arXiv subjects

Frank Filbir

Publications and source records attributed to Frank Filbir.

13 recordsLinked to original sources

arXiv2025

Fourier Inversion on the Group of Signatures

The main objective of this work is to develop a framework for Fourier analysis on the group of signatures, $G_N(\mathbb{R}^d)$. Employing Kirillov's orbit method, we define the Fourier transform on this group via irreducible unitary representations. Our main contribution is the derivation of necessary and sufficient conditions for identifying coadjoint orbits in general position of $G_N(\mathbb{R}^d)$. This enables the computation of the set of \textit{jump-indices} of generic orbits, crucial for the Fourier inversion theorem. We also obtain an explicit construction of a polarization for any linear functional in general position, which allows a concrete description of the Fourier transform of functions on $G_N(\mathbb{R}^d)$. This framework yields an explicit Fourier inversion formula for the group of signatures in arbitrary dimension. Furthermore, we show that the theoretical framework developed here extends naturally to a broader class of graded Lie groups, including the group generated by the truncated tensor algebra $T_0^N(\mathbb{R}^d)$.

math.RT↗

arXiv2024

RIS: Regularized Imaging Spectroscopy for STIX on-board Solar Orbiter

Imaging spectroscopy, i.e., the generation of spatially resolved count spectra and of cubes of count maps at different energies, is one of the main goals of solar hard X-ray missions based on Fourier imaging. For these telescopes, so far imaging spectroscopy has been realized via the generation of either count maps independently reconstructed at the different energy channels, or electron flux maps reconstructed via deconvolution of the bremsstrahlung cross-section. Our aim is to introduce the Regularized Imaging Spectroscopy method (RIS), in which regularization implemented in the count space imposes a smoothing constraint across contiguous energy channels, without the need to deconvolve the bremsstrahlung effect. STIX records imaging data computing visibilities in the spatial frequency domain. Our RIS is a sequential scheme in which part of the information coded in the image reconstructed at a specific energy channel is transferred to the reconstruction process at a contiguous channel via visibility interpolation based on Variably Scaled Kernels. In the case of STIX visibilities recorded during the November 11, 2022 flaring event, we show that RIS is able to generate hard X-ray maps whose morphology smoothly evolves from one energy channel to the contiguous one, and that from these maps it is possible to infer spatially-resolved count spectra characterized by notable numerical stability. We also show that the performances of this approach are robust with respect to both the image reconstruction method and the count energy channel utilized to trigger the sequential process. RIS is appropriate to construct image cubes from STIX visibilities that are characterized by a smooth behavior across count energies, thus allowing the generation of numerically stable (and, thus, physically reliable) local count spectra.

astro-ph.SR↗

arXiv2024

Wirtinger gradient descent methods for low-dose Poisson phase retrieval

The problem of phase retrieval has many applications in the field of optical imaging. Motivated by imaging experiments with biological specimens, we primarily consider the setting of low-dose illumination where Poisson noise plays the dominant role. In this paper, we discuss gradient descent algorithms based on different loss functions adapted to data affected by Poisson noise, in particular in the low-dose regime. Starting from the maximum log-likelihood function for the Poisson distribution, we investigate different regularizations and approximations of the problem to design an algorithm that meets the requirements that are faced in applications. In the course of this, we focus on low-count measurements. For all suggested loss functions, we study the convergence of the respective gradient descent algorithms to stationary points and find constant step sizes that guarantee descent of the loss in each iteration. Numerical experiments in the low-dose regime are performed to corroborate the theoretical observations.

math.NA↗

arXiv2023

Regularized Shannon sampling formulas related to the special affine Fourier transform

In this paper, we present new regularized Shannon sampling formulas related to the special affine Fourier transform (SAFT). These sampling formulas use localized sampling with special compactly supported window functions, namely B-spline, sinh-type, and continuous Kaiser-Bessel window functions. In contrast to the Shannon sampling series for SAFT, the regularized Shannon sampling formulas for SAFT possesses an exponential decay of the approximation error and are numerically robust in the presence of noise, if certain oversampling condition is fulfilled. Several numerical experiments illustrate the theoretical results.

math.NA↗

arXiv2023

Marcinkiewicz--Zygmund inequalities for scattered and random data on the $q$-sphere

The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz--Zygmund inequalities provide answers to both the recovery and the quadrature aspect. In this paper, we put ourselves on the $q$-dimensional sphere $\mathbb{S}^q$, and investigate how well continuous $L_p$-norms of polynomials $f$ of maximum degree $n$ on the sphere $\mathbb{S}^q$ can be discretized by positively weighted $L_p$-sum of finitely many samples, and discuss the relationship between the offset between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points $\xi_1,\ldots,\xi_N$ on $\mathbb{S}^q$, the dimension $q$, and the polynomial degree $n$.

math.NA↗

arXiv2022

The system of translates and the special affine Fourier transform

The translation operator $T^A$ associated with the special affine Fourier transform (SAFT) $\mathscr{F}_A$ is introduced from harmonic analysis point of view. The analogues of Wendel's theorem, Wiener theorem, Weiner-Tauberian theorem and Bernstein type inequality in the context of the SAFT are established. The shift invariant space $V_A$ associated with the special affine Fourier transform is introduced and studied along with sampling problems.

math.FA↗

arXiv2022

Image Recovery for Blind Polychromatic Ptychography

Ptychography is a lensless imaging technique, which considers reconstruction from a set of far-field diffraction patterns obtained by illuminating small overlapping regions of the specimen. In many cases, a distribution of light inside the illuminated region is unknown and has to be estimated along with the object of interest. This problem is referred to as blind ptychography. While in ptychography the illumination is commonly assumed to have a point spectrum, in this paper we consider an alternative scenario with non-trivial light spectrum known as blind polychromatic ptychography. Firstly, we show that non-blind polychromatic ptychography can be seen as a recovery from quadratic measurements. Then, a reconstruction from such measurements can be performed by a variant of Amplitude Flow algorithm, which has guaranteed sublinear convergence to a critical point. Secondly, we address recovery from blind polychromatic ptychographic measurements by devising an alternating minimization version of Amplitude Flow and showing that it converges to a critical point at a sublinear rate. Keywords: ptychography, phase retrieval, blind, alternating minimization, gradient descent.

math.NA↗

arXiv2022

Inverse Multislice Ptychography by Layer-wise Optimisation and Sparse Matrix Decomposition

We propose algorithms based on an optimisation method for inverse multislice ptychography in, e.g. electron microscopy. The multislice method is widely used to model the interaction between relativistic electrons and thick specimens. Since only the intensity of diffraction patterns can be recorded, the challenge in applying inverse multislice ptychography is to uniquely reconstruct the electrostatic potential in each slice up to some ambiguities. In this conceptual study, we show that a unique separation of atomic layers for simulated data is possible when considering a low acceleration voltage. We also introduce an adaptation for estimating the illuminating probe. For the sake of practical application, we finally present slice reconstructions using experimental 4D scanning transmission electron microscopy (STEM) data.

cs.IT↗

arXiv2021

Super-Resolution on the Two-Dimensional Unit Sphere

We study the problem of recovering an atomic measure on the unit 2-sphere $\mathbb{S}^2$ given finitely many moments with respect to spherical harmonics. The analysis relies on the formulation of this problem as an optimization problem on the space of bounded Borel measures on $\mathbb{S}^2$ as it was considered by Y. de Castro & F. Gamboa and E. Cand\'es & C. Fernandez-Granda. We construct a dual certificate using a kernel given in an explicit form and make a concrete analysis of the interpolation problem. Numerical examples are provided and analyzed.

math.FA↗

arXiv2020

On recovery guarantees for angular synchronization

The angular synchronization problem of estimating a set of unknown angles from their known noisy pairwise differences arises in various applications. It can be reformulated as a optimization problem on graphs involving the graph Laplacian matrix. We consider a general, weighted version of this problem, where the impact of the noise differs between different pairs of entries and some of the differences are erased completely; this version arises for example in ptychography. We study two common approaches for solving this problem, namely eigenvector relaxation and semidefinite convex relaxation. Although some recovery guarantees are available for both methods, their performance is either unsatisfying or restricted to the unweighted graphs. We close this gap, deriving recovery guarantees for the weighted problem that are completely analogous to the unweighted version.

math.OC↗

arXiv2019

A direct solver for the phase retrieval problem in ptychographic imaging

Measurements achieved with ptychographic imaging are a special case of diffraction measurements. They are generated by illuminating small parts of a sample with, e.g., a focused X-ray beam. By shifting the sample, a set of far-field diffraction patterns of the whole sample are then obtained. From a mathematical point of view those measurements are the squared modulus of the windowed Fourier transform of the sample. Thus, we have a phase retrieval problem for local Fourier measurements. A direct solver for this problem was introduced by Iwen, Viswanathan and Wang in 2016 and improved by Iwen, Preskitt, Saab and Viswanathan in 2018. Motivated by the applied perspective of ptychographic imaging, we present a generalization of this method and compare the different versions in numerical experiments. The new method proposed herein turns out to be more stable, particularly in the case of missing data.

math.NA↗

arXiv2013

Metric entropy, n-widths, and sampling of functions on manifolds

We first investigate on the asymptotics of the Kolmogorov metric entropy and nonlinear n-widths of approximation spaces on some function classes on manifolds and quasi-metric measure spaces. Secondly, we develop constructive algorithms to represent those functions within a prescribed accuracy. The constructions can be based on either spectral information or scattered samples of the target function. Our algorithmic scheme is asymptotically optimal in the sense of nonlinear n-widths and asymptotically optimal up to a logarithmic factor with respect to the metric entropy.

math.NA↗

arXiv2008

Cesàro means of Jacobi expansions on the parabolic biangle

We study Cesàro $(C,δ)$ means for two-variable Jacobi polynomials on the parabolic biangle $B=\{(x_1,x_2)\in{\mathbb R}^2:0\leq x_1^2\leq x_2\leq 1\}$. Using the product formula derived by Koornwinder & Schwartz for this polynomial system, the Cesàro operator can be interpreted as a convolution operator. We then show that the Cesàro $(C,δ)$ means of the orthogonal expansion on the biangle are uniformly bounded if $δ>α+β+1$, $α-\frac 12\geqβ\geq 0$. Furthermore, for $δ\geqα+2β+\frac 32$ the means define positive linear operators.

math.CA↗