Showing papers in "Inverse Problems in 2018"

Stable Architectures for Deep Neural Networks

Abstract: Deep neural networks have become invaluable tools for supervised machine learning, e.g. classification of text or images. While often offering superior results over traditional techniques and successfully expressing complicated patterns in data, deep architectures are known to be challenging to design and train such that they generalize well to new data. Critical issues with deep architectures are numerical instabilities in derivative-based learning algorithms commonly called exploding or vanishing gradients. In this paper, we propose new forward propagation techniques inspired by systems of ordinary differential equations (ODE) that overcome this challenge and lead to well-posed learning problems for arbitrarily deep networks. The backbone of our approach is our interpretation of deep learning as a parameter estimation problem of nonlinear dynamical systems. Given this formulation, we analyze stability and well-posedness of deep learning and use this new understanding to develop new network architectures. We relate the exploding and vanishing gradient phenomenon to the stability of the discrete ODE and present several strategies for stabilizing deep learning for very deep networks. While our new architectures restrict the solution space, several numerical experiments show their competitiveness with state-of-the-art networks.

Joint image reconstruction method with correlative multi-channel prior for x-ray spectral computed tomography

Abstract: Rapid developments in photon-counting and energy-discriminating detectors have the potential to provide an additional spectral dimension to conventional x-ray grayscale imaging. Reconstructed spectroscopic tomographic data can be used to distinguish individual materials by characteristic absorption peaks. The acquired energy-binned data, however, suffer from low signal-to-noise ratio, acquisition artifacts, and frequently angular undersampled conditions. New regularized iterative reconstruction methods have the potential to produce higher quality images and since energy channels are mutually correlated it can be advantageous to exploit this additional knowledge. In this paper, we propose a novel method which jointly reconstructs all energy channels while imposing a strong structural correlation. The core of the proposed algorithm is to employ a variational framework of parallel level sets to encourage joint smoothing directions. In particular, the method selects reference channels from which to propagate structure in an adaptive and stochastic way while preferring channels with a high data signal-to-noise ratio. The method is compared with current state-of-the-art multi-channel reconstruction techniques including channel-wise total variation and correlative total nuclear variation regularization. Realistic simulation experiments demonstrate the performance improvements achievable by using correlative regularization methods.

Nonlocal low-rank and sparse matrix decomposition for spectral CT reconstruction.

Abstract: Spectral computed tomography (CT) has been a promising technique in research and clinic because of its ability to produce improved energy resolution images with narrow energy bins. However, the narrow energy bin image is often affected by serious quantum noise because of the limited number of photons used in the corresponding energy bin. To address this problem, we present an iterative reconstruction method for spectral CT using nonlocal low-rank and sparse matrix decomposition (NLSMD), which exploits the self-similarity of patches that are collected in multi-energy images. Specifically, each set of patches can be decomposed into a low-rank component and a sparse component, and the low-rank component represents the stationary background over different energy bins, while the sparse component represents the rest of different spectral features in individual energy bins. Subsequently, an effective alternating optimization algorithm was developed to minimize the associated objective function. To validate and evaluate the NLSMD method, qualitative and quantitative studies were conducted by using simulated and real spectral CT data. Experimental results show that the NLSMD method improves spectral CT images in terms of noise reduction, artifacts suppression and resolution preservation.

Monotonicity-based electrical impedance tomography for lung imaging

Abstract: This paper presents a monotonicity-based spatiotemporal conductivity imaging method for continuous regional lung monitoring using electrical impedance tomography (EIT). The EIT data (i.e., the boundary current-voltage data) can be decomposed into pulmonary, cardiac and other parts using their different periodic natures. The time-differential current-voltage operator corresponding to the lung ventilation can be viewed as either semi-positive or semi-negative definite owing to monotonic conductivity changes within the lung regions. We used this monotonicity constraints to improve the quality of lung EIT imaging. We tested the proposed methods in numerical simulations, phantom experiments and human experiments.

Compton camera imaging and the cone transform: A brief overview

Abstract: While most of Radon transform applications to imaging involve integrations over smooth sub-manifolds of the ambient space, lately important situations have appeared where the integration surfaces are conical. Three of such applications are single scatter optical tomography, Compton camera medical imaging, and homeland security. In spite of the similar surfaces of integration, the data and the inverse problems associated with these two modalities differ significantly. In this article, we present a brief overview of the mathematics arising in Compton camera imaging. In particular, the emphasis is made on the overdetermined data and flexible geometry of the detectors. For the detailed results, as well as other approaches (e.g., smaller-dimensional data or restricted geometry of detectors) the reader is directed to the relevant publications. Only a brief description and some references are provided for the single scatter optical tomography.

A new version of the convexification method for a 1D coefficient inverse problem with experimental data

Abstract: A new version of the convexification method is developed analytically and tested numerically for a 1-D coefficient inverse problem in the frequency domain. Unlike the previous version, this one does not use the so-called "tail function", which is a complement of a certain truncated integral with respect to the wave number. Globally strictly convex cost functional is constructed with the Carleman Weight Function. Global convergence of the gradient projection method to the correct solution is proved. Numerical tests are conducted for both computationally simulated and experimental data.

Blind image fusion for hyperspectral imaging with the directional total variation

Abstract: Hyperspectral imaging is a cutting-edge type of remote sensing used for mapping vegetation properties, rock minerals and other materials. A major drawback of hyperspectral imaging devices is their intrinsic low spatial resolution. In this paper, we propose a method for increasing the spatial resolution of a hyperspectral image by fusing it with an image of higher spatial resolution that was obtained with a different imaging modality. This is accomplished by solving a variational problem in which the regularization functional is the directional total variation. To accommodate for possible mis-registrations between the two images, we consider a non-convex blind super-resolution problem where both a fused image and the corresponding convolution kernel are estimated. Using this approach, our model can realign the given images if needed. Our experimental results indicate that the non-convexity is negligible in practice and that reliable solutions can be computed using a variety of different optimization algorithms. Numerical results on real remote sensing data from plant sciences and urban monitoring show the potential of the proposed method and suggests that it is robust with respect to the regularization parameters, mis-registration and the shape of the kernel.

Iterative Updating of Model Error for Bayesian Inversion

Abstract: In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when optimization algorithms are used to find a single estimate, or to speed up Markov chain Monte Carlo (MCMC) calculations in the Bayesian framework. The use of an approximate model introduces a discrepancy, or modeling error, that may have a detrimental effect on the solution of the ill-posed inverse problem, or it may severely distort the estimate of the posterior distribution. In the Bayesian paradigm, the modeling error can be considered as a random variable, and by using an estimate of the probability distribution of the unknown, one may estimate the probability distribution of the modeling error and incorporate it into the inversion. We introduce an algorithm which iterates this idea to update the distribution of the model error, leading to a sequence of posterior distributions that are demonstrated empirically to capture the underlying truth with increasing accuracy. Since the algorithm is not based on rejections, it requires only limited full model evaluations. We show analytically that, in the linear Gaussian case, the algorithm converges geometrically fast with respect to the number of iterations when the data is finite dimensional. For more general models, we introduce particle approximations of the iteratively generated sequence of distributions; we also prove that each element of the sequence converges in the large particle limit under a simplifying assumption. We show numerically that, as in the linear case, rapid convergence occurs with respect to the number of iterations. Additionally, we show through computed examples that point estimates obtained from this iterative algorithm are superior to those obtained by neglecting the model error.

Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements

Abstract: An inverse problem to recover a space-dependent factor of a source term and an order of a time derivative in a fractional diffusion equation from final data is considered. The uniqueness and stability of the solution to this problem is proved. A direct method to regularize the problem is proposed.

Inverse random source scattering for the Helmholtz equation in inhomogeneous media

Abstract: This paper is concerned with an inverse random source scattering problem in an inhomogeneous background medium. The wave propagation is modeled by the stochastic Helmholtz equation with the source driven by additive white noise. The goal is to reconstruct the statistical properties of the random source such as the mean and variance from the boundary measurement of the radiated random wave field at multiple frequencies. Both the direct and inverse problems are considered. We show that the direct problem has a unique mild solution by a constructive proof. For the inverse problem, we derive Fredholm integral equations, which connect the boundary measurement of the radiated wave field with the unknown source function. A regularized block Kaczmarz method is developed to solve the ill-posed integral equations. Numerical experiments are included to demonstrate the effectiveness of the proposed method.

Uniqueness results on phaseless inverse acoustic scattering with a reference ball

Abstract: This paper is devoted to the uniqueness in inverse scattering problems for the Helmholtz equation with phaseless far-field data. Some novel techniques are developed to overcome the difficulty of translation invariance induced by a single incident plane wave. In this paper, based on adding a reference ball as an extra artificial impenetrable obstacle (resp. penetrable homogeneous medium) to the inverse obstacle (resp. medium) scattering system and then using superpositions of a plane wave and a fixed point source as the incident waves, we rigorously prove that the location and shape of the obstacle as well as its boundary condition or the refractive index can be uniquely determined by the modulus of far-field patterns. The reference ball technique in conjunction with the superposition of incident waves brings in several salient benefits. First, the framework of our theoretical analysis can be applied to both the inverse obstacle and medium scattering problems. Second, for inverse obstacle scattering, the underlying boundary condition could be of a general type and be uniquely determined. Third, only a single frequency is needed. Finally, it provides a very simple proof of the uniqueness.

Parameterizations for ensemble Kalman inversion

Abstract: The use of ensemble methods to solve inverse problems is attractive because it is a derivative-free methodology which is also well-adapted to parallelization. In its basic iterative form the method produces an ensemble of solutions which lie in the linear span of the initial ensemble. Choice of the parameterization of the unknown field is thus a key component of the success of the method. We demonstrate how both geometric ideas and hierarchical ideas can be used to design effective parameterizations for a number of applied inverse problems arising in electrical impedance tomography, groundwater flow and source inversion. In particular we show how geometric ideas, including the level set method, can be used to reconstruct piecewise continuous fields, and we show how hierarchical methods can be used to learn key parameters in continuous fields, such as length-scales, resulting in improved reconstructions. Geometric and hierarchical ideas are combined in the level set method to find piecewise constant reconstructions with interfaces of unknown topology.

Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales

Abstract: We study Tikhonov regularization for ill-posed non-linear operator equations in Hilbert scales. Our focus is on the interplay between the smoothness-promoting properties of the penalty and the smoothness inherent in the solution. The objective is to study the situation when the unknown solution fails to have a finite penalty value, hence when the penalty is oversmoothing. By now this case was only studied for linear operator equations in Hilbert scales. We extend those results to certain classes of non-linear problems. The main result asserts that, under appropriate assumptions, order optimal reconstruction is still possible. In an appendix we highlight that the non-linearity assumption underlying the present analysis is met for specific applications.

Mathematical models for magnetic particle imaging

Abstract: Magnetic particle imaging (MPI) is a relatively new imaging modality. The nonlinear magnetization behavior of nanoparticles in an applied magnetic field is employed to reconstruct an image of the concentration of nanoparticles. Finding a sufficiently accurate model for the particle behavior is still an open problem. For this reason the reconstruction is still computed using a measured forward operator which is obtained in a time-consuming calibration process. The state of the art model used for the imaging methodology and first model-based reconstructions relies on strong model simplifications which turned out to cause too large modeling errors. Neglecting particle-particle interactions, the forward operator can be expressed by a Fredholm integral operator of the first kind describing the inverse problem. In this article we give an overview of relevant mathematical models which have not been investigated theoretically in the context of inverse problems yet. We consider deterministic models which are based on the physical behavior including relaxation mechanisms affecting the particle magnetization. The behavior of the models is illustrated with numerical simulations for monodisperse as well as polydisperse tracer. We further motivate linear and nonlinear problems beyond the solely concentration reconstruction related to applications. This model survey complements a recent topical review on MPI [30] and builds the basis for upcoming theoretical as well as empirical investigations.

Spatial-spectral cube matching frame for spectral CT reconstruction

Abstract: Spectral computed tomography (CT) reconstructs the same scanned object from projections of multiple narrow energy windows, and it can be used for material identification and decomposition. However, the multi-energy projection dataset has a lower signal-noise-ratio (SNR), resulting in poor reconstructed image quality. To address this thorny problem, we develop a spectral CT reconstruction method, namely spatial-spectral cube matching frame (SSCMF). This method is inspired by the following three facts: i) human body usually consists of two or three basic materials implying that the reconstructed spectral images have a strong sparsity; ii) the same basic material component in a single channel image has similar intensity and structures in local regions. Different material components within the same energy channel share similar structural information; iii) multi-energy projection datasets are collected from the subject by using different narrow energy windows, which means images reconstructed from different energy-channels share similar structures. To explore those information, we first establish a tensor cube matching frame (CMF) for a BM4D denoising procedure. Then, as a new regularizer, the CMF is introduced into a basic spectral CT reconstruction model, generating the SSCMF method. Because the SSCMF model contains an L0-norm minimization of 4D transform coefficients, an effective strategy is employed for optimization. Both numerical simulations and realistic preclinical mouse studies are performed. The results show that the SSCMF method outperforms the state-of-the-art algorithms, including the simultaneous algebraic reconstruction technique, total variation minimization, total variation plus low rank, and tensor dictionary learning.

Solving ill-posed control problems by stabilized finite element methods : an alternative to Tikhonov regularization

Abstract: Tikhonov regularization is one of the most commonly used methods for the regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control pr ...

Tikhonov regularization in Hilbert scales under conditional stability assumptions

Abstract: Conditional stability estimates allow us to characterize the degree of ill-posedness of many inverse problems, but without further assumptions they are not sufficient for the stable solution in the presence of data perturbations. We here consider the stable solution of nonlinear inverse problems satisfying a conditional stability estimate by Tikhonov regularization in Hilbert scales. Order optimal convergence rates are established for a priori and a posteriori parameter choice strategies. The role of a hidden source condition is investigated and the relation to previous results for regularization in Hilbert scales is elaborated. The applicability of the results is discussed for some model problems, and the theoretical results are illustrated by numerical tests.

Joint reconstruction via coupled Bregman iterations with applications to PET-MR imaging

Abstract: Joint reconstruction has recently attracted a lot of attention, especially in the field of medical multi-modality imaging such as PET-MRI. Most of the developed methods rely on the comparison of image gradients, or more precisely their location, direction and magnitude, to make use of structural similarities between the images. A challenge and still an open issue for most of the methods is to handle images in entirely different scales, i.e. different magnitudes of gradients that cannot be dealt with by a global scaling of the data. We propose the use of generalized Bregman distances and infimal convolutions thereof with regard to the well-known total variation functional. The use of a total variation subgradient respectively the involved vector field rather than an image gradient naturally excludes the magnitudes of gradients, which in particular solves the scaling behavior. Additionally, the presented method features a weighting that allows to control the amount of interaction between channels. We give insights into the general behavior of the method, before we further tailor it to a particular application, namely PET-MRI joint reconstruction. To do so, we compute joint reconstruction results from blurry Poisson data for PET and undersampled Fourier data from MRI and show that we can gain a mutual benefit for both modalities. In particular, the results are superior to the respective separate reconstructions and other joint reconstruction methods.

Retrieval of acoustic sources from multi-frequency phaseless data

Abstract: This paper is concerned with the inverse source problem of reconstructing an unknown acoustic excitation from phaseless measurements of the radiated fields away at multiple frequencies. It is well known that the non-uniqueness issue is a major challenge associated with such an inverse problem. We develop a novel strategy to overcome this challenging problem by recovering the radiated fields via adding some reference point sources as extra artificial sources to the inverse source system. This novel reference source technique requires only a few extra data, and brings in a simple phase retrieval formula. The stability of this phase retrieval approach is rigorously analyzed. After the reacquisition of the phase information, the multi-frequency inverse source problem with recovered phase information is solved by the Fourier method, which is non-iterative, fast and easy to implement. Several numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed method.

A bilevel approach for parameter learning in inverse problems

Abstract: A learning approach for selecting regularization parameters in multi-penalty Tikhonov regularization is investigated. It leads to a bilevel optimization problem, where the lower level problem is a Tikhonov regularized problem parameterized in the regularization parameters. Conditions which ensure the existence of solutions to the bilevel optimization problem are derived, and these conditions are verified for two relevant examples. Difficulties arising from the possible lack of convexity of the lower level problems are discussed. Optimality conditions are given provided that a reasonable constraint qualification holds. Finally, results from numerical experiments used to test the developed theory are presented.

Multicompartment Magnetic Resonance Fingerprinting.

Abstract: Magnetic resonance fingerprinting (MRF) is a technique for quantitative estimation of spin- relaxation parameters from magnetic-resonance data. Most current MRF approaches assume that only one tissue is present in each voxel, which neglects intravoxel structure, and may lead to artifacts in the recovered parameter maps at boundaries between tissues. In this work, we propose a multicompartment MRF model that accounts for the presence of multiple tissues per voxel. The model is fit to the data by iteratively solving a sparse linear inverse problem at each voxel, in order to express the measured magnetization signal as a linear combination of a few elements in a precomputed fingerprint dictionary. Thresholding-based methods commonly used for sparse recovery and compressed sensing do not perform well in this setting due to the high local coherence of the dictionary. Instead, we solve this challenging sparse-recovery problem by applying reweighted-𝓁1-norm regularization, implemented using an efficient interior-point method. The proposed approach is validated with simulated data at different noise levels and undersampling factors, as well as with a controlled phantom-imaging experiment on a clinical magnetic-resonance system.

3D Compton scattering imaging and contour reconstruction for a class of Radon transforms

Abstract: Compton scattering imaging is a nascent concept arising from the current development of high-sensitive energy detectors and is devoted to exploit the scattering radiation to image the electron density of the studied medium. Such detectors are able to collect incoming photons in terms of energy. This paper introduces potential 3D modalities in Compton scattering imaging (CSI). The associated measured data are modeled using a class of generalized Radon transforms. The study of this class of operators leads to build a filtered back-projection kind algorithm preserving the contours of the sought-for function and offering a fast approach to partially solve the associated inverse problems. Simulation results including Poisson noise demonstrate the potential of this new imaging concept as well as the proposed image reconstruction approach.

Deconvolving the input to random abstract parabolic systems: a population model-based approach to estimating blood/breath alcohol concentration from transdermal alcohol biosensor data*

Abstract: The distribution of random parameters in, and the input signal to, a distributed parameter model with unbounded input and output operators for the transdermal transport of ethanol are estimated. The model takes the form of a diffusion equation with the input, which is on the boundary of the domain, being the blood or breath alcohol concentration (BAC/BrAC), and the output, also on the boundary, being the transdermal alcohol concentration (TAC). Our approach is based on the reformulation of the underlying dynamical system in such a way that the random parameters are treated as additional spatial variables. When the distribution to be estimated is assumed to be defined in terms of a joint density, estimating the distribution is equivalent to estimating a functional diffusivity in a multi-dimensional diffusion equation. The resulting system is referred to as a population model, and well-established finite dimensional approximation schemes, functional analytic based convergence arguments, optimization techniques, and computational methods can be used to fit it to population data and to analyze the resulting fit. Once the forward population model has been identified or trained based on a sample from the population, the resulting distribution can then be used to deconvolve the BAC/BrAC input signal from the biosensor observed TAC output signal formulated as either a quadratic programming or linear quadratic tracking problem. In addition, our approach allows for the direct computation of corresponding credible bands without simulation. We use our technique to estimate bivariate normal distributions and deconvolve BAC/BrAC from TAC based on data from a population that consists of multiple drinking episodes from a single subject and a population consisting of single drinking episodes from multiple subjects.

The method of fundamental solutions for computing acoustic interior transmission eigenvalues

Abstract: We analyze the method of fundamental solutions (MFS) in two different versions with focus on the computation of approximate acoustic interior transmission eigenvalues in 2D for homogeneous media. Our approach is mesh- and integration free, but suffers in general from the ill-conditioning effects of the discretized eigenoperator, which we could then successfully balance using an approved stabilization scheme. Our numerical examples cover many of the common scattering objects and prove to be very competitive in accuracy with the standard methods for PDE-related eigenvalue problems. We finally give an approximation analysis for our framework and provide error estimates, which bound interior transmission eigenvalue deviations in terms of some generalized MFS output.

Fast imaging of scattering obstacles from phaseless far-field measurements at a fixed frequency

Abstract: This paper is concerned with the inverse obstacle scattering problem with phaseless far-field data at a fixed frequency. The main difficulty of this problem is the so-called translation invariance property of the modulus of the far-field pattern or the phaseless far-field pattern generated by one plane wave as the incident field, which means that the location of the obstacle cannot be recovered from such phaseless far-field data at a fixed frequency. It was recently proved in our previous work Xu et al 2018 (SIAM J. Appl. Math. 78 1737–53) that the obstacle can be uniquely determined by the phaseless far-field patterns generated by infinitely many sets of superpositions of two plane waves with different directions at a fixed frequency if the obstacle is a priori known to be a sound-soft or an impedance obstacle with real-valued impedance function. The purpose of this paper is to develop a direct imaging algorithm to reconstruct the location and shape of the obstacle from the phaseless far-field data corresponding to infinitely many sets of superpositions of two plane waves with a fixed frequency as the incident fields. Our imaging algorithm only involves the calculation of the products of the measurement data with two exponential functions at each sampling point and is thus fast and easy to implement. Further, the proposed imaging algorithm does not need to know the type of boundary conditions on the obstacle in advance and is capable to reconstruct multiple obstacles with different boundary conditions. Numerical experiments are also carried out to illustrate that our imaging method is stable, accurate and robust to noise.

Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems

Abstract: We consider the inverse problem of recovering an unknown functional parameter u in a separable Banach space, from a noisy observation y of its image through a known possibly non-linear ill-posed map G. The data y is finite-dimensional and the noise is Gaussian. We adopt a Bayesian approach to the problem and consider Besov space priors (see Lassas et al. 2009), which are well-known for their edge-preserving and sparsity-promoting properties and have recently attracted wide attention especially in the medical imaging community. Our key result is to show that in this non-parametric setup the maximum a posteriori (MAP) estimates are characterized by the minimizers of a generalized Onsager--Machlup functional of the posterior. This is done independently for the so-called weak and strong MAP estimates, which as we show coincide in our context. In addition, we prove a form of weak consistency for the MAP estimators in the infinitely informative data limit. Our results are remarkable for two reasons: first, the prior distribution is non-Gaussian and does not meet the smoothness conditions required in previous research on non-parametric MAP estimates. Second, the result analytically justifies existing uses of the MAP estimate in finite but high dimensional discretizations of Bayesian inverse problems with the considered Besov priors.

Extended sampling method in inverse scattering

Abstract: A new sampling method for inverse scattering problems is proposed to process far field data of one incident wave. The method sets up ill-posed integral equations and uses the (approximate) solutions to reconstruct the target. In contrast to the classical linear sampling method, the kernels of the associated integral operators are the far field patterns of sound-soft balls. The measured data is moved to right hand sides of the equations, which gives the method the ability to process limited aperture data. Furthermore, a multilevel technique is employed to improve the reconstruction. Numerical examples show that the method can effectively determine the location and approximate the support with little a priori information of the unknown target.