Differential imaging is a technique to post-process images captured by ground-based telescopes during an observation campaign, in order to make exoplanets in a distant planetary system directly visible and to remove the so-called quasi-static speckles that dramatically affect detection capabilities. In order to introduce geometric diversity between the exoplanets and the quasi-static speckles, the light is split into spectral channels during the data acquisition process, producing a 4-D data cube with images recorded at many wavelengths and at many times. In this work, we propose to follow an inverse problem approach to model the as- tronomical data as the contribution of a low-rank component containing the background of quasi-static speckles and a sparse component contain- ing the exoplanets. We then formulate the resulting model as a convex non-smooth optimization model so that an accelerated proximal gradient descent can be used to solve the detection problem.