  We introduce a novel family of invariant, convex, and non-quadratic
functionals that we employ to derive regularized solutions of ill-posed linear
inverse imaging problems. The proposed regularizers involve the Schatten norms
of the Hessian matrix, computed at every pixel of the image. They can be viewed
as second-order extensions of the popular total-variation (TV) semi-norm since
they satisfy the same invariance properties. Meanwhile, by taking advantage of
second-order derivatives, they avoid the staircase effect, a common artifact of
TV-based reconstructions, and perform well for a wide range of applications. To
solve the corresponding optimization problems, we propose an algorithm that is
based on a primal-dual formulation. A fundamental ingredient of this algorithm
is the projection of matrices onto Schatten norm balls of arbitrary radius.
This operation is performed efficiently based on a direct link we provide
between vector projections onto $\ell_q$ norm balls and matrix projections onto
Schatten norm balls. Finally, we demonstrate the effectiveness of the proposed
methods through experimental results on several inverse imaging problems with
real and simulated data.
