Fourier Interpolation and Reconstruction Via Shannon-type Techniques

Abstract

Abstract In this series of papers, we present a new approach to the problems of Fourier synthesis in finite dimension (the data are complex quantities corresponding to a finite and irregular sampling of the Fourier transform of some object function). Part I concerns the principles and part II their application in aperture synthesis. Depending on what is emphasized, this method is called FIRST or WIPE: FIRST for the principles (Fourier interpolation and reconstruction via Shannon-type techniques), and WIPE for the corresponding deconvolution method (WIPE is reminiscent of CLEAN, a well-known deconvolution method in astronomy). The regularization principle of FIRST refers to the Shannon sampling formula and to theoretical considerations related to multiresolution analysis. The notions of field and resolution appear via the definition of two key spaces: the object workspace and the object representation space (a subspace of the first). The data define a function in another finite-dimensional space: the Fourier data space. The functions lying in this space take their values on a frequency list which is the concatenation of the experimental frequency list with a regularization frequency list. The latter defines a ‘virtual frequency coverage’ beyond the frequency coverage to be synthesized, up to the highest frequencies of the cubic-spline scaling functions generating the object workspace. This virtual sampling is performed at the Shannon rate corresponding to the synthesized field, which is thus involved in the definition of the inner product of the Fourier data space. The reconstructed image is then defined as the function minimizing a regularized objective functional in which the data are damped appropriately. To describe WIPE, the imaging kernel of FIRST, we then adopt a terminology derived from that of CLEAN. At each iteration of the selected constructive process (conjugate gradients, for example) WIPE compares the dusty map with the dusty map of the model. In the corresponding truncated discrete convolution, the discrete point-spread function, the dusty beam has two components: the traditional dirty beam and the regularization beam. The speed of the fast Fourier transform is a major advantage when computing large maps. The discrete Fourier transform of the dusty beam must then be computed on a grid twice as dense as that required for the discrete Fourier transform of the dusty map. This point is thoroughly examined. Besides the clarity of the principles and the advantages of WIPE over CLEAN, we also indicate how, in a more general way, FIRST can be used for analysing the results provided by other methods.