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Hessian Schatten-Norm Regularization for Linear Inverse Problems (2013)
| Citations: | 11 - 8 self |
BibTeX
@MISC{Lefkimmiatis13hessianschatten-norm,
author = {Stamatios Lefkimmiatis and John Paul Ward and Michael Unser},
title = {Hessian Schatten-Norm Regularization for Linear Inverse Problems},
year = {2013}
}
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Abstract
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, which are 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 primaldual 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 ℓq 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.
Keyphrases
linear inverse problem hessian schatten-norm regularization schatten norm ball schatten norm matrix projection tv-based reconstruction invariance property popular totalvariation wide range fundamental ingredient regularized solution non-quadratic functionals image reconstruction direct link common artifact vector projection novel family norm ball hessian operator second-order derivative index term eigenvalue optimization staircase effect proposed regularizers second-order extension ill-posed linear inverse primaldual formulation arbitrary radius hessian matrix experimental result corresponding optimization problem
