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Morphological counterparts of linear shift-invariant scale-spaces. (English) Zbl 1392.68428

Summary: 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.

MSC:

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
35Q68 PDEs in connection with computer science

Software:

SIFT
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