Springer Online Journal Archives 1860-2000
Abstract The minimum-norm least-squares solution of the phase-closure equations of an interferometric array is very stable. Furthermore, this canonical solution can be obtained by simple backprojection, each closure phase leaving its “algebraic” imprint on the corresponding baselines. More precisely, the generalized inverse of the phase-closure operator C of an n-point array is equal to its adjoint (its Hermitian transpose) divided by n: C + =C * /n. Likewise, the generalized inverse of the phase-aberration operator B is equal to B * /n. These remarkable properties, which have so far remained unnoticed, play an essential part in the algebraic analysis of phase-closure imaging, and thereby in the understanding and the treatment of the inverse problems of aperture synthesis. The applications presented in this paper concern the phase-factor restoration problem in optical interferometry and speckle imaging. We first propose a new iterative procedure for obtaining a particular least-squares solution. In the framework of this nonlinear technique, we then show how to initialize at each iteration the inner process of linear optimization. The backprojection method, which is the obvious choice in the case of weakly-redundant devices, is compared with the recursive techniques used in bispectral analysis for highly-redundant configurations. At the end of the restoration step under consideration, the phase indetermination reduces to a vector lying in the null space of the bispectral operator. The global reconstruction process is closely related to the regularization methods used for band-limited extrapolation. In this context, we outline the final hybrid procedure to be implemented, indicating how certain regularizing constraints can raise the intrinsic indeterminations related to the existence of the null space.
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