We address the problem of joint signal restoration and parameter estimation in the context of the forthcoming MUSE instrument, which will provide spectroscopic measurements of light emitted by very distant galaxies. Restoration of spectra is formulated as a linear inverse problem, accounting for the instrument response and the noise spectral variability. Estimation is considered in the setting of sparse approximation, where restoration is performed jointly with the detection of relevant patterns in the spectra. To this aim, a dictionary of elementary spectral features is designed according to astrophysical spectroscopy. Sparse estimation is considered through the minimization of a quadratic data misfit criterion with an ℓ<sup>1</sup>-norm penalization, where nonzero components are associated to the detected features. An efficient optimization strategy is proposed, based on the Iterative Coordinate Descent (ICD) principle, with accelerations that dramatically reduce the computational cost. The algorithm does not rely on fast transforms and can be applied to a wide variety of criteria if the sparsity constraint is separable. Results on simulated MUSE-like data reveal satisfactory performance in terms of denoising and detection of physically relevant spectral features. On such data, the proposed algorithm is shown to outperform both state-of-the-art gradient-based and homotopy continuation methods. Simulations with a compressed sensing-like random matrix also reveal better performance compared with usual algorithms, showing that ICD can be a powerful strategy for sparse optimization.