Class Of Inverse Problems Research Articles

Inverse problems are solvable on real number signal processing hardware

Despite the success of Deep Learning (DL) serious reliability issues such as non-robustness persist. An interesting aspect is, whether these problems arise due to insufficient tools or fundamental limitations of DL. We study this question from the computability perspective by characterizing the limits the applied hardware imposes. For this, we focus on the class of inverse problems, which, in particular, encompasses any task to reconstruct data from measurements. On digital hardware, a conceptual barrier on the capabilities of DL for solving finite-dimensional inverse problems has in fact already been derived. This paper investigates the general computation framework of Blum-Shub-Smale (BSS) machines, describing the processing and storage of arbitrary real values. Although a corresponding real-world computing device does not exist, research and development towards real number computing hardware, usually referred to by “neuromorphic computing”, has increased in recent years. In this work, we show that the framework of BSS machines does enable the algorithmic solvability of finite dimensional inverse problems. Our results emphasize the influence of the considered computing model in questions of accuracy and reliability.

Towards a new standard for seismic moment tensor inversion containing 3-D earth structure uncertainty

SUMMARY Moment tensor (MT) inversion is a classical geophysical inverse problem that infers a force-equivalent model of a seismic source from seismological observations. Like other inverse problems, the accuracy of the inversion depends on the reliability of the forward problem simulating waveforms from the source location through an Earth structural model. Apart from errors in data, the error in forward waveform simulation, also known as theory error, is a significant source of error contributing to the misfit function between the predicted and observed waveforms. Here, we set up numerical experiments to comprehensively probe the sensitivity of the linearized MT inversion to 3-D regional earth model errors, a known predominant factor of the theory error. Using the Monte Carlo method, we estimate the empirical structural covariance matrices to characterize the waveform mismatch due to the imperfect knowledge of Earth's structure. First, although the inversion accuracy deteriorates with increasing model errors, incorporating the structural covariance matrices into the misfit function improves the accuracy of inversion results for all theorized error distributions. Secondly, we propose a slightly modified form of the structural covariance matrix, which further enhances the inversion outcome. Lastly, as the true structural errors are likely spatially correlated, we highlight the importance of adequately treating the correlation into the MT inversion because of its significant impact on inversion. Overall, as a preliminary effort in quantifying 3-D structural errors on MT inversion, this study proves the computational feasibility by means of numerical experiments and will hopefully provide a way forward for future work on this topic.

Convergent Regularization in Inverse Problems and Linear Plug-and-Play Denoisers

AbstractRegularization is necessary when solving inverse problems to ensure the well-posedness of the solution map. Additionally, it is desired that the chosen regularization strategy is convergent in the sense that the solution map converges to a solution of the noise-free operator equation. This provides an important guarantee that stable solutions can be computed for all noise levels and that solutions satisfy the operator equation in the limit of vanishing noise. In recent years, reconstructions in inverse problems are increasingly approached from a data-driven perspective. Despite empirical success, the majority of data-driven approaches do not provide a convergent regularization strategy. One such popular example is given by iterative plug-and-play (PnP) denoising using off-the-shelf image denoisers. These usually provide only convergence of the PnP iterates to a fixed point, under suitable regularity assumptions on the denoiser, rather than convergence of the method as a regularization technique, thatis under vanishing noise and regularization strength. This paper serves two purposes: first, we provide an overview of the classical regularization theory in inverse problems and survey a few notable recent data-driven methods that are provably convergent regularization schemes. We then continue to discuss PnP algorithms and their established convergence guarantees. Subsequently, we consider PnP algorithms with learned linear denoisers and propose a novel spectral filtering technique of the denoiser to control the strength of regularization. Further, by relating the implicit regularization of the denoiser to an explicit regularization functional, we are the first to rigorously show that PnP with a learned linear denoiser leads to a convergent regularization scheme. The theoretical analysis is corroborated by numerical experiments for the classical inverse problem of tomographic image reconstruction.

Mine seismic source location via forward modeling of spherical waves in a multilayered horizontal or inclined medium

Seismic source location is a classic inverse problem in seismology. In mathematical physics, inverse problems have multiple natural solutions. The objective of this study was to develop a generic theory and method of seeking the true solution from multiple solutions for the location of a coal mine seismic source in an idealized velocity structure model of a coal mine with a small scale and complex geological environment. Starting from the simplest velocity structure model, the complexity of the model gradually increased, until it approached the real velocity structure model, i.e., the multilayered horizontal and inclined velocity structure model, in order to find a generic method for solving the multi-solution inverse problem of coal mine seismic source location. Specifically, the wavefront distribution equation in a two-layer horizontal medium was derived and then expanded to any multi-layer horizontal medium. Based on this equation, a positive definite nonlinear equation system was established from the perspective of any observation system. The equation system contained four unknown variables of the spatiotemporal position of the seismic source. To determine the spatiotemporal parameters of the seismic source, nonlinear equations for four stations were required. To solve the nonlinear equation system, an initial iteration value was determined. In order to reduce the difficulty of determining the initial iteration value, the variable substitution method was used to reduce the number of location parameters. By rotating the original geodetic coordinate system of the station to be parallel and orthogonal to the medium interface, the wavefront method was extended to inclined medium. In conclusion, in this study, the problem of coal mine seismic source location in a multi-layer horizontal or inclined medium was effectively solved. The method proposed in this study provides a reference for solving the true solution from multiple solutions for the location of a coal mine seismic source in small-scale coal mines with complex geological environments.

GRIDS-Net: Inverse shape design and identification of scatterers via geometric regularization and physics-embedded deep learning

This study presents a deep learning based methodology for both remote sensing and design of acoustic scatterers. The ability to determine the shape of a scatterer, either in the context of material design or sensing, plays a critical role in many practical engineering problems. This class of inverse problems is extremely challenging due to their high-dimensional, nonlinear, and ill-posed nature. To overcome these technical hurdles, we introduce a geometric regularization approach for deep neural networks (DNN) based on non-uniform rational B-splines (NURBS) and capable of predicting complex 2D scatterer geometries in a parsimonious dimensional representation. Then, this geometric regularization is combined with physics-embedded learning and integrated within a robust convolutional autoencoder (CAE) architecture to accurately predict the shape of 2D scatterers in the context of identification and inverse design problems. An extensive numerical study is presented in order to showcase the remarkable ability of this approach to handle complex scatterer geometries while generating physically-consistent acoustic fields. The study also assesses and contrasts the role played by the (weakly) embedded physics in the convergence of the DNN predictions to a physically consistent inverse design.

Qualitative inverse problems: mapping data to the features of trajectories and parameter values of an ODE model

Our recent work on linear and affine dynamical systems has laid out a general framework for inferring the parameters of a differential equation model from a discrete set of data points collected from a system being modeled. It introduced a new class of inverse problems where qualitative information about the parameters and the associated dynamics of the system is determined for regions of the data space, rather than just for isolated experiments. Rigorous mathematical results have justified this approach and have identified common features that arise for certain classes of integrable models. In this work we present a thorough numerical investigation that shows that several of these core features extend to a paradigmatic linear-in-parameters model, the Lotka–Volterra (LV) system, which we consider in the conservative case as well as under the addition of terms that perturb the system away from this regime. A central construct for this analysis is a concise representation of parameter and dynamical features in the data space that we call the P n -diagram, which is particularly useful for visualization of the qualitative dependence of the system dynamics on data for low-dimensional (small n) systems. Our work also exposes some new properties related to non-uniqueness that arise for these LV systems, with non-uniqueness manifesting as a multi-layered structure in the associated P 2-diagrams.

Deep Earth rotational seismology

SUMMARYRotational seismology opens a new avenue to study the deep interior of the Earth. Using data from the Wettzell Observatorium, Germany, where a ring laser gyroscope and a 3-component translational broadband seismometer are co-located, we report the presence of clear S, ScS and SdS signals on both rotational and translational seismograms. Using S wave arrivals, we propose a new methodology to extract information on velocity changes in the Earth mantle and we show that, by combining both translational and rotational data, we are able to solve the well known velocity-depth ambiguity inherent to classical inverse problems. The methodology is validated using ray theory and 2.5-D finite-difference synthetics. We provide a proof-of-concept showing that future studies of the Earth’s deep interior can be improved by combining translational and rotational records.

Helsinki deblur challenge 2021: Description of photographic data

Image deconvolution is a classical inverse problem that serves well as a computational test bench for reconstruction algorithms. Namely, the direct operator can be modelled in a straightforward way either by convolution or by multiplication in the frequency domain. Further, the ill-posedness of the inverse problem can be adjusted by the form of the point spread function (PSF). An open photographic dataset is described, suitable for testing practical deconvolution methods. The image material was designed and collected for the Helsinki Deblur Challenge 2021. The dataset contains pairs of images taken by two identical cameras of the same target but with different conditions. One camera is always in focus and generates sharp and low-noise images, while the other camera produces blurred and noisy photos as it is gradually more and more out of focus and has a higher ISO setting. The data is available here: https://doi.org/10.5281/zenodo.4916176

Inverse problems on low-dimensional manifolds

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we assume that the unknown belongs to a finite-dimensional manifold: this assumption arises in many real-world scenarios where natural objects have a low intrinsic dimension and belong to a certain submanifold of a much larger ambient space. We prove uniqueness and Hölder and Lipschitz stability results in this general setting, also in the case when only a finite discretization of the measurements is available. Then, a Landweber-type reconstruction algorithm from a finite number of measurements is proposed, for which we prove global convergence, thanks to a new criterion for finding a suitable initial guess.These general results are then applied to several examples, including two classical nonlinear ill-posed inverse boundary value problems. The first is Calderón’s inverse conductivity problem, for which we prove a Lipschitz stability estimate from a finite number of measurements for piece-wise constant conductivities with discontinuities on an unknown triangle. A similar stability result is then obtained for Gel’fand-Calderón’s problem for the Schrödinger equation, in the case of piece-wise constant potentials with discontinuities on a finite number of non-intersecting balls.

Convergence analysis of inexact Newton–Landweber iteration under Hölder stability

In this paper, we focus on a class of inverse problems with Lipschitz continuous Fréchet derivatives both in Hilbert spaces and Banach spaces. The convergence and convergence rate of the inexact Newton–Landweber method (INLM) for such problems are presented under some assumptions. For the inverse problems in Hilbert spaces, we revisit the convergence result and the convergence rate of the INLM under Lipschitz condition and Hölder stability. Furthermore, the INLM for nonlinear inverse problems in Banach spaces is also considered. By using a Hölder stability corresponding to the Bregman distance, we derive the convergence property and convergence rate of the method.

One-to-all lightweight Fourier channel attention convolutional neural network for speckle reconstructions.

Speckle reconstruction is a classical inverse problem in computational imaging. Inspired by the memory effect of the scattering medium, deep learning methods reveal excellent performance in extracting the correlation of speckle patterns. Nowadays, advanced models generally include more than 10M parameters and mostly pay more attention to the spatial feature information. However, the frequency domain of images also contains precise hierarchical representations. Here we propose a one-to-all lightweight Fourier channel attention convolutional neural network (FCACNN) with Fourier channel attention and the res-connected bottleneck structure. Compared with the state-of-the-art model, i.e., self-attention armed convolutional neural network (SACNN), our architecture has better feature extraction and reconstruction ability. The Pearson correlation coefficient and Jaccard index scores of FCACNN increased by at least 5.2% and 13.6% compared with task-related models. And the parameter number of the lightweight FCACNN is only 1.15M. Furthermore, the validation results show that the one-to-all model, FCACNN, has excellent generalization capability on unseen speckle patterns such as handwritten letters and Quickdraws.

Alternating step size method for solving ill-posed linear operator equations in energetic space

An alternating step size method for solving ill-posed linear operator equations is introduced in energetic space, a subspace of the Hilbert space. Focus is on the case of a positive bounded self-conjugated operator under the assumption that the error for the right hand side of the equation is available. The advantage of the energetic norm is that does not require smoothness of the exact solution x. Convergence of the new method is discussed and shown to be faster for similar error estimates. Furthermore, the conditions for convergence in the energetic norm are derived which also imply convergence in the original Hilbert space norm. The new method is demonstrated on several numerical examples of the integral equation in L20,1 space, taken from the class of inverse problems in potential theory, and convergence speed compared with that of the fixed step size method. Finally, a novel real-word application of the integral equation is proposed: deblurring averages of trial-to-trial jittered EEG responses. When applying the Kosambi–Hilbert Torsion (KTH) to align these responses, the jitter is more effectively reduced and alignment improves with the deblurred average as a reference than with the original average.

A Brief Survey on the Inverse Galois Problem

Inverse Galois problem (IGP) states whether any finite group is realizable as a Galois group over the field K. It is the question of the structure and representation of the Galois group and also questions its epimorphic images. So, it is called an inverse Galois problem. For K=ℚ (the field of rational number), it is called a classical inverse Galois problem (CIGP). This paper reviews the positive answer to the classical inverse Galois problem (CIGP) for all finite abelian groups and some finite non-abelian solvable groups. We also discuss this problem (CIGP) for some finite non-solvable groups in this paper. This problem still remains to solve, but if we find the true value of the statement ‘All subgroups of order m of the symmetric group (Sm) for all m are realizable as Galois group over ℚ’ then its truth value gives the answer of CIGP. We check this statement for m=1,2,3,4 and 5 in this paper, where we get that this statement is true. If this statement is true, then CIGP has a positive answer. But if this statement is false then CIGP has a negative answer.

Deep learning methods for inverse problems.

In this paper we investigate a variety of deep learning strategies for solving inverse problems. We classify existing deep learning solutions for inverse problems into three categories of Direct Mapping, Data Consistency Optimizer, and Deep Regularizer. We choose a sample of each inverse problem type, so as to compare the robustness of the three categories, and report a statistical analysis of their differences. We perform extensive experiments on the classic problem of linear regression and three well-known inverse problems in computer vision, namely image denoising, 3D human face inverse rendering, and object tracking, in presence of noise and outliers, are selected as representative prototypes for each class of inverse problems. The overall results and the statistical analyses show that the solution categories have a robustness behaviour dependent on the type of inverse problem domain, and specifically dependent on whether or not the problem includes measurement outliers. Based on our experimental results, we conclude by proposing the most robust solution category for each inverse problem class.

Scalar and vector tomography for the weighted transport equation with application to helioseismology

Abstract Motivated by the application to helioseismology, we demonstrate uniqueness and stability for a class of inverse problems of the weighted transport equation. Using 𝐴-analytic functions, this inverse problem is expressed as a Cauchy problem. In this form, we show that, for a finite even trigonometric polynomial weight function, the resulting system is well-conditioned numerically and permits a Carleman-like estimate with boundary terms.

Unsupervised Deep Learning Methods for Biological Image Reconstruction and Enhancement: An overview from a signal processing perspective.

Recently, deep learning approaches have become the main research frontier for biological image reconstruction and enhancement problems thanks to their high performance, along with their ultra-fast inference times. However, due to the difficulty of obtaining matched reference data for supervised learning, there has been increasing interest in unsupervised learning approaches that do not need paired reference data. In particular, self-supervised learning and generative models have been successfully used for various biological imaging applications. In this paper, we overview these approaches from a coherent perspective in the context of classical inverse problems, and discuss their applications to biological imaging, including electron, fluorescence and deconvolution microscopy, optical diffraction tomography and functional neuroimaging.

Estimating vocal tract geometry from acoustic impedance using deep neural network.

A data-driven approach using artificial neural networks is proposed to address the classic inverse area function problem, i.e., to determine the vocal tract geometry (modelled as a tube of nonuniform cylindrical cross-sections) from the vocal tract acoustic impedance spectrum. The predicted cylindrical radii and the actual radii were found to have high correlation in the three- and four-cylinder model (Pearson coefficient (ρ) and Lin concordance coefficient (ρc) exceeded 95%); however, for the six-cylinder model, the correlation was low (ρ around 75% and ρc around 69%). Upon standardizing the impedance value, the correlation improved significantly for all cases (ρ and ρc exceeded 90%).

An introduction to the mathematics of the imaging modalities using small-scaled contrast agents

In the recent years, we witness a great interest in imaging, in a wide sense, using contrast agents. One of the reasons is that many imaging modalities, as the ones related to medical sciences, suffer from several shortcomings. The most serious one is the issue of instability. Indeed, it is, nowadays, a common certainty that classical inverse problems of recovering objects from remote measurements are, mostly, highly unstable. To recover the stability, it is advised to create, whenever possible, the missing contrasts in the targets to image. In this survey paper, we follow this direction and propose an approach how to analyze mathematically the effect of the injected agents on the different fields under consideration. These contrast agents are small-sized particles modeled with materials that enjoy high contrasts as compared to the ones of the background. These two properties allow them, under critical scales of size/contrast, to create local spots when excited from far. These local spots can be remotely recovered in stable ways. The accessible information on the target are encoded in theses spots. After stating a class of such imaging modalities that enter into this framework, as the acoustic imaging, photo-acoustic imaging, optical imaging and more, we provide detailed analysis for first two modalities where the contrast agents are micro-bubbles and nano-particles respectively.

A Composite Initialization Method for Phase Retrieval

Phase retrieval is a classical inverse problem with respect to recovering a signal from a system of phaseless constraints. Many recently proposed methods for phase retrieval such as PhaseMax and gradient-descent algorithms enjoy benign theoretical guarantees on the condition that an elaborate estimate of true solution is provided. Current initialization methods do not perform well when number of measurements are low, which deteriorates the success rate of current phase retrieval methods. We propose a new initialization method that can obtain an estimate of the original signal with uniformly higher accuracy which combines the advantages of the null vector method and maximal correlation method. The constructed spectral matrix for the proposed initialization method has a simple and symmetrical form. A lower error bound is proved theoretically as well as verified numerically.

New worlds, new chemistry, new ceramics

AbstractSpace missions have documented the chemical diversity and complexity of planets in our solar system. Astronomers have found thousands of exoplanets orbiting other stars in our galaxy. Almost every star in our galaxy has one or more orbiting planets and they appear to be much more diverse than the few sampled in our solar system. This diversity implies immense chemical, ceramic, geological, and mineralogical complexity. Constrained by the limited data, understanding this range of planets poses a classic inverse problem in materials science, namely, to constrain the composition, phase relations, and evolution of a planet based on a few remote observations of its properties. The purpose of this paper is to provide a general view of the Earth and of planets in our solar system and beyond, to introduce fundamental concepts of planetary structure and chemistry, and to identify outstanding questions and opportunities from the point of view of chemistry, materials science, and ceramics. This paper is composed of two parts. First, an introduction is followed by a discussion of the structure, geophysics, and phase equilibria in the Earth, to orient the non‐geologic reader to the basic phenomena and concepts. This is followed by a loose transcription of a lecture given by Navrotsky at the (virtual) 2020 American Ceramic Society meeting. This talk stressed the evolving nature of planetary science, especially inspired by the discovery of myriads of exoplanets, highlighted the Materials of the Universe Center at Arizona State University, and briefly described and compared planets in our solar system.