Penalised, Sieved Maximum Likelihood
for Radio Interferometric Imaging
Radio interferometric imaging aims at recovering the sky intensity function from a collection of samples recorded on the ground by a network of antennas. This inverse problem is typically addressed by discrete penalty methods, where the intensity function is pixelated, e.g. approximated by step functions. In such methods, both the data-fidelity term and the discretisation step are often non-canonical to the problem at hand. In this presentation, we consider a new recovery paradigm, based on the maximum likelihood principle combined with the method of sieves. To this end, we introduce a functional data model and formulate a maximum likelihood problem at the continuous level. According to the method of sieves, we then constrain the search space to a finite dimensional Hilbert space spanned by a set of basis functions. These functions, typically shifted zonal kernels, can be thought of as generalised pixels, better suited than classical step functions for representing most physically admissible signals on the celestial sphere. To ensure identifiability when the sieve has greater dimensionality than the data-space, we futher penalise the discrete sieved likelihood problem with various penalty functionals.
We propose and compare the performance of three types of sieves: Matched, Dirichlet and L-splines. By means of representer theorems, we show that the matched and L-splines sieves are canonical to a class of continuous optimisation problems with respectively Tikhonov and generalised total variation penalties. In such cases, the discrete penalised maximum likelihood problem obtained with the method of sieves is hence equivalent to its continuous counterpart.
In practice, the penalised sieved maximum likelihood problem is solved by means of proximal gradient ascent with Nesterov's acceleration. For sieves composed of shifted zonal kernels, the shifts are chosen according to sampling theorems on the sphere or selected from an over-complete dictionary of candidate locations with Orthogonal Functional Matching Pursuit (OFMP).