Resolving phase ambiguities in the calibration of redundant interferometric arrays: implications for array design

: We provide new results enabling robust interferometric image reconstruction in the presence of unknown aperture piston variation via the technique of Redundant Spacing Calibration (RSC). The RSC technique uses redundant measurements of the same interferometric baseline with different pairs of apertures to reveal the piston variation among these pairs. In both optical and radio interferometry, the presence of phase-wrapping ambiguities in the measurements is a fundamental issue that needs to be addressed for reliable image reconstruction. In this paper, we show that these ambiguities affect recently-developed RSC phasor-based reconstruction approaches operating on the complex visibilities, as well as traditional phase-based approaches operating on their logarithm. We also derive new sufficient conditions for an interferometric array to be immune to these ambiguities in the sense that their effect can be rendered benign in image reconstruction. This property, which we call wrap-invariance, has implications for the reliability of imaging via classical three-baseline phase closures as well as generalized closures. We show that wrap-invariance is conferred upon arrays whose interferometric graph satisfies a certain cycle-free condition. For cases in which this condition is not satisfied, a simple algorithm is provided for identifying those graphcycles which prevent its satisfaction. We apply this algorithm to diagnose and correct a member of a pattern family popular in the literature.