Axel Flinth

Phase Retrieval from Gabor Measurements

Compressed sensing investigates the recovery of sparse signals from linear measurements. But often, in a wide range of applications, one is given only the absolute values (squared) of the linear measurements. Recovering such signals (not necessarily sparse) is known as the phase retrieval problem. We consider this problem in the case when the measurements are time-frequency shifts of a suitably chosen generator, i.e. coming from a Gabor frame. We prove an easily checkable injectivity condition for recovery of any signal from all

A geometrical stability condition for compressed sensing

N^2

PROMP: A sparse recovery approach to lattice-valued signals

N2 time-frequency shifts, and for recovery of sparse signals, when only some of those measurements are given.

Hierarchical Sparse Channel Estimation for Massive MIMO.

Compressed sensing deals with the recovery of sparse signals from linear measurements. Without any additional information, it is possible to recover an s-sparse signal using m > s log(d/s) measurements in a robust and stable way. Some applications provide additional information, such as on the location the support of the signal. Using this information, it is conceivable that the threshold amount of measurements can be lowered. A proposed algorithm for this task is weighted l1-minimization. Put shortly, one modifies standard 11-minimization by assigning different weights to different parts of the index set [1, ... d]. The task of choosing the weights is, however, non-trivial. This paper provides a complete answer to the question of an optimal choice of the weights. In fact, it is shown that it is possible to directly calculate unique weights that are optimal in the sense that the threshold amount of measurements needed for exact recovery is minimized. The proof uses recent results about the connection between convex geometry and compressed sensing-type algorithms.

Exact solutions of infinite dimensional total-variation regularized problems

Abstract During the last decade, the paradigm of compressed sensing has gained significant importance in the signal processing community. While the original idea was to utilize sparsity assumptions to design powerful recovery algorithms of vectors x ∈ R d , the concept has been extended to cover many other types of problems. A noteable example is low-rank matrix recovery. Many methods used for recovery rely on solving convex programs. A particularly nice trait of compressed sensing is its geometrical intuition. In recent papers, a classical optimality condition has been used together with tools from convex geometry and probability theory to prove beautiful results concerning the recovery of signals from Gaussian measurements. In this paper, we aim to formulate a geometrical condition for stability and robustness, i.e. for the recovery of approximately structured signals from noisy measurements. We will investigate the connection between the new condition with the notion of restricted singular values, classical stability and robustness conditions in compressed sensing, and also to important geometrical concepts from complexity theory. We will also prove the maybe somewhat surprising fact that for many convex programs, exact recovery of a signal x 0 immediately implies some stability and robustness when recovering signals close to x 0 .

Estimation of Angles of Arrival Through Superresolution -- A Soft Recovery Approach for General Antenna Geometries

The suboptimal performance of wavelets with regard to the approximation of multivariate data gave rise to new representation systems, specifically designed for data with anisotropic features. Some prominent examples of these are given by ridgelets, curvelets, and shearlets, to name a few.The great variety of such so-called directional systems motivated the search for a common framework, which unites many under one roof and enables a simultaneous analysis, for example with respect to approximation properties. Building on the concept of parabolic molecules, the recently introduced framework of α -molecules does in fact include the previous mentioned systems. Until now however it is confined to the bivariate setting, whereas nowadays one often deals with higher dimensional data. This motivates the extension of this unifying theory to dimensions larger than 2, put forward in this work. In particular, we generalize the central result that the cross-Gramian of any two systems of α -molecules will to some extent be localized.As an exemplary application, we investigate the sparse approximation of video signals, which are instances of 3D data. The multivariate theory allows us to derive almost optimal approximation rates for a large class of representation systems.