Neural networks for computing and denoising the continuous nonlinear Fourier spectrum in focusing nonlinear Schrödinger equation

We combine the nonlinear Fourier transform (NFT) signal processing with machine learning methods for solving the direct spectral problem associated with the nonlinear Schrödinger equation. The latter is one of the core nonlinear science models emerging in a range of applications. Our focus is on the unexplored problem of computing the continuous nonlinear Fourier spectrum associated with decaying profiles, using a specially-structured deep neural network which we coined NFT-Net. The Bayesian optimisation is utilised to find the optimal neural network architecture. The benefits of using the NFT-Net as compared to the conventional numerical NFT methods becomes evident when we deal with noise-corrupted signals, where the neural networks-based processing results in effective noise suppression. This advantage becomes more pronounced when the noise level is sufficiently high, and we train the neural network on the noise-corrupted field profiles. The maximum restoration quality corresponds to the case where the signal-to-noise ratio of the training data coincides with that of the validation signals. Finally, we also demonstrate that the NFT b-coefficient important for optical communication applications can be recovered with high accuracy and denoised by the neural network with the same architecture.

A Full-Aperture Image Synthesis Method for the Rotating Rectangular Aperture System Using Fourier Spectrum Restoration

The novel rotating rectangular aperture (RRA) system provides a good solution for space-based, large-aperture, high-resolution imaging tasks. Its imaging quality depends largely on the image synthesis algorithm, and the mainstream multi-frame deblurring approach is sophisticated and time-consuming. In this paper, we propose a novel full-aperture image synthesis algorithm for the RRA system, based on Fourier spectrum restoration. First, a numerical simulation model is established to analyze the RRA system’s characteristics and obtain the point spread functions (PSFs) rapidly. Then, each image is used iteratively to calculate the increment size and update the final restored Fourier spectrum. Both the simulation’s results and the practical experiment’s results show that our algorithm performs well in terms of objective evaluation and time consumption.

Elastic wave dispersion equation considering material and geometric nonlinearities

In this letter, we describe the propagation of longitudinal waves in one-dimensional nonlinear elastic thin rods with material nonlinearity and geometric nonlinearity. Mathematical analysis is used to derive the analytical dispersion relationship of longitudinal waves; subsequently, the numerical Fourier spectrum method is used to solve the nonlinear wave equation directly and the results are used to verify the correctness of the derived nonlinear dispersion relation.

Retrieving the refractive index of a sphere from the phase spectrum of its light-scattering profile

We studied the Fourier spectrum of the light-scattering profiles of single particles in the Rayleigh-Gans-Debye (RGD) and Wentzel–Kramers–Brillouin (WKB) approximations. In the case of a homogeneous sphere, we found the relationship between the key parameters of the spectrum (including its phase) and the sphere characteristics – both analytically and numerically in the framework of the approximations and the rigorous Lorentz–Mie theory, respectively. Based on these results, we have improved the existing spectral characterization method for spheres extending the applicability range to particles with a higher refractive index.

Efficient Fourier Single-Pixel Imaging with Gaussian Random Sampling

Fourier single-pixel imaging (FSI) is a branch of single-pixel imaging techniques. It allows any image to be reconstructed by acquiring its Fourier spectrum by using a single-pixel detector. FSI uses Fourier basis patterns for structured illumination or structured detection to acquire the Fourier spectrum of image. However, the spatial resolution of the reconstructed image mainly depends on the number of Fourier coefficients sampled. The reconstruction of a high-resolution image typically requires a number of Fourier coefficients to be sampled. Consequently, a large number of single-pixel measurements lead to a long data acquisition time, resulting in imaging of a dynamic scene challenging. Here we propose a new sampling strategy for FSI. It allows FSI to reconstruct a clear and sharp image with a reduced number of measurements. The key to the proposed sampling strategy is to perform a density-varying sampling in the Fourier space and, more importantly, the density with respect to the importance of Fourier coefficients is subject to a one-dimensional Gaussian function. The final image is reconstructed from the undersampled Fourier spectrum through compressive sensing. We experimentally demonstrate the proposed method is able to reconstruct a sharp and clear image of 256 × 256 pixels with a sampling ratio of 10%. The proposed method enables fast single-pixel imaging and provides a new approach for efficient spatial information acquisition.