Achievable Angles Between two Compressed Sparse Vectors Under Norm/Distance Constraints Imposed by the Restricted Isometry Property: A Plane Geometry Approach

The angle between two compressed sparse vectors subject to the norm/distance constraints imposed by the restricted isometry property (RIP) of the sensing matrix plays a crucial role in the studies of many compressive sensing (CS) problems. Assuming that (i) u and v are two sparse vectors separated by an angle thetha, and (ii) the sensing matrix Phi satisfies RIP, this paper is aimed at analytically characterizing the achievable angles between Phi*u and Phi*v. Motivated by geometric interpretations of RIP and with the aid of the well-known law of cosines, we propose a plane geometry based formulation for the study of the considered problem. It is shown that all the RIP-induced norm/distance constraints on Phi*u and Phi*v can be jointly depicted via a simple geometric diagram in the two-dimensional plane. This allows for a joint analysis of all the considered algebraic constraints from a geometric perspective. By conducting plane geometry analyses based on the constructed diagram, closed-form formulae for the maximal and minimal achievable angles are derived. Computer simulations confirm that the proposed solution is tighter than an existing algebraic-based estimate derived using the polarization identity. The obtained results are used to derive a tighter restricted isometry constant of structured sensing matrices of a certain kind, to wit, those in the form of a product of an orthogonal projection matrix and a random sensing matrix. Follow-up applications to three CS problems, namely, compressed-domain interference cancellation, RIP-based analysis of the orthogonal matching pursuit algorithm, and the study of democratic nature of random sensing matrices are investigated.Comment: submitted to IEEE Trans. Information Theor

Usmjereno daljinsko istraživanje

Concepts of directional remote sensing are put forward based on two-dimensional compressive sensing. Very little measured data are required to acquire and reconstruct change areas in directional remote sensing. The measured data in one-dimensional compressive sensing not only keep the energy of a sparse signal, but also inherit the sparse signal’s direction information. However, direction information can\u27t be applied to reconstruction and test of a sparse signal in one-dimensional compression sensing. The two-dimensional compressive sensing model is proposed based on sparse features of change areas in remote sensing. Moreover, a sparse signal reconstruction algorithm (two-step reconstruction method, 2SRM) is proposed based on two-dimensional compressive sensing by use of the energy and direction information. The theoretical analysis and experimental results show the signal reconstruction ability of 2SRM is stronger. SNR (Signal to Noise Ratio) and PSNR (Peak Signal to Noise Ratio) of 2SRM increase by 16.57 dB as compared with a single traditional reconstruction algorithm at most.Koncepti usmjerenog daljinskog istraživanja postavljeni su na osnovi dvodimenzionalnog kompresivnog istraživanja. Potrebno je vrlo malo mjernih podataka da bi se postigla i rekonstruirala područja promjena u usmjerenom daljinskom istraživanju. Mjerni podaci u jednodimenzionalnom kompresivnom istraživanju ne samo da čuvaju energiju rasutih signala već, također, preuzimaju informacije o smjeru rasutog signala. Informacija o smjeru se ipak ne može primijeniti za rekonstrukciju i ispitivanje rasutog signala u jednodimenzionalnom kompresivnom istraživanju. Model dvodimenzionalnog kompresivnog istraživanja predlaže se na osnovi rasutih svojstava područja promjene u daljinskom istraživanju. Osim toga, algoritam rekonstrukcije rasutog signala (metoda rekonstrukcije u dva koraka, 2SRM) predlaže se na osnovi dvodimenzionalnog kompresivnog istraživanja koristeći informacije o energiji i smjeru. Teorijska analiza i eksperimentalni rezultati pokazuju da je sposobnost rekonstrukcije signala primjenom metode 2SRM jača. SNR (vrijednost omjera signala i šuma) i PSNR (vrijednost maksimalnog omjera signala i šuma) primjenom metode 2SRM povećavaju se najviše za 16,57 dB u odnosu na pojedinačni tradicionalni algoritam rekonstrukcije

Sparse Bases and Bayesian Inference of Electromagnetic Scattering

Many approaches in CEM rely on the decomposition of complex radiation and scattering behavior with a set of basis vectors. Accurate estimation of the quantities of interest can be synthesized through a weighted sum of these vectors. In addition to basis decompositions, sparse signal processing techniques developed in the CS community can be leveraged when only a small subset of the basis vectors are required to sufficiently represent the quantity of interest. We investigate several concepts in which novel bases are applied to common electromagnetic problems and leverage the sparsity property to improve performance and/or reduce computational burden. The first concept explores the use of multiple types of scattering primitives to reconstruct scattering patterns of electrically large targets. Using a combination of isotropic point scatterers and wedge diffraction primitives as our bases, a 40% reduction in reconstruction error can be achieved. Next, a sparse basis is used to improve DOA estimation. We implement the BSBL technique to determine the angle of arrival of multiple incident signals with only a single snapshot of data from an arbitrary arrangement of non-isotropic antennas. This is an improvement over the current state-of-the-art, where restrictions on the antenna type, configuration, and a priori knowledge of the number of signals are often assumed. Lastly, we investigate the feasibility of a basis set to reconstruct the scattering patterns of electrically small targets. The basis is derived from the TCM and can capture non-localized scattering behavior. Preliminary results indicate that this basis may be used in an interpolation and extrapolation scheme to generate scattering patterns over multiple frequencies

Nearfield Acoustic Holography using sparsity and compressive sampling principles

Regularization of the inverse problem is a complex issue when using Near-field Acoustic Holography (NAH) techniques to identify the vibrating sources. This paper shows that, for convex homogeneous plates with arbitrary boundary conditions, new regularization schemes can be developed, based on the sparsity of the normal velocity of the plate in a well-designed basis, i.e. the possibility to approximate it as a weighted sum of few elementary basis functions. In particular, these new techniques can handle discontinuities of the velocity field at the boundaries, which can be problematic with standard techniques. This comes at the cost of a higher computational complexity to solve the associated optimization problem, though it remains easily tractable with out-of-the-box software. Furthermore, this sparsity framework allows us to take advantage of the concept of Compressive Sampling: under some conditions on the sampling process (here, the design of a random array, which can be numerically and experimentally validated), it is possible to reconstruct the sparse signals with significantly less measurements (i.e., microphones) than classically required. After introducing the different concepts, this paper presents numerical and experimental results of NAH with two plate geometries, and compares the advantages and limitations of these sparsity-based techniques over standard Tikhonov regularization.Comment: Journal of the Acoustical Society of America (2012

Trends in Mathematical Imaging and Surface Processing

Motivated both by industrial applications and the challenge of new problems, one observes an increasing interest in the field of image and surface processing over the last years. It has become clear that even though the applications areas differ significantly the methodological overlap is enormous. Even if contributions to the field come from almost any discipline in mathematics, a major role is played by partial differential equations and in particular by geometric and variational modeling and by their numerical counterparts. The aim of the workshop was to gather a group of leading experts coming from mathematics, engineering and computer graphics to cover the main developments

Detection of failures in antenna arrays through a Lebesgue-space approach

In this paper, a novel antenna array diagnostic approach is presented. The failures in antenna arrays are detected by means of a non-Hilbertian Lebesgue-space L-p technique to solve the underlying inverse problem. The solution of this inverse problem enables to retrieve the distribution of faulty feed excitations of the antenna under test starting from far-field measurements. The developed approach has been numerically validated. Simulations concern planar arrays where different rates and distributions of failures have been tested. Results show good capabilities in detecting damaged regions in the analyzed scenarios