Reconstruction Formula Research Articles

Donoho-Logan large sieve principles for modulation and polyanalytic Fock spaces

We obtain estimates for the Lp-norm of the short-time Fourier transform (STFT) for functions in modulation spaces, providing information about the concentration on a given subset of R2, leading to deterministic guarantees for perfect reconstruction using convex optimization methods. More precisely, we obtain large sieve inequalities of the Donoho-Logan type, but instead of localizing the signals in regions T×W of the time-frequency plane using the Fourier transform to intertwine time and frequency, we localize the representation of the signals in terms of the short-time Fourier transform in sets Δ with arbitrary geometry. At the technical level, since there is no proper analogue of Beurling's extremal function in the STFT setting, we introduce a new method, which rests on a combination of an argument similar to Schur's test with an extension of Seip's local reproducing formula to general Hermite windows. When the windows are Hermite functions, we obtain local reproducing formulas for polyanalytic Fock spaces which lead to explicit large sieve constant estimates and, as a byproduct, to a reconstruction formula for f∈L2(R) from its STFT values on arbitrary discs.

Reconstruction in the Calderón problem on conformally transversally anisotropic manifolds

We show that a continuous potential q can be constructively determined from the knowledge of the Dirichlet–to–Neumann map for the Schrödinger operator −Δg+q on a conformally transversally anisotropic manifold of dimension ≥3, provided that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [12]. A crucial role in our reconstruction procedure is played by a constructive determination of the boundary traces of suitable complex geometric optics solutions based on Gaussian beams quasimodes concentrated along non-tangential geodesics on the transversal manifold, which enjoy uniqueness properties. This is achieved by applying the simplified version of the approach of [33] to our setting. We also identify the main space introduced in [33] with a standard Sobolev space on the boundary of the manifold. Another ingredient in the proof of our result is a reconstruction formula for the boundary trace of a continuous potential from the knowledge of the Dirichlet–to–Neumann map.

On the Lipschitz stability of inverse nodal problem for Dirac system

Inverse nodal problem on Dirac operator is determination problem of the parameters in the boundary conditions, number m and potential function V by using a set of nodal points of a component of two component vector eigenfunctions as the given spectral data. In this study, we solve a stability problem using nodal set of vector eigenfunctions and show that the space of all V functions is homeomorphic to the partition set of all space of asymptotically equivalent nodal sequences induced by an equivalence relation. Moreover, we give a reconstruction formula for the potential function as a limit of a sequence of functions and associated nodal data of one component of vector eigenfunction. Our technique depends on the explicit asymptotic expressions of the nodal parameters and, it is basically similar to [1, 2] which is given for Sturm-Liouville and Hill's operators, respectively.

On a class of gl(n) ⊗ gl(n)-valued classical r-matrices and separation of variables

We consider a problem of separation of variables for the Lax-integrable Hamiltonian systems governed by gl(n) ⊗ gl(n)-valued classical r-matrices r(u, v). We report on a class of classical non-skew-symmetric non-dynamical gl(n) ⊗ gl(n)-valued r-matrices rJ(u, v) labeled by arbitrary anisoropy matrix J ∈ gl(n) for which the “magic recipe” of Sklyanin [Prog. Theor. Phys., 118, 35 (1995)] in the theory of variable separation is applicable. An example of n = 3 corresponding to gl(3) ⊗ gl(3)-valued r-matrices is elaborated in detail. For the case of the r-matrices rJ(u, v) and n = 3, the coordinates of separation, the reconstruction formulas, and the Abel-type equations are explicitly written for the different types of matrices J.

Fractional Stockwell transform: Theory and applications

The Stockwell transform (ST), which is an invertible time-frequency spectral localization technique, uniquely combines elements of wavelet transform and short-time Fourier transform. However, its signal analysis ability is restricted in the time-frequency plane. In this paper, the novel fractional Stockwell transform (FRST) is proposed to address this problem, based on the ST and fractional Fourier transform (FRFT). It displays the time and fractional-frequency information jointly in the time-fractional-frequency plane. More importantly, it has explicit physical interpretation and usefulness for practical applications. First, the theories of continuous FRST are presented in detail, including its definition, basic properties and time-fractional-frequency analysis. Next, the discrete version for the transform, together with its reconstruction formula is considered. Further, the FRST in the realm of almost periodic functions is explored to expand its theory in the persistent signal space. Lastly, we investigate and discuss several applications based on FRST, including the time-frequency of chirp signals and the detection of disturbed chirp signals. Simulations are presented to verify the rationality and effectiveness of the FRST.

Nonconforming discretizations of convex minimization problems and precise relations to mixed methods

This article discusses nonconforming finite element methods for convex minimization problems and systematically derives dual mixed formulations. Duality relations lead to simple error estimates that avoid an explicit treatment of nonconformity errors. A reconstruction formula provides the discrete solution of the dual problem via a simple postprocessing procedure which implies a strong duality relation and is of interest in a posteriori error estimation. The framework applies to differentiable and nonsmooth problems, examples include p-Laplace, total-variation regularized, and obstacle problems. Numerical experiments illustrate advantages of nonconforming over standard conforming methods.

Bipartite quantum measurements with optimal single-sided distinguishability

We analyse orthogonal bases in a composite N×N Hilbert space describing a bipartite quantum system and look for a basis with optimal single-sided mutual state distinguishability. This condition implies that in each subsystem the N2 reduced states form a regular simplex of a maximal edge length, defined with respect to the trace distance. In the case N=2 of a two-qubit system our solution coincides with the elegant joint measurement introduced by Gisin. We derive explicit expressions of an analogous constellation for N=3 and provide a general construction of N2 states forming such an optimal basis in HN⊗HN. Our construction is valid for all dimensions for which a symmetric informationally complete (SIC) generalized measurement is known. Furthermore, we show that the one-party measurement that distinguishes the states of an optimal basis of the composite system leads to a local quantum state tomography with a linear reconstruction formula. Finally, we test the introduced tomographical scheme on a complete set of three mutually unbiased bases for a single qubit using two different IBM machines.

Wavelet transforms associated with the index Whittaker transform

The continuous wavelet transform (CWT) associated with the index Whittaker transform is defined and discussed using its convolution theory. Existence theorem and reconstruction formula for CWT are obtained. Moreover, composition of CWT is discussed, and its Plancherel's and Parseval's relations are also derived. Further, the discrete version of this wavelet transform and its reconstruction formula are given. Furthermore, certain properties of the discrete Whittaker wavelet transform are discussed.

On the Quaternionic Short-Time Fourier and Segal\u2013Bargmann Transforms

In this paper, we study a special one-dimensional quaternion short-time Fourier transform (QSTFT). Its construction is based on the slice hyperholomorphic Segal–Bargmann transform. We discuss some basic properties and prove different results on the QSTFT such as Moyal formula, reconstruction formula and Lieb’s uncertainty principle. We provide also the reproducing kernel associated with the Gabor space considered in this setting.

INVERTIBILITY OF GENERALIZED BESSEL MULTIPLIERS IN HILBERT C ∗ -MODULES

In this note, a general version of Bessel multipliers in Hilbert $C^*$-modules is presented and then, many results obtained for multipliers are extended. Also the conditions for invertibility of generalized multipliers are investigated in details. The invertibility of multipliers is very important because it helps us to obtain more reconstruction formula.

Piecewise linear interface-capturing volume-of-fluid method in axisymmetric cylindrical coordinates

The paper describes a novel implementation of the piecewise linear interface-capturing volume-of-fluid method (PLIC-VOF) in axisymmetric cylindrical coordinates. The principal innovative feature involved in this work is that both the forward and inverse reconstruction problems are solved analytically, resulting in an appreciable speed-up in computing time in comparison with an iterative approach. All reconstruction formulae are introduced explicitly, and an example illustrating their derivation is included for clarity. The numerical implementation of the PLIC-VOF interface tracking method developed here is described in detail, as well as its coupling with the 3D incompressible Navier-Stokes solver PSI-BOIL, which features a finite-volume approach based on a fixed, rectangular grid. This coupling includes a method to calculate the surface tension force within an axisymmetric VOF framework by means of height functions. The method is first verified to ensure its correct implementation in the code, and to evaluate its performance, and several advection tests are employed to demonstrate successful solution of the basic transport problem. It is shown that convergence to equilibrium may be achieved for static problems, involving the control of parasitic currents, for a variety of grid arrangements, and for different material properties. Several axisymmetric dam-break and rising bubble problems, for which high-quality measured data are available, are also presented to serve as validation tests. In all cases, very good agreement of simulation results with experimental data has been recorded, and the dynamics of the problem well reproduced.

A new general class of RC association models: Estimation and main properties

The paper introduces a new class of row-column (RC) association models for contingency tables by allowing the user to select both the scale on which interactions are measured as in Kateri and Papaioannou (1994) and the type of logit (local, global, continuation) suitable for the row and column classification variables as in Bartolucci and Forcina (2002). These choices determine the matrix of interactions which is subjected to rank constraints. Examples are provided where the new models fit substantially better than traditional ones; an intuitive explanation for this behavior is outlined and supported by numerical investigations. An extension of the optimality property in Kateri and Papaioannou (1994) is derived, leading to a general representation theorem and reconstruction formulas for the joint probabilities. These results are the key to show that, given marginal logits, the generalized interactions introduced in this paper determine uniquely the bivariate distribution; in addition, for each pair of logit types, we establish which kind of positive association is implied by our extended interactions being non negative. Quick model selection within this wide class can be performed by an efficient algorithm for computing maximum likelihood estimates which allows for additional linear constraints both on marginal logits and interactions.

Multi-windowed vertex-frequency analysis for signals on undirected graphs

The recent emerging graph signal processing technologies have been widely applied to analyze signals defined on irregular domains, e.g., data collected from social networks, sensor networks, or transportation systems. Vertex frequency analysis, especially the windowed graph Fourier transform, is one of the most important tools for graph signal analysis and representations. Nevertheless, with a selected window function, it is rather challenging to construct tight frames via the windowed graph Fourier transform. To facilitate the construction of tight frames, in this paper, we consider multi-windowed graph Fourier transforms to develop novel vertex frequency analysis methods. Firstly, under the multi-windowed setting, tight graph Fourier frames are elaborately constructed to fulfill technical demands in different application scenarios. The canonical dual frames of the multi-windowed graph Fourier frames are investigated to establish the reconstruction formulas of graph signals. Additionally, we propose shift multi-windowed graph Fourier frames by directly using the shift operators, e.g., the adjacency matrix. The related tight frames, dual frames and their constructions are also discussed. Experimental results show that the proposed two types of frames can efficiently extract vertex-frequency features of synthetic graph signals. Furthermore, anomaly data can also be detected by these frames.

Explicit geometric construction of sparse inverse mass matrices for arbitrary tetrahedral grids

The geometric reinterpretation of the Finite Element Method (FEM) shows that Raviart–Thomas and Nédélec mass matrices map from degrees of freedoms (DoFs) attached to geometric elements of a tetrahedral grid to DoFs attached to the barycentric dual grid. The algebraic inverses of the mass matrices map DoFs attached to the barycentric dual grid back to DoFs attached to the corresponding primal tetrahedral grid, but they are of limited practical use since they are dense.In this paper we present a new geometric construction of sparse inverse mass matrices for arbitrary tetrahedral grids and possibly inhomogeneous and anisotropic materials, debunking the conventional wisdom that the barycentric dual grid prohibits a sparse representation for inverse mass matrices. In particular, we provide a unified framework for the construction of both edge and face mass matrices and their sparse inverses. Such a unifying principle relies on novel geometric reconstruction formulas, from which, according to a well-established design strategy, local mass matrices are constructed as the sum of a consistent and a stabilization part. A major difference with the approaches proposed so far is that the consistent part is defined geometrically and explicitly, that is, without the necessity of computing the inverses of local matrices. This provides a sensible speedup and an easier implementation. We use these new sparse inverse mass matrices to discretize a three-dimensional Poisson problem, providing the comparison between the results obtained by various formulations on a benchmark problem with analytical solution.

Stability Estimates for Phase Retrieval from Discrete Gabor Measurements

Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that in the deterministic setting phase retrieval from frame coefficients is always unstable in infinite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016) and possibly severely ill-conditioned in finite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016). Recently, it has also been shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable under a more relaxed semi-global phase recovery regime based on atoll functions (Alaifari in Found Comput Math 19(4):869–900, 2019). In finite dimensions, we present first evidence that this semi-global reconstruction regime allows one to do phase retrieval from measurements of bandlimited signals induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales like a low order polynomial in the space dimension. To this end, we utilise reconstruction formulae which have become common tools in recent years (Bojarovska and Flinth in J Fourier Anal Appl 22(3):542–567, 2016; Eldar et al. in IEEE Signal Process Lett 22(5):638–642, 2014; Li et al. in IEEE Signal Process Lett 24(4):372–376, 2017; Nawab et al. in IEEE Trans Acoust Speech Signal Process 31(4):986–998, 1983).

Source reconstruction and stability via boundary control of abstract viscoelastic systems

We study the inverse source problem for a class of viscoelastic systems from a single boundary measurement in a general spatial dimension. We give specific reconstruction formula and stability estimate for the source in terms of the boundary measurement. Our approach relies on the exact boundary controllability of the corresponding viscoelastic systems for which we also provide a new proof based on a modification of the well-known moment method.

Reconstruction of spline spectra-signals from generalized sinc function by finitely many samples

Reconstruction of signals by their Fourier (transform) samples is investigated in many mathematical/engineering problems such as the inverse Radon transform and optical diffraction tomography. This paper concerns on the reconstruction of spline-spectra signals in $$V(\hbox {sinc}_{a})$$ by finitely many Fourier samples, where $$\hbox {sinc}_{a}$$ is the generalized sinc function. There are two main results on this topic. When the spectra knots are known, the exact reconstruction formula conducted by finitely many Fourier samples is established in the first main theorem. When the spectra knots are unknown, in the second main theorem we establish the approximations to the spline-spectra signals also by finitely many Fourier samples. Numerical simulations are conducted to check the efficiency of the approximation.

Compressed Sensing Photoacoustic Tomography Reduces to Compressed Sensing for Undersampled Fourier Measurements

Photoacoustic tomography (PAT) is an emerging imaging modality that aims at measuring the high-contrast optical properties of tissues by means of high-resolution ultrasonic measurements. The interaction between these two types of waves is based on the thermoacoustic effect. In recent years, many works have investigated the applicability of compressed sensing to PAT, in order to reduce measuring times while maintaining a high reconstruction quality. However, in most cases, theoretical guarantees are missing. In this work, we show that in many measurement setups of practical interest, compressed sensing PAT reduces to compressed sensing for undersampled Fourier measurements. This is achieved by applying known reconstruction formulae in the case of the free-space model for wave propagation, and by applying the theories of Riesz bases and nonuniform Fourier series in the case of the bounded domain model. Extensive numerical simulations illustrate and validate the approach.

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>

Composition of wavelet transforms and wave packet transform involving Kontorovich-Lebedev transform

The main objective of this paper is to study the composition of continuous Kontorovich-Lebedev wavelet transform (KL-wavelet transform) and wave packet transform (WPT) based on the Kontorovich-Lebedev transform (KL-transform). Estimates for KL-wavelet and KL-wavelet transform are obtained, and Plancherel?s relation for composition of KL-wavelet transform and WPT-transform are derived. Reconstruction formula for WPT associated to KL-transform is also deduced and at the end Calderon?s formula related to KL-transform using its convolution property is obtained.