Phase Recovery Guarantees from Designed Coded Diffraction Patterns in Optical Imaging

Phase retrieval is an inverse problem that consists in estimating a scene from diffraction intensities. This problem appears in optical imaging, which has three main diffraction zones where the measurements can be acquired, i.e., near, middle and far. Recent works have theoretically solved this inverse problem for the far zone, creating redundancy in the measurement process by including a coded aperture, which allows to modulate the scene and acquire coded diffraction patterns (CDP). However, in the state-of-the-art, the PR problem has not been theoretically studied for CDP at the near and middle zones. Moreover, the structure of the coded aperture is selected at random, leading to suboptimal estimations. Indeed, some of the coding elements employed in the literature are unfeasible because they increase the power of the scene in the modulation process. This paper provides theoretical guarantees for the recovery of a scene from CDP acquired at the three diffraction zones using admissible modulations. Based on the theoretical results, it is established that the image reconstruction quality directly depends on the coded aperture structure; therefore, a design strategy is proposed. In fact, when the scene can be sparsely represented in some basis, its support can be better estimated for a suitable choice of the coding elements in the modulation process. Experimental results show that the scene is successfully recovered by using designed coded apertures with up to 40% less measurements compared to non-designed ensembles.