Exploiting Structural Complexity for Robust and Rapid Hyperspectral Imaging

This paper presents several strategies for spectral de-noising of hyperspectral images and hypercube reconstruction from a limited number of tomographic measurements. In particular we show that the non-noisy spectral data, when stacked across the spectral dimension, exhibits low-rank. On the other hand, under the same representation, the spectral noise exhibits a banded structure. Motivated by this we show that the de-noised spectral data and the unknown spectral noise and the respective bands can be simultaneously estimated through the use of a low-rank and simultaneous sparse minimization operation without prior knowledge of the noisy bands. This result is novel for for hyperspectral imaging applications. In addition, we show that imaging for the Computed Tomography Imaging Systems (CTIS) can be improved under limited angle tomography by using low-rank penalization. For both of these cases we exploit the recent results in the theory of low-rank matrix completion using nuclear norm minimization

ROBUST PRINCIPAL COMPONENT ANALYSIS: THEORETICAL ASPECTS AND ALGORITHMIC COMPARATIVE EVALUATION FOR DIMENSIONALITY REDUCTION

Στην παρούσα διπλωματική εργασία εξετάζεται το κατά πόσο η ευρέως γνωστή ανάλυση κύριων συνιστωσών ως μια μέθοδος μείωσης της διάστασης μπορεί να καταστεί εύρωστη απέναντι σε ακραίες τιμές / παρατηρήσεις, και αν κάτι τέτοιο είναι δυνατό ποιο αλγοριθμικό σχήμα από τη βιβλιογραφία αποτελεί την καλύτερη επιλογή. Αρχικά, παρουσιάζεται η κλασική ανάλυση κύριων συνιστωσών, οι βασικές της ιδέες, εκείνες οι ιδιότητες-κλειδιά της οι οποίες την έχουν καταστήσει τόσο δημοφιλή, τα πλεονεκτήματά της καθώς και τα μειονεκτήματα αυτής. Στη συνέχεια, γίνεται μνεία στα βασικά θεωρητικά αποτελέσματα που αφορούν στην πιθανότητα η ανάλυση κύριων συνιστωσών να καταστεί εύρωστη απέναντι σε ακραίες τιμές, καθώς επίσης και σε μερικές ενδιαφέρουσες εφαρμογές της πραγματικής ζωής όπου κάτι τέτοιο θα ήταν αρκετά χρήσιμο. Ακολούθως, λαμβάνει χώρα μια αναλυτική παρουσίαση των πιο διάσημων αλγοριθμικών σχημάτων που σχεδιάστηκαν ώστε να αντιμετωπίσουν αυτό το πρόβλημα, ακολουθούμενη από μία συγκριτική ανάλυση μεταξύ τους η οποία εδράζεται σε ευρέως χρησιμοποιούμενες μετρικές ποιότητας σε αυτό το επιστημονικό πεδίο. Τέλος, εξετάζεται μια μελέτη-περίπτωσης προερχόμενη από το πεδίο της επεξεργασίας εικόνας, ώστε από τη μία πλευρά να αποτιμηθεί η επίδοση των υπό μελέτη αλγορίθμων σε “δυσκολότερες” πειραματικές συνθήκες, από την άλλη δε πλευρά να διερευνηθεί η πρακτική χρησιμότητά τους σε ρεαλιστικές εφαρμογές.In the present master thesis we examine the question of whether the PCA method for dimensionality reduction could become robust vis-à-vis gross errors, and if so which algorithmic scheme from the literature would be the best choice. In the beginning, we present the classical PCA method, its main ideas, those key properties that have made it so popular, its advantages and its disadvantages. Afterwards, we state the main theoretical results concerning the possibility of robustyfying the PCA method, as well as some interesting applications of real life in which a robust PCA method could prove extremely useful. Subsequently, a detailed presentation of the most popular algorithmic schemes designed to tackle this problem takes place, followed by a respective comparative analysis among them based on widely used quality metrics used in this scientific field. Finally, a case-study inspired by the field of image processing is examined, in order on the one hand to evaluate the performance of the algorithmic schemes studied in the present thesis under tougher experimental circumstances, as well as on the other hand to examine their practical use in realistic applications

On Grid Compressive Sampling for Spherical Field Measurements in Acoustics

We derive a compressive sampling method for acoustic field reconstruction using field measurements on a predefined spherical grid that has theoretically guaranteed relations between signal sparsity, measurement number, and reconstruction accuracy. This method can be used to reconstruct band-limited spherical harmonic or Wigner $D$-function series (spherical harmonic series are a special case) with sparse coefficients. Contrasting typical compressive sampling methods for Wigner $D$-function series that use arbitrary random measurements, the new method samples randomly on an equiangular grid, a practical and commonly used sampling pattern. Using its periodic extension, we transform the reconstruction of a Wigner $D$-function series into a multi-dimensional Fourier domain reconstruction problem. We establish that this transformation has a bounded effect on sparsity level and provide numerical studies of this effect. We also compare the reconstruction performance of the new approach to classical Nyquist sampling and existing compressive sampling methods. In our tests, the new compressive sampling approach performs comparably to other guaranteed compressive sampling approaches and needs a fraction of the measurements dictated by the Nyquist sampling theorem. Moreover, using one-third of the measurements or less, the new compressive sampling method can provide over 20 dB better denoising capability than oversampling with classical Fourier theory.Comment: 19 pages 14 figure

Adaptive MIMO Radar for Target Detection, Estimation, and Tracking

We develop and analyze signal processing algorithms to detect, estimate, and track targets using multiple-input multiple-output: MIMO) radar systems. MIMO radar systems have attracted much attention in the recent past due to the additional degrees of freedom they offer. They are commonly used in two different antenna configurations: widely-separated: distributed) and colocated. Distributed MIMO radar exploits spatial diversity by utilizing multiple uncorrelated looks at the target. Colocated MIMO radar systems offer performance improvement by exploiting waveform diversity. Each antenna has the freedom to transmit a waveform that is different from the waveforms of the other transmitters. First, we propose a radar system that combines the advantages of distributed MIMO radar and fully polarimetric radar. We develop the signal model for this system and analyze the performance of the optimal Neyman-Pearson detector by obtaining approximate expressions for the probabilities of detection and false alarm. Using these expressions, we adaptively design the transmit waveform polarizations that optimize the target detection performance. Conventional radar design approaches do not consider the goal of the target itself, which always tries to reduce its detectability. We propose to incorporate this knowledge about the goal of the target while solving the polarimetric MIMO radar design problem by formulating it as a game between the target and the radar design engineer. Unlike conventional methods, this game-theoretic design does not require target parameter estimation from large amounts of training data. Our approach is generic and can be applied to other radar design problems also. Next, we propose a distributed MIMO radar system that employs monopulse processing, and develop an algorithm for tracking a moving target using this system. We electronically generate two beams at each receiver and use them for computing the local estimates. Later, we efficiently combine the information present in these local estimates, using the instantaneous signal energies at each receiver to keep track of the target. Finally, we develop multiple-target estimation algorithms for both distributed and colocated MIMO radar by exploiting the inherent sparsity on the delay-Doppler plane. We propose a new performance metric that naturally fits into this multiple target scenario and develop an adaptive optimal energy allocation mechanism. We employ compressive sensing to perform accurate estimation from far fewer samples than the Nyquist rate. For colocated MIMO radar, we transmit frequency-hopping codes to exploit the frequency diversity. We derive an analytical expression for the block coherence measure of the dictionary matrix and design an optimal code matrix using this expression. Additionally, we also transmit ultra wideband noise waveforms that improve the system resolution and provide a low probability of intercept: LPI)