  We propose a new joint image reconstruction method by recovering edge
directly from observed data. More specifically, we reformulate joint image
reconstruction with vectorial total-variation regularization as an $l_1$
minimization problem of the Jacobian of the underlying multi-modality or
multi-contrast images. Derivation of data fidelity for Jacobian and
transformation of noise distribution are also detailed. The new minimization
problem yields an optimal $O(1/k^2)$ convergence rate, where $k$ is the
iteration number, and the per-iteration cost is low thanks to the close-form
matrix-valued shrinkage. We conducted numerical tests on a number
multi-contrast magnetic resonance image (MRI) datasets, which show that the
proposed method significantly improves reconstruction efficiency and accuracy
compared to the state-of-the-arts.
