Lattice Pseudo-Random Arrays and Next-Generation Coded Aperture Imaging Systems

Imaging using high-energy radiation with a spectrum ranging from x-ray to γ-ray has found many applications, including high-energy astronomy and medical imaging. In these wavelengths, imaging using lenses is not possible since the rays cannot be refracted or reflected, and hence cannot be focused. An alternative technique to do imaging in this spectrum is to use classical single-pinhole cameras, in which the lenses are replaced with a tiny pinhole. The problem in these cameras is that the pinholes pass a low intensity light, while for imaging purposes, a much stronger light is needed. Simply increasing the number of the pinholes in an arbitrary manner cannot solve this problem, as it increases the signal-to-noise ratio (SNR) at the expense of decreased resolution of the image.

How to increase the number of the pinholes while maintaining a good resolution of the image? 

Coded aperture imaging (CAI) was introduced to address this challenge. A coded aperture is a grating or grid that casts a coded image on a plane of detectors by blocking and unblocking the light in a known pattern. The coded image is then correlated with a decoding array in order to reconstruct the original image. The deployment of pinholes and the decoding array are usually jointly designed to make the reconstruction perfect or near-perfect. An illustration is given below. 

The state-of-the-art uniformly redundant array-based CAI designs were based on binary alphabet. They are closely related with pseudo-random arrays. As a result, the size and shape of arrays are quite restricted by special algebraic properties. However, with the development in the hardware technology, implementation of the spatial phase modulators can be possible for ultra-high frquency rays, which can lead to realizable complex-valued physical masks  (by courtesy of MIT Lincoln Laboratory). In that case, if both coding and decoding systems use such masks, an analog reconstruction (which is preferred over digital one) could be achieved. Therefore, we are now free to consider the elements of an aperture to be unimodular complex numbers.  We naturally ask the question:

Given a broader range of alphabets, is it possible to design wider classes of CAI systems, flexible in both size and shape?

Moreover, it there a methodology to design CAI that can adapt to any particular physical aperture mask, such that the theoretically largest SNR gain can be achieved?

In this work, we give a positive answer. We introduce an overarching concept, theory, and methodology to design new types of coded aperture imaging systems. By several demonstrating examples, we show how to jointly design user-specific patterns of pinholes and the associated decoding arrays such that perfect reconstruction is possible (in the sense of no systematic error). Specially, we provide an interesting example to show how to theoretically achieve infinite SNR with only a finite alphabet (of phase modulators) on two dimensional hexagonal lattices.

Our proposal is based on a natural extension of classical pseudo-random arrays. The new theory not only cultivates more flexible designs of imaging systems, but also brings new theoretical insights to difference set in combinatorics. 

Jie Ding, Mohammad Noshad, Vahid Tarokh, “Complementary Lattice Arrays for Coded Aperture Imaging”. pdf

Image sources: https://goo.gl/eDlpZc & https://goo.gl/VCHZoy