Principle of Fredholm image reconstruction in the vignetting zone of an externally occulted solar coronagraph

Abstract

Aims. This study is carried out in the context of data processing of the solar coronagraph ASPIICS of the future formation-flying mission Proba-3, which is expected to provide images of the corona very close to the limb. There will be a transition zone of the order of 100 arcsec close to the limb, where the telescope aperture suffers a strong vignetting by the external occulter (a disc of 1.42 m at 144 m). The instrument response in this region will vary rapidly both in shape and in integrated intensity, the latter being particular to the external occultation. The aim of this paper is to propose a technique to recover as much as possible of the image of the corona very close to the limb in the vignetting zone. Methods. The object image relationship in this zone is not defined by the usual convolution but by the more general Fredholm integral of the first kind. Theoretical aspects of the problem are detailed in the context of a matrix formalism for the inversion of the Fredholm integral, formalism that we maintain up to the end of the numerical simulations, which is specific to the present work. The iterative Richardson-Lucy algorithm, specially written for the non-constant integrated intensity of the responses is used here for reconstruction. A study of the effect of noise on a photodetected image is made. Results. An important part of the work consisted in calculating the elements of the transfer matrix between the object and the image for a simulation on a small region of size 100 × 100 arcsec sampled over 128 × 128 pixels. This is obtained propagating the light through the system using a previously published approach. On a toy object, the reconstruction is excellent down to about 60 arcsec from the limb, corresponding to a vignetting of 50%. The drawback is that the recovery of a N × N object requires the handling of a N2 × N2 matrix, i.e. a 16384 × 16384 transfer matrix here. However, taking into account radial symmetries of the experiment, we propose the use of a transformation from Cartesian to polar coordinates which allows to apply the same procedure all around the sun as for a small region.