Dual descent regularization algorithms in variable exponent Lebesgue spaces for imaging

We consider one-step iterative algorithms to solve ill-posed inverse problems in the framework of variable exponent Lebesgue spaces L-p(.). These unconventional spaces are particular (non-Hilbertian) Banach spaces which can induce adaptive local regularization in the resolution of inverse problems. We first study gradient descent iteration schemes in Banach spaces, where the classical Riesz theorem does not hold and, consequently, primal and dual spaces are no longer isometrically isomorphic. In particular, we prove that gradient methods in Banach spaces can be fully explained and understood in the context of proximal operator theory, with appropriate norm or Bregman distances as proximity measure, which shows a deep connection between regularization iterative schemes and convex optimization. We review the key concept of duality map, and provide an explicit formula of the duality map for the space L-p(.). Then we apply the Landweber and the Conjugate Gradient methods, extended to Banach setting, to solve deblurring imaging problems in L-p(.) and propose an effective strategy to select the point-wise variable exponent function p(.). Our numerical tests show the advantages of considering variable exponent Lebesgue spaces w.r.t. both the standard L-2 Hilbert and the constant exponent Lebesgue space L-p, in terms of both reconstruction quality and convergence speed.