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We introduce a multi-scale energy formulation for plug and play (PnP) imagerecovery. The main highlight of the proposed framework is energy formulation,where the log prior of the distribution is learned by a convolutional neuralnetwork (CNN) module. The energy formulation enables us to introduceoptimization algorithms with guaranteed convergence, even when the CNN moduleis not constrained as a contraction. Current PnP methods, which do not oftenhave well-defined energy formulations, require a contraction constraint thatrestricts their performance in challenging applications. The energy and thecorresponding score function are learned from reference data using denoisingscore matching, where the noise variance serves as a smoothness parameter thatcontrols the shape of the learned energy function. We introduce a multi-scaleoptimization strategy, where a sequence of smooth approximations of the trueprior is used in the optimization process. This approach improves theconvergence of the algorithm to the global minimum, which translates toimproved performance. The preliminary results in the context of MRI show thatthe multi-scale energy PnP framework offers comparable performance to unrolledalgorithms. Unlike unrolled methods, the proposed PnP approach can work witharbitrary forward models, making it an easier option for clinical deployment.In addition, the training of the proposed model is more efficient from a memoryand computational perspective, making it attractive in large-scale (e.g., 4D)settings.

Multi-Scale Energy (MuSE) plug and play framework for inverse problems