Multifrequency interferometric imaging with intensity-only measurements

We consider the problem of coherent imaging using intensity-only measurements. The main challenge in intensity-only imaging is recovering phase information that is not directly available in the data, but is essential for coherent image reconstruction. Imaging without phases arises in many applications such as crystallography, ptychography and optics where images are formed from the spectral intensities.

The earliest and most widely used methods for imaging with intensity-only measurements are alternating projection algorithms. The basic idea is to project the iterates on the intensity data sequentially in both the real and the Fourier spaces. Although these algorithms are very efficient for reconstructing the missing phases in the data, and performance is often good in practice, they do not always converge to the true, missing phases. This is especially so if strong constrains or prior information about the object to be imaged, such as spatial support and non-negativity, are not reliably available.

Rather than using phase retrieval methods, we propose a different approach in which well-designed illumination strategies exploit the spatial and frequency diversity inherent in the problem. These illumination strategies allow for the recovery of interferometric data that contain relative phase information which is all that is needed to reconstruct a so-called holographic image. There is no need for phase reconstruction in this approach. Moreover, we show that this methodology leads to holographic images that suffer no loss of resolution compared with those that use full phase information. This is so when
the media through which the probing signals propagate are assumed to be homogeneous.

We also consider inhomogeneous media where wavefront distortions can arise. In such media the incoherence in the recovered interferometric data can be reduced by restricting them to small spatial and frequency offsets. Using an efficient implementation of this restriction process we obtain holographic images with a somewhat reduced resolution compared to the homogeneous medium case.

The robustness of our approach will be explored with numerical simulations carried out in an optical (digital) microscopy imaging regime.