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The inverse problems of image reconstruction in Very Long Baseline Interferometry are very particular myopic deconvolution problems. The real and imaginary parts of the logarithm of the transfer function lie in the ranges of two remarkable operators: the amplitude aberration operator B? and the phase aberration operator B?. The projections onto the ranges of B? and B? are very simple operators which can be written in closed form. When the data are “spectral quantities” such as visibility measurements, the constraints on the transfer function (the calibration constraints) can therefore be easily imposed. In the case of homogeneous arrays, this leads to a better efficiency of the calibration techniques developed by Schwab and Cornwell. When the data are “higher-order spectral quantities” such as those provided, for example, by averaged complex closure terms, it is easy to extract from these data a spectral information quite similar to that given by visibility measurements. This modifies the very principle of the reconstruction methods used today in such situations, for example, the method of Readhead and Wilkinson. This new approach results from the particular algebraic structures of interferometry in higher-order spectral analysis. To exhibit these structures we introduce four compilation operators, among which: the phase compilation operator C? (the phase closure operator of order p > 3), and the alternate amplitude compilation operator C? (the amplitude closure operator of even order p > 4). All these operators have remarkable properties. For example, the generalized inverses of the closure operators of order p are given by very simple backprojection relations. According to these remarkable algebraic structures, a higher-order spectral information ? (with the appropriate subscript) yields a spectral information ? defined up to a function lying in the range of B. In the corresponding reconstruction process, all the closure data are taken into account with the same weight (not only those corresponding to a set of independent closure relations). The interest of homogenous arrays is thus reinforced. It is also clear that the quality of a higher-order spectral information ? only depends on that of its projection onto the range of the phase or amplitude closure operator. This remark is essential for the choice of the closure order. All these nice “homological” structures, and their implications in optical interferometry, have been taken into account in our image-reconstruction algorithms.
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