Search Results (171)

In this paper, the authors investigate a partial inverse spectral problem for Sturm–Liouville operators with frozen arguments on star-shaped graphs. The problem is to reconstruct the potential on one edge from the known potentials on the other edges together with two sequences of eigenvalues from a prescribed spectral set. The proposed approach is constructive. First, the characteristic function associated with the given spectral data is constructed, allowing the unknown potential contribution to be isolated. The potential is then recovered by expanding the resulting expressions in an appropriate Riesz basis and solving a corresponding system of linear equations. Based on established uniqueness results, this procedure yields a constructive numerical algorithm. Numerical examples demonstrate reliable reconstruction for both smooth and piecewise continuous potentials, providing a practical scheme for frozen-argument problems on star graphs. Full article

In this paper, the half-inverse spectral problem for energy-dependent Sturm–Liouville problems (that is, differential pencils), defined on interval $[0,\pi ]$ with the potential functions $p,q$ being a priori known on the subinterval $[0,\pi /2]$, is considered. We provide a method for the unique reconstruction of the two potential functions on $[\pi /2,\pi ]$ and the boundary condition at $x=\pi$ by using one full spectrum. Consequently, based on the reconstruction method, we also provide a necessary and sufficient condition under which the existence of the quadratic pencil of differential operators is unique. Full article

Chlorophyll serves as a crucial indicator for crop growth monitoring and reflects the health status of crops. Hyperspectral remote sensing technology, leveraging its advantages of repeated observations and high-throughput analysis, provides an effective approach for non-destructive chlorophyll monitoring. However, determining the optimal spectral scale remains the primary bottleneck constraining the widespread application of hyperspectral remote sensing in crop chlorophyll estimation: excessively fine spectral scale readily introduces redundant information, leading to dramatically increased data dimensions and reduced computational efficiency; conversely, overly coarse spectral scale risks losing critical spectral features such as absorption peaks and reflection troughs, thereby compromising model accuracy. Therefore, establishing an appropriate spectral scale that effectively preserves spectral feature information while maintaining computational efficiency is crucial for enhancing the accuracy and practicality of chlorophyll remote sensing estimation. To address this, this study proposes a three-dimensional analytical framework integrating “spectral scale—machine learning algorithm—crop growth stage” to systematically solve the scale optimization problem. Ground-truth measurements and hyperspectral data from five growth stages of winter wheat in Fengyang County, Anhui Province, were collected. Spectral bands sensitive to chlorophyll were analyzed, and four modeling methods—Ridge Regression (RR), K-Nearest Neighbors (KNN), Random Forest (RF), and Support Vector Regression (SVR)—were employed to integrate data from different spectral scales with respective bandwidths of 2, 3, 5, 7, 10, 20, and 50 nanometers (nm). The results evaluated the response characteristics of raw band reflectance to chlorophyll values and its impact on machine learning-based chlorophyll estimation across different spectral scales. Results indicate: (1) Canopy spectra significantly correlated with winter wheat chlorophyll primarily reside in the red and red-edge bands; (2) For single-scale analysis, larger spectral scales (10, 20 nm) enhance monitoring accuracy compared to 1 nm high-resolution data, while medium and small scales (5, 7 nm) may degrade accuracy due to redundant noise introduction. (3) Integrating growth stages, spectral scales, and machine learning revealed optimal monitoring accuracy during the jointing and heading stages using 1–5 nm spectral scales combined with the KNN algorithm. For the booting, flowering, and grain filling stages, the highest accuracy was achieved using 20–50 nm spectral scales combined with either the KNN or RF algorithm. The results indicate that high-precision chlorophyll inversion for winter wheat does not rely on a single fixed model or scale, but rather on the dynamic adaptation of the “scale-model-growth stage” triad. The proposed systematic framework not only provides a theoretical basis for chlorophyll monitoring using multi-platform remote sensing data, but also offers methodological support for future crop-sensing sensor design and data processing strategy optimization. Full article

In the development of quantum mechanics in the 1920s, both matrix mechanics (developed by Born, Heisenberg and Jordon) and wave mechanics (developed by Schrödinger) prevailed. These early attempts corresponded to the quantum mechanics of particles. Matrix mechanics was found to lead directly to the Schrödinger equation, and the Schrödinger equation could be used to derive the alternative problem for matrix mechanics. Later emphasis lay on the development of the dynamics of fields, where the classical field equations were quantized (see, for example, Weinberg). Today, quantum field theory is one of the most successful physical theories ever developed. The symmetry between particle and wave mechanics is exploited herein. One of the important properties of quantum mechanics is that it is linear, leading to some confusion about how to treat the problem of nonlinear classical field equations. In the present paper we address the case of classical nonlinear soliton equations which are exactly integrable in terms of the periodic/quasiperiodic inverse scattering transform. This means that all physical spectral solutions of the soliton equations can be computed exactly for these specific boundary conditions. Unfortunately, such solutions are highly nonlinear, leading to difficulties in solving the associated quantum mechanical problems. Here we find a strategy for developing the quantum mechanical solutions for soliton dynamics. To address this difficulty, we apply a recently derived result for soliton equations, i.e., that all solutions can be written as quasiperiodic Fourier series. This means that soliton equations, in spite of their nonlinear solutions, are perfectly linearizable with quasiperiodic boundary conditions, the topic of finite gap theory, i.e., the inverse scattering transform with periodic/quasiperiodic boundary conditions. We then invoke the result that soliton equations are Hamiltonian, and we are able to show that the generalized coordinates and momenta also have quasiperiodic Fourier series, a generalized linear superposition law, which is valid in the case of nonlinear, integrable classical dynamics and is here extended to quantum mechanics. Hamiltonian dynamics with the quasiperiodicity of inverse scattering theory thus leads to matrix mechanics. This completes the main theme of our paper, i.e., that classical, nonlinear soliton field equations, linearizable with quasiperiodic Fourier series, can always be quantized in terms of matrix mechanics. Thus, the solitons and their nonlinear interactions are given an explicit description in quantum mechanics. Future work will be formulated in terms of the associated Schrödinger equation. Full article

In the present work, we aim to study the inverse problem of recovering an unknown spatial source term in a multi-term time-fractional diffusion equation involving the fractional Laplacian. The forward problem is first analyzed in appropriate fractional Sobolev spaces, establishing the existence, uniqueness, and regularity of solutions. Exploiting the spectral representation of the solution and properties of multinomial Mittag–Leffler functions, we prove uniqueness and derive a stability estimate for the spatial source term from finaltime observations. The inverse problem is then formulated as a Tikhonov regularized optimization problem, for which existence, uniqueness, and strong convergence of the regularized minimizer are rigorously established. On the computational side, we propose an efficient reconstruction algorithm based on the conjugate gradient method, with temporal discretization via an L¹-type scheme for Caputo derivatives and spatial discretization using a Galerkin approach adapted to the nonlocal fractional Laplacian. Numerical experiments confirm the accuracy and robustness of the proposed method in reconstructing the unknown source term. Full article

Integrated optical filters are fundamental and indispensable components of silicon photonics, which enhance the data throughput of high-demand communication networks. Grating-assisted filters have been widely used due to the merits they offer: flat top, low crosstalk, and no FSR. In this paper, we report an inverse-designed narrow-band silicon Bragg grating filter that unites lateral-misalignment apodization with cooperative particle swarm optimization (CPSO). The initial coupling-coefficient profile of the filter is first yielded by a layer-peeling algorithm (LPA). Subsequently, the final structure is designed by CPSO to approach the desired spectral response. The filter is fabricated on a 220 nm silicon-on-insulator platform. The measured results exhibit 3.39 nm bandwidth, 19.34 dB side lobe suppression ratio (SLSR), and 1.75 dB insertion loss. The proposed design method effectively solves the problem of excessively high side lobes in uniform gratings and LPA-designed gratings when designing narrow-bandwidth filters. Full article

The advent of multichannel ground-penetrating radar systems capable of acquiring multiview, multistatic, and multifrequency data is offering new possibilities to improve subsurface imaging performance. However, this raises the need for reconstruction approaches capable of handling such sophisticated configurations and the resulting increase in the data volume. Therefore, the challenge lies in identifying proper measurement configurations that balance image quality with the complexity and duration of data acquisition. As a contribution to this topic, the present paper focuses on a measurement system working in reflection mode and composed of an array of antennas, consisting of a transmitting antenna and several receiving antennas, whose spatial offset is comparable to the probing wavelength. Therefore, for each position of the transmitting antenna, a single-view/multistatic configuration is considered. The imaging task is solved by adopting a linear microwave tomographic approach, which provides a qualitative reconstruction of the investigated scenario. In particular, a 3D inverse scattering problem is tackled for an isotropic, homogeneous, lossless, and non-magnetic medium under the Born approximation, considering both single- and multi-frequency data. A preliminary analysis, referring to a 3D free-space reference scenario, is performed in terms of the spectral content of the scattering operator and the system’s point spread function. Finally, an experimental validation under laboratory conditions is presented in order to verify the expected imaging capability of the inversion approach. Full article

Soil organic matter (SOM) is essential for ecosystem health and agricultural productivity. Accurate prediction of SOM content is critical for modern agricultural management and sustainable soil use. Existing digital soil mapping (DSM) models, when processing temporal data, primarily focus on modeling the changes in input data across successive time steps. However, they do not adequately model the relationships among different input variables, which hinders the capture of complex data patterns and limits the accuracy of predictions. To address this problem, this paper proposes a novel deep learning model, 2-Channel Network (2C-Net), leveraging sequential multi-temporal remote sensing images to improve SOM prediction. The network separates input data into temporal and spatial data, processing them through independent temporal and spatial channels. Temporal data includes multi-temporal Sentinel-2 spectral reflectance, while spatial data consists of environmental covariates including climate and topography. The Multi-sequence Feature Fusion Module (MFFM) is proposed to globally model spectral data across multiple bands and time steps, and the Diverse Convolutional Architecture (DCA) extracts spatial features from environmental data. Experimental results show that 2C-Net outperforms the baseline model (CNN-LSTM) and mainstream machine learning model for DSM, with R² = 0.524, RMSE = 0.884 (%), MAE = 0.581 (%), and MSE = 0.781 (%)². Furthermore, this study demonstrates the significant importance of sequential spectral data for the inversion of SOM content and concludes the following: for the SOM inversion task, the bare soil period after tilling is a more important time window than other bare soil periods. 2C-Net model effectively captures spatiotemporal features, offering high-accuracy SOM predictions and supporting future DSM and soil management. Full article

We address the inverse spectral problem for a PT-symmetric Dirac operator with discontinuity conditions imposed along the entire real axis—a configuration that has not been explicitly solved in prior literature. Our approach constructs fundamental solutions via convergent recursive series expansions and establishes their linear independence through a constant Wronskian. We derive explicit formulas for transmission and reflection coefficients, assemble them into a PT-symmetric scattering matrix, and demonstrate how both spectral and scattering data uniquely determine the underlying complex-valued, discontinuous potentials. Unlike classical treatments, which assume smoothness or limited discontinuities, our framework handles full-axis discontinuities within a non-Hermitian setting, proving uniqueness and providing a constructive recovery algorithm. This method not only generalizes existing inverse scattering theory to PT-symmetric discontinuous operators but also offers direct applicability to optical waveguides, metamaterials, and quantum field models where gain–loss mechanisms and zero-width resonances are critical. Full article

Spectral computed tomography (SCT) enables material decomposition, artifact reduction, and contrast enhancement, leveraging symmetry principles across its technical framework to enhance material differentiation and image quality. However, its nonlinear data acquisition process involving noise and scatter leads to a highly ill-posed inverse problem. To address this, we propose a dual-domain iterative reconstruction network that combines joint learning reconstruction with physical process modeling, which also uses the symmetric complementary properties of the two domains for optimization. A dedicated physical module models the SCT forward process to ensure stability and accuracy, while a residual-to-residual strategy reduces the computational burden of model-based iterative reconstruction (MBIR). Our method, which won the AAPM DL-Spectral CT Challenge, achieves high-accuracy material decomposition. Extensive evaluations also demonstrate its robustness under varying noise levels, confirming the method’s generalizability. This integrated approach effectively combines the strengths of physical modeling, MBIR, and deep learning. Full article

High-quality multispectral LiDAR (MSL) data are crucial for land cover (LC) classification. However, the Titan MSL system encounters challenges of inconsistent spatial–spectral information due to its unique scanning and data saving method, restricting subsequent classification accuracy. Existing spectral reconstruction methods often require empirical parameter settings and involve high computational costs, limiting automation and complicating application. To address this problem, we introduce the dual attention spectral optimization reconstruction network (DossaNet), leveraging an attention mechanism and spatial–spectral information. DossaNet can adaptively adjust weight parameters, streamline the multispectral point cloud acquisition process, and integrate it into classification models end-to-end. The experimental results show the following: (1) DossaNet exhibits excellent generalizability, effectively recovering accurate LC spectra and improving classification accuracy. Metrics across the six classification models show some improvements. (2) Compared with the method lacking spectral reconstruction, DossaNet can improve the overall accuracy (OA) and average accuracy (AA) of PointNet++ and RandLA-Net by a maximum of 4.8%, 4.47%, 5.93%, and 2.32%. Compared with the inverse distance weighted (IDW) and k-nearest neighbor (KNN) approach, DossaNet can improve the OA and AA of PointNet++ and DGCNN by a maximum of 1.33%, 2.32%, 0.86%, and 2.08% (IDW) and 1.73%, 3.58%, 0.28%, and 2.93% (KNN). The findings further validate the effectiveness of our proposed method. This method provides a more efficient and simplified approach to enhancing the quality of multispectral point cloud data. Full article

The quantitative characterization of near-surface heterogeneity using ground-penetrating radar (GPR) is an important but challenging task. The estimation of subsurface geostatistical parameters from surface-based common-offset GPR reflection data has so far relied upon a Monte-Carlo-type inversion approach. This allows for a comprehensive exploration of the parameter space and provides some measure of uncertainty with regard to the inferred results. However, the associated computational costs are inherently high. To alleviate this problem, we present an alternative deep-learning-based technique, that, once trained in a supervised context, allows us to perform the same task in a highly efficient manner. The proposed approach uses a convolutional neural network (CNN), which is trained on a vast database of autocorrelations obtained from synthetic GPR images for a comprehensive range of stochastic subsurface models. An important aspect of the training process is that the synthetic GPR data are generated using a computationally efficient approximate solution of the underlying physical problem. This strategy effectively addresses the notorious challenge of insufficient training data, which frequently impedes the application of deep-learning-based methods in applied geophysics. Tests on a wide range of realistic synthetic GPR data generated using a finite-difference time-domain (FDTD) solution of Maxwell’s equations, as well as a comparison with the results of the traditional Monte Carlo approach on a pertinent field dataset, confirm the viability of the proposed method, even in the presence of significant levels of data noise. Our results also demonstrate that typical mismatches between the dominant frequencies of the analyzed and training data can be readily alleviated through simple spectral shifting. Full article

The article presents a new method of passive dynamic weighing of vehicles based on the registration of seismic signals that occur when wheels pass through strips specially applied to the road surface. Signal processing is carried out using spectral methods, including fast Fourier transform, consistent filtering, and regularization methods for solving inverse problems. Special attention is paid to the use of linear-frequency-modulated signals, which make it possible to distinguish the responses of individual axes even when superimposed. Field tests were carried out on a real section of the road, during which signals from vehicles of various classes were recorded using eight geophones. The average error in determining the speed of 1.2 km/h and the weight of 8.7% was experimentally achieved, while the correct determination of the number of axles was 96.5%. The results confirm the high accuracy and sustainability of the proposed approach with minimal implementation costs. It is shown that this system can be scaled up for use in intelligent transport systems and applied in real traffic conditions without the need to intervene in the design of the roadway. Full article

The outdoor spectral detection of winter jujube quality is affected by complex ambient light and surface heterogeneity, resulting in limited inversion accuracy. To address this problem, this study proposes a correction method for outdoor spectral inversion based on the bidirectional polarization reflectance distribution function (BPDF) model. It was used to enhance the detection accuracy of water content and soluble solid (SSC) content of winter jujube. Experimentally, 900–1750 nm hyperspectral data of ripe winter jujube samples were collected at non-polarization and 0°, 45°, 90°, and 135° polarization azimuths. The spectra were inverted using four semi-empirical BPDF models, Nadal–Breon, Litvinov, Maignan and Xie–Cheng, and the corrected spectra were obtained by mean fusion. The quality prediction models are subsequently combined with the competitive adaptive reweighting algorithm (CARS) and partial least squares (PLS). The results showed that the modified spectra significantly optimized the prediction performance. The prediction set correlation coefficients (Rp) of the water content and SSC models were improved by 10–30% compared with the original spectra. The percentage of models with RPIQ values greater than 2 increased from 40% to 60%. Among them, the Litvinov model performs outstandingly in the direction of no polarization and 135° polarization, with the highest Rp of 0.8829 for water content prediction and RPIQ of 2.54. The Xie–Cheng model has an RPIQ of 2.64 for SSC prediction at 90° polarization, which shows the advantage of sensitivity to the deeper constituents. The different models complemented each other in multi-polarization scenarios. The Nadal–Breon model was suitable for epidermal reflection-dominated scenarios, and the Maignan model efficiently coupled epidermal and internal moisture characteristics through the moisture sensitivity index. The study verifies the effectiveness of the spectral correction method based on the BPDF model for outdoor quality detection of winter jujube, which provides a new path for the spectral detection of agricultural products in complex environments. In the future, it is necessary to further optimize the dynamic adjustment mechanism of the model parameters and improve the ability of environmental interference correction by combining multi-source data fusion. Full article

This study introduces a data-driven and machine learning (ML)-based methodology for converting the encounter wave frequency spectrum to the original wave spectrum, a critical process for navigating vessels with forward speed in various control and adjustment missions. The spectral conversion from the encounter- to original-frequency domain faces challenges under certain wave conditions due to the non-uniqueness of the inverse problem. To resolve these challenges, the authors developed an artificial neural network (ANN) model that transforms the encounter-frequency spectrum into the original wave spectrum at a given vessel speed and wave direction. The model was trained and validated using a large dataset mapped from various JONSWAP wave spectra to the corresponding encounter-frequency spectra for various vessel speeds and wave parameters. The hyperparameters of the ANN model were subsequently tested and optimized. The results demonstrate that the ANN model can effectively predict the original wave spectrum with high accuracy, as evidenced by a favorable R2 value and error distribution analysis. This approach not only enhances the reliability of wave spectrum estimation during maritime navigation but also broadens the capability of real-time operational controls and adjustments. Full article