  Statistical dependencies among wavelet coefficients are commonly represented
by graphical models such as hidden Markov trees(HMTs). However, in linear
inverse problems such as deconvolution, tomography, and compressed sensing, the
presence of a sensing or observation matrix produces a linear mixing of the
simple Markovian dependency structure. This leads to reconstruction problems
that are non-convex optimizations. Past work has dealt with this issue by
resorting to greedy or suboptimal iterative reconstruction methods. In this
paper, we propose new modeling approaches based on group-sparsity penalties
that leads to convex optimizations that can be solved exactly and efficiently.
We show that the methods we develop perform significantly better in
deconvolution and compressed sensing applications, while being as
computationally efficient as standard coefficient-wise approaches such as
lasso.
