Journal of Mathematical Imaging and Vision

, Volume 56, Issue 2, pp 352–366 | Cite as

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces



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.


Scale-space Mathematical morphology Partial differential equations Pseudodifferential equations Convex analysis Cramér transform Infimal convolution 



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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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Mathematical Image Analysis Group, Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany

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