Underdetermined Inverse Problem Research Articles

Hybrid optimization schemes for solving the piezoresistive inversion problem in self-sensing materials

Materials with electrically conductive nanofillers have the ability to ‘sense’ changes to their mechanical state. When these materials are deformed, the embedded nanofiller networks are disturbed causing a measurable change in the electrical conductivity of the material. This self-sensing property, known as piezoresistivity, has been leveraged in numerous engineering venues. Although this property has been thoroughly explored, prevailing self-sensing techniques provide little-to-no information about the underlying mechanical state of the material, such as the displacement and strain. This information must be indirectly obtained from the conductivity change. This limitation exists because obtaining mechanics from conductivity is an under-determined inverse problem with many possible mathematically feasible solutions. Previous work in this area used metaheuristic algorithms and imposed mechanics-based constraints to solve the piezoresistive inversion problem. Although this approach was successful, it was computationally inefficient due to the stochastic search process and the need to perform multiple searches to find a converged solution. To overcome this limitation, we herein propose a hybrid optimization scheme for solving the piezoresistive inversion problem. This scheme is implemented in two steps. In the first step, a metaheuristic algorithm performs a single search for a suitable solution to the inverse problem. In the second step, a gradient descent algorithm searches for the final solution using the solution from the previous step as the starting point. We explore different norms for the fitness function of the metaheuristic search and demonstrate using experimental data that the proposed hybrid optimization scheme can accurately and efficiently calculate displacements and strains from conductivity changes. This exploration significantly advances the state of the art by enabling computationally efficient and highly accurate predictions of full-field mechanical condition in self-sensing materials for the first time, thereby paving the way for greater use of these principles in practice.

Lightweight super-resolution multimode fiber imaging with regularized linear regression

Super-resolution multimode fiber imaging provides the means to image samples quickly with compact and flexible setups finding many applications from biology and medicine to material science and nanolithography. Typically, fiber-based imaging systems suffer from low spatial resolution and long measurement times. State-of-the-art computational approaches can achieve fast super-resolution imaging through a multimode fiber probe but currently rely on either per-sample optimised priors or large data sets with subsequent long training and image reconstruction times. This unfortunately hinders any real-time imaging applications. Here we present an ultimately fast non-iterative algorithm for compressive image reconstruction through a multimode fiber. The proposed approach helps to avoid many constraints by determining the prior of the target distribution from a simulated set and solving the under-determined inverse matrix problem with a mathematical closed-form solution. We have demonstrated theoretical and experimental evidence for enhanced image quality and sub-diffraction spatial resolution of the multimode fiber optical system.

Neural network analysis of neutron and X-ray reflectivity data incorporating prior knowledge.

Due to the ambiguity related to the lack of phase information, determining the physical parameters of multilayer thin films from measured neutron and X-ray reflectivity curves is, on a fundamental level, an underdetermined inverse problem. This ambiguity poses limitations on standard neural networks, constraining the range and number of considered parameters in previous machine learning solutions. To overcome this challenge, a novel training procedure has been designed which incorporates dynamic prior boundaries for each physical parameter as additional inputs to the neural network. In this manner, the neural network can be trained simultaneously on all well-posed subintervals of a larger parameter space in which the inverse problem is underdetermined. During inference, users can flexibly input their own prior knowledge about the physical system to constrain the neural network prediction to distinct target subintervals in the parameter space. The effectiveness of the method is demonstrated in various scenarios, including multilayer structures with a box model parameterization and a physics-inspired special parameterization of the scattering length density profile for a multilayer structure. In contrast to previous methods, this approach scales favourably when increasing the complexity of the inverse problem, working properly even for a five-layer multilayer model and a periodic multilayer model with up to 17 open parameters.

Inverse Analysis of Strata in Seepage Field Based on Regularization Method and Geostatistics Theory

During the construction of underground engineering, the prediction of groundwater distribution and rock body permeability is essential for evaluating the safety of the project and guiding subsequent design and construction. This article proposes an objective function that solves an underdetermined inverse analysis problem based on the least-squares theory and regularization method and uses geostatistics theory and the variogram function to describe the spatial characteristics of the actual engineering system. It also establishes an optimization model of the analysis stratum seepage field and puts forward the method of using on-site test observation data to solve the stratum penetration coefficient. Relying on the foundation pit project of the Lingshanwei Station of Qingdao Metro, the on-site pumping and packer permeability test was conducted for different strata venues in the foundation pit, and the on-site water-head observation value was obtained. Physical detection of the influence area of foundation pit excavation confirms the correctness of the model from the region and verifies the accuracy of the model on the value through the on-site pumping test. Results show that the accuracy of the use of this objective function to solve the underdetermined inverse problem is above 85%, which proves the effectiveness of the method. The stratigraphic geological information obtained by the inverse analysis model provides an important basis for engineering design and security construction.

Wavelet-Improved Score-Based Generative Model for Medical Imaging.

The score-based generative model (SGM) has demonstrated remarkable performance in addressing challenging under-determined inverse problems in medical imaging. However, acquiring high-quality training datasets for these models remains a formidable task, especially in medical image reconstructions. Prevalent noise perturbations or artifacts in low-dose Computed Tomography (CT) or under-sampled Magnetic Resonance Imaging (MRI) hinder the accurate estimation of data distribution gradients, thereby compromising the overall performance of SGMs when trained with these data. To alleviate this issue, we propose a wavelet-improved denoising technique to cooperate with the SGMs, ensuring effective and stable training. Specifically, the proposed method integrates a wavelet sub-network and the standard SGM sub-network into a unified framework, effectively alleviating inaccurate distribution of the data distribution gradient and enhancing the overall stability. The mutual feedback mechanism between the wavelet sub-network and the SGM sub-network empowers the neural network to learn accurate scores even when handling noisy samples. This combination results in a framework that exhibits superior stability during the learning process, leading to the generation of more precise and reliable reconstructed images. During the reconstruction process, we further enhance the robustness and quality of the reconstructed images by incorporating regularization constraint. Our experiments, which encompass various scenarios of low-dose and sparse-view CT, as well as MRI with varying under-sampling rates and masks, demonstrate the effectiveness of the proposed method by significantly enhanced the quality of the reconstructed images. Especially, our method with noisy training samples achieves comparable results to those obtained using clean data. Our code at https://zenodo.org/record/8266123.

Curve Fitting Based Minimum Norm Estimation (CFB-MNE) for motor unit spatial localization using high density surface electromyogram signals

The purpose of the study is the real time spatial localization of single motor units using simulated Motor Unit Action Potential dictionary from a cylindrical muscle volume conductor model by means of a state-of-the-art curve fitting method. This Curve Fitting Based Minimum Norm Estimation (CFB-MNE) was made possible by using minimum norm estimation methods to solve an underdetermined inverse problem knowing produced HD-sEMG signals. Specific data extracted from the inverse problem of a pre-determined number of simulated motor units were used to create a 3D curve, which can be used as a fitting curve to anticipate the unknown location of motor units. Results show that the proposed algorithms succeeded in accurately localizing motor units with varying noise levels in a fast duration (less than 100 ms). From all the simulations, a mean root mean square error of maximum 1 mm was recorded for the localization of the depth (less than 1 mm) and a negligible error for the angular position (less than 1°). The proposed work is an innovative algorithm that aids in non-invasively localizing motor units within a muscle in real time. Further efforts are planned to manage MUAP superposition and type. Final applications are related to prosthetic control, rehabilitation guidance or aging monitoring.

Fast electrical impedance tomography based on sparse Bayesian learning

Electrical impedance tomography (EIT) is a severely under-determined and ill-posed inverse problem. Therefore, based on the compressed sensing theory and the block sparse Bayesian learning (BSBL) model, a novel accelerated EIT sparse imaging algorithm is presented. The key feature of the proposed algorithm is that the convex–concave procedure (CCP) method is used to optimize the non-convex cost function of the block sparse Bayesian learning model which improves the convergence speed and reduces the time complexity of the algorithm. The proposed method can adaptively explore and utilize the inter-block sparsity and intra-block structural correlation of non-sparse signals without any a priori information, thereby sparse imaging is performed on non-sparse physiological data effectively. The results comparing with other methods show that the proposed algorithm not only reconstruct inclusions with different shapes and conductivity applicably and effectively but also reduce the running time of the algorithm.

High frequency conductivity decomposition by solving physically constraint underdetermined inverse problem in human brain

The developed magnetic resonance electrical properties tomography (MREPT) can visualize the internal conductivity distribution at Larmor frequency by measuring the B1 transceive phase data from magnetic resonance imaging (MRI). The recovered high-frequency conductivity (HFC) value is highly complex and heterogeneous in a macroscopic imaging voxel. Using high and low b-value diffusion weighted imaging (DWI) data, the multi-compartment spherical mean technique (MC-SMT) characterizes the water molecule movement within and between intra- and extra-neurite compartments by analyzing the microstructures and underlying architectural organization of brain tissues. The proposed method decomposes the recovered HFC into the conductivity values in the intra- and extra-neurite compartments via the recovered intra-neurite volume fraction (IVF) and the diffusion patterns using DWI data. As a form of decomposition of intra- and extra-neurite compartments, the problem to determine the intra- and extra-neurite conductivity values from the HFC is still an underdetermined inverse problem. To solve the underdetermined problem, we use the compartmentalized IVF as a criterion to decompose the electrical properties because the ion-concentration and mobility have different characteristics in the intra- and extra-neurite compartments. The proposed method determines a representative apparent intra- and extra-neurite conductivity values by changing the underdetermined equation for a voxel into an over-determined minimization problem over a local window consisting of surrounding voxels. To suppress the noise amplification and estimate a feasible conductivity, we define a diffusion pattern distance to weight the over-determined system in the local window. To quantify the proposed method, we conducted a simulation experiment. The simulation experiments show the relationships between the noise reduction and the spatial resolution depending on the designed local window sizes and diffusion pattern distance. Human brain experiments (five young healthy volunteers and a patient with brain tumor) were conducted to evaluate and validate the reliability of the proposed method. To quantitatively compare the results with previously developed methods, we analyzed the errors for reconstructed extra-neurite conductivity using existing methods and indirectly verified the feasibility of the proposed method.

Self-Supervised Coordinate Projection Network for Sparse-View Computed Tomography

Sparse-view Computed Tomography (SVCT) has great potential for decreasing patient radiation exposure dose during scanning. In this work, we propose a Self-supervised COordinate Projection nEtwork (SCOPE) to reconstruct the artifact-free CT image from the acquired SV sinogram by solving the inverse problem of tomography imaging. To solve the under-determined inverse imaging problem, we first introduce an implicit neural representation (INR) network to constrain the solution space via image continuity prior. And inspired by the relationship between linear algebra and inverse problems, we propose a novel re-projection strategy to generate a dense view sinogram from the initial solution, which significantly improves the rank of the linear equation system and produces a more stable CT image solution space. Specially, the desired CT image is represented as an implicit function of the two-dimensional spatial coordinate to directly approximate the SV sinogram through the CT imaging forward model. Then, a dense-view sinogram is generated from the fine-trained INR network. The final CT reconstruction is reconstructed by applying Filtered Back Projection (FBP) to the generated dense-view sinogram. Additionally, we integrate the recent hash encoding into our SCOPE model, which efficiently accelerates the model training process. We evaluate SCOPE in parallel and fan X-ray beam SVCT reconstruction tasks. Our experiment results demonstrate that the re-projection strategy significantly improves the image reconstruction quality (+3 dB for PSNR at least). The proposed SCOPE model provides state-of-the-art reconstruction results compared to two latest INR-based methods and two well-popular supervised DL methods for the SV CT image reconstruction.

Solving Inverse Problems With Deep Neural Networks - Robustness Included?

In the past five years, deep learning methods have become state-of-the-art in solving various inverse problems. Before such approaches can find application in safety-critical fields, a verification of their reliability appears mandatory. Recent works have pointed out instabilities of deep neural networks for several image reconstruction tasks. In analogy to adversarial attacks in classification, it was shown that slight distortions in the input domain may cause severe artifacts. The present article sheds new light on this concern, by conducting an extensive study of the robustness of deep-learning-based algorithms for solving underdetermined inverse problems. This covers compressed sensing with Gaussian measurements as well as image recovery from Fourier and Radon measurements, including a real-world scenario for magnetic resonance imaging (using the NYU-fastMRI dataset). Our main focus is on computing adversarial perturbations of the measurements that maximize the reconstruction error. A distinctive feature of our approach is the quantitative and qualitative comparison with total-variation minimization, which serves as a provably robust reference method. In contrast to previous findings, our results reveal that standard end-to-end network architectures are not only resilient against statistical noise, but also against adversarial perturbations. All considered networks are trained by common deep learning techniques, without sophisticated defense strategies.

Deep learning-based solvability of underdetermined inverse problems in nonlinear ultrasonic characterization of micro damages

Detection and evaluation of micro-damages in the early stages of engineering failure are crucial for various industrial structures to ensure their safety and prevent further catastrophic accidents. The nonlinear ultrasonic technique (NUT) has gained increasing popularity and recognition for breaking through the detection sensitivity limit upon micro-damages that usually are invisible to conventional linear techniques. However, it remains an ongoing challenge to quantitatively characterize micro-damages using NUT due to great difficulties in fully modeling the complicated interaction mechanism between the nonlinear ultrasonic waves and micro-damages. This work presents a data-driven perspective for solving multiparameter underdetermined inverse problems that are at the core of NUT, while allowing by-passing the creation of high-fidelity physics-based models. Nonlinear Lamb wave measurements with group-velocity mismatching are conducted to introduce both size and localization information of damages to the assembled dataset. A nonlinearity-aware discrete wavelet transform-bidirectional long short-term memory network is proposed to directly process nonlinear ultrasonic responses to automatically model latent nonlinear dynamics, thus establishing the complex mapping between the nonlinear ultrasonic signals and the multi-dimensional damage features. In particular, an attempt is made to augment the physical explainability of the proposed deep learning approach through a frequency component importance analysis. The trained network enables accurate and explainable predictions of length and localization of closed cracks and robustness against varying degrees of noise. Our work paves a promising and practical way to promote the transformation of NUT from the qualitative analysis for accurate and efficient quantitative prediction.

Seismic data reconstruction based on low dimensional manifold model

Seismic data reconstruction is an essential and yet fundamental step in seismic data processing workflow, which is of profound significance to improve migration imaging quality, multiple suppression effect, and seismic inversion accuracy. Regularization methods play a central role in solving the under-determined inverse problem of seismic data reconstruction. In this paper, a novel regularization approach is proposed, the low dimensional manifold model (LDMM), for reconstructing the missing seismic data. Our work relies on the fact that seismic patches always occupy a low dimensional manifold. Specifically, we exploit the dimension of the seismic patches manifold as a regularization term in the reconstruction problem, and reconstruct the missing seismic data by enforcing low dimensionality on this manifold. The crucial procedure of the proposed method is to solve the dimension of the patches manifold. Toward this, we adopt an efficient dimensionality calculation method based on low-rank approximation, which provides a reliable safeguard to enforce the constraints in the reconstruction process. Numerical experiments performed on synthetic and field seismic data demonstrate that, compared with the curvelet-based sparsity-promoting L1-norm minimization method and the multichannel singular spectrum analysis method, the proposed method obtains state-of-the-art reconstruction results.

A New Model of Jupiter's Magnetic Field at the Completion of Juno's Prime Mission

AbstractA spherical harmonic model of the magnetic field of Jupiter is obtained from vector magnetic field observations acquired by the Juno spacecraft during 32 of its first 33 polar orbits. These Prime Mission orbits sample Jupiter's magnetic field nearly uniformly in longitude (∼11° separation) as measured at equator crossing. The planetary magnetic field is represented with a degree 30 spherical harmonic and the external field is approximated near the origin with a simple external spherical harmonic of degree 1. Partial solution of the underdetermined inverse problem using generalized inverse techniques yields a model (“JRM33”) of the planetary magnetic field with spherical harmonic coefficients reasonably well determined through degree and order 13. Useful information regarding the field extends through degree 18, well fit by a Lowes' spectrum with a dynamo core radius of 0.81 Rj, presumably the outer radius of the convective metallic hydrogen region. This new model provides a most detailed view of a planetary dynamo and evidence of advection of the magnetic field by deep zonal winds in the vicinity of the Great Blue Spot (GBS), an isolated and intense patch of flux near Jupiter's equator. Comparison of the JRM33 and JRM09 models suggests secular variation of the field in the vicinity of the GBS during Juno's nearly 5 years of operation in orbit about Jupiter. The observed secular variation is consistent with the penetration of zonal winds to a depth of ∼3,500 km where a flow velocity of ∼0.04 ms−1 is required to match the observations.

Sparse Zernike Fitting for Dynamic LAS Tomographic Images of Temperature and Water Vapor Concentration

Laser absorption spectroscopy (LAS) tomography yields cross-sectional images of flame temperature and gas molar concentration in combustions. The total laser absorbance of a specific gas, such as water vapor at each point in the region of interest, was obtained from local normalized second harmonics on limited detectors. These detectors restrict the spatial resolution during image reconstructions through typical iterative methods. A sparse Zernike fitting method was introduced for LAS tomography, as the temperature in real physical reality is smoothly distributed. The distributions of total absorbance were sparsely fit as finite Zernike polynomials, and the coefficients were mostly zero or about zero. The number of coefficients was much smaller than that of pixels, i.e., unknowns in iterative methods. The underdetermined inverse problem was effectively alleviated, and a higher spatial resolution was achieved. A fast convergence and effective method for optimal step selection was also introduced for the iterative solution of the sparse model. The effectiveness of the proposed method was validated by both simulated and experimental data for temperature and concentration imaging. Performance comparisons were made with the typical iterative methods, such as SART and Landweber methods; temporal variations of reconstructed distributions for dynamical flames were captured with higher signal-to-noise ratios. Both the fundamental frequencies of 8.797 Hz without acoustic excitation and 12.7 Hz in the case of 150-Hz excitation were successfully captured, and tomographic images agreed well with the dynamical flame evolutions.

The Advantage Analysis of 3-D Inversion of Airborne Gravity Gradiometry Data With Larger Sampling Interval

Airborne gravity gradiometry is an effective tool for its high efficiency and sensitivity to the interesting buried targets, which has been increasingly introduced in mineral and petroleum exploration. When compared to gravimetry, full tensor gradient (FTG) measurements provide higher resolution and signal-to-noise ratio. In this article, we build and test models with different sampling intervals and compare the performances of individual components or their combinations in minimum structure inversion to evaluate the benefits of FTG combinations. As part of a theoretical study, the singular value spectrum (SVS) and depth-resolution plot (DRP) are examined to determine how much information and resolution different datasets provide in the underdetermined inverse problem. The synthetic examples show that integrating more components in inversion improves the resolution of the recovered model, and the inversion result of gravity or individual component cannot accurately restore the distribution of anomalous bodies. With the increasing sampling interval, joint inversion of FTG data still provides enough information corresponding to the source density distribution, and the limit survey interval is tested through models. In addition, we use a real survey over the Vinton dome as a case study and discover that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V_{\mathrm {zz}}$ </tex-math></inline-formula> component fails to reconstruct the caprock model at coarse line spacing, whereas FTG combinations recover the acceptable geometry flawlessly. It also reflects the benefit of FTG combinations for inversion processing, which reduces the interference of measured noise on inversion results and allows for longer sampling intervals during the survey, effectively lowering the cost of an airborne survey and improving exploration efficiency

Adaptive L p Reconstruction Method for Synthetic Aperture Interferometric Radiometer

The brightness temperature reconstruction of synthetic aperture interferometric radiometers (SAIRs) is an underdetermined and ill-posed inverse problem. The classical Hilbert-space regularization methods have often the disadvantages of the over-smoothness effects and the oscillations. In this paper, an adaptive Lp reconstruction method in Lebesgue spaces is introduced to obtain the accurate brightness temperature map in SAIRs. The proposed approach can adaptively adjust the exponent p between 1.2 and 2 according to the region of the map to be re-constructed. The experimental results show that the Lp reconstruction method can effectively reduce over-smoothness and Gibbs oscillations, and better improve the accuracy of the inverse results, compared to the traditional minimum-norm and band-limited regularization methods.

Hierarchical regularization of solution ambiguity in underdetermined inverse and optimization problems

Estimating modeling parameters based on a prescribed optimization target requires to solve an inverse problem, which is commonly ill-posed. Consequently, either infinitely many or no solutions may exist, depending on whether the system is under- or overdetermined, and whether it is consistent or inconsistent. This paper focuses on scenarios where the solution is ambiguous and infinitely many combinations of possible parameter values can accurately achieve the optimization target. Selecting the most suitable solution requires incorporating additional constraints into the model, which is achieved by regularizing the inverse problem. However, common regularization approaches require the specification of a priori unknown regularization hyperparameters that are difficult and tedious to obtain, and can have a large impact on the result.Here, a novel strategy to reduce the ambiguity of such inverse problems is presented, ensuring that the primary optimization target is always reached accurately. To further reduce the solution space, additional constraints are included, until the optimal modeling parameters are found. Importantly, the required regularization parameters have a direct physical meaning and can be derived sequentially, starting from an initial guess that can be obtained conveniently by solving the system without regularization.By considering several illustrative examples, the applicability of the method is demonstrated, and its potential for various comparable inverse problems is highlighted.

Assimilation of Images via Dictionary Learning-Based Sparsity Regularization Strategy: An Application for Retrieving Fluid Flows

In this work, we propose a structure sparsity regularization strategy in the framework of 4-D variational data assimilation (4-D Var). In meteorology and oceanography, the number of unknown model variables is far fewer than that of image observations, often leading to solve an underdetermined nonlinear inverse problem. In recent years, the l¹-norm-based sparsity regularization approach has attracted great attention in the field of 4-D Var because of its data structure preservation and noise suppression. To avoid little underlying physical priors considered, we introduce a widely used dictionary learning (DL) method to adaptively derive an efficient sparse approximation via learning a basis from a given dataset. For our target application of estimating sea surface flows, we consider a DL sparsity constraint on the variable of flow vorticity due to its rich spatial variation related to flows evolution. A novel anisotropic regularization method combined with fluid dynamics characteristics could overcome magnitude underestimation and staircase artifacts appearing in the gradient regularization-based 4-D Var method. The split Bregman iteration with fast convergence property is employed to solve the l¹+l² nonsmooth minimization problem. The promising fluid flows estimation performance in real test cases (assimilation of image sequences collected from CORIOLIS experimental turntable) demonstrates the efficiency of our approach.

Unsupervised deep learning with higher-order total-variation regularization for multidimensional seismic data reconstruction

In 3D marine seismic acquisition, the seismic wavefield is not sampled uniformly in the spatial directions. This leads to a seismic wavefield consisting of irregularly and sparsely populated traces with large gaps between consecutive sail lines, especially in the near offsets. The problem of reconstructing the complete seismic wavefield from a subsampled and incomplete wavefield is formulated as an underdetermined inverse problem. We have investigated unsupervised deep learning based on a convolutional neural network for multidimensional wavefield reconstruction of irregularly populated traces defined on a regular grid. Our network is based on an encoder-decoder architecture with an overcomplete latent representation, including appropriate regularization penalties to stabilize the solution. We proposed a combination of penalties, which consists of the [Formula: see text]-norm penalty on the network parameters, and a first- and second-order total-variation penalty on the model. We determined the performance of our method on broadband synthetic data and field data represented by constant-offset gathers from a source-over-cable data set from the Barents Sea. In the field data example, we compare the results to a full production flow from a contractor company, which is based on a 5D Fourier interpolation approach. In this example, our approach displays improved reconstruction of the wavefield with less noise in the sparse near offsets compared with the industry approach, which leads to improved structural definition of the near offsets in the migrated sections.

3D denoised completion network for deep single-pixel reconstruction of hyperspectral images.

Single-pixel imaging acquires an image by measuring its coefficients in a transform domain, thanks to a spatial light modulator. However, as measurements are sequential, only a few coefficients can be measured in the real-time applications. Therefore, single-pixel reconstruction is usually an underdetermined inverse problem that requires regularization to obtain an appropriate solution. Combined with a spectral detector, the concept of single-pixel imaging allows for hyperspectral imaging. While each channel can be reconstructed independently, we propose to exploit the spectral redundancy between channels to regularize the reconstruction problem. In particular, we introduce a denoised completion network that includes 3D convolution filters. Contrary to black-box approaches, our network combines the classical Tikhonov theory with the deep learning methodology, leading to an explainable network. Considering both simulated and experimental data, we demonstrate that the proposed approach yields hyperspectral images with higher quantitative metrics than the approaches developed for grayscale images.