Scanning-free compressive reconstruction of object motion with sub-pixel accuracy

Abstract

Sub-pixel movement detection is an under-sampling problem. The basic idea for successful detection is to spread out the information over a larger sampling region. Diffraction provides a natural way to spread out the information; however, conventional digital holographic methods are not effective for extracting sub-pixel accuracy. Here we show how to apply compressive reconstruction to the same problem effectively. Compressed sensing is a new framework to systematically find highly accurate solutions to an under-sampled linear system. To guarantee the accuracy of reconstruction result, compressed sensing requires that the unknown has to be sparse in some predetermined basis. In our study, for the one dimensional sub-pixel movement detection, we propose to use the derivative operator as the sparsifying basis. We implemented the derivative operator to the hologram and applied a sparsity constraint on the object derivative space for compressive holography. Together with spectrum domain zero-padding, our compressive algorithm allows for sub-pixel accuracy edge localization. The extension to the 2D case is not trivial. It has been shown that the spiral phase mask can serve as an approximate 2D derivative operator in the Fourier domain. In this case, we implemented spiral phase filtering in the hologram spectrum domain. By applying cross-correlation between reconstructions for consecutive subpixel movements, sub-pixel movement was successfully detected.