Stabilized Reconstruction in Signal and Image Processing

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Abstract
A new strategy for signal and image reconstruction with incomplete information and partial constraints is defined. The main purpose of the first paper (part I) is to specify the conditions under which it is possible to extrapolate or to interpolate, in some region W, the Fourier transform of a function having its support in some bounded region V. With the aid of some very simple geometrical concepts it is shown, in particular, why spectral interpolation is possible to some extent, whereas infinite extrapolation is forbidden. At the cost of a less full representation of the object, it is therefore natural to formulate the deconvolution problems in terms of (weighted) spectral interpolation with partial extrapolation. The corresponding regularization principle, which is compared with that of Tikhonov, proves to be intimately related to the notion of resolution limit of the reconstruction process. In this context, the smallest eigenvalue of the operator to be inverted can be estimated analytically before setting the reconstruction process in motion. The corresponding analysis is developed by stressing the essential results directly involved in the interactive numerical implementation presented in part II: stability conditions, number of degrees of freedom of the reconstruction process, etc. All the related technical aspects are presented in closed form.