  The analysis of ocean surface is widely performed using synthetic aperture
radar (SAR) imagery as it yields information for wide areas under challenging
weather conditions, during day or night, etc. Speckle noise constitutes however
the main reason for reduced performance in applications such as classification,
ship detection, target tracking and so on. This paper presents an investigation
into the despeckling of SAR images of the ocean that include ship wake
structures, via sparse regularisation using the Cauchy proximal operator. We
propose a closed-form expression for calculating the proximal operator for the
Cauchy prior, which makes it applicable in generic proximal splitting
algorithms. In our experiments, we simulate SAR images of moving vessels and
their wakes. The performance of the proposed method is evaluated in comparison
to the L1 and TV norm regularisation functions. The results show a superior
performance of the proposed method for all the utilised images generated.
