Hessian Schatten-Norm Regularization for Linear Inverse Problems

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. The...

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Bibliographic Details
Published in:IEEE transactions on image processing 2013-05, Vol.22 (5), p.1873-1888
Main Authors: Lefkimmiatis, S., Ward, J. P., Unser, M.
Format: Article
Language:English
Subjects:
Algorithms
Applied sciences
Diagnostic Imaging
Eigenvalue optimization
Exact sciences and technology
Face - anatomy & histology
Hessian operator
Humans
Image processing
Image Processing, Computer-Assisted - methods
Image reconstruction
Imaging
Information, signal and communications theory
Inverse problems
Linear programming
matrix projections
Minimization
Models, Theoretical
Schatten norms
Signal processing
Telecommunications and information theory
Vectors
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Summary: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 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 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.
ISSN:1057-7149
1941-0042
DOI:10.1109/TIP.2013.2237919