Next generation radio observatories such as the MWA, LWA, LOFAR, CARMA and SKA provide a number of challenges for interferometric data analysis. These challenges include heterogeneous arrays, direction-dependent instrumental gain, and refractive and
scintillating atmospheric conditions. From the analysis perspective, this means that calibration solutions can not be described using a single complex gain per antenna. In this paper we use the optimal map-making formalism developed for CMB analyses to extend traditional interferometric radio analysis techniques--removing the assumption of a single complex gain per antenna and allowing more complete descriptions of the instrumental and atmospheric conditions. Due to the similarity with holographic mapping of radio antenna surfaces, we call this extended analysis approach software holography. The resulting analysis algorithms are computationally efficient, unbiased, and optimally sensitive. We show how software holography can be used to solve some of the challenges of next generation observations, and how more familiar analysis techniques can be derived as limiting cases.
