Reconstruction of Binary Functions and Shapes from Incomplete Frequency Information

The characterization of a binary function by partial frequency information is considered. We show that it is possible to reconstruct binary signals from incomplete frequency measurements via the solution of a simple linear optimization problem. We further prove that if a binary function is spatially structured (e.g. a general black-white image or an indicator function of a shape), then it can be recovered from very few low frequency measurements in general. These results would lead to efficient methods of sensing, characterizing and recovering a binary signal or a shape as well as other applications like deconvolution of binary functions blurred by a low-pass filter. Numerical results are provided to demonstrate the theoretical arguments.