Optimization of penalized criteria for image restoration. Application to sparse spike train deconvolution in ultrasonic imaging

The solution to many image restoration and reconstruction problems is often defined as the minimizer of a penalized criterion that accounts simultaneously for the data and the prior. This thesis deals more specifically with the minimization of edge-preserving penalized criteria. We focus on algorithms for large-scale problems. The minimization of penalized criteria can be addressed using a half-quadratic approach (HQ). Converging HQ algorithms have been proposed. However, their numerical cost is generally too high for large-scale problems. An alternative is to implement inexact HQ algorithms. Nonlinear conjugate gradient algorithms can also be considered using scalar HQ algorithms for the line search (NLCG+HQ1D). Some issues on the convergence of the aforementioned algorithms remained open until now. In this thesis we : – Prove the convergence of inexact HQ algorithms and NLCG+HQ1D. – Point out strong links between HQ algorithms and NLCG+HQ1D. – Experimentally show that inexact HQ algorithms and NLCG+HQ1D perform better than exact HQ algorithms, for an image deconvolution test problem. – Apply the penalized approach to a deconvolution problem in the field of ultrasonic imaging for nondestructive testing.