Ill-posed Nonlinear Inverse Problem Research Articles

Oracle-Net for Nonlinear Compressed Sensing in Electrical Impedance Tomography Reconstruction Problems

Sparse recovery principles play an important role in solving many nonlinear ill-posed inverse problems. We investigate a variational framework with learned support estimation for compressed sensing sparse reconstructions, where the available measurements are nonlinear and possibly corrupted by noise. A graph neural network, named Oracle-Net, is proposed to predict the support from the nonlinear measurements and is integrated into a regularized recovery model to enforce sparsity. The derived nonsmooth optimization problem is then efficiently solved through a constrained proximal gradient method. Error bounds on the approximate solution of the proposed Oracle-based optimization are provided in the context of the ill-posed Electrical Impedance Tomography problem (EIT). Numerical solutions of the EIT nonlinear inverse reconstruction problem confirm the potential of the proposed method which improves the reconstruction quality from undersampled measurements, under sparsity assumptions.

Bouligand–Newton type methods for non-smooth ill-posed problems

Abstract We consider Newton-type methods for solving nonlinear ill-posed inverse problems in Hilbert spaces where the forward operators are not necessarily G\^{a}teaux differentiable. Modifications are proposed with the non-existing Fr'{e}chet derivatives replaced by a family of bounded linear operators satisfying suitable properties. These bounded linear operators can be constructed by the Bouligand subderivatives which are defined as limits of Fr'{e}chet derivatives of the forward operator in differentiable points. The Bouligand subderivative mapping in general is not continuous unless the forward operator is G\^{a}teaux differentiable which
introduces challenges for convergence analysis of the corresponding Bouligand-Newton type methods. In this paper we will show that, under the discrepancy principle, these Bouligand-Newton type methods are iterative regularization methods of optimal order. Numerical results for an inverse problem arising from a non-smooth semi-linear elliptic equation are presented to test the performance of the methods.

An accelerated inexact Newton regularization scheme with a learned feature-selection rule for non-linear inverse problems

With computational inverse problems, it is desirable to develop an efficient inversion algorithm to find a solution from measurement data through a mathematical model connecting the unknown solution and measurable quantity based on the first principles. However, most of mathematical models represent only a few aspects of the physical quantity of interest, and some of them are even incomplete in the sense that one measurement corresponds to many solutions satisfying the forward model. In this paper, in light of the recently developed iNETT method in (2023 Inverse Problems 39 055002), we propose a novel iterative regularization method for efficiently solving non-linear ill-posed inverse problems with potentially non-injective forward mappings and (locally) non-stable inversion mappings. Our approach integrates the inexact Newton iteration, the non-stationary iterated Tikhonov regularization, the two-point gradient acceleration method, and the structure-free feature-selection rule. The main difficulty in the regularization technique is how to design an appropriate regularization penalty, capturing the key feature of the unknown solution. To overcome this difficulty, we replace the traditional regularization penalty with a deep neural network, which is structure-free and can identify the correct solution in a huge null space. A comprehensive convergence analysis of the proposed algorithm is performed under standard assumptions of regularization theory. Numerical experiments with comparisons with other state-of-the-art methods for two model problems are presented to show the efficiency of the proposed approach.

Error estimates for simplified Levenberg–Marquardt method for nonlinear ill-posed operator equations in Hilbert Spaces

Abstract In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 𝒟 ⁢ ( F ) ⊂ 𝖴 → 𝖸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 𝖴 {\mathsf{U}} and 𝖸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 𝒟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions.

[formula omitted] sparsity regularization for nonlinear ill-posed problems

In this paper, the α‖⋅‖ℓ1−β‖⋅‖ℓ2 sparsity regularization with parameters α≥β≥0 is studied for nonlinear ill-posed inverse problems. The well-posedness of the regularization is investigated. Compared to the case where α>β≥0, the results for the case α=β>0 are weaker due to the lack of coercivity and Radon-Riesz property of the regularization term. Under certain conditions on the nonlinearity of F, sparsity is shown for every minimizer of the α‖⋅‖ℓ1−β‖⋅‖ℓ2 regularized inverse problem. Moreover, for the case α>β≥0, convergence rates O(δ12) and O(δ) are proved for the regularized solution toward a sparse exact solution, under different yet commonly adopted conditions on the nonlinearity of F. The iterative soft thresholding algorithm is shown to be useful to solve the α‖⋅‖ℓ1−β‖⋅‖ℓ2 regularized problem for nonlinear ill-posed equations. Numerical results illustrate the efficiency of the proposed method.

Application of the Two-sided difference quotient in the solution of nonlinear Ill-posed inverse self-adjoint elliptic problem

When we use a discretization by finite differences, to solve differential equations we find problems at the border of the domain of the solution. If the solution is also immersed in a ill-posed inverse problem; we can find very bad solutions. In this paper we apply a discretization of two - sided difference quotients method to solve Ill-posed inverse self-adjoint elliptic problem [Kirsch(2011)]. Some numerical examples showing the effectiveness of this method and we will use mollification techniques to smooth the solutions.

Information bottleneck in peptide conformation determination by x-ray absorption spectroscopy

We apply a recently developed technique utilizing machine learning for statistical analysis of computational nitrogen K-edge spectra of aqueous triglycine. This method, the emulator-based component analysis, identifies spectrally relevant structural degrees of freedom from a data set filtering irrelevant ones out. Thus tremendous reduction in the dimensionality of the ill-posed nonlinear inverse problem of spectrum interpretation is achieved. Structural and spectral variation across the sampled phase space is notable. Using these data, we train a neural network to predict the intensities of spectral regions of interest from the structure. These regions are defined by the temperature-difference profile of the simulated spectra, and the analysis yields a structural interpretation for their behavior. Even though the utilized local many-body tensor representation implicitly encodes the secondary structure of the peptide, our approach proves that this information is irrecoverable from the spectra. A hard x-ray Raman scattering experiment confirms the overall sensibility of the simulated spectra, but the predicted temperature-dependent effects therein remain beyond the achieved statistical confidence level.

Greybox: A hybrid algorithm for direct estimation of tracer kinetic parameters from undersampled DCE-MRI data.

A variety of deep learning-based and iterative approaches are available to predict Tracer Kinetic (TK) parameters from fully sampled or undersampled dynamic contrast-enhanced (DCE) MRI data. However, both the methods offer distinct benefits anddrawbacks. To propose a hybrid algorithm (named as 'Greybox'), using both model- as well as DL-based, for solving a multi-parametric non-linear inverse problem of directly estimating TK parameters from undersampled DCE MRI data, which is invariant to undersamplingrate. The proposed algorithm was inspired by plug-and-play algorithms used for solving linear inverse imaging problems. This technique was tested for its effectiveness in solving the nonlinear ill-posed inverse problem of generating 3D TK parameter maps from four-dimensional (4D; Spatial + Temporal) retrospectively undersampled k-space data. The algorithm learns a deep learning-based prior using UNET to estimate the and parameters based on the Patlak pharmacokinetic model, and this trained prior was utilized to estimate the TK parameter maps using an iterative gradient-based optimization scheme. Unlike the existing DL models, this network is invariant to the undersampling rate of the input data. The proposed method was compared with the total variation-based direct reconstruction technique on brain, breast, and prostate DCE-MRI datasets for various undersampling rates using the Radial Golden Angle (RGA) scheme. For the breast dataset, an indirect estimation using the Fast Composite Splitting algorithm was utilized for comparison. Undersampling rates of 8 , 12 and 20 were used for the experiments, and the results were compared using the PSNR and SSIM as metrics. For the breast dataset of 10 patients, data from four patients were utilized for training (1032 samples), two for validation (752 samples), and the entire volume of four patients for testing. Similarly, for the prostate dataset of 18 patients, 10 patients were utilized for training (720 samples), five for validation (216 samples), and the whole volume of three patients for testing. For the brain dataset of nineteen patients, ten patients were used for training (3152 samples), five for validation (1168 samples), and the whole volume of four patients for testing. Statistical tests were also conducted to assess the significance of the improvement inperformance. The experiments showed that the proposed Greybox performs significantly better than other direct reconstruction methods. The proposed algorithm improved the estimated and in terms of the peak signal-to-noise ratio by up to 3 dB compared to other standard reconstructionmethods. The proposed hybrid reconstruction algorithm, Greybox, can provide state-of-the-art performance in solving the nonlinear inverse problem of DCE-MRI. This is also the first of its kind to utilize convolutional neural network-based encodings as part of the plug-and-play priors to improve the performance of the reconstructionalgorithm.

Sparsity promoting reconstructions via hierarchical prior models in diffuse optical tomography

Diffuse optical tomography (DOT) is a severely ill-posed nonlinear inverse problem that seeks to estimate optical parameters from boundary measurements. In the Bayesian framework, the ill-posedness is diminished by incorporating a priori information of the optical parameters via the prior distribution. In case the target is sparse or sharp-edged, the common choice as the prior model are non-differentiable total variation and $ \ell^1 $ priors. Alternatively, one can hierarchically extend the variances of a Gaussian prior to obtain differentiable sparsity promoting priors. By doing this, the variances are treated as unknowns allowing the estimation to locate the discontinuities.In this work, we formulate hierarchical prior models for the nonlinear DOT inverse problem using exponential, standard gamma and inverse-gamma hyperpriors. Depending on the hyperprior and the hyperparameters, the hierarchical models promote different levels of sparsity and smoothness. To compute the MAP estimates, the previously proposed alternating algorithm is adapted to work with the nonlinear model. We then propose an approach based on the cumulative distribution function of the hyperpriors to select the hyperparameters. We evaluate the performance of the hyperpriors with numerical simulations and show that the hierarchical models can improve the localization, contrast and edge sharpness of the reconstructions.

Nonlinear ill-posed problem in low-dose dental cone-beam computed tomography

Abstract This paper describes the mathematical structure of the ill-posed nonlinear inverse problem of low-dose dental cone-beam computed tomography (CBCT) and explains the advantages of a deep learning-based approach to the reconstruction of computed tomography images over conventional regularization methods. This paper explains the underlying reasons why dental CBCT is more ill-posed than standard computed tomography. Despite this severe ill-posedness, the demand for dental CBCT systems is rapidly growing because of their cost competitiveness and low radiation dose. We then describe the limitations of existing methods in the accurate restoration of the morphological structures of teeth using dental CBCT data severely damaged by metal implants. We further discuss the usefulness of panoramic images generated from CBCT data for accurate tooth segmentation. We also discuss the possibility of utilizing radiation-free intra-oral scan data as prior information in CBCT image reconstruction to compensate for the damage to data caused by metal implants.

A high order discontinuous Galerkin method for the recovery of the conductivity in Electrical Impedance Tomography

In this work, we develop an efficient high order discontinuous Galerkin (DG) method for solving the Electrical Impedance Tomography (EIT). EIT is a highly nonlinear ill-posed inverse problem where the interior conductivity of an object is recovered from the surface measurements of voltage and current flux. We first propose a new optimization problem based on the recovery of the conductivity from the Dirichlet-to-Neumann map to minimize the mismatch between the predicted current and the measured current on the boundary. And we further prove the existence of the minimizer. Numerically the optimization problem is solved by a third order DG method with quadratic polynomials. Numerical results for several two-dimensional problems with both single and multiple inclusions are demonstrated to show the high accuracy and efficiency of the proposed high order DG method. Analysis and computation for discontinuous conductivities are also studied in this work.

Cure d’éventration périnéale par voie ouverte abdominale (avec vidéo)

In this work, we develop an efficient high order discontinuous Galerkin (DG) method for solving the Electrical Impedance Tomography (EIT). EIT is a highly nonlinear ill-posed inverse problem where the interior conductivity of an object is recovered from the surface measurements of voltage and current flux. We first propose a new optimization problem based on the recovery of the conductivity from the Dirichlet-to-Neumann map to minimize the mismatch between the predicted current and the measured current on the boundary. And we further prove the existence of the minimizer. Numerically the optimization problem is solved by a third order DG method with quadratic polynomials. Numerical results for several two-dimensional problems with both single and multiple inclusions are demonstrated to show the high accuracy and efficiency of the proposed high order DG method. Analysis and computation for discontinuous conductivities are also studied in this work.

Generalized Inexact Newton-Landweber Iteration for Possibly Non-Smooth Inverse Problems in Banach Spaces

In this paper, we consider a generalized inexact Newton-Landweber iteration to solve nonlinear ill-posed inverse problems in Banach spaces, where the forward operator might not be Gâteaux differentiable. The method is designed with non-smooth convex penalty terms, including L1-like and total variation-like penalty functionals, to capture special features of solutions such as sparsity and piecewise constancy. Furthermore, the inaccurate inner solver is incorporated into the minimization problem in each iteration step. Under some assumptions, based on ε-subdifferential, we establish the convergence analysis of the proposed method. Finally, some numerical simulations are provided to illustrate the effectiveness of the method for solving both smooth and non-smooth nonlinear inverse problems.

Microwave Near-Field Dynamical Tomography of Thorax at Pulmonary and Cardiovascular Activity.

The developed near-field microwave diagnostics of dynamical lung tomography provide information about variations of air and blood content depth structure in the processes of breathing and heart beating that are unattainable for other available methods. The method of dynamical pulse 1D tomography (profiling) is based on solving the corresponding nonlinear ill-posed inverse problem in the extremely complicated case of the strongly absorbing frequency-dispersive layered medium with the dual regularization method-a new Lagrange approach in the theory of ill-posed problems. This method has been realized experimentally by data of bistatic measurements with two electrically small bow-tie antennas that provide a subwavelength resolution. The proposed methods of 3D lung tomography based on the multisensory pulse, multifrequency, or multi-base measurements are based on solving the corresponding integral equations in the Born approximation. The experimental 3D tomography of lung air content was obtained by the results of the multiple 1D pulse profiling by pulse measurements in several grid points over the planar square region of the thorax. Additionally, the possible applicability of multifrequency measurements of scattered harmonic signals in the monitoring of lungs was demonstrated by four-frequency measurements in the process of breathing. The results demonstrated the feasibility of the proposed control in the diagnosis of some lung diseases.

Deep-plug-and-play proximal Gauss-Newton method with applications to nonlinear, ill-posed inverse problems

In this paper we propose a proximal Gauss-Newton method for the penalized nonlinear least squares optimization problem arising from regularization of ill-posed nonlinear inverse problems. By exploiting the modular structure that characterizes the proximal-type methods, we plug in a pre-trained graph neural net denoiser in place of the standard proximal map. This allows to mould the prior on the data. An encoder-decoder Graph U-Net architecture is proposed as denoiser, which works on unstructured data; its mathematical formulation is derived to analyse the Liptschitz condition. With the intent of showing the benefits of applying deep Plug-and-Play reconstructions, we consider as an exemplar application, the nonlinear Electrical Impedance Tomography, a promising non-invasive imaging technique mathematically formulated as a highly nonlinear ill-posed inverse problem.

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.

SAR-CGAN: Improved generative adversarial network for EIT reconstruction of lung diseases

The image reconstruction of electrical impedance tomography (EIT) is a nonlinear ill-posed inverse problem, and the reconstructed images tend to have artifacts due to noise in the measurement system. The shallow neural networks are difficult to fully express the nonlinear relationship of the EIT imaging problems. Deep neural networks have demonstrated great potential to remove artifacts from initial conductivity images caused by system noise. In the paper, a novel deep learning method, namely, conditional generative adversarial network with soft-attention gates (AGs) and residual learning (SAR-CGAN), is proposed to solve the above problems of EIT imaging. SAR-CGAN consists of generator and discriminator. Specifically, the AGs are employed by the generator to improve sensitivity and accuracy for lung EIT imaging by suppressing feature activation in irrelevant EIT region. And residual learning framework is introduced in the discriminator to address the problem of model degradation and easily enjoy accuracy gains from greatly increased depth. Extensive confirmatory and comparative experiments with three kinds of lung disease are conducted and compared with CNN, DnCNN, Attention U-Net and CGAN methods. The imaging results indicate that SAR-CGAN method can accurately recover the position and boundaries of lesions in lung images and effective reconstruction can be accomplished for the new conductivity distribution and noisy samples.

Multi-exponential model to describe pressure-dependent P- and S-wave velocities and its use to estimate the crack aspect ratio

We present new quantitative model describing the pressure dependence of acoustic P- and S-wave velocities. Assuming that a variety of individual mechanisms or defects (such as cracks, pore collapse and grain crushing) can contribute to the pressure-dependent change of the wave velocity, we order a characteristic pressure to all of them and allow a series of exponential terms in the description of the (P- and S-waves) velocity-pressure function. We estimate the parameters of the multi-exponential rock physical model in inversion procedures using laboratory measured P- and S-wave velocity data. As is known, the conventional damped least squares method gives acceptable results only when one or two individual mechanisms are assumed. Increasing the number of exponential terms leads to highly nonlinear ill-posed inverse problem. Due to this reason, we develop the spectral inversion method (SIM) in which the velocity amplitudes (the spectral lines in the characteristic pressure spectrum) are only considered as unknowns. The characteristic pressures (belonging to the velocity amplitudes) are excluded from the set of inversion unknowns, instead, they are defined in a set of fixed positions equidistantly distributed in the actual interval of the independent variable (pressure). Through this novel linear inversion method, we estimate the parameters of the multi-exponential rock physical model using laboratory measured P- and S-wave velocity data. The characteristic pressures are related to the closing pressures of cracks which are described by well-known rock mechanical relationships depending on the aspect ratio of elliptical cracks. This gives the possibility to estimate the aspect ratios in terms of the characteristic pressures.

Porosity reconstruction based on Biot elastic model of porous media by homotopy perturbation method

Fluid-saturated porous media are two-phase media, which are composed of solid and liquid phases. Biot theory for fluid-saturated porous media holds that underground media are composed of porous elastic solid and compressible viscous fluid filled with pore space. Compared with the single-phase media theory, the fluid-saturated porous media theory can describe the subsurface media more precisely, and the elastic wave equations in the fluid-saturated porous media contain more parameters used to describe the formation properties. Therefore, fluid-saturated porous media theory is widely used in geophysical exploration, seismic engineering, and other fields.This paper considers the porosity reconstruction problem of the elastic wave equations in the fluid-saturated porous media based on Biot theory, which is a typically nonlinear ill-posed inverse problem from mathematical viewpoint. The proposed method is a novel iteration regularization scheme based on the homotopy perturbation technique. To verify the validity and applicability, numerical experiments of two-dimensional and three-dimensional porosity models have been carried out. Numerical results illustrate that this method can overcome the numerical instability and are robust to data noise in the reconstruction procedure. Furthermore, compared with the classical regularized Gauss-Newton method, the homotopy perturbation method greatly widens the convergence region while keeping the fast convergence rate.

Mixed scale dense convolutional networks for x-ray phase contrast imaging

X-ray in-line phase contrast imaging relies on the measurement of Fresnel diffraction intensity patterns due to the phase shift and the attenuation induced by the object. The recovery of phase and attenuation from one or several diffraction patterns is a nonlinear ill-posed inverse problem. In this work, we propose supervised learning approaches using mixed scale dense (MS-D) convolutional neural networks to simultaneously retrieve the phase and the attenuation from x-ray phase contrast images. This network architecture uses dilated convolutions to capture features at different image scales and densely connects all feature maps. The long range information in images becomes quickly available, and greater receptive field size can be obtained without losing resolution. This network architecture seems to account for the effect of the Fresnel operator very efficiently. We train the networks using simulated data of objects consisting of either homogeneous components, characterized by a fixed ratio of the induced refractive phase shifts and attenuation, or heterogeneous components, consisting of various materials. We also train the networks in the image domain by applying a simple initial reconstruction using the adjoint of the Fréchet derivative. We compare the results obtained with the MS-D network to reconstructions using U-Net, another popular network architecture, as well as to reconstructions using the contrast transfer function method, a direct phase and attenuation retrieval method based on linearization of the direct problem. The networks are evaluated using simulated noisy data as well as images acquired at NanoMAX (MAX IV, Lund, Sweden). In all cases, large improvements of the reconstruction errors are obtained on simulated data compared to the linearized method. Moreover, on experimental data, the networks improve the reconstruction quantitatively, improving the low-frequency behavior and the resolution.