Randomized Signal Processing with Continuous Frames

This paper focuses on signal processing tasks in which the signal is transformed from the signal space to a higher dimensional coefficient space (also called phase space) using a continuous frame, processed in the coefficient space, and synthesized to an output signal. We show how to approximate such methods, termed phase space signal processing methods, using a Monte Carlo method. As opposed to standard discretizations of continuous frames, based on sampling discrete frames from the continuous system, the proposed Monte Carlo method is directly a quadrature approximation of the continuous frame. We show that the Monte Carlo method allows working with highly redundant continuous frames, since the number of samples required for a certain accuracy is proportional to the dimension of the signal space, and not to the dimension of the phase space. Moreover, even though the continuous frame is highly redundant, the Monte Carlo samples are spread uniformly, and hence represent the coefficient space more faithfully than standard frame discretizations.

Compressive Coded Rotating Mirror Camera for High-Speed Imaging

We develop a novel compressive coded rotating mirror (CCRM) camera to capture events at high frame rates in passive mode with a compact instrument design at a fraction of the cost compared to other high-speed imaging cameras. Operation of the CCRM camera is based on amplitude optical encoding (grey scale) and a continuous frame sweep across a low-cost detector using a motorized rotating mirror system which can achieve single pixel shift between adjacent frames. Amplitude encoding and continuous frame overlapping enable the CCRM camera to achieve a high number of captured frames and high temporal resolution without making sacrifices in the spatial resolution. Two sets of dynamic scenes have been captured at up to a 120 Kfps frame rate in both monochrome and colored scales in the experimental demonstrations. The obtained heavily compressed data from the experiment are reconstructed using the optimization algorithm under the compressive sensing (CS) paradigm and the highest sequence depth of 1400 captured frames in a single exposure has been achieved with the highest compression ratio of 368 compared to other CS-based high-speed imaging technologies. Under similar conditions the CCRM camera is 700× faster than conventional rotating mirror based imaging devices and could reach a frame rate of up to 20 Gfps.

Lower Semi-frames, Frames, and Metric Operators

This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.