ML estimation of wavelet regularization hyperparameters in inverse problems

In this paper we are interested in regularizing hyperparameter estimation by maximum likelihood in inverse problems with wavelet regularization. One parameter per subband will be estimated by gradient ascent algorithm. We have to face with two main difficulties: i) sampling the a posteriori image distribution to compute the gradient; ii) choosing a suited step-size to ensure good convergence properties. We first show that introducing an auxiliary variable makes the sampling feasible using classical Metropolis-Hastings algorithm and Gibbs sampler. Secondly, we propose an adaptive step-size selection and a line-search strategy to improve the gradient-based method. Good performances of the proposed approach are demonstrated on both synthetic and real data.