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30th European Signal Processing Conference (EUSIPCO)

作者： Altmann, Yoann Heriot Watt Univ Sch Engn & Phys Sci Edinburgh Midlothian Scotland

ISBN: (纸本)9789082797091

In this paper, we investigate and compare approximate Bayesian methods for high-dimensional linear inverse problems where sparsity-promoting prior distributions can be used to regularized the inference process. In particular, we investigate fully factorized priors which lead to multimodal and potentially non-smooth posterior distributions such as Bernoulli-Gaussian priors. In addition to the most traditional variational Bayes framework based on mean-field approximation, we compare different implementations of power expectation-propagation (EP) in terms of estimation of the posterior means and marginal variances, using fully factorized approximations. The different methods are compared using low-dimensional examples and we then discuss the potential benefits of power EP for image restoration. These preliminary results tend to confirm that in the case of Gaussian likelihoods, EP generally provides more reliable marginal variances while power EP offers more flexibility for generalised linear inverse problems.

关键词： Approximate Bayesian inference Expectation-Propagation linear inverse problems Variational Bayes Sparsity

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OVERCOMING MEASUREMENT INCONSISTENCY IN DEEP LEARNING FOR linear inverse problems: APPLICATIONS IN MEDICAL IMAGING

OVERCOMING MEASUREMENT INCONSISTENCY IN DEEP LEARNING FOR LI...

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IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

作者： Vella, Marija Mota, Joao F. C. Heriot Watt Univ Inst Sensors Signals & Syst Edinburgh Midlothian Scotland

ISBN: (纸本)9781728176055

The remarkable performance of deep neural networks (DNNs) currently makes them the method of choice for solving linear inverse problems. They have been applied to super-resolve and restore images, as well as to reconstruct MR and CT images. In these applications, DNNs invert a forward operator by finding, via training data, a map between the measurements and the input images. It is then expected that the map is still valid for the test data. This framework, however, introduces measurement inconsistency during testing. We show that such inconsistency, which can be critical in domains like medical imaging or defense, is intimately related to the generalization error. We then propose a framework that post-processes the output of DNNs with an optimization algorithm that enforces measurement consistency. Experiments on MR images show that enforcing measurement consistency via our method can lead to large gains in reconstruction performance.

关键词： Neural networks linear inverse problems medical imaging optimization total variation

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DEEP LEARNING FOR linear inverse problems USING THE PLUG-AND-PLAY PRIORS FRAMEWORK

DEEP LEARNING FOR LINEAR INVERSE PROBLEMS USING THE PLUG-AND...

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IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

作者： Chen, Wei Wipf, David Rodrigues, Miguel Beijing Jiaotong Univ State Key Lab Rail Traff Control & Safety Beijing Peoples R China Amazon AI Res Lab Shanghai Peoples R China UCL Dept Elect & Elect Engn London England

ISBN: (纸本)9781728176055

linear inverse problems appear in many applications, where different algorithms are typically employed to solve each inverse problem. Nowadays, the rapid development of deep learning (DL) provides a fresh perspective for solving the linear inverse problem: a number of well-designed network architectures results in state-of-the-art performance in many applications. In this overview paper, we present the combination of the DL and the Plug-and-Play priors (PPP) framework, showcasing how it allows solving various inverse problems by leveraging the impressive capabilities of existing DL based denoising algorithms. Open challenges and potential future directions along this line of research are also discussed.

关键词： Deep learning linear inverse problems plug-and-play priors

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Rethinking Sketching as Sampling: Efficient Approximate Solution to linear inverse problems

Rethinking Sketching as Sampling: Efficient Approximate Solu...

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IEEE Global Conference on Signal and Information Processing (GlobalSIP)

作者： Gama, Fernando Marques, Antonio G. Mateos, Gonzalo Ribeiro, Alejandro Univ Penn Dept Elect & Syst Engn Philadelphia PA 19104 USA King Juan Carlos Univ Dept Signal Theory & Comms Madrid Spain Univ Rochester Dept Elect & Comp Engn Rochester NY 14627 USA

ISBN: (纸本)9781509045457

Sampling and reconstruction of bandlimited graph signals have well-appreciated merits for dimensionality reduction, affordable storage, and online processing of streaming network data. However, these parsimonious signals are oftentimes encountered with high-dimensional linear inverse problems. Hence, interest shifts from reconstructing the signal itself towards instead approximating the input to a prescribed linear operator efficiently. In this context, we propose a novel sampling scheme that leverages the bandlimitedness of the output as well as the transformation whose input we wish to approximate. We formulate problems to jointly optimize sample selection and a sketch of the target inverse mapping, so when the latter is affordably applied to the sampled output signal, the result is close to the desired input. The developed sampling plus reduced-complexity processing pipeline is particularly useful for streaming data, where the linear transform has to be applied fast and repeatedly to successive response signals.

关键词： Sketching sampling graph signal processing streaming linear inverse problems

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Two-step algorithms for linear inverse problems with non-quadratic regularization

Two-step algorithms for linear inverse problems with non-qua...

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IEEE International Conference on Image Processing (ICIP 2007)

作者： Bioucas-Dias, Jose M. Figueiredo, Mario A. T. Univ Tecn Lisboa Inst Telecommun Inst Super Tecn Lisbon Portugal

ISBN: (纸本)9781424414369

Iterative shrinkage/thresholding (IST) algorithms have been recently proposed to handle high-dimensional convex optimization problems arising in image inverse problems (namely deconvolution) under non-quadratic regularization (e.g., total variation or sparsity inducing regularizers, on wavelet representations). The convergence speed of IST algorithms depends heavily on the nature of the direct operator, being very slow when this operator is severely ill-conditioned. In this paper, we introduce a two-step version of IST (termed 21ST, pronounced "twist") showing much faster convergence for strongly ill-conditioned operators. We give theoretical results concerning the convergence behavior of 21ST and show its effectiveness for wavelet-based and total variation image deconvolution.

关键词： image restoration deblurring linear inverse problems wavelets total variation iterative algorithms

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A Spectrum-based Regularization Approach to linear inverse problems: Models, Learned Parameters and Algorithms

A Spectrum-based Regularization Approach to Linear Inverse P...

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作者： Castanon, Jorge Castanon Alberto Rice University

学位级别：Ph.D.

In this thesis, we study the problem of recovering signals, in particular images, that approximately satisfy severely ill-conditioned or underdetermined linear systems. For example, such a linear system may represent a set of under-sampled and noisy linear measurements. It is well-known that the quality of the recovery critically depends on the choice of an appropriate regularization model that incorporates prior information about the target solution. Two of the most successful regularization models are the Tikhonov and Total Variation (TV) models, each of which is used in a wide range of applications. We design and investigate a class of spectrum-based models that generalize and improve upon both the Tikhonov and the TV methods, as well as their combinations or so-called hybrids. The proposed models contain "spectrum parameters" that are learned from training data sets through solving optimization problems. This parameter-learning feature gives these models the flexibility to adapt to desired target solutions. We devise efficient algorithms for all the proposed models and conduct comprehensive numerical experiments to evaluate their performance as compared to established models. Numerical results show a generally superior quality in recovered images by our approach from under-sampled linear measurements. Using the proposed algorithms, one can often obtain much improved quality at a moderate increase in computational time.

关键词： numerical optimization regularization linear inverse problems machine learning image recovery compressive sensing

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An Optimal Multi-Vector Iterative Algorithm in a Krylov Subspace for Solving the Ill-Posed linear inverse problems

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CMC-COMPUTERS MATERIALS & CONTINUA 2013年第2期33卷 175-198页

作者： Liu, Chein-Shan Natl Taiwan Univ Dept Civil Engn Taipei 10764 Taiwan

An optimal m-vector descent iterative algorithm in a Krylov subspace is developed, of which the m weighting parameters are optimized from a properly defined objective function to accelerate the convergence rate in solving an ill-posed linear problem. The optimal multi-vector iterative algorithm (OMVIA) is convergent fast and accurate, which is verified by numerical tests of several linear inverse problems, including the backward heat conduction problem, the heat source identification problem, the inverse Cauchy problem, and the external force recovery problem. Because the OMVIA has a good filtering effect, the numerical results recovered are quite smooth with small error, even under a large noise up to 10%.

关键词： linear inverse problems Ill-posed linear equations system Optimal multi-vector iterative algorithm (OMVIA) Future cone Invariant-manifold Krylov subspace method

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Direct Sequential Simulation for spherical linear inverse problems

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COMPUTERS & GEOSCIENCES 2022年 160卷

作者： Otzen, Mikkel Finlay, Christopher C. Hansen, Thomas Mejer DTU Space Centrifugevej 356 DK-2800 Lyngby Denmark Aarhus Univ Hoegh Guldbergs Gade 2Bldg 1671 DK-8000 Aarhus C Denmark

We present a method for obtaining efficient probabilistic solutions to geostatistical and linear inverse problems in spherical geometry. Our Spherical Direct Sequential Simulation (SDSSIM) framework combines information from possibly noisy observations, that provide either point information on the model or are related to the model by a linear averaging kernel, and statistics derived from a-priori information. It generates realizations from marginal posterior probability distributions of model parameters that are not limited to be Gaussian. We avoid the restriction to Cartesian geometry built into many existing geostatistical simulation codes, and work instead with grids in spherical geometry relevant to problems in Earth and Space sciences. We demonstrate our scheme using a synthetic example, showing that it produces realistic posterior realizations consistent with the known solution while fitting observations within their uncertainty and reproducing the distribution of model parameters and covariance statistics of a-priori models. Secondly, we present an application to real satellite observations, estimating the posterior probability distribution for the geomagnetic field at the core-mantle boundary. Our results reproduce well-known features of the core-mantle boundary magnetic field, and also allow probabilistic investigations of the magnetic field morphology. Small length scale features in the posterior realizations are not determined by the observations but match the covariance statistics extracted from geodynamo simulations. The framework presented here represents a step towards more general approaches to probabilistic inversion in spherical geometry.

关键词： Spherical sequential simulation linear inverse problems Spherical geometry Geomagnetism Geophysical methods Earth observation

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An optimal tri-vector iterative algorithm for solving ill-posed linear inverse problems

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inverse problems IN SCIENCE AND ENGINEERING 2013年第4期21卷 650-681页

作者： Liu, Chein-Shan Natl Taiwan Univ Dept Civil Engn Taipei 10764 Taiwan

A future cone in the Minkowski space, defined in terms of the square-norm of the residual vector for an ill-posed linear system to be solved, is used to derive an optimal tri-vector descent system of nonlinear ordinary differential equations (ODEs). Then, a simple Euler scheme is used to generate an iterative algorithm from these ODEs, of which the two parameters appeared are optimized from a properly defined merit function to accelerate the convergence speed in solving the ill-posed linear systems. The optimal tri-vector iterative algorithm (OTVIA) is fast convergent and accurate, which is proven by numerical tests of inverse problems, including the backward heat conduction problem, the Calderon inverse problem and the inverse Cauchy problems. By defining a suitable convergence rate, we assess the convergence speeds of OTVIA and the conjugate gradient method (CGM), which reveal that the performance of OTVIA is better than the CGM. Also by comparing the OTVIA with the generalized minimal residual method (GMRES), we observe that the OTVIA is better than the GMRES.

关键词： linear inverse problems ill-posed linear equations system conjugate gradient method generalized minimal residual method optimal tri-vector iterative algorithm invariant-manifold

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Optimal indirect estimation for linear inverse problems with discretely sampled functional data

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inverse problems 2021年第12期37卷 125001-125001页

作者： Pricop-Jeckstadt, Mihaela Univ Politehn Bucuresti Splaiul Independentii Nr 313 RO-060042 Bucharest Romania

Optimal mean estimation from noisy independent pathes of a stochastic process that are indirectly observed is an issue of great interest in functional inverse problems. In this paper, minimax rates of convergence for a class of linear inverse problems with correlated noise, general source conditions and various degrees of ill-posedness are proven in a continuous setting, when the pathes are entirely observed, and in a projected framework. The phase transition phenomenon characteristic to the functional data analysis appears also here and the thresholds that separate the sparse and the dense data set scenarios are computed for different smoothness conditions. The common design proves to be a special case of the independent design in view of the interpretation of the sampling properties via s-numbers and the price to pay for the data correlation turns out to be high. Finally. numerical experiments involving Abel's integral operator illustrate the goodness-of-fit of the Tikhonov estimator in various scenarios reflecting the common and independent design as well as sparse and dense sampling.

关键词： optimality minimax rates functional data linear inverse problems Tikhonov regularization Abel's integral operator

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