Ill-posed Inverse Problem Research Articles

Semantic regularization of electromagnetic inverse problems

Solving ill-posed inverse problems typically requires regularization based on prior knowledge. To date, only prior knowledge that is formulated mathematically (e.g., sparsity of the unknown) or implicitly learned from quantitative data can be used for regularization. Thereby, semantically formulated prior knowledge derived from human reasoning and recognition is excluded. Here, we introduce and demonstrate the concept of semantic regularization based on a pre-trained large language model to overcome this vexing limitation. We study the approach, first, numerically in a prototypical 2D inverse scattering problem, and, second, experimentally in 3D and 4D compressive microwave imaging problems based on programmable metasurfaces. We highlight that semantic regularization enables new forms of highly-sought privacy protection for applications like smart homes, touchless human-machine interaction and security screening: selected subjects in the scene can be concealed, or their actions and postures can be altered in the reconstruction by manipulating the semantic prior with suitable language-based control commands.

DETERMINATION OF HELIUM PERMEABILITY OF SILICA MICROSPHERES

This work is devoted to the study of helium permeability of hollow glass silica microspheres. To conduct the study, an experimental stand was prepared to obtain kinetic sorption curves for helium at given values of pressure and temperature. In the course of coordinating experiments and analyzing the helium sorption curves of silica microspheres in the temperature range 21:5–110:0 C, permeability parameters were obtained using a mathematical model: permeability and specific sorption volume. A new method is proposed for determining the distribution of microspheres according to characteristic permeabilities based on solving the inverse ill-posed problem for the integral equation. The obtained values are in good agreement with the results obtained using a previously developed mathematical model.

Electromagnetic Source Imaging via a Data-Synthesis-Based Convolutional Encoder-Decoder Network.

Electromagnetic source imaging (ESI) requires solving a highly ill-posed inverse problem. To seek a unique solution, traditional ESI methods impose various forms of priors that may not accurately reflect the actual source properties, which may hinder their broad applications. To overcome this limitation, in this article, a novel data-synthesized spatiotemporally convolutional encoder-decoder network (DST-CedNet) method is proposed for ESI. The DST-CedNet recasts ESI as a machine learning problem, where discriminative learning and latent-space representations are integrated in a CedNet to learn a robust mapping from the measured electroencephalography/magnetoencephalography (E/MEG) signals to the brain activity. In particular, by incorporating prior knowledge regarding dynamical brain activities, a novel data synthesis strategy is devised to generate large-scale samples for effectively training CedNet. This stands in contrast to traditional ESI methods where the prior information is often enforced via constraints primarily aimed for mathematical convenience. Extensive numerical experiments as well as analysis of a real MEG and epilepsy EEG dataset demonstrate that the DST-CedNet outperforms several state-of-the-art ESI methods in robustly estimating source signals under a variety of source configurations.

Accelerated motion correction with deep generative diffusion models.

The aim of this work is to develop a method to solve the ill-posed inverse problem of accelerated image reconstruction while correcting forward model imperfections in the context of subject motion during MRI examinations. The proposed solution uses a Bayesian framework based on deep generative diffusion models to jointly estimate a motion-free image and rigid motion estimates from subsampled and motion-corrupt two-dimensional (2D) k-space data. We demonstrate the ability to reconstruct motion-free images from accelerated two-dimensional (2D) Cartesian and non-Cartesian scans without any external reference signal. We show that our method improves over existing correction techniques on both simulated and prospectively accelerated data. We propose a flexible framework for retrospective motion correction of accelerated MRI based on deep generative diffusion models, with potential application to other forward model corruptions.

The Application of Piecewise Regularization Reconstruction to the Calibration of Strain Beams.

Standard beams are mainly used for the calibration of strain sensors using their load reconstruction models. However, as an ill-posed inverse problem, the solution to these models often fails to converge, especially when dealing with dynamic loads of different frequencies. To overcome this problem, a piecewise Tikhonov regularization method (PTR) is proposed to reconstruct dynamic loads. The transfer function matrix is built both using the denoised excitations and the corresponding responses. After singular value decomposition (SVD), the singular values are divided into submatrices of different sizes by utilizing a piecewise function. The regularization parameters are solved by optimizing the piecewise submatrices. The experimental result shows that the MREs of the PTR method are 6.20% at 70 Hz and 5.86% at 80 Hz. The traditional Tikhonov regularization method based on GCV exhibits MREs of 28.44% and 29.61% at frequencies of 70 Hz and 80 Hz, respectively, whereas the L-curve-based approach demonstrates MREs of 29.98% and 18.42% at the same frequencies. Furthermore, the PREs of the PTR method are 3.54% at 70 Hz and 3.73% at 80 Hz. The traditional Tikhonov regularization method based on GCV exhibits PREs of 27.01% and 26.88% at frequencies of 70 Hz and 80 Hz, respectively, whereas the L-curve-based approach demonstrates PREs of 29.50% and 15.56% at the same frequencies. All in all, the method proposed in this paper can be extensively applied to load reconstruction across different frequencies.

An interactive method based on multi-objective optimization for limited-angle CT reconstruction

Objective. Limited-angle x-ray computed tomography (CT) is a typical ill-posed inverse problem, leading to artifacts in the reconstructed image due to the incomplete projection data. Most iteration CT reconstruction methods involve optimization for a single object. This paper explores a multi-objective optimization model and an interactive method based on multi-objective optimization to suppress the artifacts of limited-angle CT. Approach. The model includes two objective functions on the dual domain within the data consistency constraint. In the interactive method, the structural similarity index measure (SSIM) is regarded as the value function of the decision maker (DM) firstly. Secondly, the DM arranges the objective functions of the multi-objective optimization model to be optimized according to their absolute importance. Finally, the SSIM and the simulated annealing (SA) method help the DM choose the desirable reconstruction image by improving the SSIM value during the iteration process. Main results. Simulation and real data experiments demonstrate that the artifacts can be suppressed by the proposed method, and the results were superior to those reconstructed by the other three reconstruction methods in preserving the edge structure of the image. Significance. The proposed interactive method based on multi-objective optimization shows some potential advantages over classical single object optimization methods.

A TomoSAR regularization-based method for height change detection in urban areas

Over the past years, urban change detection has always been a challenging issue for researchers. Due to the two-dimensional intrinsic property of satellite imageries and the inherent less intuitive interpretability of radar images, height change detection and SAR imagery-based change detection have been even more complicated tasks. However, TomoSAR-based height change detection is a novel topic on which only a few research works have ever been realized. This study explores the viability of tomographic-based change detection in urban areas, tackling the common challenges encountered in 2D change detection with SAR images. In contrast with the standard TomoSAR methods that usually suffer from the ill-posed inverse problem, the TomoSAR regularization-based methods are considered powerful aids for reconstructing the reflectivity function of the observed scenes. The TomoSAR regularization-based method utilizes the Constrained Least Squares (CLS) and the Weighted Constrained Least Squares (WCLS) as two efficient strategies for coming up with a well-posed solution to the prevalent ill-posed TomoSAR problem. They can assist SAR tomography to deal with the possible impairing issues arising from low numbers and the distribution of baselines. This advantage helps enhance the accuracy of the results. The experimental results are investigated in terms of completeness and correctness measures to evaluate the feasibility and reliability of the proposed method. The proposed method for height change detection has been performed on two datasets acquired by the TerraSAR-X sensor in 2013 and the Sentinel-1 sensor in 2022 over Tehran City, Iran. The final results demonstrate that the proposed algorithm can detect height changes with good feasibility and reliability in terms of completeness and correctness of better than 84% and 77%, respectively.

Poisson image deblurring with frame-based nonconvex regularization

Poisson image deblurring, which aims to restore the latent image from its blurred and noisy observation, has drawn significant attention in image processing. Due to its ill-posed nature, enhancing image quality often involves incorporating a well-defined prior to effectively regularize the ill-posed inverse problem. Building upon the framelet system, we propose a frame-based nonconvex regularization method for Poisson image deblurring. The method is formulated by combining a data-fitting term with the difference of two norms, namely ℓ1 and ℓ2, on the latent image. We solve the optimization problem by combining the difference of convex functions algorithm (DCA) with the alternating direction method of multipliers (ADMM), establishing its convergence. We further employ a simulated annealing procedure and show that proposed algorithm almost certainly converges to a global minimum. Two different approaches are employed to handle the frame-based ℓ1 norm within the ADMM framework. In particular, the frame-based nonconvex regularization method is also considered for the blind Poisson problem. An effective recovery model and its algorithm are presented. Experimental results demonstrate the effectiveness of our proposed models compared with other models in terms of quantitative metrics and visual quality.

High-resolution acoustic imaging method based on equivalent source method and reweighted l1 minimization

Sparse regularization has been successfully applied to equivalent source method (ESM) in order to improve the acoustic imaging resolution. However, the application is not always feasible, especially at low frequencies. To overcome the problem, this paper proposes a high-resolution acoustic imaging method. In this method, reweighted l1 minimization is introduced to ESM to deal with the ill-posed inverse problems. Then the obtained equivalent source strengths are used to locate the sound sources. Compared to the sparse regularization-based ESM, the proposed method can provide a low side lobe and higher spatial resolution of acoustic imaging. Meanwhile, by arranging equivalent sources in three-dimensional space, the proposed method can also realize the acoustic imaging in three-dimensional sound field with high resolution. The results of the simulation and experiment demonstrate the validations.

A subject-specific unsupervised deep learning method for quantitative susceptibility mapping using implicit neural representation

Quantitative susceptibility mapping (QSM) is an MRI-based technique that estimates the underlying tissue magnetic susceptibility based on phase signal. Deep learning (DL)-based methods have shown promise in handling the challenging ill-posed inverse problem for QSM reconstruction. However, they require extensive paired training data that are typically unavailable and suffer from generalization problems. Recent model-incorporated DL approaches also overlook the non-local effect of the tissue phase in applying the source-to-field forward model due to patch-based training constraint, resulting in a discrepancy between the prediction and measurement and subsequently suboptimal QSM reconstruction. This study proposes an unsupervised and subject-specific DL method for QSM reconstruction based on implicit neural representation (INR), referred to as INR-QSM. INR has emerged as a powerful framework for learning a high-quality continuous representation of the signal (image) by exploiting its internal information without training labels. In INR-QSM, the desired susceptibility map is represented as a continuous function of the spatial coordinates, parameterized by a fully-connected neural network. The weights are learned by minimizing a loss function that includes a data fidelity term incorporated by the physical model and regularization terms. Additionally, a novel phase compensation strategy is proposed for the first time to account for the non-local effect of tissue phase in data consistency calculation to make the physical model more accurate. Our experiments show that INR-QSM outperforms traditional established QSM reconstruction methods and the compared unsupervised DL method both qualitatively and quantitatively, and is competitive against supervised DL methods under data perturbations.

Identifying synergies between VLBI and STIX imaging

Context. Reconstructing an image from noisy, sparsely sampled Fourier data is an ill-posed inverse problem that occurs in a variety of subjects within science, including data analysis for Very Long Baseline Interferometry (VLBI) and the Spectrometer/Telescope for Imaging X-rays (STIX) with respect to solar observations. The need for high-resolution, high-fidelity imaging fosters the active development of a range of novel imaging algorithms in a variety of different algorithmic settings. However, despite these ongoing, parallel developments, such synergies remain unexplored. Aims. We study, for the first time, the synergies between the data analysis for the STIX instrument and VLBI. In particular, we compare the methodologies that have been developed in both fields and evaluate their potential. In this way, we identify key trends in the performance of several algorithmic ideas and draw recommendations for the future spending of resources in the study and implementation of novel imaging algorithms. Methods. To this end, we organized a semi-blind imaging challenge with data sets and source structures that are typical for sparse VLBI, specifically in the context of the Event Horizon Telescope (EHT) as well as STIX observations. We used 17 different algorithms from both communities, from six different imaging frameworks, in the challenge, making this work the largest scale code comparison for STIX and VLBI to date. Results. We identified strong synergies between the two communities, as proven by the success of the imaging methods proposed for STIX in imaging VLBI data sets and vice versa. Novel imaging methods outperform the standard CLEAN algorithm significantly in every test case. Improvements over the performance of CLEAN offer deeper updates to the inverse modeling pipeline necessary or, consequently, the possibility to replace inverse modeling with forward modeling. Entropy-based methods and Bayesian methods perform best on STIX data. The more complex imaging algorithms utilizing multiple regularization terms (recently proposed for VLBI) add little to no additional improvements for STIX. However, they do outperform the other methods on EHT data, which correspond to a larger number of angular scales. Conclusions. This work demonstrates the great synergy between the STIX and VLBI imaging efforts and the great potential for common developments. The comparison identifies key trends on the efficacy of specific algorithmic ideas for the VLBI and the STIX setting that may evolve into a roadmap for future developments.

TSS-ConvNet for electrical impedance tomography image reconstruction

Objective. The objective of this study was to propose a novel data-driven method for solving ill-posed inverse problems, particularly in certain conditions such as time-difference electrical impedance tomography for detecting the location and size of bubbles inside a pipe. Approach. We introduced a new layer architecture composed of three paths: spatial, spectral, and truncated spectral paths. The spatial path processes information locally, whereas the spectral and truncated spectral paths provide the network with a global receptive field. This unique architecture helps eliminate the ill-posedness and nonlinearity inherent in the inverse problem. The three paths were designed to be interconnected, allowing for an exchange of information on different receptive fields with varied learning abilities. Our network has a bottleneck architecture that enables it to recover signal information from noisy redundant measurements. We named our proposed model truncated spatial-spectral convolutional neural network (TSS-ConvNet). Main results. Our model demonstrated superior accuracy with relatively high resolution on both simulation and experimental data. This indicates that our approach offers significant potential for addressing ill-posed inverse problems in complex conditions effectively and accurately. Significance. The TSS-ConvNet overcomes the receptive field limitation found in most existing models that only utilize local information in Euclidean space. We trained the network on a large dataset covering various configurations with random parameters to ensure generalization over the training samples.

Estimating Nodal Admittance Matrix for Ill-Posed Inverse Power Flow Problem in Power Grid

Estimating Nodal Admittance Matrix for Ill-Posed Inverse Power Flow Problem in Power Grid

Using multiobjective optimization to reconstruct interferometric data

Context. In very long baseline interferometry (VLBI), signals recorded at multiple antennas are combined to form a sparsely sampled virtual aperture with an effective diameter set by the largest separation between the antennas. Due to the sparsity of the sampled aperture, VLBI imaging constitutes an ill-posed inverse problem. Various algorithms have been employed to deal with the VLBI imaging, including the recently proposed multiobjective evolutionary algorithm by decomposition (MOEA/D) described in the first paper of this series. Aims. Among the approaches to the reconstruction of the image features in total intensity from sparsely sampled visibilities, extensions to the polarimetric and the temporal domain are of great interest for the VLBI community in general and the Event Horizon Telescope Collabroration (EHTC) in particular. Based on the success of MOEA/D in presenting an alternative claim of the image structure in a unique, fast, and largely unsupervised way, we study the extension of MOEA/D to polarimetric and time dynamic reconstructions in this paper. Methods. To this end, we utilized the multiobjective, evolutionary framework introduced for MOEA/D, but added the various penalty terms specific to total intensity imaging time-variable and polarimetric variants, respectively. We computed the Pareto front (the sample of all non-dominated solutions) and identified clusters of close proximities. Results. We tested MOEA/D with synthetic data sets that are representative for the main science targets and instrumental configuration of the EHTC and its possible successors. We successfully recovered the polarimetric and time-dynamic signature of the ground truth movie (even with relative sparsity) and a set of realistic data corruptions. Conclusions. MOEA/D has been successfully extended to polarimetric and time-dynamic reconstructions and, specifically, in a setting that would be expected for the EHTC. It offers a unique alternative and independent claim to the already existing methods, along with a number of additional benefits, namely: it is the first method that effectively explores the problem globally and compared to regularized maximum likelihood (RML) methods. Thus, it waives the need for parameter surveys. Hence, MOEA/D is a novel, useful tool to characterize the polarimetric and dynamic signatures in a VLBI data set robustly with a minimal set of user-based choices. In a consecutive work, we will address the last remaining limitation for MOEA/D (the number of pixels and numerical performance), so that MOEA/D can firmly solidify its place within the VLBI data reduction pipeline.

Noise-Free Optimization in Early Training Steps for Image Super-resolution

Recent deep-learning-based single image super-resolution (SISR) methods have shown impressive performance whereas typical methods train their networks by minimizing the pixel-wise distance with respect to a given high-resolution (HR) image. However, despite the basic training scheme being the predominant choice, its use in the context of ill-posed inverse problems has not been thoroughly investigated. In this work, we aim to provide a better comprehension of the underlying constituent by decomposing target HR images into two subcomponents: (1) the optimal centroid which is the expectation over multiple potential HR images, and (2) the inherent noise defined as the residual between the HR image and the centroid. Our findings show that the current training scheme cannot capture the ill-posed nature of SISR and becomes vulnerable to the inherent noise term, especially during early training steps. To tackle this issue, we propose a novel optimization method that can effectively remove the inherent noise term in the early steps of vanilla training by estimating the optimal centroid and directly optimizing toward the estimation. Experimental results show that the proposed method can effectively enhance the stability of vanilla training, leading to overall performance gain. Codes are available at github.com/2minkyulee/ECO.

Q-compensated image-domain least-squares reverse time migration through preconditioned point-spread functions

Image-domain least-squares reverse time migration (IDLSRTM) through point-spread functions (PSFs) has been proven to be a feasible approach to improve the spatial resolution and amplitude fidelity of reflection images recovered by reverse time migration (RTM). However, it usually ignores the earth’s Q-effects, which may lead to an unfocused reflection image with an undesired spatial resolution. In this paper, we develop a Q-compensated IDLSRTM approach (denoted as Q-IDLSRTM) through PSFs, in which we use the viscoacoustic wave equation based on the generalized standard linear solid model to simulate inherent subsurface attenuation and the linear inversion to compensate for the amplitude attenuation. The PSFs are estimated by a round of modeling-migration computation and spatial interpolation on the fly. There are two key points in the developed Q-IDLSRTM approach. The first is that we must apply the deblurring filter as a preconditioner to compensate for the attenuation of image amplitude of PSFs and RTM in a viscoacoustic medium, before the iterative solution. The preconditioned PSFs and RTM images can help us to construct a less ill-posed image-domain inverse problem that can produce an improved image quality and a faster convergence rate, compared with the conventional Q-IDLSRTM approach without the deblurring filter. The second key point is that we can impose the L1-norm constraint and total variation regularization on the reflection image to stabilize the solution of the ill-posed inverse problem. Several 2D and 3D experiments verify that the developed approach can achieve better imaging quality in terms of amplitude fidelity and spatial resolution relative to the conventional Q-IDLSRTM and acoustic IDLSRTM approaches.

Data-driven solutions of ill-posed inverse problems arising from doping reconstruction in semiconductors

ABSTRACT The non-destructive estimation of doping concentrations in semiconductor devices is of paramount importance for many applications ranging from crystal growth to defect and inhomogeneity detection. A number of technologies (such as LBIC, EBIC and LPS) have been developed which allow the detection of doping variations via photovoltaic effects. The idea is to illuminate the sample at several positions and detect the resulting voltage drop or current at the contacts. We model a general class of such photovoltaic technologies by ill-posed global and local inverse problems based on a drift-diffusion system which describes charge transport in a self-consistent electrical field. The doping profile is included as a parametric field. To numerically solve a physically relevant local inverse problem, we present three approaches, based on least squares, multilayer perceptrons, and residual neural networks. Our data-driven methods reconstruct the doping profile for a given spatially varying voltage signal induced by a laser scan along the sample's surface. The methods are trained on synthetic data sets which are generated by finite volume solutions of the forward problem. While the linear least square method yields an average absolute error around 10%, the nonlinear networks roughly halve this error to 5%.

Integrating the PROSAIL and SVR Models to Facilitate the Inversion of Grassland Aboveground Biomass: A Case Study of Zoigê Plateau, China

Grasslands play a vital role in the global ecosystem. Efficient and reproducible methods for estimating the grassland aboveground biomass (AGB) are crucial for understanding grassland growth, promoting sustainable development, and assessing the carbon cycle. Currently, the available methods are limited by their computational inefficiency, model transfer, and sampling scale. Therefore, in this study, the estimation of grassland AGB over a large area was achieved by coupling the PROSAIL model with the support vector machine regression (SVR) method. The ill-posed inverse problem of the PROSAIL model was mitigated through kernel-based regularization using the SVR model. The Zoigê Plateau was used as the case study area, and the results demonstrated that the estimated biomass accurately reproduced the reference AGB map generated by zooming in on on-site measurements (R2 = 0.64, RMSE = 43.52 g/m2, RRMSE = 15.13%). The estimated AGB map also maintained a high fitting accuracy with field sampling data (R2 = 0.69, RMSE = 44.07 g/m2, RRMSE = 14.21%). Further, the generated time-series profiles of grass AGB for 2022 were consistent with the trends in local grass growth dynamics. The proposed method combines the advantages of the PROSAIL model and the regression algorithm, reduces the dependence on field sampling data, improves the universality and repeatability of grassland AGB estimation, and provides an efficient approach for grassland ecosystem construction and planning.

Deep-unrolling architecture for image-domain least-squares migration

Deep-image prior (DIP) is a novel approach to solving ill-posed inverse problems whose solution is parameterized with an untrained deep neural network and cascaded with the forward modeling operator. A key component to the success of such a method is represented by the choice of the network architecture, which must act as a natural prior to the inverse problem at hand and provide a strong inductive bias toward the desired solution. Inspired by the close link between neural networks and iterative algorithms in classical optimization, we apply an unrolled version of the gradient descent (GD) algorithm as our DIP network architecture, denoted as the deep-unrolling (DU) architecture. Each layer of the unrolled network is comprised of two parts: the first part corresponds to the GD step of the data-fidelity term, whereas the second part, formed by a six-layer convolutional neural network (CNN), plays the role of a regularizer in the associated objective function. The developed DU architecture is applied to the problem of image-domain least-squares migration (IDLSM) to invert migrated seismic images for their underlying reflectivity and is denoted as DU-IDLSM. As such, the DU architecture parameterizes the reflectivity, and the input of each layer of the unrolled network is the reflectivity at the previous layer. Similar to the classical DIP approach, the parameters of the DU architecture are optimized in an unsupervised fashion by minimizing the data misfit function itself. Through experiments with a part of the Sigsbee2A model and a marine field data set, we test the effectiveness of the DU-IDLSM approach and highlight two key benefits. First, the DU architecture can effectively regularize the inversion process, resulting in reflectivity estimates with fewer artifacts and higher image resolution than those produced by conventional IDLSM approaches. Second, we indicate that by including dropout layers in the CNN architecture, DU-IDLSM can produce a qualitative measure of the uncertainty associated with the least-squares migration process.

Local uncertainty quantification for 3-D time-domain full-waveform inversion with ensemble Kalman filters: application to a North Sea OBC data set

SUMMARY Full-waveform inversion (FWI) has emerged as the state-of-the art high resolution seismic imaging technique, both in seismology for global and regional scale imaging and in the industry for exploration purposes. While gaining in popularity, FWI, at an operational level, remains a heavy computational process involving the repeated solution of large-scale 3-D wave propagation problems. For this reason it is a common practice to focus the interpretation of the results on the final estimated model. This is forgetting FWI is an ill-posed inverse problem in a high dimensional space for which the solution is intrinsically non-unique. This is the reason why being able to qualify and quantify the uncertainty attached to a model estimated by FWI is key. To this end, we propose to extend at an operational level the concepts introduced in a previous study related to the coupling between ensemble Kalman filters (EnKFs) and FWI. These concepts had been developed for 2-D frequency-domain FWI. We extend it here to the case of 3-D time-domain FWI, relying on a source subsampling strategy to assimilate progressively the data within the Kalman filter. We apply our strategy to an ocean bottom cable field data set from the North Sea to illustrate its feasibility. We explore the convergence of the filter in terms of number of elements, and extract variance and covariance information showing which part of the model are well constrained and which are not. Analysing the variance helps to gain insight on how well the final estimated model is constrained by the whole FWI workflow. The variance maps appears as the superposition of a smooth trend related to the geometrical spreading and a high resolution trend related to reflectors. Mapping lines of the covariance (or correlation matrix) to the model space helps to gain insight on the local resolution. Through a wave propagation analysis, we are also able to relate variance peaks in the model space to variance peaks in the data space. Compared to other posterior-covariance approximation scheme, our combination between EnKF and FWI is intrinsically scalable, making it a good candidate for exploiting the recent exascale high performance computing machines.