  Lensless in-line holography is a simple, portable, and cost-effective method
of imaging especially for the biomedical microscopy applications. We propose a
multiplicative gradient descent optimization based method to obtain multi-depth
imaging from a single hologram acquired in this imaging system. We further
extend the method to achieve phase imaging from a single hologram.
Negative-log-likelihood functional with the assumption of poisson noise has
been used as the cost function to be minimized. The ill-posed nature of the
problem is handled by the sparse regularization and the upper-bound constraint.
The gradient descent optimization requires calculation of the partial
derivative of the cost function with respect to a given estimate of the object.
A method of obtaining this quantity for holography in both the cases of real
object and complex object has been shown. The reconstruction method has been
validated using extensive simulation and experimental studies. The comparison
with the previously established iterative shrinkage/thresholding algorithm
based compressive holography shows that the proposed method has the following
advantages: significantly faster convergence rate, better reconstructed image
quality and the ability to perform phase imaging.
