Sparse phase retrieval of structured signals by Prony's method

The phase retrieval problem consists in the recovery of a complex-valued signal from the magnitudes of its Fourier transform. Restricting ourselves to the case of sparse structured signals f, which can be represented as a linear combination of N arbitrary translations of a given generator function, we show that almost all f can be recovered from 𝒪 (N²) intensity measurements |ℱ[f](ω)| up to trivial ambiguities. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of Adaptive Fourier Series by Sparse Approximation of Exponential Sums

Journal of Fourier Analysis and Applications - In this paper, we study the convergence of adaptive Fourier sums for real-valued $$2\pi $$ -periodic functions. For this purpose, we approximate the...     

A deterministic sparse FFT algorithm for vectors with small support

In this paper we consider the special case where a signal x\({\in }\,\mathbb {C}^{N}\) is known to vanish outside a support interval of length m < N. If the support length m of x or a good bound of it is a-priori known we derive a sublinear deterministic algorithm to compute x from its discrete Fourier transform \(\widehat {\mathbf x}\,{\in }\,\mathbb {C}^{N}\). In case of exact Fourier measurements we require only \({\mathcal O}\)(m\(\log \)m) arithmetical operations. For noisy measurements, we propose a stable \({\mathcal O}\)(m\(\log \)N) algorithm.     

Ambiguities in one-dimensional phase retrieval of structured functions

Phase retrieval means that we wish to recover a complex-valued signal (discrete or continuous) from the magnitudes of its Fourier transform. Here we restrict ourselves to the recovery of structured functions, e.g. linear spline functions with equidistant knots. First, a complete characterization of the occurring ambiguities is presented. Moreover, we investigate additionally given moduli of the signal itself regarding their ability to reduce the set of ambiguities. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-scale symbol and autocorrelation symbol for B-splines with multiple knots

The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two-scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spline interpolation problem for spline functions of degree 2m+1 and defectr, first considered by Lipow and Schoenberg [9], is equivalent to the Riesz basis property of our B-splines with degreem and defectr. In this way we obtain a new, simple proof for the assertion that the cardinal Hermite spline interpolation problem in [9] has a unique solution.     

An Area Preserving Projection from the Regular Octahedron to the Sphere

In this paper, we propose an area preserving bijective map from the regular octahedron to the unit sphere ${\mathbb{S}^2}$ , both centered at the origin. The construction scheme consists of two steps. First, each face F _i of the octahedron is mapped to a curved planar triangle ${\mathcal{T}_i}$ of the same area. Afterwards, each ${\mathcal{T}_i}$ is mapped onto the sphere using the inverse Lambert azimuthal equal area projection with respect to a certain point of ${\mathbb{S}^2}$ . The proposed map is then used to construct uniform and refinable grids on a sphere, starting from any triangular uniform and refinable grid on the triangular faces of the octahedron.     

Sparse fast DCT for vectors with one-block support

Numerical Algorithms - In this paper, we present a new fast and deterministic algorithm for the inverse discrete cosine transform of type II that reconstructs the vector $\mathbf {x}\in \mathbb...     

Iterative Phase Retrieval with Sparsity Constraints

We consider the problem of iterative phase retrieval in near-field X-ray propagation imaging. Single-distance measurements ask for strong a priori information about the specimen that are not always accessible, e.g. the specimen's support. We propose to exploit sparsity of real-valued cartoon-like images using soft-thresholding of shearlet coefficients. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

How many Fourier samples are needed for real function reconstruction?

In this paper we present some new results on the reconstruction of structured functions by a small number of equidistantly distributed Fourier samples. In particular, we show that real spline functions of order m with non-uniform knots containing N terms can be uniquely reconstructed by only m+N Fourier samples. Further, linear combinations of N non-equispaced shifts of a known low-pass function Φ can be reconstructed by N+1 Fourier samples. In the bivariate case, we consider the problem of function recovering by a small amount of Fourier samples on different lines through the origin. Our methods are based on the Prony method. The proofs given in this paper are constructive. Some numerical examples show the applicability of the proposed approach.