Ill-posed 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.

Global weak solvability in a self-consistent chemotaxis-Navier–Stokes system involving Dirichlet boundary conditions for the signal

The self-consistent chemotaxis-Navier–Stokes system with nonlinear diffusion [Formula: see text] is considered in a bounded domain [Formula: see text] with smooth boundary. Compared to the previously most-studied chemotaxis-fluid system proposed in [I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA. 102 (2005) 2277–2282], the coupling in this system is stronger and more nonlinear. When the system is accompanied by homogeneous boundary conditions of no-flux type for [Formula: see text] and [Formula: see text], and of Dirichlet type for [Formula: see text], a quasi-Lyapunov structure provides sufficient regularity features to facilitate a basic existence theory. However, if we change the boundary condition of the signal to [Formula: see text] with a given non-negative function [Formula: see text][Formula: see text], then the Dirichlet boundary condition imposed here seems to destroy the quasi-Lyapunov structure. Despite this, we shall find a new energy structure and prove that for suitably regular initial data, the assumption [Formula: see text] is sufficient for the global existence and boundedness of the weak solution. To the best of our knowledge, this is the first work on the global well-posedness problem of the self-consistent chemotaxis-fluid system involving Dirichlet boundary conditions for the signal.

A sequence decomposition genetic algorithm for petrophysical inversion with spatial correlation

Predicting petrophysical properties based on seismic attributes is essential for accurate subsurface characterization. Typically, a forward model is first established to relate petrophysical properties to geophysical measurements, and then the petrophysical properties are inverted from actual measurements. Imposing prior constraints of spatial relationship is also desirable to regularize the ill-posed inverse problem. However, petrophysical inversion involving spatial correlation presents a high-dimensional optimization problem with multiple local extrema. This problem can be challenging to solve, particularly when discrete variables (such as lithofacies) and continuous variables (such as porosity and fluid saturation) are considered together. Therefore, we propose a global optimization algorithm that utilizes Markov chain decomposition and a genetic algorithm for joint discrete-continuous petrophysical inversion with spatial correlation. The innovation of this algorithm is to spatially decouple the sequence optimization problem into multiple low-dimensional sub-problems and overcome the initial value dependence to converge to the global optimal solution. Furthermore, an approximation operation is introduced to expedite the optimization algorithm. Numerical experiments demonstrate that after algorithmic acceleration, the spatially correlated inversion requires the same computational time as a simple spatially uncorrelated inversion and achieves significantly better accuracy than the latter. Finally, we apply the method to a case study in East China, where lithofacies, clay content, porosity, and water saturation are inferred from the data of wave velocities and density. The inversion results around the wells generally match the well logging data, verifying the feasibility of the new method.

Feasibility of acousto-electric tomography

Abstract In acousto-electric tomography (AET) the goal is to reconstruct the electric conductivity in a domain from electrostatic boundary measurements of corresponding currents and voltages, while the domain is perturbed by a time-dependent acoustic wave, thus taking advantage of the acousto-electric effect. We approach the AET reconstruction in two steps: First, the interior power density is obtained from boundary measurements by solving a linear inverse and ill-posed problem; second, the interior conductivity is reconstructed from the power density by solving a non-linear and well-posed problem. Mathematically these inverse problems are fairly well understood, and reconstruction methods work well on synthetic data. This is in contrast to experimental findings. An effect can indeed be observed and data can be collected. However, the acousto-electric coupling is very weak, and consequently, the change in the measured voltage due to the acoustic perturbation might be too small compared to the background noise for viable reconstructions. 

In this paper, we take one step towards understanding the feasibility of AET. We provide an in-silico model of the coupled physics scenario based on standard models for the individual phenomena. Moreover, we formulate and implement numerically a full reconstruction method for the inverse problem via the two steps. We perform computational experiments with realistically chosen parameters from the context of medical imaging. The focus is on understanding the role of the acousto-electric coupling parameter and the signal-to-noise ratio (SNR). The critical signal strength is analyzed and the omnipresent Johnson-Nyquist noise is estimated. We obtain both positive and negative findings; we can reconstruct features even under severe noise conditions, but we also find that the SNR one is likely to face in practice is too low to obtain useful reconstructions.

Dynamic CBCT imaging using prior model-free spatiotemporal implicit neural representation (PMF-STINR).

Dynamic cone-beam computed tomography (CBCT) can capture high-spatial-resolution, time-varying images for motion monitoring, patient setup, and adaptive planning of radiotherapy. However, dynamic CBCT reconstruction is an extremely ill-posed spatiotemporal inverse problem, as each CBCT volume in the dynamic sequence is only captured by one or a few X-ray projections, due to the slow gantry rotation speed and the fast anatomical motion (e.g., breathing). 
Approach: We developed a machine learning-based technique, prior-model-free spatiotemporal implicit neural representation (PMF-STINR), to reconstruct dynamic CBCTs from sequentially acquired X-ray projections. PMF-STINR employs a joint image reconstruction and registration approach to address the under-sampling challenge, enabling dynamic CBCT reconstruction from singular X-ray projections. Specifically, PMF-STINR uses spatial implicit neural representations to reconstruct a reference CBCT volume, and it applies temporal INR to represent the intra-scan dynamic motion with respect to the reference CBCT to yield dynamic CBCTs. PMF-STINR couples the temporal INR with a learning-based B-spline motion model to capture time-varying deformable motion during the reconstruction. Compared with the previous methods, the spatial INR, the temporal INR, and the B-spline model of PMF-STINR are all learned on the fly during reconstruction in a one-shot fashion, without using any patient-specific prior knowledge or motion sorting/binning. 
Main results: PMF-STINR was evaluated via digital phantom simulations, physical phantom measurements, and a multi-institutional patient dataset featuring various imaging protocols (half-fan/full-fan, full sampling/sparse sampling, different energy and mAs settings, etc.). The results showed that the one-shot learning-based PMF-STINR can accurately and robustly reconstruct dynamic CBCTs and capture highly irregular motion with high temporal (~0.1s) resolution and sub-millimeter accuracy.
Significance: PMF-STINR can reconstruct dynamic CBCTs and solve the intra-scan motion problem from conventional 3D CBCT scans without using any prior anatomical/motion model or motion sorting/binning. It can be a promising tool for motion management by offering richer motion information than traditional 4D-CBCTs.

[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 towards 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.

Schwarz Problem for Model Partial Differential Equations with One Complex Variable

This paper investigates the Schwarz problem. Initially, the focus lies on analyzing the problem for the first, second orders. Subsequently, attention shifts towards studying the same problem for equations of higher order. In the realm of second-order equations, the Schwarz problem is specifically examined for some operators; Laplace, Bitsadze and its complex conjugate. The findings demonstrate that the Schwarz problem for an n-order equation, when equipped with solely one boundary condition, exhibits an infinite number of solutions. However, by incorporating additional boundary conditions, it becomes feasible to obtain a unique solution for problem concerning n-order equations, effectively rendering it a well-posed problem.

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.

Image denoising and model-independent parameterization for IVIM MRI

Objective. To improve intravoxel incoherent motion imaging (IVIM) magnetic resonance Imaging quality using a new image denoising technique and model-independent parameterization of the signal versus b-value curve. Approach. IVIM images were acquired for 13 head-and-neck patients prior to radiotherapy. Post-radiotherapy scans were also acquired for five of these patients. Images were denoised prior to parameter fitting using neural blind deconvolution, a method of solving the ill-posed mathematical problem of blind deconvolution using neural networks. The signal decay curve was then quantified in terms of several area under the curve (AUC) parameters. Improvements in image quality were assessed using blind image quality metrics, total variation (TV), and the correlations between parameter changes in parotid glands with radiotherapy dose levels. The validity of blur kernel predictions was assessed by the testing the method's ability to recover artificial ‘pseudokernels’. AUC parameters were compared with monoexponential, biexponential, and triexponential model parameters in terms of their correlations with dose, contrast-to-noise (CNR) around parotid glands, and relative importance via principal component analysis. Main results. Image denoising improved blind image quality metrics, smoothed the signal versus b-value curve, and strengthened correlations between IVIM parameters and dose levels. Image TV was reduced and parameter CNRs generally increased following denoising. AUC parameters were more correlated with dose and had higher relative importance than exponential model parameters. Significance. IVIM parameters have high variability in the literature and perfusion-related parameters are difficult to interpret. Describing the signal versus b-value curve with model-independent parameters like the AUC and preprocessing images with denoising techniques could potentially benefit IVIM image parameterization in terms of reproducibility and functional utility.

Improvement of Aerosol Coarse-Mode Detection through Additional Use of Infrared Wavelengths in the Inversion of Arctic Lidar Data

An Nd:YAG-based Raman lidar provides a mature technology to derive profiles of the optical properties of aerosols over a wide altitude range. However, the derivation of micro-physical parameters is an ill-posed problem. Hence, increasing the information content of lidar data is desirable. Recently, ceilometers and wind lidar systems, both operating in the near-infrared region, have been successfully employed in aerosol research. In this study, we demonstrate that the inclusion of additional backscatter coefficients from these two latter instruments clearly improves the inversion of micro-physical parameters such as volume distribution function, effective radius, or single-scattering albedo. We focus on the Arctic aerosol and start with the typical volume distribution functions of Arctic haze and boreal biomass burning. We forward calculate the optical coefficients that the lidar systems should have seen and include or exclude the backscatter coefficients of the ceilometer (910 nm) and wind lidar data (1500 nm) to analyze the value of these wavelengths in their ability to reproduce the volume distribution function, which may be mono- or bimodal. We found that not only the coarse mode but also the properties of the accumulation mode improved when the additional wavelengths were considered. Generally, the 1500 nm wavelength has greater value in correctly reproducing the aerosol properties.

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.

Temporal windows of unconscious processing cannot easily be disrupted.

Conscious perception is preceded by long periods of unconscious processing. These periods are crucial for analyzing temporal information and for solving the many ill-posed problems of vision. An important question is what starts and ends these windows and how they may be interrupted. Most experimental paradigms do not offer the methodology required for such investigation. Here, we used the sequential metacontrast paradigm, in which two streams of lines, expanding from the center to the periphery, are presented, and participants are asked to attend to one of the motion streams. If several lines in the attended motion stream are offset, the offsets are known to integrate mandatorily and unconsciously, even if separated by up to 450ms. Using this paradigm, we here found that external visual objects, such as an annulus, presented during the motion stream, do not disrupt mandatory temporal integration. Thus, if a window is started once, it appears to remain open even in the presence of disruptions that are known to interrupt visual processes normally. Further, we found that interrupting the motion stream with a gap disrupts temporal integration but does not terminate the overall unconscious processing window. Thus, while temporal integration is key to unconscious processing, not all stimuli in the same processing window are integrated together. These results strengthen the case for unconscious processing taking place in windows of sensemaking, during which temporal integration occurs in a flexible and perceptually meaningful manner.

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.

Inversion with stratigraphy-guided deep learning

Property estimation in seismic exploration traditionally relies on seismic inversion, which is an ill-posed problem. However, recent advances in deep learning (DL), specifically supervised neural networks, indicate promise for accuracy improvements. Building upon this, “stratigraphy-guided deep learning” (SGDL) is a novel method that encodes stratigraphic units as discrete features within the DL model. Our primary objective is to evaluate SGDL in a scenario with available geologic data and field data calibration, such as well tops and horizons. We conduct a case study predicting porosity and acoustic impedance from poststack seismic data. Robustness evaluations demonstrate a 20% average enhancement in the correlation for acoustic impedance across 10 test wells from the Volve data set. We find that SGDL inversion outperforms traditional and other DL methods, reaching a 91% correlation for one benchmark blind well. These results offer compelling evidence that the incorporation of stratigraphic units as features in the DL model contributes to further enhancing the accuracy of property estimation. In summary, SGDL represents a novel approach that integrates DL with geologic data, offering significant enhancements in property estimation accuracy within the field of seismic inversion.

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.

Orbital polarimetric tomography of a flare near the Sagittarius A* supermassive black hole

AbstractThe interaction between the supermassive black hole at the centre of the Milky Way, Sagittarius A*, and its accretion disk occasionally produces high-energy flares seen in X-ray, infrared and radio. One proposed mechanism that produces flares is the formation of compact, bright regions that appear within the accretion disk and close to the event horizon. Understanding these flares provides a window into accretion processes. Although sophisticated simulations predict the formation of these flares, their structure has yet to be recovered by observations. Here we show a three-dimensional reconstruction of an emission flare recovered from Atacama Large Millimeter/Submillimeter Array light curves observed on 11 April 2017. Our recovery shows compact, bright regions at a distance of roughly six times the event horizon. Moreover, it suggests a clockwise rotation in a low-inclination orbital plane, consistent with prior studies by GRAVITY and the Event Horizon Telescope. To recover this emission structure, we solve an ill-posed tomography problem by integrating a neural three-dimensional representation with a gravitational model for black holes. Although the recovery is subject to, and sometimes sensitive to, the model assumptions, under physically motivated choices, our results are stable and our approach is successful on simulated data.

Towards a Reliable Uncertainty Quantification in Residual Stress Measurements with Relaxation Methods: Finding Average Residual Stresses is a Well-Posed Problem

Towards a Reliable Uncertainty Quantification in Residual Stress Measurements with Relaxation Methods: Finding Average Residual Stresses is a Well-Posed Problem

A CGLS-based method for solving magnetic moments of hybrid-model magnetic targets

In order to solve the problem of magnetic moment estimation of magnetic targets, the inverse problem of magnetic moment estimation was constructed based on the hybrid model of ellipsoid and magnetic dipole array. In order to solve the ill-posed problem of the magnetic moment estimation equations, the method was designed to estimate the magnetic moment parameters of the ellipsoid and magnetic dipole array, and the improved discrepancy principle and the maximum chi-square distribution stop criterion were introduced to improve the semi-convergence behavior of the conjugate gradient least squares (CGLS) method. Through simulation examples of magnetic target magnetic moment estimation and ship model measurement data, the performance of two CGLS methods, Tikhonov algorithm and stepwise regression method, were compared and analyzed from four aspects: relative error of magnetic moment estimation, relative error of magnetic field fitting, relative error of magnetic field extrapolation, and computational time complexity. The CGLS method has the advantages of high accuracy of magnetic moment estimation, high immunity of magnetic moment estimation to interference, high accuracy of magnetic field fitting and extrapolation, and low computational time complexity.

A defect opening profile reconstruction method based on multidirectional magnetic flux leakage detection

Magnetic flux leakage (MFL) is a popular technique for detecting defects. The opening profile reconstruction is one of the most popular concerns. The commonly used unidirectional magnetization-based methods easily lead to incomplete MFL information to reconstruct the opening profile. The superimposition of MFL signals from different directions would cause information cancellation, making it difficult for the existing data processing methods to reconstruct. To solve the issues, this study constructs a model-based method to inverse defect opening profiles. It builds the forward equations from one magnetic permeability to multidirectional MFL fields. Then the pseudo-inversion with Tikhonov regularization is applied as the inversion algorithm. The algorithm associates the forward equations and finds the solution that can fulfill them simultaneously. Such a method can combine multidirectional MFL information without cancellation. It reduces the ill-posed problems that exist in unidirectional inversion and makes the reconstructed profile complete and accurate. The comparison experiments have been carried out, and the results have proved the high accuracy of the proposed method.