Hermite-Gaussian functions for image synthesis

The advent of high fidelity observatory interferometers such ALMA, invites new opportunities in algorithmic research for image processing. Interferometer telescopes perform irregular sampling on the Fourier transform of the sky images called "visibilities." The inverse problem resolution is called "Image Synthesis". This inverse problem is "Ill-posed" and thus regularization technique should be applied. State-of-the-art image synthesis algorithm (e.g. CLEAN) do provide an inversion of data in Fourier space to produce sky images. Current algorithm are based on spatial regular grid for image processing requiring interpolation for estimating values on grid from values on arbitrary sampling points. Numeric implementation also considers a set of basis function for representing 2-d field. We propose to use a non-local representation for the inverse problem. Our method is based on Hermite-Gaussian (HG) functions. HG functions are a complete set of eigenvectors for the Fourier operator, which we use for representing the problem. The HG set also approximately diagonalize the image's expansion coefficient correlation matrix. We present a proof of concept expressed on images.