Signal reconstruction from interferometric measurements under sensing constraints

We develop a framework in which the problem of signal reconstruction from interferometric measurements amounts to one of basis analysis, with measurements that are linear in the basis coefficients. We leverage a generalized interferometry approach to enable the reconstruction of signals represented in different bases of arbitrary domains. Our framework is unifying in that it applies whether the sought-after information concerns the input signal or a sample object, as well as in settings where we have no control over the relative delays of the two paths of the interferometer. While the linear transformation underlying the measurement system has only a limited number of degrees of freedom set by the constrained sensing structure, we show that compressive signal recovery is achievable without introducing any additional randomization to the measurement setup. We establish performance guarantees under constrained sensing by proving that the transformation satisfies sufficient conditions for successful reconstruction. Also, we propose two control policies to guide the collection of informative measurements given prior knowledge about the constrained sensing structure. By minimizing the mutual coherence of the sampled measurements, the controlled approach is shown to yield further gains in sample size complexity.