  A fruitful approach for solving signal deconvolution problems consists of
resorting to a frame-based convex variational formulation. In this context,
parallel proximal algorithms and related alternating direction methods of
multipliers have become popular optimization techniques to approximate
iteratively the desired solution. Until now, in most of these methods, either
Lipschitz differentiability properties or tight frame representations were
assumed. In this paper, it is shown that it is possible to relax these
assumptions by considering a class of non necessarily tight frame
representations, thus offering the possibility of addressing a broader class of
signal restoration problems. In particular, it is possible to use non
necessarily maximally decimated filter banks with perfect reconstruction, which
are common tools in digital signal processing. The proposed approach allows us
to solve both frame analysis and frame synthesis problems for various noise
distributions. In our simulations, it is applied to the deconvolution of data
corrupted with Poisson noise or Laplacian noise by using (non-tight) discrete
dual-tree wavelet representations and filter bank structures.
