Unified survey of Fourier synthesis methodologies

P. Maréchal, E. Anterrieu, A. Lannes
CNRS

Session ID: T11.04   Type: oral

Abstract:

The Fourier synthesis methodologies play a key role in aperture synthesis. It is therefore essential to have a unified understanding of the regularized Fourier synthesis techniques that have been implemented so far: WIPE with (or without) positivity constraint, cross-entropy, etc. We have recently shown that the related reconstruction criteria can be derived from a unique principle: the Principle of Maximum Entropy on the Mean (PMEM). For example, the (Tikhonov-type) regularization principle of WIPE results from the choice of a prior probability measure  penalizing the high-frequency components of the reconstructed image. (Note that a similar regularization operation is performed in CLEAN, but a posteriori.) By proceeding in a similar way, but with a Poissonian measure , the PMEM then yields the generalized cross-entropy regularizer. In both cases, the precise definition of  must take into account the region in which the `image to be reconstructed' is strictly positive: the image support. For example, the support provided by the matching pursuit process of WIPE (a refined version of that provided by CLEAN) can be used for defining a cross-entropy regularizer in which the prior knowledge of the image to be reconstructed reduces to a continuous representation of the characteristic function of this support, at the selected resolution level. The results thus obtained are then very similar to those of WIPE: the best possible fit to the data is achieved with a good control of robustness. Such results are rather unexpected from an entropy-based method! This shows, if need be, that these methods should not be developed independently. The strategy adopted in WIPE, which constructs the image support progressively, while using a regularization principle based on the concept of resolution, proves to be particularly well suited to the problems of aperture synthesis. Its multiresolution extension is also very promising.

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