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Discretization Error Analysis for Tikhonov Regularization
 ANALYSIS AND APPLICATIONS
, 2011
"... We study the discretization of inverse problems defined by a Carleman operator. In particular we develop a discretization strategy for this class of inverse problems and we give a convergence analysis. Learning from examples as well as the discretization of integral equations can be analysed in our ..."
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Cited by 11 (8 self)
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We study the discretization of inverse problems defined by a Carleman operator. In particular we develop a discretization strategy for this class of inverse problems and we give a convergence analysis. Learning from examples as well as the discretization of integral equations can be analysed in our setting.
Estimation of optimal PDEbased denoising in the SNR sense
 IEEE Trans. Image Processing
, 2004
"... Abstract — This paper is concerned with finding the best PDEbased denoising process, out of a set of possible ones. We focus either on finding the proper weight of the fidelity term in the energy minimization formulation, or on determining the optimal stopping time of a nonlinear diffusion process. ..."
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Cited by 11 (5 self)
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Abstract — This paper is concerned with finding the best PDEbased denoising process, out of a set of possible ones. We focus either on finding the proper weight of the fidelity term in the energy minimization formulation, or on determining the optimal stopping time of a nonlinear diffusion process. A necessary condition for achieving maximal SNR is stated, based on the covariance of the noise and the residual part. We provide two practical alternatives for estimating this condition, by observing that the filtering of the image and the noise can be approximated by a decoupling technique, with respect to the weight or time parameters. Our automatic algorithm obtains quite accurate results on a variety of synthetic and natural images, including piecewise smooth and textured ones. We assume that the statistics of the noise were previously estimated. No apriori knowledge regarding the characteristics of the clean image is required. A theoretical analysis is carried out, where several SNR performance bounds are established for the optimal strategy and for a widely used method, wherein the variance of the residual part equals the variance of the noise. I.
Estimation of the optimal variational parameter via SNR analysis
 In ScaleSpace ’05
, 2005
"... Abstract. We examine the problem of finding the optimal weight of the fidelity term in variational denoising. Our aim is to maximize the signal to noise ratio (SNR) of the restored image. A theoretical analysis is carried out and several bounds are established on the performance of the optimal strat ..."
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Cited by 9 (4 self)
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Abstract. We examine the problem of finding the optimal weight of the fidelity term in variational denoising. Our aim is to maximize the signal to noise ratio (SNR) of the restored image. A theoretical analysis is carried out and several bounds are established on the performance of the optimal strategy and a widely used method, wherein the variance of the residual part equals the variance of the noise. A necessary condition is set to achieve maximal SNR. We provide a practical method for estimating this condition and show that the results are sufficiently accurate for a large class of images, including piecewise smooth and textured images. 1
Initial temperature reconstruction for a nonlinear heat equation: application to radiative and conductive heat transfer
 5th International Conference on Inverse Problems in Engineering: Theory and Practice, volume III, pages P02: 1–8
, 2005
"... Consider a cooling process described by a nonlinear heat equation. We are interested to recover the initial temperature from temperature measurements which are available on a part of the boundary for some time. Up to now even for the linear heat equation such problem has been usually studied as a no ..."
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Cited by 2 (1 self)
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Consider a cooling process described by a nonlinear heat equation. We are interested to recover the initial temperature from temperature measurements which are available on a part of the boundary for some time. Up to now even for the linear heat equation such problem has been usually studied as a nonlinear illposed operator equation, and regularization methods involving Frechet derivatives have been applied. We propose a fast derivativefree iterative method. Numerical results are presented for the glass cooling process, where nonlinearity appears due to radiation.
Regularized FixedPoint Iterations for Nonlinear Inverse Problems
, 2005
"... In this paper we introduce a derivativefree, iterative method for solving nonlinear illposed problems F x = y, where instead of y noisy data yδ with �y − yδ � ≤ δ are given and F: D(F) ⊆ X → Y is a nonlinear operator between Hilbert spaces X and Y. This method is defined by splitting the operator ..."
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Cited by 1 (1 self)
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In this paper we introduce a derivativefree, iterative method for solving nonlinear illposed problems F x = y, where instead of y noisy data yδ with �y − yδ � ≤ δ are given and F: D(F) ⊆ X → Y is a nonlinear operator between Hilbert spaces X and Y. This method is defined by splitting the operator F into a linear part A and a nonlinear part G, such that F = A + G. Then iterations are organized as Auk+1 = yδ − Guk. In the context of illposed problems we consider the situation when A does not have a bounded inverse, thus each iteration needs to be regularized. Under some conditions on the operators A and G we study the behavior of the iteration error. We obtain its stability with respect to the iteration number k as well as the optimal convergence rate with respect to the noise level δ, provided that the solution satisfies a generalized source condition. As an example, we consider an inverse problem of initial temperature reconstruction for a nonlinear heat equation, where the nonlinearity appears due to radiation effects. The obtained iteration error in the numerical results has the theoretically expected behavior. The theoretical assumptions are illustrated by a computational experiment.
ReducedBasis Approximations and A Posteriori Error Bound for Nonaffine and Nonlinear Partial Differential Equations: Application to Inverse Analysis
, 2005
"... ..."
Efficient FFTbased Circulant Embedding Method and Subspace Algorithms for Wideband Singlecarrier Equalization
"... In singlecarrier transmission systems, wider bandwidth allows higher data rate by transmitting narrower pulses. For wireless applications, however, this means that the effective channel response is longer and the number of significant taps increases. This results in higher computational burden at ..."
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In singlecarrier transmission systems, wider bandwidth allows higher data rate by transmitting narrower pulses. For wireless applications, however, this means that the effective channel response is longer and the number of significant taps increases. This results in higher computational burden at the receiver. In this paper, we first review some wellknown equalization methods and their shortcomings. Then we will propose the use of the circulant embedding method and the CG algorithm as efficient equalizers that are specifically well suited in dealing with long delay spread channels. These methods take advantage of the low computational complexity of the FFT algorithm, resulting in the same overall computational complexity of N log(N). Furthermore, when the CG algorithm is used correctly, it may perform better than the MMSE equalizer at a much lower cost.
Fast InSAR Multichannel Phase Unwrapping for DEM Generation
"... Abstract—In this paper, a method to solve the multichannel phase unwrapping problem is presented. MAP approach together with Markov Random Fields have proved to be effective, allowing to restore the uniqueness of the solution without introducing external constraints to regularize the problem. The id ..."
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Abstract—In this paper, a method to solve the multichannel phase unwrapping problem is presented. MAP approach together with Markov Random Fields have proved to be effective, allowing to restore the uniqueness of the solution without introducing external constraints to regularize the problem. The idea is to develop a fast algorithm to unwrap the interferometric phase in the multichannel configuration, which is, in the main time, able to provide the global optimum solution. To reach this target, an a priori model based on Total Variation is used together with optimization algorithm based on graphcut technique. The proposed approach has been tested both on simulated and real data. The obtained results show the effectiveness of our approach. I.
Estimation of Optimal PDEBased Denoising in the SNR Sense
"... Abstract—This paper is concerned with finding the best partial differential equationbased denoising process, out of a set of possible ones. We focus on either finding the proper weight of the fidelity term in the energy minimization formulation or on determining the optimal stopping time of a nonli ..."
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Abstract—This paper is concerned with finding the best partial differential equationbased denoising process, out of a set of possible ones. We focus on either finding the proper weight of the fidelity term in the energy minimization formulation or on determining the optimal stopping time of a nonlinear diffusion process. A necessary condition for achieving maximal SNR is stated, based on the covariance of the noise and the residual part. We provide two practical alternatives for estimating this condition by observing that the filtering of the image and the noise can be approximated by a decoupling technique, with respect to the weight or time parameters. Our automatic algorithm obtains quite accurate results on a variety of synthetic and natural images, including piecewise smooth and textured ones. We assume that the statistics of the noise were previously estimated. No a priori knowledge regarding the characteristics of the clean image is required. A theoretical analysis is carried out, where several SNR performance bounds are established for the optimal strategy and for a widely used method, wherein the variance of the residual part equals the variance of the noise. Index Terms—Image denoising, nonlinear diffusion, signaltonoise ratio (SNR), totalvariation, variational image processing. I.
unknown title
"... This is a preliminary version. Do not circulate! Finite element approach to clustering of multidimensional time series∗ ..."
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This is a preliminary version. Do not circulate! Finite element approach to clustering of multidimensional time series∗