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Multi-scale Arithmetization of Linear Transformations

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Abstract

A constructive nonstandard interpretation of a multiscale affine transformation scheme is presented. It is based on the \(\Omega \)-numbers of Laugwitz and Schmieden and on the discrete model of the real line of Reeb and Harthong. In this setting, the nonstandard version of the Euclidean affine transformation gives rise to a sequence of quasi-linear transformations over integer spaces, allowing integer-only computations.

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Notes

  1. In [5, 17], it is stated that \(\forall x\in \mathscr {H}\!\mathscr {R}_{\omega }, x\ne _\omega 0, \exists y\in \mathscr {H}\!\mathscr {R}_{\omega }\) such that \(x\times _{\omega }y=_\omega 1\), but the relation \(\ne _\omega \) is not the negation of the relation \(=_\omega \) (in [17], the implementation in the Coq system relies on the intuitionistic logic, so it does not make use of the law of the excluded middle). In this paper, the structure \((\mathscr {H}\!\mathscr {R}_{\omega },+,\times _{\omega })\) needs not be a field, just a ring and we make no use of the relation \(\ne _\omega \) so the assertions can be understood in both classical and intuitionistic logics.

  2. Given an equivalence relation, a section is the choice of a representative, while the converse operation is called projection.

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  1. ICube, CNRS, University of Strasbourg, 300 Bd Sébastien Brant - CS 10413, 67412, Illkirch, France

    Loïc Mazo

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Mazo, L. Multi-scale Arithmetization of Linear Transformations. J Math Imaging Vis 61, 432–442 (2019). https://doi.org/10.1007/s10851-018-0853-6

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