Sampling innovations

Sampling theory has prospered extensively in the last century. The elegant mathematics and the vast number of applications are the reasons for its popularity. The applications involved in this thesis are in signal processing and communications and call out to mathematical notions in Fourier theory, spectral analysis, basic linear algebra, spline and wavelet theory. This thesis is divided in two parts. Chapters 2 and 3 consider uniform sampling of non-bandlimited signals and Chapters 4, 5, and 6 treat different irregular sampling problems. In the first part we address the problem of sampling signals that are not bandlimited but are characterized as having a finite number of degrees of freedom per unit of time. These signals will be called signals with a finite rate of innovation. We show that these signals can be uniquely represented given a sufficient number of samples obtained using an appropriate sampling kernel. The number of samples must be greater or equal to the degrees of freedom of the signal; in other words, the sampling rate must be greater or equal to the rate of innovation of the signal. In particular, we derive sampling schemes for periodic and finite length streams of Diracs and piecewise polynomial signals using the sinc, the differentiated sinc and the Gaussian kernels. Sampling and reconstruction of piecewise bandlimited signals and filtered piecewise polynomials is also considered. We also derive local reconstruction schemes for infinite length piecewise polynomials with a finite "local" rate of innovation using compact support kernels such as splines. Numerical experiments on all of the reconstruction schemes are shown. The first topic of the second part of this thesis is the irregular sampling problem of bandlimited signals with unknown sampling instances. The locations of the irregular set of samples are found by treating the problem as a combinatorial optimization problem. Search methods for the locations are described and numerical simulations on a random set and a jittery set of locations are made. The second topic is the periodic nonuniform sampling problem of bandlimited signals. The irregular set of samples involved has a structure which is irregular yet periodic. We develop a fast scheme that reduces the complexity of the problem by exploiting the special pattern of the locations. The motivation for developing a fast scheme originates from the fact that the periodic nonuniform set was also considered in the sampling with unknown locations problem and that a fast search method for the locations was sought. Finally, the last topic is the irregular sampling of signals that are linearly and nonlinearly approximated using Fourier and wavelet bases. We present variants of the Papoulis Gerchberg algorithm which take into account the information given in the approximation of the signal. Numerical experiments are presented in the context of erasure correction

Autonomous 3D Exploration of Large Structures Using an UAV Equipped with a 2D LIDAR

This paper addressed the challenge of exploring large, unknown, and unstructured industrial environments with an unmanned aerial vehicle (UAV). The resulting system combined well-known components and techniques with a new manoeuvre to use a low-cost 2D laser to measure a 3D structure. Our approach combined frontier-based exploration, the Lazy Theta* path planner, and a flyby sampling manoeuvre to create a 3D map of large scenarios. One of the novelties of our system is that all the algorithms relied on the multi-resolution of the octomap for the world representation. We used a Hardware-in-the-Loop (HitL) simulation environment to collect accurate measurements of the capability of the open-source system to run online and on-board the UAV in real-time. Our approach is compared to different reference heuristics under this simulation environment showing better performance in regards to the amount of explored space. With the proposed approach, the UAV is able to explore 93% of the search space under 30 min, generating a path without repetition that adjusts to the occupied space covering indoor locations, irregular structures, and suspended obstaclesUnión Europea Marie Sklodowska-Curie 64215Unión Europea MULTIDRONE (H2020-ICT-731667)Uniión Europea HYFLIERS (H2020-ICT-779411

Performance of Linear Field Reconstruction Techniques with Noise and Uncertain Sensor Locations

We consider a wireless sensor network, sampling a bandlimited field, described by a limited number of harmonics. Sensor nodes are irregularly deployed over the area of interest or subject to random motion; in addition sensors measurements are affected by noise. Our goal is to obtain a high quality reconstruction of the field, with the mean square error (MSE) of the estimate as performance metric. In particular, we analytically derive the performance of several reconstruction/estimation techniques based on linear filtering. For each technique, we obtain the MSE, as well as its asymptotic expression in the case where the field number of harmonics and the number of sensors grow to infinity, while their ratio is kept constant. Through numerical simulations, we show the validity of the asymptotic analysis, even for a small number of sensors. We provide some novel guidelines for the design of sensor networks when many parameters, such as field bandwidth, number of sensors, reconstruction quality, sensor motion characteristics, and noise level of the measures, have to be traded off

Bayesian inference of CMB gravitational lensing

The Planck satellite, along with several ground based telescopes, have mapped the cosmic microwave background (CMB) at sufficient resolution and signal-to-noise so as to allow a detection of the subtle distortions due to the gravitational influence of the intervening matter distribution. A natural modeling approach is to write a Bayesian hierarchical model for the lensed CMB in terms of the unlensed CMB and the lensing potential. So far there has been no feasible algorithm for inferring the posterior distribution of the lensing potential from the lensed CMB map. We propose a solution that allows efficient Markov Chain Monte Carlo sampling from the joint posterior of the lensing potential and the unlensed CMB map using the Hamiltonian Monte Carlo technique. The main conceptual step in the solution is a re-parameterization of CMB lensing in terms of the lensed CMB and the "inverse lensing" potential. We demonstrate a fast implementation on simulated data including noise and a sky cut, that uses a further acceleration based on a very mild approximation of the inverse lensing potential. We find that the resulting Markov Chain has short correlation lengths and excellent convergence properties, making it promising for application to high resolution CMB data sets of the future