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Rotationally Invariant Time–Frequency Scattering Transforms

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Abstract

The focus of this paper is on the mathematical construction of a transform that is invariant to a finite rotation group and is stable to small perturbations. A key step in our theory lies in the construction of directionally sensitive functions that are partially generated by rotations. We call such a family a rotational uniform covering frame and by studying rotations of the frame, we derive the desired operator, which we call the rotational Fourier scattering transform. We prove that the transformation is rotationally invariant to a finite rotation group, is bounded above and below, is non-expansive, and contracts small translations and additive diffeomorphisms. To address the numerical aspects of this theory, we also construct digital versions of the frame and show how to faithfully truncate the transform. We also discuss connections between this new family of directional representations with previously constructed ones.

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Acknowledgements

This work was supported in part by the LTS through Maryland Procurement Office, NSF DMS grant 1738003, Defense Threat Reduction Agency grant HDTRA 1-13-1-0015, and by the Army Research Office Grants W911 NF-15-1-0112 and W911 NF-16-1-0008. Weilin Li was supported by the National Science Foundation Grant DMS-1440140 while in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.

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  1. Department of Mathematics, University of Maryland, College Park, USA

    Wojciech Czaja

  2. Courant Institute of Mathematical Sciences, New York University, New York, USA

    Weilin Li

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  1. Wojciech Czaja
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  2. Weilin Li
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Correspondence to Weilin Li.

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Communicated by Thomas Strohmer.

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Czaja, W., Li, W. Rotationally Invariant Time–Frequency Scattering Transforms. J Fourier Anal Appl 26, 4 (2020). https://doi.org/10.1007/s00041-019-09705-w

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  • DOI: https://doi.org/10.1007/s00041-019-09705-w

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