  In this paper, we investigate the minimax properties of Stein block
thresholding in any dimension $d$ with a particular emphasis on $d=2$. Towards
this goal, we consider a frame coefficient space over which minimaxity is
proved. The choice of this space is inspired by the characterization provided
in \cite{BorupNielsen} of family of smoothness spaces on $\mathbb{R}^d$, a
subclass of so-called decomposition spaces \cite{Feichtinger}. These smoothness
spaces cover the classical case of Besov spaces, as well as smoothness spaces
corresponding to curvelet-type constructions. Our main theoretical result
investigates the minimax rates over these decomposition spaces, and shows that
our block estimator can achieve the optimal minimax rate, or is at least
nearly-minimax (up to a $\log$ factor) in the least favorable situation.
Another contribution is that the minimax rates given here are stated for a
general noise sequence model in the transform coefficient domain beyond the
usual i.i.d. Gaussian case. The choice of the threshold parameter is
theoretically discussed and its optimal value is stated for some noise models
such as the (non-necessarily i.i.d.) Gaussian case. We provide a simple, fast
and a practical procedure. We also report a comprehensive simulation study to
support our theoretical findings. The practical performance of our Stein block
denoising compares very favorably to the BLS-GSM state-of-the art denoising
algorithm on a large set of test images. A toolbox is made available for
download on the Internet to reproduce the results discussed in this paper.
