  We introduce multi-scale energy models to learn the prior distribution of
images, which can be used in inverse problems to derive the Maximum A
Posteriori (MAP) estimate and to sample from the posterior distribution.
Compared to the traditional single-scale energy models, the multi-scale
strategy improves the estimation accuracy and convergence of the MAP algorithm,
even when it is initialized far away from the solution. We propose two kinds of
multi-scale strategies: a) the explicit (e-MuSE) framework, where we use a
sequence of explicit energies, each corresponding to a smooth approximation of
the original negative log-prior, and b) the implicit (i-MuSE), where we rely on
a single energy function whose gradients at different scales closely match the
corresponding e-MuSE gradients. Although both schemes improve convergence and
accuracy, the e-MuSE MAP solution depends on the scheduling strategy, including
the choice of intermediate scales and exit conditions. In contrast, the i-MuSE
formulation is significantly simpler, resulting in faster convergence and
improved performance. We compare the performance of the proposed MuSE models in
the context of Magnetic Resonance (MR) image recovery. The results demonstrate
that the multi-scale framework yields a MAP reconstruction comparable in
quality to the End-to-End (E2E) trained models, while being relatively
unaffected by the changes in the forward model. In addition, the i-MuSE scheme
also allows the generation of samples from the posterior distribution, enabling
us to estimate the uncertainty maps.
