Compressive Sensing of Analog Signals Using Discrete Prolate Spheroidal Sequences

Compressive sensing (CS) has recently emerged as a framework for efficiently capturing signals that are sparse or compressible in an appropriate basis. While often motivated as an alternative to Nyquist-rate sampling, there remains a gap between the discrete, finite-dimensional CS framework and the problem of acquiring a continuous-time signal. In this paper, we attempt to bridge this gap by exploiting the Discrete Prolate Spheroidal Sequences (DPSS's), a collection of functions that trace back to the seminal work by Slepian, Landau, and Pollack on the effects of time-limiting and bandlimiting operations. DPSS's form a highly efficient basis for sampled bandlimited functions; by modulating and merging DPSS bases, we obtain a dictionary that offers high-quality sparse approximations for most sampled multiband signals. This multiband modulated DPSS dictionary can be readily incorporated into the CS framework. We provide theoretical guarantees and practical insight into the use of this dictionary for recovery of sampled multiband signals from compressive measurements

Eigenvalue estimates for Fourier concentration operators on two domains

We derive eigenvalue estimates for concentration operators associated with the discrete Fourier transform and two concentration domains satisfying certain regularity conditions. These conditions are met, for example, when the discrete domain, contained in a lattice, is obtained by discretization of a suitably regular domain in the Euclidean space. As a limit, we obtain eigenvalue estimates for Fourier concentration operators associated with two suitably regular domains in the Euclidean space. Our results cover for the first time non-convex and non-symmetric concentration models in the spatial and frequency domains, as demanded by numerous applications that exploit the expected approximate low dimensionality of the modeled phenomena. The proofs build on Israel's work on one dimensional intervals [arXiv: 1502.04404v1]. The new ingredients are the use of redundant wave-packet expansions and a dyadic decomposition argument to obtain Schatten norm estimates for Hankel operators.Comment: Extended discussion of application

Information Theoretic Methods For Biometrics, Clustering, And Stemmatology

This thesis consists of four parts, three of which study issues related to theories and applications of biometric systems, and one which focuses on clustering. We establish an information theoretic framework and the fundamental trade-off between utility of biometric systems and security of biometric systems. The utility includes person identification and secret binding, while template protection, privacy, and secrecy leakage are security issues addressed. A general model of biometric systems is proposed, in which secret binding and the use of passwords are incorporated. The system model captures major biometric system designs including biometric cryptosystems, cancelable biometrics, secret binding and secret generating systems, and salt biometric systems. In addition to attacks at the database, information leakage from communication links between sensor modules and databases is considered. A general information theoretic rate outer bound is derived for characterizing and comparing the fundamental capacity, and security risks and benefits of different system designs. We establish connections between linear codes to biometric systems, so that one can directly use a vast literature of coding theories of various noise and source random processes to achieve good performance in biometric systems. We develop two biometrics based on laser Doppler vibrometry: LDV) signals and electrocardiogram: ECG) signals. For both cases, changes in statistics of biometric traits of the same individual is the major challenge which obstructs many methods from producing satisfactory results. We propose a ii robust feature selection method that specifically accounts for changes in statistics. The method yields the best results both in LDV and ECG biometrics in terms of equal error rates in authentication scenarios. Finally, we address a different kind of learning problem from data called clustering. Instead of having a set of training data with true labels known as in identification problems, we study the problem of grouping data points without labels given, and its application to computational stemmatology. Since the problem itself has no true answer, the problem is in general ill-posed unless some regularization or norm is set to define the quality of a partition. We propose the use of minimum description length: MDL) principle for graphical based clustering. In the MDL framework, each data partitioning is viewed as a description of the data points, and the description that minimizes the total amount of bits to describe the data points and the model itself is considered the best model. We show that in synthesized data the MDL clustering works well and fits natural intuition of how data should be clustered. Furthermore, we developed a computational stemmatology method based on MDL, which achieves the best performance level in a large dataset

Joint reconstruction of the mass distributions of galaxy clusters from gravitational lensing and thermal gas

We focus on the reconstruction of mass distributions of the massive galaxy clusters, which are the largest gravitationally bound objects in the Universe. An approach to determine the masses of clusters is based on the effects of gravitational lensing. Estimating errors induced by this method is crucial but computationally expensive. We present a novel approach to estimate analytically the errors made by reconstructions which use weak-lensing information. As galaxy clusters host a large amount of intracluster medium they provide a multitude of observables. We present a new method to infer the lensing potential from two of these: signals of the thermal Sunyaev-Zel’dovich effect and the emission of X-rays due to thermal bremsstrahlung. By assuming that the gas is in hydrostatic equilibrium and follows a polytropic equation of state, we link these observables to the gravitational potential, which is then projected along the line-of-sight to infer the lensing potential. For this we deproject the observables by means of the Richardson-Lucy algorithm. We test our method on clusters with analytic profiles, a numerical simulation and on the galaxy cluster RXJ1347. Our efforts are the first steps towards a non-parametric algorithm for a joint cluster reconstruction. By taking all possible cluster observables into account, mass distributions of clusters will be determined more accurately