We discuss the ultimate quantum limits in compressed sensing, a new technique which allows for accurate reconstruction of so-called “sparse” signals even when the sampling rate is far below the Nyquist limit.
Compressed Sensing (CS) is a new method of signal and image processing which allows for exact recovery of an image from a number of samples much smaller than that required by the Nyquist/Shannon theorem. Compressed Sensing uses a priori information about the object called “sparsity”, which means that only a small number of image samples are nonzero. We have analyzed the superresolution behavior of CS taking into account the quantum fluctuations in the image. Our analysis allows to characterize the ultimate capabilities of CS imposed by the quantum nature of the light.
Compressive sensing utilizes sparsity to realize efficient image reconstruction. It is a valuable processing technique when cost, power, technology or computational overhead are limited or high. In the quantum domain technology usually limits efficient acquisition of weak or fragile signals. I will discuss the basics of information theory, compression, and compressive sensing. I will then discuss our recent work in compressive sensing. The topics of discussion include low-flux laser Radar, photonic phase transitions, high resolution biphoton ghost imaging, Ghost object tracking, 3D object tracking and high dimensional entanglement characterization. I will touch lightly on our current work of rapid wavefunction reconstruction and wavefront sensing. As an example (shown below), we were able efficiently and rapidly reconstruct high dimensional joint probability functions of biphotons in momentum and position. With conventional raster scanning this process would take approximately a year, but using double-pixel compressive sensing, the pictures were acquired in a few hours with modest flux.