  The blind structured illumination microscopy (SIM) strategy proposed in
(Mudry et al., 1992) is fully re-founded in this paper, unveiling the central
role of the sparsity of the illumination patterns in the mechanism that drives
super-resolution in the method. A numerical analysis shows that the resolving
power of the method can be further enhanced with optimized one-photon or
two-photon speckle illuminations. A much improved numerical implementation is
provided for the reconstruction problem under the image positivity constraint.
This algorithm rests on a new preconditioned proximal iteration faster than
existing solutions, paving the way to 3D and real-time 2D reconstruction
