverse Problems
- Hettlichy, Rundell
- 1996
(Show Context)
Citation Context ... problem can be written in the abbreviated form F(u) = y δ , (33) where the data y δ have the same meaning as in (2). Another inverse problem for the operator F was considered by Hettlich and Rundell =-=[22]-=-, who used iterative methods for recovering the indicator function u = χD of a star-shaped domain D ⊂ R 2 . This simpler problem corresponds to that of recovering the shape of a domain D using the kno... |

18 |
Least squares and bounded variation regularization with nondifferentiable functionals
- Nashed, Scherzer
- 1998
(Show Context)
Citation Context ...ution. □ In the following two theorems we present the main convergence and stability results. The proofs use classical techniques from the analysis of Tikhonov-type regularization methods (see, e.g., =-=[1, 15, 33, 40]-=- or[14, chapter 10]). Theorem 8 (convergence for exact data). Assume that we have exact data, i.e., yδ = y and β>0. For every α>0,let ( z1 α ,z2 α ,φ1 α ,φ2 ) α denote a minimizer of ˆGα on the set of... |

16 | E and Rappaport C 2000 A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets Inverse Problems 16 - Dorn, Miller |

15 |
Kunisch K and Neubauer A 1989 Convergence rates for Tikhonov regularization of nonlinear ill-posed problems Inverse Problems 5 523–40
- Engl
(Show Context)
Citation Context ...ution. □ In the following two theorems we present the main convergence and stability results. The proofs use classical techniques from the analysis of Tikhonov-type regularization methods (see, e.g., =-=[1, 15, 33, 40]-=- or[14, chapter 10]). Theorem 8 (convergence for exact data). Assume that we have exact data, i.e., yδ = y and β>0. For every α>0,let ( z1 α ,z2 α ,φ1 α ,φ2 ) α denote a minimizer of ˆGα on the set of... |

15 | Sparse matrix computations arising in distributed parameter identification
- Vogel
- 1999
(Show Context)
Citation Context ... piecewise constant. The interests in detecting discontinuous solutions for inverse problems have existed for a long time. Total variation regularization and other techniques have been studied so far =-=[1, 7, 9, 21, 52]-=-. Recently, level set-type of methods, cf [35], have become popular due to their superior performance for many real-life inverse problems related to discontinuous functions, see [3, 4, 8, 10–13, 17, 1... |

9 | Inverse source problems Mathematics Surveys Monographs vol 34 - Isakov - 1989 |

8 | Reconstruction by level sets of n-ary scattering obstacles - Litman - 2005 |

6 |
2006. “Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models
- Chan, Esedoglu, et al.
(Show Context)
Citation Context ...lving (1) combined with a ROF total variation regularization. The Rudin–Osher–Fatemi (ROF) model as well as its several variations are quite well-known PDEs-based methods in image analysis (cf, e.g., =-=[5, 38]-=-). It is worth noting that ROF-type models in image analysis involve the minimization of functionals defined in BV(�) and, in this context, well posedness (the existence of minimizers) is well known [... |

6 | Lesselier D 2006 Level set methods for inverse scattering - Dorn |

5 |
Vogel C R 1994 Analysis of bounded variation penalty methods for ill-posed problems
- Acar
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Citation Context |

5 | Kunisch K and Li Z 2001 Level-set function approach to an inverse interface problem Inverse Problems 17 - Ito |

4 | Level set method for optimal shape design of MRAM core - Melicher, Cimrák, et al. - 2008 |

4 | A projection-based level-set approach to enhance conductivity anomaly reconstruction in electrical resistance tomography - Miled, Miller - 2007 |

4 | Doel K and Ascher U M 2006 On level set regularization for highly ill-posed distributed parameter estimation problems - den |

3 | E and Huang H 2006 On effective methods for implicit piecewise smooth surface recovery - Ascher, Haber |

3 |
2005), Electrical impedance tomography using level set representation and total variational regularization
- Chung, Chan, et al.
(Show Context)
Citation Context ...l set methods for inverse problems. Compared with other multiple level-set approaches, there are several features of the method proposed here. 19Inverse Problems 25 (2009) 035004 A DeCezaro et al In =-=[6, 10, 45]-=-, multiple level-set methods have been used for tomography and PET imaging. The difference here is to add an extra regularization term to the minimization functional and this has avoided the re-initia... |

3 |
I and Vogel C R 1989 Well-posedness and convergence of some regularization methods for nonlinear ill-posed problems Inverse Problems 5
- Seidman
(Show Context)
Citation Context ...ution. □ In the following two theorems we present the main convergence and stability results. The proofs use classical techniques from the analysis of Tikhonov-type regularization methods (see, e.g., =-=[1, 15, 33, 40]-=- or[14, chapter 10]). Theorem 8 (convergence for exact data). Assume that we have exact data, i.e., yδ = y and β>0. For every α>0,let ( z1 α ,z2 α ,φ1 α ,φ2 ) α denote a minimizer of ˆGα on the set of... |

2 |
Tai X 2003, Identification of discontinuous coefficients in elliptic problems using total variation regularization
- Chan
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Citation Context |

2 |
Tai X C 2004 Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients
- Chan
(Show Context)
Citation Context ...surable subsets �j ⊂ �,j = 1,...,N and constants {c1,c2,...,cN} such that |�| = ∑ j |�j | and u(x) = cj a.e. in �j . In order to represent the unknown function u, we use a multiple level-set approach =-=[8, 45, 51]-=-. This is done as follows: first we introduce the H 1 functions {φj } p j=1 , where p ∈ N is the smallest integer satisfying 2p � N (for simplicity of the presentation we shall assume in this paper th... |

2 | 2007 Multi-phase permittivity reconstruction in electrical capacitance tomography by level-set methods - Fang |

2 |
O and Leitão A 2005 Analysis of regularization methods for the solution of ill-posed problems involving discontinuous operators
- Frühauf, Scherzer
(Show Context)
Citation Context ...(x) > 0}, 2Inverse Problems 25 (2009) 035004 A DeCezaro et al �2 ={x ∈ �; φ 1 (x) > 0,φ 2 (x) < 0}, �3 ={x ∈ �; φ 1 (x) < 0,φ 2 (x) > 0}, �4 ={x ∈ �; φ 1 (x) < 0,φ 2 (x) < 0}. As already observed in =-=[18]-=-, the operator H maps H 1 (�) onto the space V := {u∈ L ∞ (�)|u = χ� ′,�′ ⊂ � measurable, H n−1 (∂� ′ )<∞}, (4) where Hn−1 (D) denotes the (n − 1)-dimensional Hausdorff measure of the set D. Therefore... |

2 | Tai X 2006 A variant of the level set method and applications to image segmentation - Lie, Lysaker |

2 | Identification of a core from boundary data - Ring - 1995 |

2 | W.R.B.: 2005, Nonlinear image reconstruction for electrical capacitance tomography using experimental data - Soleimani, Lionheart - 1987 |

2 | Lionheart W R B and Dorn O 2006 Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data Inverse Problems Sci - Soleimani |

2 | Vese L A 2007 A piecewise-constant binary model for electrical impedance tomography Inverse Problems and Imaging - Tanushev |

2 | Doel K and Ascher U M 2007 Dynamic level set regularization for large distributed parameter estimation problems Inverse Problems 23 - den |

1 | Kress R 2007 Nonlinear integral equations for the inverse electrical impedance problem Inverse Problems 23 - Eckel |

1 | Jadamba B and Khan A A 2008 Equation error approach for elliptic inverse problems with an application to the identification of lame parameters Inverse Problems Sci - Gockenbach |

1 | Khan A A 2007 An abstract framework for elliptic inverse problems: part 1. An output least-squares approach - Gockenbach |

1 | et al 2006 Phase boundary estimation in electrical resistance tomography with weighted multi-layered neural networks and front point approach Meas - Kim |

1 | et al 2007 Moving interfacial boundary estimation in stratified flow of two immiscible liquids using electrical resistance tomography Meas - Kim |

1 | Tai X C 2007 Piecewise constant level set method for multiphase motion Int - Li |

1 | de Groen P and Sahli H 2008 A nonlinear iterative reconstruction and analysis approach to shape-based approximate electromagnetic tomography - Naik, Eriksson |

1 | Tai X C, Aanonsen S I and Espedal M 2007 A binary level set model for elliptic inverse problems with discontinuous coefficients Int - Nielsen |

1 |
Ring W 2007 A mumford-shah level-set approach for the inversion and segmentation of x-ray tomography data
- Ramlau
(Show Context)
Citation Context ...ntify multiple phase problems. The level-set method we use here is still related to the traditional level-set function of [35] which is continuous, but may not be a distance function. The approach to =-=[36]-=- can be used for a multi-region detection problem. The technique of [36] is to use shape sensitivity to get the decent directions for curves and then just uses the level-set method to move the curve i... |

1 | et al 2006 A three-dimensional inverse finite-element method applied to experimental eddy-current imaging data - Soleimani |

1 | C and Li H 2007 A piecewise constant level set method for elliptic inverse problems - Tai |

1 | V and Vauhkonen M 2006 Free-surface and admittivity estimation in electrical impedance tomography Int - Tossavainen, Kolehmainen |