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Semi-Inner-Products for Convex Functionals and Their Use in Image Decomposition

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Abstract

Semi-inner-products in the sense of Lumer are extended to convex functionals. This yields a Hilbert-space like structure to convex functionals in Banach spaces. In particular, a general expression for semi-inner-products with respect to one homogeneous functionals is given. Thus one can use the new operator for the analysis of total variation and higher order functionals like total-generalized-variation. Having a semi-inner-product, an angle between functions can be defined in a straightforward manner. It is shown that in the one homogeneous case the Bregman distance can be expressed in terms of this newly defined angle. In addition, properties of the semi-inner-product of nonlinear eigenfunctions induced by the functional are derived. We use this construction to state a sufficient condition for a perfect decomposition of two signals and suggest numerical measures which indicate when those conditions are approximately met.

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Acknowledgments

The author would like to thank the support of the Israel Science Foundation (ISF), Grant 2097/15.

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Authors and Affiliations

  1. Department of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, 32000, Israel

    Guy Gilboa

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  1. Guy Gilboa
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Correspondence to Guy Gilboa.

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Gilboa, G. Semi-Inner-Products for Convex Functionals and Their Use in Image Decomposition. J Math Imaging Vis 57, 26–42 (2017). https://doi.org/10.1007/s10851-016-0661-9

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