Binary Phase Retrieval of Cosine Transforms via Local Curvature Minimization

Abstract

Cosine transforms see frequent use in image and video compression due to their ease of computation and high energy compaction. This has sparked interest within the optics community in computing cosine transforms optically for compression. When imaging in the Fourier plane to capture a cosine transform, one encounters a phase retrieval problem: optical fields have both magnitude and phase, but cameras only capture the magnitude of the field. Cosine transforms restrict the phase retrieval problem to a binary domain as opposed to the unit circle, but general phase retrieval solutions do not leverage this. In this work, we demonstrate that sign errors in phase retrieval for cosine transforms result in a significant increase in the magnitude of the Hessian of the transform at the error. Motivated by this, we develop an algorithm to solve this binary phase retrieval problem by minimizing the curvature of the reconstructed cosine transform. We demonstrate via computational experiments that given the magnitude of the cosine transform of an image, we can consistently reconstruct the original image within a multiscale structural similarity index of 0.95. We show that the solve time grows approximately linearly with the number of pixels solved.