Multi-Incidence Holographic Profilometry for Large Gradient Surfaces with Sub-Micron Focusing Accuracy

Surface reconstruction for micro-samples with large discontinuities using digital holography is a challenge. To overcome this problem, multi-incidence digital holographic profilometry (MIDHP) has been proposed. MIDHP relies on the numerical generation of the longitudinal scanning function (LSF) for reconstructing the topography of the sample with large depth and high axial resolution. Nevertheless, the method is unable to reconstruct surfaces with large gradients due to the need of: (i) high precision focusing that manual adjustment cannot fulfill and (ii) preserving the functionality of the LSF that requires capturing and processing many digital holograms. In this work, we propose a novel MIDHP method to solve these limitations. First, an autofocusing algorithm based on the comparison of shapes obtained by the LSF and the thin tilted element approximation is proposed. It is proven that this autofocusing algorithm is capable to deliver in-focus plane localization with submicron resolution. Second, we propose that wavefield summation for the generation of the LSF is carried out in Fourier space. It is shown that this scheme enables a significant reduction of arithmetic operations and can minimize the number of Fourier transforms needed. Hence, a fast generation of the LSF is possible without compromising its accuracy. The functionality of MIDHP for measuring surfaces with large gradients is supported by numerical and experimental results.

Maximisers for Strichartz inequalities on the torus

We study the existence of maximisers for a one-parameter family of Strichartz inequalities on the torus. In general, maximising sequences can fail to be precompact in L 2 ( T ) , and maximisers can fail to exist. We provide a sufficient condition for precompactness of maximising sequences (after translation in Fourier space), and verify the existence of maximisers for a range of values of the parameter. Maximisers for the Strichartz inequalities correspond to stable, periodic (in space and time) solutions of a model equation for optical pulses in a dispersion-managed fiber.

Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors

We show that the Randomized Benchmarking (RB) protocol is a convolution amenable to Fourier space analysis. By adopting the mathematical framework of Fourier transforms of matrix-valued functions on groups established in recent work from Gowers and Hatami \cite{GH15}, we provide an alternative proof of Wallman's \cite{Wallman2018} and Proctor's \cite{Proctor17} bounds on the effect of gate-dependent noise on randomized benchmarking. We show explicitly that as long as our faulty gate-set is close to the targeted representation of the Clifford group, an RB sequence is described by the exponential decay of a process that has exactly two eigenvalues close to one and the rest close to zero. This framework also allows us to construct a gauge in which the average gate-set error is a depolarizing channel parameterized by the RB decay rates, as well as a gauge which maximizes the fidelity with respect to the ideal gate-set.

X-ray powder diffraction in education. Part I. Bragg peak profiles

A collection of scholarly scripts dealing with the mathematics and physics of peak profile functions in X-ray powder diffraction has been written using the Wolfram language in Mathematica. Common distribution functions, the concept of convolution in real and Fourier space, instrumental aberrations, and microstructural effects are visualized in an interactive manner and explained in detail. This paper is the first part of a series dealing with the mathematical description of powder diffraction patterns for teaching and education purposes.

Uniform decay rates of a Bresse thermoelastic system in the whole space

In this paper, we investigate the decay properties of the thermoelastic Bresse system in the whole space. We consider many cases depending on the parameters of the model and we establish new decay rates. We need to mention here that, in some cases we don’t have the regularity-loss phenomena as in the previous works in the literature. To prove our results, we use the energy method in the Fourier space to build a very delicate Lyapunov functionals that give the desired results.

Learning Deeper Non-Monotonic Networks by Softly Transferring Solution Space

Different from popular neural networks using quasiconvex activations, non-monotonic networks activated by periodic nonlinearities have emerged as a more competitive paradigm, offering revolutionary benefits: 1) compactly characterizing high-frequency patterns; 2) precisely representing high-order derivatives. Nevertheless, they are also well-known for being hard to train, due to easily over-fitting dissonant noise and only allowing for tiny architectures (shallower than 5 layers). The fundamental bottleneck is that the periodicity leads to many poor and dense local minima in solution space. The direction and norm of gradient oscillate continually during error backpropagation. Thus non-monotonic networks are prematurely stuck in these local minima, and leave out effective error feedback. To alleviate the optimization dilemma, in this paper, we propose a non-trivial soft transfer approach. It smooths their solution space close to that of monotonic ones in the beginning, and then improve their representational properties by transferring the solutions from the neural space of monotonic neurons to the Fourier space of non-monotonic neurons as the training continues. The soft transfer consists of two core components: 1) a rectified concrete gate is constructed to characterize the state of each neuron; 2) a variational Bayesian learning framework is proposed to dynamically balance the empirical risk and the intensity of transfer. We provide comprehensive empirical evidence showing that the soft transfer not only reduces the risk of non-monotonic networks on over-fitting noise, but also helps them scale to much deeper architectures (more than 100 layers) achieving the new state-of-the-art performance.

An FFT-based approach for Bloch wave analysis: application to polycrystals

A method based on the Fast Fourier Transform is proposed to obtain the dispersion relation of acoustic waves in heterogeneous periodic media with arbitrary microstructures. The microstructure is explicitly considered using a voxelized Representative Volume Element (RVE). The dispersion diagram is obtained solving an eigenvalue problem for Bloch waves in Fourier space. To this aim, two linear operators representing stiffness and mass are defined through the use of differential operators in Fourier space. The smallest eigenvalues are obtained using the implicitly restarted Lanczos and the subspace iteration methods, and the required inverse of the stiffness operator is done using the conjugate gradient with a preconditioner. The method is used to study the propagation of acoustic waves in elastic polycrystals, showing the strong effect of crystal anistropy and polycrystaline texture on the propagation. It is shown that the method combines the simplicity of classical Fourier series analysis with the versatility of Finite Elements to account for complex geometries proving an efficient and general approach which allows the use of large RVEs in 3D.

Sparsity-promoting approach to polarization analysis of seismic signals in the time-frequency domain

In this paper, we present a new approach to the TF-domain PA methods. More precisely, we provide an in-detailed discussion on rearranging the eigenvalue decomposition polarization analysis (EDPA) formalism in the frequency domain to obtain the frequency-dependent polarization properties from the Fourier coefficients owing to the Fourier space orthogonality. Then, by extending the formulation to the TF-domain and incorporating sparsity-promoting time-frequency representation (SP-TFR), we alleviate the limited resolution when estimating the TFdomain polarization parameters. The final details of the technique are to apply an adaptive sparsity-promoting time-frequency filtering (SP-TFF) to extract and filter different phases of the seismic wave. By processing earthquake waveforms, we show that by combining amplitude, directivity, and rectilinearity attributes on the sparse TF-domain polarization map of the signal, we are able to extract or filter different phases of seismic waves.