Search

A recent trend in the signal/image processing literature is the optimization of Fourier sampling schemes for specific datasets of signals. In this paper, we explain why choosing optimal non Cartesian Fourier sampling patterns is a difficult nonconvex problem by bringing to light two optimization issues. The first one is the existence of a combinatorial number of spurious minimizers for a generic class of signals. The second one is a vanishing gradient effect for the high frequencies. We conclude the paper by showing how using large datasets can mitigate the first effect and illustrate experimentally the benefits of using stochastic gradient algorithms with a variable metric. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse random source problem for a stochastic time fractional diffusion equation, where the source is assumed to be driven by a Gaussian random field. The direct problem is shown to be well-posed by examining the well-posedness and regularity of the solution for the equivalent stochastic two-point boundary value problem in the frequency domain. For the inverse problem, the Fourier modulus of the diffusion coefficient of the random source is proved to be uniquely determined by the variance of the Fourier transform of the boundary data. As a phase retrieval for the inverse problem, the phaselift method with random masks is applied to recover the diffusion coefficient from its Fourier modulus. Numerical experiments are reported to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We prove a Hölder-logarithmic stability estimate for the problem of finding a sufficiently regular compactly supported function v on from its Fourier transform given on [−r, r]d. This estimate relies on a Hölder stable continuation of from [−r, r]d to a larger domain. The related reconstruction procedures are based on truncated series of Chebyshev polynomials. We also give an explicit example showing optimality of our stability estimates. [ABSTRACT FROM AUTHOR]

This paper is dedicated to design a direct sampling method of inverse electromagnetic scattering problems, which uses multi-frequency sparse backscattering far field data for reconstructing the boundaries of perfectly conducting obstacles. We show that the smallest strip containing the unknown object can be approximately determined by the multi-frequency backscattering far field data at two opposite observation directions. The proof is based on the Kirchhoff approximation and Fourier transform. Such a strip is then reconstructed by an indicator, which is the absolute value of an integral of the product of the data and some properly chosen function over the frequency interval. With the increase of the number of the backscattering data, the location and shape of the underlying object can be reconstructed. Numerical examples are conducted to show the validity and robustness of the proposed sampling method, even with moderately sized frequencies. The numerical examples also show that the concave part of the underlying object can be well reconstructed, and the different connected components of the underlying object can be well separated. [ABSTRACT FROM AUTHOR]

One of the most powerful approaches to imaging at the nanometer length scale is coherent diffraction imaging using x-ray sources. For amorphous (non-crystalline) samples, raw data collected in the far-field can be interpreted as the modulus of the two-dimensional continuous Fourier transform of the unknown object. The goal is then to recover the phase through computational means by exploiting prior information about the sample (such as its support), after which the unknown object can be visualized at high resolution. While many algorithms have been proposed for this phase retrieval problem, careful analysis of its well-posedness has received relatively little attention. In this paper, we show that the problem is, in general, not well-posed and describe some of the underlying issues that are responsible for the ill-posedness. We then show how this analysis can be used to develop experimental protocols that lead to better conditioned inverse problems. [ABSTRACT FROM AUTHOR]

Let be a compact group and let. We define the non-unique games (NUG) problem as finding to minimize. We introduce a convex relaxation of the NUG problem to a semidefinite program (SDP) by taking the Fourier transform of fij over. The NUG framework can be seen as a generalization of the little Grothendieck problem over the orthogonal group and the unique games problem and includes many practically relevant problems, such as the maximum likelihood estimator to registering bandlimited functions over the unit sphere in d-dimensions and orientation estimation of noisy cryo-electron microscopy (cryo-EM) projection images. We implement an SDP solver for the NUG cryo-EM problem using the alternating direction method of multipliers (ADMM). Numerical study with synthetic datasets indicate that while our ADMM solver is slower than existing methods, it can estimate the rotations more accurately, especially at low signal-to-noise ratio (SNR). [ABSTRACT FROM AUTHOR]

An ill-posed Cauchy problem for a 3D elliptic partial differential equation with variable coefficients is considered. A well-posed quasi-boundary-value (QBV) problem is given to approximate it. Some stability estimates are given. For the numerical implementation, a large sparse system is obtained from discretizing the QBV problem using the finite difference method. A left-preconditioned generalized minimum residual method is used to solve the large system effectively. For the preconditioned system, a fast solver using the fast Fourier transform is given. Numerical results show that the method works well. [ABSTRACT FROM AUTHOR]

The reconstruction of a function on the rotation group from mean values along all geodesics is an overdetermined problem, i.e. it is sufficient to know the mean values for a three-dimensional subset of all geodesics on the rotation group. In this paper we give a Fourier slice theorem for the restricted problem. Based on the Fourier slice theorem and fast Fourier transforms on the rotation group and the sphere we introduce a fast algorithm for the forward transform. Analyzing the inverse problem we come up with an exact inversion formula for bandlimited functions on the rotation group. Unfortunately this inversion formula turns out to be extremely ill conditioned. Therefore we introduce an iterative approach which makes use of regularization and the fast algorithm for the forward transform. Numerical experiments indicate the applicability of our algorithms. [ABSTRACT FROM AUTHOR]

Fourier phasing is the problem of retrieving Fourier phase information from Fourier intensity data. The standard Fourier phase retrieval (without a mask) is known to have many solutions which cause the standard phasing algorithms to stagnate and produce wrong or inaccurate solutions. In this paper Fourier phase retrieval is carried out with the introduction of a randomly fabricated mask in measurement and reconstruction. Highly probable uniqueness of solution, up to a global phase, was previously proved with exact knowledge of the mask. Here the uniqueness result is extended to the case where only rough information about the mask’s phases is assumed. The exponential probability bound for uniqueness is given in terms of the uncertainty-to-diversity ratio of the unknown mask. New phasing algorithms alternating between the object update and the mask update are systematically tested and demonstrated to have the capability of recovering both the object and the mask (within the object support) simultaneously, consistent with the uniqueness result. Phasing with a phase-uncertain mask is shown to be robust with respect to the correlation in the mask as well as the Gaussian and Poisson noises. [ABSTRACT FROM AUTHOR]

This paper presents a new joint inversion approach to shape-based inverse problems. Given two sets of data from distinct physical models, the main objective is to obtain a unified characterization of inclusions within the spatial domain of the physical properties to be reconstructed. Although our proposed method generally applies to many types of inverse problems, the main motivation here is to characterize subsurface contaminant source zones by processing down-gradient hydrological data and cross-gradient electrical resistance tomography observations. Inspired by Newton's method for multi-objective optimization, we present an iterative inversion scheme in which descent steps are chosen to simultaneously reduce both data-model misfit terms. Such an approach, however, requires solving a non-smooth convex problem at every iteration, which is computationally expensive for a pixel-based inversion over the whole domain. Instead, we employ a parametric level set technique that substantially reduces the number of underlying parameters, making the inversion computationally tractable. The performance of the technique is examined and discussed through the reconstruction of source zone architectures that are representative of dense non-aqueous phase liquid (DNAPL) contaminant release in a statistically homogenous sandy aquifer. In these examples, the geometric configuration of the DNAPL mass is considered along with additional information about its spatial variability within the contaminated zone, such as the identification of low and high saturation regions. Comparison of the reconstructions with the true DNAPL architectures highlights the superior performance of the model-based technique and joint inversion scheme. [ABSTRACT FROM AUTHOR]

Consider a two-dimensional backward problem for a time-fractional diffusion process, which can be considered as image de-blurring where the blurring process is assumed to be slow diffusion. In order to avoid the over-smoothing effect for object image with edges and to construct a fast reconstruction scheme, the total variation regularizing term and the data residual error in the frequency domain are coupled to construct the cost functional. The well posedness of this optimization problem is studied. The minimizer is sought approximately using the iteration process for a series of optimization problems with Bregman distance as a penalty term. This iteration reconstruction scheme is essentially a new regularizing scheme with coupling parameter in the cost functional and the iteration stopping times as two regularizing parameters. We give the choice strategy for the regularizing parameters in terms of the noise level of measurement data, which yields the optimal error estimate on the iterative solution. The series optimization problems are solved by alternative iteration with explicit exact solution and therefore the amount of computation is much weakened. Numerical implementations are given to support our theoretical analysis on the convergence rate and to show the significant reconstruction improvements. [ABSTRACT FROM AUTHOR]

We develop fast discrete Fourier transforms (and their adjoints) from a square in space to a disk in the Fourier domain. Since our new transforms are not unitary, we develop a fast inversion algorithm and derive corresponding estimates that allow us to avoid iterative methods typically used for inversion. We consider the eigenfunctions of the corresponding band-limiting and space-limiting operator to describe spaces on which these new transforms can be inverted and made useful. In the process, we construct polar grids which provide quadratures and interpolation with controlled accuracy for functions band-limited within a disk. For rapid computation of the involved trigonometric sums we use the unequally spaced fast Fourier transform, thus yielding fast algorithms for all new transforms. We also introduce polar grids motivated by linearized scattering problems which are obtained by discretizing a family of circles. These circles are generated by using a single circle passing through the origin and rotating this circle with the origin as a pivot. For such grids, we provide a fast algorithm for interpolation to a near optimal grid in the disk, yielding an accurate adjoint transform and inversion algorithm. [ABSTRACT FROM AUTHOR]

The squared eigenfunctions of the spectral problem associated with the Camassa-Holm equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the Camassa-Holm hierarchy as a generalized Fourier transform. The main result of this work is the derivation of the completeness relation for the squared solutions of the Camassa-Holm spectral problem. We show that all the fundamental properties of the Camassa-Holm equation such as the integrals of motion, the description of the equations of the whole hierarchy and their Hamiltonian structures can be naturally expressed making use of the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions. [ABSTRACT FROM AUTHOR]

One of the challenges in single particle reconstruction in cryo-electron microscopy is to find a three-dimensional model of a molecule using its two-dimensional noisy projection-images. In this paper, we propose a robust ‘angular reconstitution’ algorithm for molecules with n-fold cyclic symmetry, that estimates the orientation parameters of the projections-images. Our suggested method utilizes self common lines which induce identical lines within the Fourier transform of each projection-image. We show that the location of self common lines admits quite a few favorable geometrical constraints, thus allowing to detect them even in a noisy setting. In addition, for molecules with higher order rotational symmetry, our proposed method exploits the fact that there exist numerous common lines between any two Fourier transformed projection-images of such molecules, thus allowing to determine their relative orientation even under high levels of noise. The efficacy of our proposed method is demonstrated using numerical experiments conducted on simulated and experimental data. [ABSTRACT FROM AUTHOR]

Estimation of the discrete-time Fourier transform (DTFT) at points of a finite domain arises in many imaging applications. A new approach to this task, the golden angle linogram Fourier domain (GALFD), is presented, together with a computationally fast and accurate tool, named golden angle linogram evaluation (GALE), for approximating the DTFT at points of a GALFD. A GALFD resembles a linogram Fourier domain (LFD), which is efficient and accurate. A limitation of linograms is that embedding an LFD into a larger one requires many extra points, at least doubling the domain’s cardinality. The GALFD, on the other hand, allows for incremental inclusion of relatively few data points. Approximation error bounds and floating point operations counts are presented to show that GALE computes accurately and efficiently the DTFT at the points of a GALFD. The ability to extend the data collection in small increments is beneficial in applications such as magnetic resonance imaging. Experiments for simulated and for real-world data are presented to substantiate the theoretical claims. The mathematical analysis, algorithms, and software developed in the paper are equally suitable to other angular distributions of rays and therefore we bring the benefits of linograms to arbitrary radial patterns. [ABSTRACT FROM AUTHOR]

In many practical situations, the recovery of information about some phenomenon of interest f  reduces to performing deconvolution on indirect measurements , corresponding to the convolution of f  with a known kernel (point spread function) p . However, in practice, only discrete measurements of g will be available. Consequently, the construction of discrete approximations to f  reduces to deriving discrete versions of . How this is done depends crucially on what is assumed about the properties of the kernel p . Here, it is assumed that p  is symmetric and integrable, and that the Fourier transform of p  is strictly positive. Different discrete versions are obtained depending on how the discretisation of is performed. After establishing convergence of the truncated schemes, we discuss the underlying difficulties reflecting their ill-posedness and detail how approximations have to be treated with due care. Some of the technical issues underlying the arguments are treated in the appendix. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the interior inverse source problem for the Helmholtz equation with attenuation in the plane from boundary Cauchy data of multiple frequencies when the source term is assumed to be compactly supported in an arbitrary domain Ω with sufficiently smooth boundary. The main goal of this paper is to understand the dependence of increasing stability on the attenuation factor or constant damping. Using Fourier transform with respect to the wave numbers, explicit bounds for the analytic continuation and Hankel function and exact observability and Carleman estimates for the wave equation led us to our goal which is an increasing stability estimates with larger wave numbers interval. [ABSTRACT FROM AUTHOR]

In isotropic homogeneous media, far fields of time-harmonic electromagnetic waves radiated by compactly supported volume currents, and elastic waves radiated by compactly supported body force densities can be modelled in very similar fashions. Both are projected restricted Fourier transforms of vector-valued source terms. In this work we generalize two types of uncertainty principles recently developed for far fields of scalar-valued time-harmonic waves in Griesmaier and Sylvester (2017 SIAM J. Appl. Math. 77 154–80) to this vector-valued setting. These uncertainty principles yield stability criteria and algorithms for splitting far fields radiated by collections of well-separated sources into the far fields radiated by individual source components, and for the restoration of missing data segments. We discuss proper regularization strategies for these inverse problems, provide stability estimates based on the new uncertainty principles, and comment on reconstruction schemes. A numerical example illustrates our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we will study the increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain Ω with a sufficiently smooth boundary. Using the Fourier transform in the frequency domain, bounds for the Hankel functions and for scattering solutions in the complex plane, improving bounds for the analytic continuation, and the exact observability for the wave equation led us to our goals which are a sharp uniqueness and increasing stability estimate when the wave number interval is growing. [ABSTRACT FROM AUTHOR]

We are concerned with time-dependent inverse source problems in elastodynamics. The source term is supposed to be the product of a spatial function and a temporal function with compact support. We present frequency-domain and time-domain approaches to show uniqueness in determining the spatial function from wave fields on a large sphere over a finite time interval. The stability estimate of the temporal function from the data of one receiver and the uniqueness result using partial boundary data are proved. Our arguments rely heavily on the use of the Fourier transform, which motivates inversion schemes that can be easily implemented. A Landweber iterative algorithm for recovering the spatial function and a non-iterative inversion scheme based on the uniqueness proof for recovering the temporal function are proposed. Numerical examples are demonstrated in both two and three dimensions. [ABSTRACT FROM AUTHOR]

We investigate an inverse problem in time-frequency localization: the approximation of the symbol of a time-frequency localization operator from partial spectral information by the method of accumulated spectrograms (the sum of the spectrograms corresponding to large eigenvalues). We derive a sharp bound for the rate of convergence of the accumulated spectrogram, improving on recent results. [ABSTRACT FROM AUTHOR]

Seismic data interpolation and denoising plays a key role in seismic data processing. These problems can be understood as sparse inverse problems, where the desired data are assumed to be sparsely representable within a suitable dictionary. In this paper, we present a new method based on a data-driven tight frame (DDTF) of Kronecker type (KronTF) that avoids the vectorization step and considers the multidimensional structure of data in a tensor-product way. It takes advantage of the structure contained in all different modes (dimensions) simultaneously. In order to overcome the limitations of a usual tensor-product approach we also incorporate data-driven directionality. The complete method is formulated as a sparsity-promoting minimization problem. It includes two main steps. In the first step, a hard thresholding algorithm is used to update the frame coefficients of the data in the dictionary; in the second step, an iterative alternating method is used to update the tight frame (dictionary) in each different mode. The dictionary that is learned in this way contains the principal components in each mode. Furthermore, we apply the proposed KronTF to seismic interpolation and denoising. Examples with synthetic and real seismic data show that the proposed method achieves better results than the traditional projection onto convex sets method based on the Fourier transform and the previous vectorized DDTF methods. In particular, the simple structure of the new frame construction makes it essentially more efficient. [ABSTRACT FROM AUTHOR]

Consider the incidence of a time-harmonic electromagnetic plane wave onto a biperiodic dielectric grating, where the surface is assumed to be a small and smooth perturbation of a plane. The diffraction is modeled as a transmission problem for Maxwell’s equations in three dimensions. This paper concerns the inverse diffraction problem which is to reconstruct the grating surface from either the diffracted field or the transmitted field. A novel approach is developed to solve the challenging nonlinear and ill-posed inverse problem. The method requires only a single incident field and is realized via the fast Fourier transform. Numerical results show that it is simple, fast, and stable to reconstruct biperiodic dielectric grating surfaces with super-resolved resolution. [ABSTRACT FROM AUTHOR]

The problem of recovering a signal of finite duration from a piece of its Fourier transform was solved at Bell Labs in the 1960’s, by exploiting a ‘miracle’: a certain naturally appearing integral operator commutes with an explicit differential one. Here we show that this same miracle holds in a matrix valued version of the same problem. [ABSTRACT FROM AUTHOR]

We consider the optimal recovery of the th degree of the Laplacian value on a function from the information on its k-plane transform, measured with an error. We present the error of the optimal recovery and the set of optimal methods on classes with the bounded th degree of the Laplacian, where . As a consequence, we give one inequality for the norms of the degree of the Laplace operator and the k-plane transform. Particular cases include new inversion methods and inequalities for the classical Radon and x-ray transforms. [ABSTRACT FROM AUTHOR]

We consider the problem of locating an obstacle in a waveguide from time domain measurements of causal waves. More precisely, we assume that we are given the scattered field due to point sources placed on a surface located inside the waveguide away from the obstacle, where the scattered field is measured on the same surface. From this multi-static scattering data we wish to determine the position and shape of an obstacle in the waveguide. To deal with this inverse problem, we adapt and analyze the time domain linear sampling method. This involves proving new time domain estimates for the forward problem, as well as analyzing several time domain operators arising in the inversion scheme. We also implement the inversion algorithm and provide numerical results in two-dimensions using synthetic data. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of recovering the (transformed) relaxation spectrum h from the (transformed) loss modulus g by inverting the integral equation , where denotes convolution, using Fourier transforms. We are particularly interested in establishing properties of h, having assumed that the Fourier transform of g has entire extension to the complex plane. In the setting of square integrable functions, we demonstrate that the Paley–Wiener theorem cannot be used to show the existence of non-trivial relaxation spectra with compact support. We prove a stronger result for tempered distributions: there are no non-trivial relaxation spectra with compact support. Finally we establish necessary and sufficient conditions for the relaxation spectrum h to be strictly positive definite. [ABSTRACT FROM AUTHOR]

The ill-posedness of the attenuated Radon transform is a challenging issue in practice due to the Poisson noise and the high level of attenuation. The investigation of the smoothing properties of the underlying operator is essential for developing a stable inversion. In this paper, we consider the framework of Sobolev spaces and derive analytically a reconstruction algorithm based on the method of the approximate inverse. The derived method inherits the efficiency and stability of the approximate inverse and supplies a method of extraction of contours. These algorithms appear to be efficient for an attenuation of human body type. However, for higher attenuations the ill-posedness increases exponentially what deteriorates accordingly the quality of reconstructions. Nevertheless, a high attenuation map affects less the contour extraction of a high contrast function and so can be neglected. This leads to simplifying the proposed method and circumvents in this case the artifacts due to the attenuation as attested by simulation results. [ABSTRACT FROM AUTHOR]

This paper concerns thermoacoustic tomography and photoacoustic tomography, two couple-physics imaging modalities that attempt to combine the high resolution of ultrasound and the high contrast capabilities of electromagnetic waves. We give sufficient conditions to recover both the sound speed of the medium being probed and the source. [ABSTRACT FROM AUTHOR]

In this article, we consider the limited angle problem in computed x-ray tomography. A common practice is to use the filtered back projection with the limited data. However, this practice has been shown to introduce the artifacts along some straight lines. In this work, we characterize the strength of these artifacts. [ABSTRACT FROM AUTHOR]

We construct a variational formulation for the problem of interpolating seismic data in the case of missing traces. We assume that we have derivative information available at the traces. The variational problem is essentially the minimization of the integral over the smallest eigenvalue of the structure tensor associated with the interpolated data. This has the physical meaning of penalizing the local presence of more than one direction in the interpolation. The variational problem is used to justify the solutions of a non-standard anisotropic diffusion problem as reasonable interpolated images. We show existence and uniqueness for this type of anisotropic diffusion. In particular, the uniqueness property is important as it guarantees that the solution can be obtained by the numerical schemes we propose. [ABSTRACT FROM AUTHOR]

Recent advances in technology have enabled the combination of positron emission tomography (PET) with magnetic resonance imaging (MRI). These PET-MRI scanners simultaneously acquire functional PET and anatomical or functional MRI data. As function and anatomy are not independent of one another the images to be reconstructed are likely to have shared structures. We aim to exploit this inherent structural similarity by reconstructing from both modalities in a joint reconstruction framework. The structural similarity between two modalities can be modelled in two different ways: edges are more likely to be at similar positions and/or to have similar orientations. We analyse the diffusion process generated by minimizing priors that encapsulate these different models. It turns out that the class of parallel level set priors always corresponds to anisotropic diffusion which is sometimes forward and sometimes backward diffusion. We perform numerical experiments where we jointly reconstruct from blurred Radon data with Poisson noise (PET) and under-sampled Fourier data with Gaussian noise (MRI). Our results show that both modalities benefit from each other in areas of shared edge information. The joint reconstructions have less artefacts and sharper edges compared to separate reconstructions and the ℓ2-error can be reduced in all of the considered cases of under-sampling. [ABSTRACT FROM AUTHOR]

We consider a topological integral transform of Bessel (concentric isospectral sets) type and Fourier (hyperplane isospectral sets) type, using the Euler characteristic as a measure. These transforms convert constructible {\mathbb Z}-valued functions to continuous {\mathbb R}-valued functions over {\mathbb R}^n. Core contributions include: the definition of the topological Bessel transform, a relationship in terms of the logarithmic blowup of the topological Fourier transform and a novel Morse index formula for the transforms. We then apply the theory to problems of target reconstruction from enumerative sensor data, including localization and shape discrimination. This last application utilizes an extension of spatially variant apodization to mitigate sidelobe phenomena. [ABSTRACT FROM AUTHOR]

This paper deals with image restoration problems where the data are nonuniform samples of the Fourier transform of the unknown object. We study the inverse problem in both semidiscrete and fully discrete formulations, and our analysis leads to an optimization problem involving the minimization of the data discrepancy under nonnegativity constraints. In particular, we show that such a problem is equivalent to a deconvolution problem in the image space. We propose a practical algorithm, based on the gradient projection method, to compute a regularized solution in the discrete case. The key point in our deconvolution-based approach is that the fast Fourier transform can be employed in the algorithm implementation without the need of preprocessing the data. A numerical experimentation on simulated and real data from the NASA RHESSI mission is also performed. [ABSTRACT FROM AUTHOR]

We consider a near-field inverse scattering problem for the wave equation: find the shape of a Dirichlet scattering object from time domain measurements of scattered waves. For this time-domain inverse problem, we propose a linear sampling method, a well-known technique for corresponding frequency domain inverse scattering problems. The problem setting and the algorithm incorporate two basic features. First, the data for the method consist of measurements of causal waves, that is, of waves that vanish before some moment in time. Second, the inversion algorithm directly works on the time-domain data without using a Fourier transformation. The first point is related to the applications we have in mind, which include for instance ground-penetrating radar imaging. The second feature allows us to naturally incorporate multiple (in fact, a continuum of) frequencies in the inversion algorithm. Consequently, it offers the potential of improving the quality of the reconstruction compared to frequency domain methods working with a single frequency. We demonstrate this potential by several numerical examples. [ABSTRACT FROM AUTHOR]

In this paper we consider the numerical solution of a phase retrieval problem for a compactly supported, linear spline f: {\bb R} \to {\bb C} with the Fourier transform {\skew5\hat{f}} , where values of |f| and |{\skew5\hat{f}}| at finitely many equispaced nodes are given. The unknown phases of complex spline coefficients fulfil a well-structured system of nonlinear equations. Thus the phase reconstruction leads to a nonlinear inverse problem, which is solved by a multilevel strategy and iterative Tikhonov regularization. The multilevel strategy concentrates the main effort of the solution of the phase retrieval problem in the coarse, less expensive levels and provides convenient initial guesses at the next finer level. On each level, the corresponding nonlinear system is solved by an iteratively regularized Gauss-Newton method. The multilevel strategy is motivated by convergence results of IRGN. This method is applicable to a wide range of examples as shown in several numerical tests for noiseless and noisy data. [ABSTRACT FROM AUTHOR]

This paper is concerned with deconvolution from error or blurring densities whose Fourier transforms have isolated zeros and show oscillatory behaviour; unlike conventional approaches where the Fourier transform decays about monotonously. We introduce specific estimation procedures based on local polynomial approximation in the Fourier domain. Under combined moment and smoothness conditions, we are able to improve the convergence rates compared to existing methods in density deconvolution. The corresponding minimax theory is derived. In compactly supported models as in signal deconvolution and Berkson regression, nearly optimal rates are achieved under conditions which are significantly weaker than those assumed in earlier papers. [ABSTRACT FROM AUTHOR]