Finding a point in the intersection of two closed convex sets is a common problem in image processing and other areas. Projections onto convex sets (POCS) is a standard algorithm for finding such a point. Dykstra's projection algorithm is a well known alternative that finds the point in the intersection closest to a given point. Yet another lesser known alternative is the alternating direction method of multipliers (ADMM) that can be used for both purposes. In this paper we discuss the differences in the convergence of these algorithms in image processing problems. The ADMM applied to finding an arbitrary point in the intersection is much faster than POCS and any algorithm for finding the nearest point in the intersection.
Imaging through scattering media finds applications in diverse fields from biomedicine to autonomous driving. However, interpreting the resulting images is difficult due to blur caused by the scattering of photons within the medium. Transient information, captured with fast temporal sensors, can be used to significantly improve the quality of images acquired in scattering conditions. Photon scattering, within a highly scattering media, is well modeled by the diffusion approximation of the Radiative Transport Equation (RTE). Its solution is easily derived which can be interpreted as a Spatio-Temporal Point Spread Function (STPSF). In this paper, we first discuss the properties of the ST-PSF and subsequently use this knowledge to simulate transient imaging through highly scattering media. We then propose a framework to invert the forward model, which assumes Poisson noise, to recover a noise-free, unblurred image by solving an optimization problem.