  The compressed sensing (CS) framework leverages the sparsity of MR images to
reconstruct from undersampled acquisitions. CS reconstructions involve one or
more regularization parameters that weigh sparsity in transform domains against
fidelity to acquired data. While parameter selection is critical for
reconstruction quality, the optimal parameters are subject and dataset
specific. Thus, commonly practiced heuristic parameter selection generalizes
poorly to independent datasets. Recent studies have proposed to tune parameters
by estimating the risk of removing significant image coefficients. Line
searches are performed across the parameter space to identify the parameter
value that minimizes this risk. Although effective, these line searches yield
prolonged reconstruction times. Here, we propose a new self-tuning CS method
for multi-coil multi-acquisition reconstructions. The proposed method uses
computationally efficient projections onto epigraph sets of the $l_1$ and
total-variation norms to simultaneously achieve parameter selection and
regularization. In vivo demonstrations are provided for balanced steady-state
free precession, time-of-flight, and T1-weighted imaging. The proposed method
achieves nearly an order of magnitude improvement in computational efficiency
over line-search methods while maintaining near-optimal parameter selection.
