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Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

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Abstract

It is well-known that there are striking analogies between linear shift-invariant systems and morphological systems for image analysis. So far, however, the relations between both system theories are mainly understood on a pure convolution / erosion level. A formal connection on the level of differential or pseudodifferential equations and their induced scale-spaces is still missing. The goal of our paper is to close this gap. We present a simple and fairly general dictionary that allows to translate any linear shift-invariant evolution equation into its morphological counterpart and vice versa. It is based on a scale-space representation by means of the symbol of its (pseudo)differential operator. Introducing a novel transformation, the Cramér–Fourier transform, puts us in a position to relate the symbol to the structuring function of a morphological scale-space of Hamilton–Jacobi type. As an application of our general theory, we derive the morphological counterparts of many linear shift-invariant scale-spaces, such as the Poisson scale-space, \(\alpha \)-scale-spaces, summed \(\alpha \)-scale-spaces, relativistic scale-spaces, and their anisotropic variants. Our findings are illustrated by experiments.

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Acknowledgments

Our research on this topic has been initiated by discussions between Joachim Weickert and Bernd Sturmfels (Berkeley) during the DEMAIN Conference in Münster (Sept. 30–Oct. 2, 2014). It is a pleasure to thank Angela Stevens (Münster) for organising this highly inspiring interdisciplinary conference.

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  1. Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, Campus E1.7, 66041, Saarbrücken, Germany

    Martin Schmidt & Joachim Weickert

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Correspondence to Martin Schmidt.

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Schmidt, M., Weickert, J. Morphological Counterparts of Linear Shift-Invariant Scale-Spaces. J Math Imaging Vis 56, 352–366 (2016). https://doi.org/10.1007/s10851-016-0646-8

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