Fourier-Bessel Basis Research Articles

Exact modeling of power spectrum multipole through spherical Fourier-Bessel basis

Exact modeling of power spectrum multipole through spherical Fourier-Bessel basis

FAST EXPANSION INTO HARMONICS ON THE DISK: A STEERABLE BASIS WITH FAST RADIAL CONVOLUTIONS.

We present a fast and numerically accurate method for expanding digitized images representing functions on [-1, 1]2 supported on the disk in the harmonics (Dirichlet Laplacian eigenfunctions) on the disk. Our method, which we refer to as the Fast Disk Harmonics Transform (FDHT), runs in operations. This basis is also known as the Fourier-Bessel basis, and it has several computational advantages: it is orthogonal, ordered by frequency, and steerable in the sense that images expanded in the basis can be rotated by applying a diagonal transform to the coefficients. Moreover, we show that convolution with radial functions can also be efficiently computed by applying a diagonal transform to the coefficients.

Fast principal component analysis for cryo-electron microscopy images.

Principal component analysis (PCA) plays an important role in the analysis of cryo-electron microscopy (cryo-EM) images for various tasks such as classification, denoising, compression, and ab initio modeling. We introduce a fast method for estimating a compressed representation of the 2-D covariance matrix of noisy cryo-EM projection images affected by radial point spread functions that enables fast PCA computation. Our method is based on a new algorithm for expanding images in the Fourier-Bessel basis (the harmonics on the disk), which provides a convenient way to handle the effect of the contrast transfer functions. For N images of size L × L, our method has time complexity O(NL3 + L4) and space complexity O(NL2 + L3). In contrast to previous work, these complexities are independent of the number of different contrast transfer functions of the images. We demonstrate our approach on synthetic and experimental data and show acceleration by factors of up to two orders of magnitude.

Radial recombination for rigid rotational alignment of images and volumes

A common task in single particle electron cryomicroscopy (cryo-EM) is the rigid alignment of images and/or volumes. In the context of images, a rigid alignment involves estimating the inner product between one image of N × N pixels and another image that has been translated by some displacement and rotated by some angle γ. In many situations the number of rotations γ considered is large (e.g. ), while the number of translations considered is much smaller (e.g. ). In these scenarios a naive algorithm requires operations to calculate the array of inner products for each image pair. This computation can be accelerated by using a Fourier–Bessel basis and the fast Fourier transform, requiring only operations per image pair. We propose a simple data driven compression algorithm to further accelerate this computation, which we refer to as the ‘radial SVD’. Our approach involves linearly recombining the different rings of the original images (expressed in polar coordinates), taking advantage of the singular value decomposition (SVD) to both compress the images and optimize a certain measure of angular discriminability. When aligning multiple images to multiple targets, the complexity of our approach is per image pair, where H is the rank of the SVD used in the compression above. A very similar strategy can be used to accelerate volume alignment, using a spherical harmonic based compression, which we will refer to as a ‘degree SVD’. The advantage gained by these approaches depends on the ratio between H and N; the smaller H is the better. In many applications H can be quite a bit smaller than N while still maintaining accuracy. We present numerical results in a cryo-EM application demonstrating that the radial and degree SVD can help save a factor of 5–10 or more for both image and volume alignment.

Joint analyses of 2D CMB lensing and 3D galaxy clustering in the spherical Fourier-Bessel basis

Cross-correlating cosmic microwave background (CMB) lensing and galaxy clustering has been shown to greatly improve the constraints on the local primordial non-Gaussianity (PNG) parameter $f_{\rm NL}$ by reducing sample variance and also parameter degeneracies. To model the full use of the 3D information of galaxy clustering, we forecast $f_{\rm NL}$ measurements using the decomposition in the spherical Fourier-Bessel (SFB) basis, which can be naturally cross-correlated with 2D CMB lensing in spherical harmonics. In the meantime, such a decomposition would also enable us to constrain the growth rate of structure, a probe of gravity, through the redshift-space distortion (RSD). As a comparison, we also consider the tomographic spherical harmonic (TSH) analysis of galaxy samples with different bin sizes. Assuming galaxy samples that mimic a few future surveys, we perform Fisher forecasts using linear modes for $f_{\rm NL}$ and the growth rate exponent $\gamma$, marginalized over standard $\Lambda$ cold dark matter ($\Lambda$CDM) cosmological parameters and two nuisance parameters that account for clustering bias and magnification bias. Compared to TSH analysis using only one bin, SFB analysis could improve $\sigma(f_{\rm NL})$ by factors 3 to 12 thanks to large radial modes. With future wide-field and high-redshift photometric surveys like the LSST, the constraint $\sigma(f_{\rm NL}) < 1$ could be achieved using linear angular multipoles up to $\ell_{\rm min}\simeq 20$. Compared to using galaxy auto-power spectra only, joint analyses with CMB lensing could improve $\sigma(\gamma)$ by factors 2 to 5 by reducing degeneracies with other parameters, especially the clustering bias. For future spectroscopic surveys like the DESI or $\textit{Euclid}$, using linear scales, $\gamma$ could be constrained to $3\,\%$ precision assuming the GR fiducial value.

Convergence analysis of various factorization rules in the Fourier-Bessel basis for solving Maxwell equations using modal methods.

The choice of factorization rule can strongly affect the convergence of solutions to Maxwell equations based on the orthogonal expansion of electromagnetic fields. While this issue has already been investigated thoughtfully for the Fourier basis (plane-wave expansion), for other bases it has not yet received much attention. Although there are works showing that, in the case of the Fourier-Bessel basis (cylindrical-wave expansion), the use of an inverse factorization rule can provide faster convergence than Laurent's rule, these works neglect the fact that other rules are also possible. Here, I mathematically demonstrate four different factorization rules for solving Maxwell equations in cylindrical coordinates using the Fourier-Bessel expansion in both infinite and finite domains. I compare their convergence for a step-index fiber (which has a known exact solution and thus enables the absolute numerical error to be determined), as well as for several VCSEL structures. I show that the cylindrical-wave expansion differs from the plane-wave expansion and that the application of an inverse factorization rule for the electric field component perpendicular to the discontinuities can result in deterioration of numerical convergence. Finally, I identify the factorization rule that gives the fastest convergence of the modal method using the Fourier-Bessel basis.

Power balance and efficiency of metasurface antennas

This paper presents two methods for the efficient evaluation of the power balance in circular metasurface (MTS) antennas implementing arbitrary modulated surface impedances on a grounded dielectric slab. Both methods assume the surface current in the homogenized MTS to be known. The first technique relies on the surface current expansion with Fourier-Bessel basis functions (FBBF) and proceeds by integration of the Poynting vector on a closed surface. The second method is based on the evaluation of the residue of the electric field spectrum at the surface-wave (SW) pole, and is demonstrated by using a current expansion in Gaussian ring basis functions (GRBF). The surface current expansions can be directly obtained either by analyzing the antenna with a Method of Moments (MoM) tool for homogenized MTSs based on FBBF or GRBF, or derived by a projection process. From there, the power contributions, namely the total power delivered by the feed, the radiated power, the SW power, and the Ohmic power losses in the dielectric are computed. Several efficiency metrics are presented and discussed: tapering efficiency, conversion efficiency, loss factor, and diffraction factor. Since the MTS apertures at hand are leaky-wave (LW) antennas, the designer must find a compromise between the aperture efficiency and the conversion efficiency. This requires accurate and fast computational techniques for the efficiency. The present paper demonstrates for the first time that the efficiency of MTS antenna devices can be accurately evaluated in a few minutes. The compromise that should be made during the design process between the tapering efficiency and the conversion efficiency is highlighted. The impact on the efficiency of isotropic versus anisotropic MTS, uniform versus non-uniform modulation index, is analyzed. An excellent agreement is obtained between both approaches, commercial software, and experimental data.

Factorization of the translation kernel for fast rigid image alignment

An important component of many image alignment methods is the calculation of inner products (correlations) between an image of pixels and another image translated by some shift and rotated by some angle. For robust alignment of an image pair, the number of considered shifts and angles is typically high, thus the inner product calculation becomes a bottleneck. Existing methods, based on fast Fourier transforms (FFTs), compute all such inner products with computational complexity per image pair, which is reduced to if only distinct shifts are needed. We propose to use a factorization of the translation kernel (FTK), an optimal interpolation method which represents images in a Fourier–Bessel basis and uses a rank-H approximation of the translation kernel via an operator singular value decomposition (SVD). Its complexity is per image pair. We prove that , where 2W is the magnitude of the maximum desired shift in pixels and is the desired accuracy. For fixed W this leads to an acceleration when is large, such as when sub-pixel shift grids are considered. Finally, we present numerical results in an electron cryomicroscopy application showing speedup factors of 3–10 with respect to the state of the art.

Multibeam and Beam Scanning With Modulated Metasurfaces

Multibeam and beam-scanning capabilities of metasurface (MTS) antennas using multiple feeds are investigated. The MTS synthesis is performed by direct inversion of an electric field integral equation (EFIE) obtained after expanding the unknown equivalent impedance profile into Fourier-Bessel basis functions. Two approaches are explored. The first one assumes a priori a discrete azimuthal symmetry in the impedance profile, so as to constrain the solution to a subspace which automatically provides multiple beams when illuminated with feeds regularly arranged along azimuth. In the second approach, there are not a priori assumptions on the impedance profile, but the systems of equations corresponding to each beam are stacked and solved simultaneously in the least-squares sense. This second approach can also be used to obtain polarization diversity. More importantly, it also enables continuous beam scanning. The latter functionality is achieved through the generation of two embedded patterns in a common azimuthal window with opposite phase slopes, followed by a continuous phasing of the two feed points. Various designs are presented in this article. All the results are validated with the method of moments (MoM).

Method of Moments Simulation of Modulated Metasurface Antennas With a Set of Orthogonal Entire-Domain Basis Functions

A family of orthogonal and entire-domain basis functions (named Fourier-Bessel) is proposed for the analysis of circular modulated metasurface (MTS) antennas. In the structures at hand, the MTS is accounted for in the electric field integral equation (EFIE) as a sheet transition impedance boundary condition on the top of a grounded dielectric slab. The closed-form Hankel transform of the Fourier-Bessel basis functions (FBBFs) allows one to use a spectral domain formulation in the method-of-moments (MoM) solution of the EFIE. Moreover, these basis functions are fully orthogonal, which implies that they are able to represent the global evolution of the current distribution in a compact form. FBBFs also present a better filtering capability of their spectrum compared to other well-known orthogonal families such as the Zernike functions. The obtained MoM matrix is sparse and compact, and it is thus very well-conditioned and can be efficiently computed and inverted. The numerical results based on the proposed decomposition are presented and compared with those based on the use of the Gaussian ring basis functions and with the full-wave analysis of MTS antennas implemented with small printed elements. A very good agreement is observed.

Fourier–Bessel Basis Functions for the Analysis of Elliptical Domain Metasurface Antennas

An approach based on the method of moments and using a set of stretched Fourier–Bessel basis functions (FBBFs) is presented for the analysis of elliptical domain metasurfaces (MTSs). This shape is preferred to the traditional circular domain, when a better control of the aperture efficiency is required. The proposed formulation makes use of a harmonic decomposition of the substrate dyadic Green's function to efficiently compute the reaction integrals. It is shown that the computation time associated with the sheet impedance matrix calculation is almost not affected by the stretching of the FBBFs. The obtained formulation is efficient since the orthogonality of the FBBFs is preserved and well adapted to an MTS optimization process. The simulation results are compared with measurements of a realistic MTS antenna implemented with subwavelength printed patches.

Science-driven 3D data compression

Photometric redshift surveys map the distribution of matter in the Universe through the positions and shapes of galaxies with poorly resolved measurements of their radial coordinates. While a tomographic analysis can be used to recover some of the large-scale radial modes present in the data, this approach suffers from a number of practical shortcomings, and the criteria to decide on a particular binning scheme are commonly blind to the ultimate science goals. We present a method designed to separate and compress the data into a small number of uncorrelated radial modes, circumventing some of the problems of standard tomographic analyses. The method is based on the Karhunen-Lo\`{e}ve transform (KL), and is connected to other 3D data compression bases advocated in the literature, such as the Fourier-Bessel decomposition. We apply this method to both weak lensing and galaxy clustering. In the case of galaxy clustering, we show that the resulting optimal basis is closely associated with the Fourier-Bessel basis, and that for certain observables, such as the effects of magnification bias or primordial non-Gaussianity, the bulk of the signal can be compressed into a small number of modes. In the case of weak lensing we show that the method is able to compress the vast majority of the signal-to-noise into a single mode, and that optimal cosmological constraints can be obtained considering only three uncorrelated KL eigenmodes, considerably simplifying the analysis with respect to a traditional tomographic approach.

SPHERICAL HARMONIC ANALYSES OF INTENSITY MAPPING POWER SPECTRA

ABSTRACT Intensity mapping is a promising technique for surveying the large-scale structure of our universe from z = 0 to z ∼ 150, using the brightness temperature field of spectral lines to directly observe previously unexplored portions of our cosmic timeline. Examples of targeted lines include the 21 cm hyperfine transition of neutral hydrogen, rotational lines of carbon monoxide, and fine-structure lines of singly ionized carbon. Recent efforts have focused on detections of the power spectrum of spatial fluctuations, but have been hindered by systematics such as foreground contamination. This has motivated the decomposition of data into Fourier modes perpendicular and parallel to the line of sight, which has been shown to be a particularly powerful way to diagnose systematics. However, such a method is well-defined only in the limit of a narrow-field, flat-sky approximation. This limits the sensitivity of intensity mapping experiments, as it means that wide surveys must be separately analyzed as a patchwork of smaller fields. In this paper, we develop a framework for analyzing intensity mapping data in a spherical Fourier–Bessel basis, which incorporates curved sky effects without difficulty. We use our framework to generalize a number of techniques in intensity mapping data analysis from the flat sky to the curved sky. These include visibility-based estimators for the power spectrum, treatments of interloper lines, and the “foreground wedge” signature of spectrally smooth foregrounds.

Fast Steerable Principal Component Analysis.

Cryo-electron microscopy nowadays often requires the analysis of hundreds of thousands of 2-D images as large as a few hundred pixels in each direction. Here, we introduce an algorithm that efficiently and accurately performs principal component analysis (PCA) for a large set of 2-D images, and, for each image, the set of its uniform rotations in the plane and their reflections. For a dataset consisting of n images of size L × L pixels, the computational complexity of our algorithm is O(nL3 + L4), while existing algorithms take O(nL4). The new algorithm computes the expansion coefficients of the images in a Fourier-Bessel basis efficiently using the nonuniform fast Fourier transform. We compare the accuracy and efficiency of the new algorithm with traditional PCA and existing algorithms for steerable PCA.

Galaxy clustering in 3D and modified gravity theories

We study Modified Gravity (MG) theories by modelling the redshifted matter power spectrum in a spherical Fourier-Bessel (sFB) basis. We use a fully non-linear description of the real-space matter power-spectrum and include the lowest-order redshift-space correction (Kaiser effect), taking into account some additional non-linear contributions. Ignoring relativistic corrections, which are not expected to play an important role for a shallow survey, we analyse two different modified gravity scenarios, namely the generalised Dilaton scalar-tensor theories and the $f({R})$ models in the large curvature regime. We compute the 3D power spectrum ${\cal C}^s_{\ell}(k_1,k_2)$ for various such MG theories with and without redshift space distortions, assuming precise knowledge of background cosmological parameters. Using an all-sky spectroscopic survey with Gaussian selection function $\varphi(r)\propto \exp(-{r^2 / r^2_0})$, $r_0 = 150 \, h^{-1} \, {\textrm{Mpc}}$, and number density of galaxies $\bar {\textrm{N}} =10^{-4}\;{\textrm{Mpc}}^{-3}$, we use a $\chi^2$ analysis, and find that the lower-order $(\ell \leq 25)$ multipoles of ${\cal C}^s_\ell(k,k')$ (with radial modes restricted to $k < 0.2 \, h \,{\textrm{Mpc}}^{-1}$) can constraint the parameter $f_{R_0}$ at a level of $2\times 10^{-5} (3\times 10^{-5})$ with $3 \sigma$ confidence for $n=1(2)$. Combining constraints from higher $\ell > 25$ modes can further reduce the error bars and thus in principle make cosmological gravity constraints competitive with solar system tests. However this will require an accurate modelling of non-linear redshift space distortions. Using a tomographic $\beta(a)$-$m(a)$ parameterization we also derive constraints on specific parameters describing the Dilaton models of modified gravity.

Tonal Noise Prediction of a Turbofan with Heterogeneous Stator and Bifurcations

The interaction of wakes generated by a fan with the outlet guide vanes occurring at the blade-passing frequency and its harmonics is mainly responsible for aeroengine tonal noise emission in approach conditions. Conventional rotor–stator interaction models assume axisymmetric rows and quasi-annular ducts. However, the stator of new engines is characterized by nonidentical vanes (so-called heterogeneous outlet guide vane) and integrates two internal bifurcations up to the nacelle outlet. These new technologies invalidate the existing tools adopted by engine manufacturers at the design stage. For this reason, hybrid methodologies based on a three-dimensional unsteady Reynolds-averaged Navier–Stokes simulation considering the complete geometry of a modern Snecma engine model are investigated in this paper. Unsteady Reynolds-averaged Navier–Stokes simulation output data are postprocessed using different integral methods to assess the impact of the heterogeneity on angular mode distribution (compared to an idealized homogeneous stator) and the effect of bifurcations on downstream propagation. In particular, the Goldstein formulation is extended to loading sources distributed over nonidentical vanes. The harmonic sound power predictions derived from loading noise equations are also compared to a direct calculation of the acoustic intensity obtained by extracting the perturbation fields over prescribed cross sections behind the outlet guide vanes. This method requires a suited spatial filtering process in order to reduce the contribution of vortical waves known for polluting the acoustic field. A modal decomposition of the extracted perturbations on a Fourier–Bessel basis is also proposed to avoid this difficulty. Finally, computational fluid dynamics solutions and acoustic predictions compare reasonably with the available experimental data issued from a half-scale turbofan test.

Mixing and transport of metals by gravitational instability-driven turbulence in galactic discs

Metal production in galaxies traces star formation, and is highly concentrated toward the centers of galactic discs. This suggests that galaxies should have inhomogeneous metal distributions with strong radial gradients, but observations of present-day galaxies show only shallow gradients with little azimuthal variation, implying the existence of a redistribution mechanism. We study the role of gravitational instability-driven turbulence as a mixing mechanism by simulating an isolated galactic disc at high resolution, including metal fields treated as passive scalars. Since any cylindrical field can be decomposed into a sum of Fourier-Bessel basis functions, we set up initial metal fields characterized by these functions and study how different modes mix. We find both shear and turbulence contribute to mixing, but the mixing strongly depends on the symmetries of the mode. Non-axisymmetric modes have decay times smaller than the galactic orbital period because shear winds them up to small spatial scales, where they are erased by turbulence. The decay timescales for axisymmetric modes are much greater, though for all but the largest-scale inhomogeneities the mixing timescale is still short enough to erase chemical inhomogeneities over cosmological times. These different timescales provide an explanation for why galaxies retain metallicity gradients while there is almost no variation at a fixed radius. Moreover, the comparatively long timescales required for mixing axisymmetric modes may explain the greater diversity of metallicity gradients observed in high redshift galaxies as compared to local ones: these systems have not yet reached equilibrium between metal production and diffusion.

Theoretical examination of the slot channel waveguide configured in a cylindrically symmetric dielectric ring profile

It has recently been experimentally demonstrated that slot channel waveguides, configured in cylindrical space, can support high azimuthal order modes similar to whispering-gallery modes. This paper presents a mode solver based on Maxwell׳s vector wave equation for the electric field cast into an eigenvalue problem using a Fourier–Bessel basis function space. The modal frequencies and field profiles of the high azimuthal order slot-channel-whispering-gallery (SCWG) modes are computed for a set of nanometer spaced silicon rings supported by oxide. The computations show, that in addition to the traditionally observed, lowest order mode, the structure may support higher order SCWG modes. We complete the analysis by computing structures response as an ambient medium index of refraction sensor which achieves over 400nm per RIU sensitivity.

Harmonics in the dark-matter sky: directional detection in the Fourier-Bessel basis

Details about the velocity distribution of weakly interacting massive particle (WIMP) dark matter in our galaxy may be revealed by nuclear-recoil detectors with directional sensitivity. Previous studies have assumed that the velocity distribution takes a simple functional form characterized by a small number of parameters. More recent work has shown that basis-function expansions may allow for more general parameterization; such an approach has been considered for both the one-dimensional speed and momentum distributions, and also for three-dimensional velocity distributions obeying certain equilibrium conditions. In this work, I extend this basis-function approach to allow for arbitrary velocity distributions by working in the Fourier-Bessel basis, deriving an analytic expression for the directional recoil spectrum. Such an approach is completely general, and may be useful if the velocity distribution is too complex to be characterized by simple functional forms or is not completely virialized. Results concerning the three-dimensional Radon transform of the Fourier-Bessel basis functions may be of general interest for tomographic applications.

Whispering-gallery modes of μ-optic silicon bottle resonator examined using a Fourier–Bessel eigen-state approach

The whispering-gallery mode space of a cylindrically symmetric silicon micro-optic bottle resonator with an axial Gaussian dielectric profile is theoretically examined. Maxwell’s wave equation for the magnetic field, expressed in cylindrical coordinates, is converted into an eigenvalue problem using a Fourier–Bessel basis space to expand the inverse dielectric and field components. The order of the eigen-matrix is reduced using symmetry arguments and azimuthal order segmentation such that the matrix can be solved on a desktop PC. The eigen-space reveals that several states can be confined in the bottle region, whereas another set of states extends the axial length of the structure similar to ridge waveguide modes. The hollow core bottle is also presented as a design modification. The micro-optic bottle resonator structure size makes it directly compatible with silicon-based photonics increasing its potential as a key component in sensing, communication, and other applications.