Source Reconstruction Problem Research Articles

EEG dynamic source imaging using a regularized optimization with spatio-temporal constraints.

One of the most important needs in neuroimaging is brain dynamic source imaging with high spatial and temporal resolution. EEG source imaging estimates the underlying sources from EEG recordings, which provides enhanced spatial resolution with intrinsically high temporal resolution. To ensure identifiability in the underdetermined source reconstruction problem, constraints on EEG sources are essential. This paper introduces a novel method for estimating source activities based on spatio-temporal constraints and a dynamic source imaging algorithm. The method enhances time resolution by incorporating temporal evolution of neural activity into a regularization function. Additionally, two spatial regularization constraints based on and norms are applied in the transformed domain to address both focal and spread neural activities, achieved through spatial gradient and Laplacian transform. Performance evaluation, conducted quantitatively using synthetic datasets, discusses the influence of parameters such as source extent, number of sources, correlation level, and SNR level on temporal and spatial metrics. Results demonstrate that the proposed method provides superior spatial and temporal reconstructions compared to state-of-the-art inverse solutions including STRAPS, sLORETA, SBL, dSPM, and MxNE. This improvement is attributed to the simultaneous integration of transformed spatial and temporal constraints. When applied to a real auditory ERP dataset, our algorithm accurately reconstructs brain source time series and locations, effectively identifying the origins of auditory evoked potentials. In conclusion, our proposed method with spatio-temporal constraints outperforms the state-of-the-art algorithms in estimating source distribution and time courses.

Forward and Inverse Source Reconstruction Problems for the Equations of Vibrations of a Rectangular Plate

Forward and Inverse Source Reconstruction Problems for the Equations of Vibrations of a Rectangular Plate

Forward and Inverse Source Reconstruction Problems for the Equations of Vibrations of a Rectangular Plate

For the equation of vibrations of a rectangular plate, the initial-boundary value and inverse problems of finding the right-hand side (the source of vibrations) are studied. Solutions of the problems are constructed explicitly as sums of series, and the corresponding uniqueness and existence theorems are proved. When substantiating the existence of a solution to the inverse problem of determining the factor on the right-hand side, which depends on spatial coordinates, the problem of small denominators of two natural variables arises, for which estimates of the separation from zero with the corresponding asymptotics are established. These estimates made it possible to prove the existence theorem for this problem in the class of regular solutions by imposing certain smoothness conditions on the given boundary functions.

A Genetic Algorithm for Pointwise Source Reconstruction by the Method of Fundamental Solutions

Inverse source reconstruction problems offer great potential for applications of interest to engineering, such as the identification of polluting sources, and to medicine, such as electroencephalography, to cite at least two relevant examples. From a mathematical point of view, the identification of a concentrated source (intensity and location) corresponds to the identification of the centroid (location) and size (intensity) of a distributed source. On the other hand, from a numerical point of view, it is observed that the use of domain discretization methods is intrinsically associated with the introduction of numerical noise in reconstruction algorithms, which is strongly inadvisable since inverse problems are reckoned to be ill-posed. The objective of this work is to explore, in the context of a Poisson problem and taking into account a numerical point of view, a new reconstruction algorithm based on the method of fundamental solutions, where a source point adequately represents the pointwise source within the domain. The inverse problem is reformulated as an optimization problem solved through a genetic algorithm. Finally, numerical examples are performed to analyze the accuracy of the proposed algorithm for two and three dimensions.

Sparsity-enhanced equivalent source method for acoustic source reconstruction via the Generalized Minimax-Concave penalty

The Equivalent Source Method (ESM) is a powerful technique for acoustic source reconstruction, which has been widely used in noise control and machinery fault detection. Because the inverse acoustic source reconstruction problem is typically ill-conditioned, regularization techniques are often adopted in ESM to achieve meaningful solutions. Based on the sparse distribution assumption of acoustic sources, sparse regularization can be utilized in ESM to promote the spatial resolution of reconstructed results. Existing sparse ESMs often adopt convex l1 norm or nonconvex lp norm as the regularization terms. However, l1 norm-regularized ESM suffers from an insufficient sparsity-inducing problem, which decreases the spatial resolution and reconstruction accuracy. Meanwhile, reconstructed results of lp norm-regularized ESM are often unstable due to multiple local minimas. In this paper, a sparsity-enhanced ESM is proposed, which introduces a nonconvex Generalized Minimax-Concave (GMC) penalty into ESM as the regularization term. The GMC penalty can enhance the sparsity of solutions and simultaneously maintain the convexity of the overall objective function in acoustic source reconstruction. Consequently, the GMC penalty-regularized ESM can improve the spatial resolution and reconstruction accuracy of the reconstructed results. To solve the acoustic source reconstruction problem rapidly, a computation framework based on Alternating Direction Method of Multipliers (ADMM) is derived. Simulation studies indicate that the proposed method can improve the acoustic source reconstruction accuracy in a wide frequency band, compared with existing l1 and lp norm-regularized ESMs. Two experimental cases further verify that the proposed sparsity-enhanced ESM can effectively reconstruct the acoustic sources with high spatial resolution and reconstruction accuracy at higher frequencies.

Detection of time-varying heat sources using an analytic forward model

We present a simple, analytic point source model for both static and time-varying point-like heat sources and the resulting temperature profile that solves the heat equation in dimension three. Simple algorithms to detect the location and spectral content of these sources are developed and numerically tested using Finite Element Mesh simulations. The resulting framework for heat source reconstruction problems, which are ill-posed inverse problems, seems promising and warrants for future research. Possible fields of application for our work are material testing, to detect manufacturing defects, and medical imaging to detect abnormal health conditions.

Numerical modeling of the 2015 Illapel Tsunami source by the inversion of DART buoys data

The problem of the tsunami source reconstruction from the remote measurements of an arriving waves in the deep-water tsunamimeters is considered as an inverse problem of mathematical physics in the class of the ill-posed problems. The latter determines the inevitable instability of the numerical solution. We propose a method providing more control on the numerical instability of the solution. The DART buoys data allows one to apply the linear shallow-water theory for the wave propagation and the r-solution method for the tsunami waveform inversion. The applied approach is independent from the earthquake source characteristics, because the only observed tsunami waveforms and a roughly estimated tsunami source area are used. A good coincidence of observed and computed tsunami waveforms was a criterion of the quality of the inversion. In this paper we study the 16 September 2015 Illapel (Chile) event and conduct a reconstruction of a tsunami source.

Sparse Lens Inversion Technique (SLIT): lens and source separability from linear inversion of the source reconstruction problem

Strong gravitational lensing offers a wealth of astrophysical information on the background source it affects, provided the lensed source can be reconstructed as if it was seen in the absence of lensing. In the present work, we illustrate how sparse optimisation can address the problem. As a first step towards a full free-form-lens-modelling technique, we consider linear inversion of the lensed source under sparse regularisation and joint deblending from the lens light profile. The method is based on morphological component analysis, assuming a known mass model. We show with numerical experiments that representing the lens and source light using an undecimated wavelet basis allows us to reconstruct the source and to separate it from the foreground lens at the same time. Both the source and lens light have a non-analytic form, allowing for the flexibility needed in the inversion to represent arbitrarily small and complex luminous structures in the lens and source. In addition, sparse regularisation avoids over-fitting the data and does not require the use of an adaptive mesh or pixel grid. As a consequence, our reconstructed sources can be represented on a grid of very small pixels. Sparse regularisation in the wavelet domain also allows for automated computation of the regularisation parameter, thus minimising the impact of the arbitrary choice of initial parameters. Our inversion technique for a fixed mass distribution can be incorporated into future lens-modelling techniques iterating over the lens mass parameters.

Source time reversal (STR) method for linear elasticity

We study the problem of source reconstruction for a linear elasticity problem applied to seismicity induced by mining. We assume the source is written as a variable separable function f(x)g(t). We first present a simple proof a local decay result for elasticity in the case of homogeneous media. We then extend the source time reversal method, originally developed for acoustic waves, to an elastic system of waves. Additionally, we present a fast reconstruction implementation for large data sets. This is especially useful in the elastic case, in which the numerical cost is higher than in fluid acoustics. We complement this work with several 2D and 3D numerical experiments and an analysis of the results.

Optoacoustic inversion via convolution kernel reconstruction in the paraxial approximation and beyond

In this article we address the numeric inversion of optoacoustic signals to initial stress profiles. Therefore we study a Volterra integral equation of the second kind that describes the shape transformation of propagating stress waves in the paraxial approximation of the underlying wave-equation. Expanding the optoacoustic convolution kernel in terms of a Fourier-series, a best fit to a pair of observed near-field and far-field signals allows to obtain a sequence of expansion coefficients that describe a given “apparative” setup. The resulting effective kernel is used to solve the optoacoustic source reconstruction problem using a Picard-Lindelöf correction scheme. We verify the validity of the proposed inversion protocol for synthetic input signals and explore the feasibility of our approach to also account for the shape transformation of signals beyond the paraxial approximation including the inversion of experimental data stemming from measurements on melanin doped PVA hydrogel tissue phantoms.

An Algorithm for Source Reconstruction in Nonlinear Shallow-Water Equations

A numerical algorithm is proposed to solve the source reconstruction problem for a system of nonlinear shallow-water equations using the dynamics of water surface perturbation measured at a finite number of spatial points and/or over a part of the surface at a fixed time. The combined inverse problem under study is reduced to the minimization of an objective functional characterizing the quadratic deviation of simulated data from measured data (a misfit function). An explicit expression for the gradient of the misfit function is obtained. The direct and conjugate problems within the framework of shallow-water equations are solved by the finite volume method. The numerical results are analyzed and compared with experimental data.

Hierarchical multiscale Bayesian algorithm for robust MEG/EEG source reconstruction

In this paper, we present a novel hierarchical multiscale Bayesian algorithm for electromagnetic brain imaging using magnetoencephalography (MEG) and electroencephalography (EEG). In particular, we present a solution to the source reconstruction problem for sources that vary in spatial extent. We define sensor data measurements using a generative probabilistic graphical model that is hierarchical across spatial scales of brain regions and voxels. We then derive a novel Bayesian algorithm for probabilistic inference with this graphical model. This algorithm enables robust reconstruction of sources that have different spatial extent, from spatially contiguous clusters of dipoles to isolated dipolar sources. We compare the new algorithm with several representative benchmarks on both simulated and real brain activities. The source locations and the correct estimation of source time courses used for the simulated data are chosen to test the performance on challenging source configurations. In simulations, performance of the novel algorithm shows superiority to several existing benchmark algorithms. We also demonstrate that the new algorithm is more robust to correlated brain activity present in real MEG and EEG data and is able to resolve distinct and functionally relevant brain areas with real MEG and EEG datasets.

On the inverse problem of source reconstruction from coherence measurements.

We consider an inverse source problem for partially coherent light propagating in the Fresnel regime. The data are the coherence of the field measured away from the source. The reconstruction is based on a minimum residue formulation, which uses the authors' recent closed-form approximation formula for the coherence of the propagated field. The developed algorithms require a small data sample for convergence and yield stable inversion by exploiting information in the coherence as opposed to intensity-only measurements. Examples with both simulated and experimental data demonstrate the ability of the proposed approach to simultaneously recover complex sources in different planes transverse to the direction of propagation.

On the inverse problem of source reconstruction from coherence measurements

We consider an inverse source problem for partially coherent light propagating in the Fresnel regime. The data are the coherence of the field measured away from the source. The reconstruction is based on a minimum residue formulation, which uses the authors’ recent closed-form approximation formula for the coherence of the propagated field. The developed algorithms require a small data sample for convergence and yield stable inversion by exploiting information in the coherence as opposed to intensity-only measurements. Examples with both simulated and experimental data demonstrate the ability of the proposed approach to simultaneously recover complex sources in different planes transverse to the direction of propagation.

Energy Dependent Source Reconstructions via Explicit Formulations of the Adjoint Particles Flux

An analytical solution to the discrete ordinates approximation of the adjoint to the multigroup transport equation in one-dimensional slab geometry is developed in this work, in order to be used in source estimation problems. The solution is firstly tested in a source-detector problem, where explicit expressions are derived to approximate the absorption rate of particles of internal detectors. Noisy data generated from readings of internal detectors were then used, along with the adjoint formulation, in an iterative process to solve an energy-dependent inverse problem of source reconstruction in the cases of polynomials and piecewise constant sources. The approach is shown to be fast and accurate.

Numerical and Experimental Assessment of Source Reconstruction for Very Near-Field Measurements With an Array of $H$ -Field Probes

This paper presents a novel formulation of the source reconstruction problem that aims at increasing the accuracy of near-field (NF) to far-field (FF) transformations in a particular measurement arrangement, where the NF data are rapidly acquired by a planar array of H-field probes located close to a ground plane. A source reconstruction algorithm is proposed for obtaining the near electric and magnetic fields as well as the FF radiation pattern of the antenna under test after filtering the effects of the currents induced on the ground plane backing the probe array. This algorithm exploits a system of integral equations that, after discretization according to the method of moments, enables backprojecting the measured fields on equivalent sources distributed over an arbitrary 3-D surface. Different integral equations are considered for that purpose. The accuracy provided by four different integral formulations is investigated from measured and synthetic experimental data and compared with the standard plane wave spectrum reconstruction technique, making it possible to arrive at a conclusion as to which one to use in order to make the final analysis.

Application of Inverse Source Reconstruction to Conformal Antennas Synthesis

The inverse source problem for a 2-D geometry is considered in the circular geometry. The mathematical relationship between the radiated pattern and the source current is investigated by the appropriate singular value decomposition for different 2-D planar and curved shapes. The spectral analysis allows discussing the number of degrees of freedom of each geometry and reveals a peculiar common passband behavior of the radiation operator. For some geometries, this can significantly affect the inverse source reconstruction problem by reducing the class of sources that can be stably reconstructed, and therefore, it can impact the achievable resolution. In addition, when the source is made of an arc of circumference, the number of degrees of freedom is reduced. The numerical results show how this affects the class of patterns that can be radiated and, consequently, be of interest in conformal sources antenna synthesis problems.

Effective one-dimensional approach to the source reconstruction problem of three-dimensional inverse optoacoustics.

The direct problem of optoacoustic signal generation in biological media consists of solving an inhomogeneous three-dimensional (3D) wave equation for an initial acoustic stress profile. In contrast, the more defiant inverse problem requires the reconstruction of the initial stress profile from a proper set of observed signals. In this article, we consider an effectively 1D approach, based on the assumption of a Gaussian transverse irradiation source profile and plane acoustic waves, in which the effects of acoustic diffraction are described in terms of a linear integral equation. The respective inverse problem along the beam axis can be cast into a Volterra integral equation of the second kind for which we explore here efficient numerical schemes in order to reconstruct initial stress profiles from observed signals, constituting a methodical progress of computational aspects of optoacoustics. In this regard, we explore the validity as well as the limits of the inversion scheme via numerical experiments, with parameters geared toward actual optoacoustic problem instances. The considered inversion input consists of synthetic data, obtained in terms of the effectively 1D approach, and, more generally, a solution of the 3D optoacoustic wave equation. Finally, we also analyze the effect of noise and different detector-to-sample distances on the optoacoustic signal and the reconstructed pressure profiles.

A Sparse Dictionary Learning Framework to Discover Discriminative Source Activations in EEG Brain Mapping

Electroencephalography (EEG) source analysis is one of the most important noninvasive human brain imaging tools that provides millisecond temporal accuracy. However, discovering essential activated brain sources associated with different brain status is still a challenging problem. In this study, we propose for the first time that the ill-posed EEG inverse problem can be formulated and solved as a sparse over-complete dictionary learning problem. In particular, a novel supervised sparse dictionary learning framework was developed for EEG source reconstruction. A revised version of discriminative K-SVD (DK-SVD) algorithm is exploited to solve the formulated supervised dictionary learning problem. As the proposed learning framework incorporated the EEG label information of different brain status, it is capable of learning a sparse representation that reveal the most discriminative brain activity sources among different brain states. Compared to the state-of-the-art EEG source analysis methods, proposed sparse dictionary learning framework achieved significant superior performance in both computing speed and accuracy for the challenging EEG source reconstruction problem through extensive numerical experiments. More importantly, the experimental results also validated that the proposed sparse learning framework is effective to discover the discriminative task-related brain activation sources, which shows the potential to advance the high resolution EEG source analysis for real-time non-invasive brain imaging research.

Energy-Constrained Distributed Learning and Classification by Exploiting Relative Relevance of Sensors’ Data

We consider the problem of communicating data from energy-constrained distributed sensors. To reduce energy requirements, we go beyond the source reconstruction problem classically addressed, and focus on the problem where the recipient wants to perform supervised learning and classification on the data received from the sensors. Restricting our attention to a noiseless communication setting under simplistic Gaussian source assumptions, we study supervised learning and classification under total energy limitations. The energy constraints are modeled in two ways: 1) a linear scaling and 2) an exponential scaling of energy with number of bits used for compression at sensors. We first assume that the underlying parameters for Gaussian distributions have already been learned, and obtain (with linear scaling, reverse-waterfilling-type) strategies for allocating energy, and thus, bits, across different sensors under these two models. Intuitively, these strategies allocate larger rates and energies to sensors that are more “relevant” for the classification goal. These strategies are used to obtain an achievable bound on the tradeoff between energy and error-probability (classification risk). We then provide an algorithm for learning the distribution-parameters of the sensor-data under energy constraints to arrive at high-reliability energy-allocation strategies, while enabling the energy-allocation algorithm to backtrack when the underlying distributions change, or when there is noise in sensed data that can push the algorithm toward a local minimum. Finally, we provide numerical results on energy-savings for classification of simulated data as well as neural data acquired from electrocorticography (ECoG) experiments.