Sampling and Super-resolution of Sparse Signals Beyond the Fourier Domain

Recovering a sparse signal from its low-pass projections in the Fourier domain is a problem of broad interest in science and engineering and is commonly referred to as super-resolution. In many cases, however, Fourier domain may not be the natural choice. For example, in holography, low-pass projections of sparse signals are obtained in the Fresnel domain. Similarly, time-varying system identification relies on low-pass projections on the space of linear frequency modulated signals. In this paper, we study the recovery of sparse signals from low-pass projections in the Special Affine Fourier Transform domain (SAFT). The SAFT parametrically generalizes a number of well known unitary transformations that are used in signal processing and optics. In analogy to the Shannon's sampling framework, we specify sampling theorems for recovery of sparse signals considering three specific cases: (1) sampling with arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels and, (3) recovery from Gabor transform measurements linked with the SAFT domain. Our work offers a unifying perspective on the sparse sampling problem which is compatible with the Fourier, Fresnel and Fractional Fourier domain based results. In deriving our results, we introduce the SAFT series (analogous to the Fourier series) and the short time SAFT, and study convolution theorems that establish a convolution--multiplication property in the SAFT domain.Comment: 42 pages, 3 figures, manuscript under revie

Speckle pattern interferometry : vibration measurement based on a novel CMOS camera

A digital speckle pattern interferometer based on a novel custom complementary metaloxide- semiconductor (CMOS) array detector is described. The temporal evolution of the dynamic deformation of a test object is measured using inter-frame phase stepping. The flexibility of the CMOS detector is used to identify regions of interest with full-field time averaged measurements and then to interrogate those regions with time-resolved measurements sampled at up to 7 kHz. The maximum surface velocity that can be measured and the number of measurement points are limited by the frame rate and the data transfer rate of the detector. The custom sensor used in this work is a modulated light camera (MLC), whose pixel design is still based on the standard four transistor active pixel sensor (APS), but each pixel has four large independently shuttered capacitors that drastically boost the well capacity from that of the diode alone. Each capacitor represents a channel which has its own shutter switch and can either be operated independently or in tandem with others. The particular APS of this camera enables a novel approach in how the data are acquired and then processed. In this Thesis we demonstrate how, at a given frame rate and at a given number of measurement points, the data transfer rate of our system is increased if compared to the data transfer rate of a system using a standard approach. Moreover, under some assumptions, the gain in system bandwidth doesn’t entail any reduction in the maximum surface velocity that can be reliably measured with inter-frame phase stepping

The Affine Uncertainty Principle, Associated Frames and Applications in Signal Processing

Uncertainty relations play a prominent role in signal processing, stating that a signal can not be simultaneously concentrated in the two related domains of the corresponding phase space. In particular, a new uncertainty principle for the affine group, which is directly related to the wavelet transform has lead to a new minimizing waveform. In this thesis, a frame construction is proposed which leads to approximately tight frames based on this minimizing waveform. Frame properties such as the diagonality of the frame operator as well as lower and upper frame bounds are analyzed. Additionally, three applications of such frame constructions are introduced: inpainting of missing audio data, detection of neuronal spikes in extracellular recorded data and peak detection in MALDI imaging data