Explicit Reconstruction Formulas Research Articles

Stable Gabor Phase Retrieval in Gaussian Shift-Invariant Spaces via Biorthogonality

We study the phase reconstruction of signals f belonging to complex Gaussian shift-invariant spaces V∞(φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V^\\infty (\\varphi )$$\\end{document} from spectrogram measurements |Gf(X)|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|{\\mathcal {G}} f(X)|$$\\end{document} where G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {G}}$$\\end{document} is the Gabor transform and X⊆R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X \\subseteq {{\\mathbb {R}}}^2$$\\end{document}. An explicit reconstruction formula will demonstrate that such signals can be recovered from measurements located on parallel lines in the time-frequency plane by means of a Riesz basis expansion. Moreover, connectedness assumptions on |f| result in stability estimates in the situation where one aims to reconstruct f on compacts intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements (Grohs and Liehr in Injectivity of Gabor phase retrieval from lattice measurements. Appl. Comput. Harmon. Anal. 62, 173–193 (2023)) we prove a sampling result on the stable approximation from finitely many spectrogram samples. The resulting algorithm provides a provably stable and convergent approximation technique. In addition, it constitutes a method of approximating signals in function spaces beyond V∞(φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V^\\infty (\\varphi )$$\\end{document}, such as Paley–Wiener spaces.

Weighted random sampling and reconstruction in general multivariate trigonometric polynomial spaces

The set of sampling and reconstruction in trigonometric polynomial spaces will play an important role in signal processing. However, in many applications, the frequencies in trigonometric polynomial spaces are not all integers. In this paper, we consider the problem of weighted random sampling and reconstruction of functions in general multivariate trigonometric polynomial spaces. The sampling set is randomly selected on a bounded cube with a probability distribution. We obtain that with overwhelming probability, the sampling inequality holds and the explicit reconstruction formula succeeds for all functions in the general multivariate trigonometric polynomial spaces when the sampling size is sufficiently large.

Quantum tomography and the quantum Radon transform

<p style='text-indent:20px;'>A general framework for the tomographical description of states, that includes, among other tomographical schemes, the classical Radon transform, quantum state tomography and group quantum tomography, in the setting of <inline-formula><tex-math id="M1">\begin{document}$ C^* $\end{document}</tex-math></inline-formula>-algebras is presented. Given a <inline-formula><tex-math id="M2">\begin{document}$ C^* $\end{document}</tex-math></inline-formula>-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on <inline-formula><tex-math id="M3">\begin{document}$ C^* $\end{document}</tex-math></inline-formula>-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform.</p><p style='text-indent:20px;'>The abstract theory is realized by using dynamical systems, that is, groups represented on <inline-formula><tex-math id="M4">\begin{document}$ C^* $\end{document}</tex-math></inline-formula>-algebra. Using a fiducial state and the corresponding GNS construction, explicit expressions for tomograms associated with states defined by density operators on the corresponding Hilbert spade are obtained. In particular a general quantum version of the classical definition of the Radon transform is presented. The theory is completed by proving that if the representation of the group is square integrable, the representation itself defines a dual tomographic map and explicit reconstruction formulas are obtained by making a judiciously use of the theory of frames. A few significant examples are discussed that illustrates the use and scope of the theory.</p>

Exact reconstruction formulas for plastic strain distribution in the surface-treated plate and their applications

Surface-treated bodies are considered under the assumption that the plastic strain distribution is uniform along the surface and changes only with depth. The elastic problem of the surface-treated plate is solved. The obtained solution is used to derive the explicit exact reconstruction formulas for distributions of main tangential components of the plastic strain tensor. It has been proved that the deflection of the surface-treated strip does not depend on the width of the strip. These results are intended for use in destructive testing methods and for numerical prediction of strength. The following initial data were used: (a) the dependence of the deflection on the thickness of the removed material layer obtained by the Davidenkov method, (b) the dependence of the tangential residual stress components on the depth, and (c) the dependence of the surface tangential stress components or elastic strain tangential components on the thickness of the removed material layer obtained by the X-ray diffraction method. The obtained formulas were analyzed and approved for all types of initial data, wherefore two experiments were conducted. A way to incorporate the plastic strain distribution into a model of a machine part of complex geometry to take into account the residual stress field in fatigue, wear, cracking, the and corrosion strength predictions is presented.

Reconstruction from convolution random sampling in local shift invariant spaces

In this paper, we consider the problem of reconstructing functions in local multiply generated shift invariant spaces from convolution random samples. The sampling set is randomly chosen with one kind of probability distribution over a bounded cube and the available data are the sampled convolution of the original function on the sampling set. We obtain an explicit reconstruction formula. This reconstruction formula succeeds with overwhelming probability when the sampling size is sufficiently large.

Parameter identification in a structured population model

We study parameter identification problems in a structured population model without mutations. Given measurements of the total population size or critical points of the population, we aim to recover its growth rate, death rate or initial distribution. We present uniqueness results under suitable assumptions and present counterexamples when these assumptions are violated. Our results are supplemented by numerical studies, either based on Tikhonov regularization or the use of explicit reconstruction formulas.

On an inverse source problem for the Biot equations in electro-seismic imaging**The research of YG was supported in part by NSFC grant 11871140, 11571065,11671071, JJKH20180006KJ, JLSTDP20190201154JC, 20160520094JH and FRFCU2412017FZ005. The research of YY was partly supported by the NSF Grant DMS-1715178, the AMS-Simons travel grant, and the start-up fund from the Michigan State University.

Electro-seismic imaging is a novel hybrid imaging modality in geophysical exploration where electromagnetic wave and seismic wave are coupled in porous medium. This paper concerns with an inverse source problem for the Biot’s equations arising in electro-seismic imaging. Using the time reversal method, we derive an explicit reconstruction formula, which immediately gives the uniqueness and stability of the reconstructed source.

High‐order ridge reconstruction for more accurate signal estimate from time‐frequency representations

Ridge reconstruction (RR) method is one of the most commonly used ways for non-stationary signal reconstruction from time-frequency representations. However, this method leads to a large reconstruction error when dealing with strongly amplitude-modulated and frequency-modulated (AM-FM) signals. To tackle this problem, in this Letter the authors propose a new and powerful reconstruction approach, named high-order RR, by considering the high-order derivative information of amplitude and phase of the signal. In particular, an explicit reconstruction formula for second-order RR is derived. Simulation results demonstrate the superiority of the proposed approach in reconstruction accuracy when addressing AM-FM signals, as well as the stability for the selection of window width.

Intertwining wavelets or multiresolution analysis on graphs through random forests

Abstract We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph. Our approach relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov intertwining relation to provide a subgraph structure and a filter bank leading to a wavelet basis of the set of functions. Our construction involves two parameters q and q ′ . The first one controls the mean number of kept vertices in the downsampling, while the second one is a tuning parameter between space localization and frequency localization. We provide an explicit reconstruction formula, bounds on the reconstruction operator norm and on the error in the intertwining relation, and a Jackson-like inequality. These bounds lead to recommend a way to choose the parameters q and q ′ . We illustrate the method by numerical experiments.

Separation of Variables in the Anisotropic Shottky–Frahm Model

We construct separated coordinates for the completely anisotropic Shottky–Frahm model on an arbitrary coadjoint orbit of SO(4). We find explicit reconstruction formulas expressing dynamical variables in terms of the separation coordinates and write the equations of motion in the Abel-type form.

A time reversal algorithm in acoustic media with Dirac measure approximations

This article is devoted to the study of a photoacoustic tomography model, where one is led to consider the solution of the acoustic wave equation with a source term writing as a separated variables function in time and space, whose temporal component is in some sense close to the derivative of the Dirac distribution at t = 0. This models a continuous wave laser illumination performed during a short interval of time. We introduce an algorithm for reconstructing the space component of the source term from the measure of the solution recorded by sensors during a time T all along the boundary of a connected bounded domain. It is based at the same time on the introduction of an auxiliary equivalent Cauchy problem allowing to derive explicit reconstruction formula and then to use of a deconvolution procedure. Numerical simulations illustrate our approach. Finally, this algorithm is also extended to elasticity wave systems.

Efficient regularization with wavelet sparsity constraints in photoacoustic tomography

In this paper, we consider the reconstruction problem of photoacoustic tomography (PAT) with a flat observation surface. We develop a direct reconstruction method that employs regularization with wavelet sparsity constraints. To that end, we derive a wavelet-vaguelette decomposition (WVD) for the PAT forward operator and a corresponding explicit reconstruction formula in the case of exact data. In the case of noisy data, we combine the WVD reconstruction formula with soft-thresholding, which yields a spatially adaptive estimation method. We demonstrate that our method is statistically optimal for white random noise if the unknown function is assumed to lie in any Besov-ball. We present generalizations of this approach and, in particular, we discuss the combination of PAT-vaguelette soft-thresholding with a total variation (TV) prior. We also provide an efficient implementation of the PAT-vaguelette transform that leads to fast image reconstruction algorithms supported by numerical results.

Reconstruction formulas for photoacoustic imaging in attenuating media

In this paper we study the problem of photoacoustic inversion in a weakly attenuating medium. We present explicit reconstruction formulas in such media and show that the inversion based on such formulas is moderately ill-posed. Moreover, we present a numerical algorithm for imaging and demonstrate in numerical experiments the feasibility of this approach.

Identification of heat transfer coefficient through linearization: explicit solution and approximation

We derive an explicit reconstruction formula for the heat flux on the internal surface of a duct based on external temperature data. With the heat flux at hand we determine the internal convective heat transfer coefficient. Technically, by determining explicitly the singular system of a compact operator and assuming noise-free data, the heat flux is expressed as an infinite singular value expansion, whereas for noisy data, the truncated expansion is used as regularized solution. An error bound is derived for the case where the truncation parameter is determined by the Morozov’s discrepancy principle. The method has high flexibility in that it is readily adapted to the finite data case without any substantial modification. Moreover, it does not require any linear system solver and it is therefore easy to implement and extremely inexpensive when compared to existing heat transfer coefficient estimation techniques. The method is illustrated by means of numerical results using both synthetic and experimental data.

Harmonic Factorization and Reconstruction of the Elasticity Tensor

In this paper, we propose a factorization of a fourth-order harmonic tensor into second-order tensors. We obtain moreover explicit equivariant reconstruction formulas, using second-order covariants, for transverse isotropic and orthotropic harmonic fourth-order tensors, and for trigonal and tetragonal harmonic fourth-order tensors up to a cubic fourth order covariant remainder.

An explicit reconstruction method for magnetic resonance electrical property tomography based on the generalized Cauchy formula

This paper presents an explicit reconstruction formula for magnetic resonance electrical property tomography (MREPT). We derive a Dbar problem from the time-harmonic Maxwell equations under the assumptions that , , and , where the z-axis is parallel to the body axis, is the measured magnetic field, and . Then, by using the generalized Cauchy formula, the electrical conductivity and permittivity are explicitly expressed in terms of their boundary values and the measured magnetic field. We also propose an iterative algorithm based on the explicit reconstruction formula without the assumption that . Numerical simulations show that the proposed methods can reconstruct the electrical properties even with noisy data.

From Low- to High-Dimensional Moments Without Magic

We aim to compute the first few moments of a high-dimensional random vector from the first few moments of a number of its low-dimensional projections. To this end, we identify algebraic conditions on the set of low-dimensional projectors that yield explicit reconstruction formulas. We also provide a computational framework, with which suitable projectors can be derived by solving an optimization problem. Finally, we show that randomized projections permit approximate recovery.

A source reconstruction algorithm for the Stokes system from incomplete velocity measurements

We consider the inverse problem of determining the spatial dependence of a source of the form in the Stokes system defined in , assuming that is known and is divergence-free. The only available observation is a single internal measurement of the velocity and the acceleration, for which one of its components is missing. Under adequate hypothesis on σ we prove uniqueness of this inverse problem and we establish an explicit reconstruction formula. This formula provides the spectral coefficients fk of the source f in terms of a family of null controls for the corresponding adjoint system indexed by .

Near-field imaging of biperiodic surfaces for elastic waves

This paper is concerned with the direct and inverse scattering of elastic waves by biperiodic surfaces in three dimensions. The surface is assumed to be a small and smooth perturbation of a rigid plane. Given a time-harmonic plane incident wave, the direct problem is to determine the displacement field of the elastic wave for a given surface; the inverse problem is to reconstruct the surface from the measured displacement field. The direct problem is shown to have a unique weak solution by studying its variational formulation. Moreover, an analytic solution is deduced by using the transformed field expansion method and the convergence is established for the power series solution. A local uniqueness is proved for the inverse problem. An explicit reconstruction formula is obtained and implemented by using the fast Fourier transform. The error estimate is derived for the reconstructed surface function, and it provides an insight on the trade-off among resolution, accuracy, and stability of the solution for the inverse problem. Numerical results show that the method is effective to reconstruct biperiodic scattering surfaces with subwavelength resolution.

Inversion of the Attenuated Geodesic X-Ray Transform over Functions and Vector Fields on Simple Surfaces

We derive explicit reconstruction formulas for the attenuated geodesic X-ray transform over functions and, in the case of nonvanishing attenuation, vector fields, on a class of simple Riemannian surfaces with boundary. These formulas partly rely on new explicit approaches to construct continuous right-inverses for backprojection operators (and, in turn, holomorphic integrating factors), which were previously unavailable in a systematic form. The reconstruction of functions is presented in two ways, the latter being motivated by numerical considerations and successfully implemented in the last section of the paper (before the appendix). Constructing the aforementioned right-inverses requires that certain Fredholm equations, first appearing in [L. Pestov and G. Uhlmann, Int. Math. Res. Not., 2004 (2004), pp. 4331--4347], be invertible. Whether this last condition reduces the applicability of the overall approach to a strict subset of simple surfaces remains open at present.