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I've come across the following claim in a paper of Mallat:

"High frequency instabilities [of a signal representation] to deformations can be avoided by grouping frequencies into dyadic packets in $\mathbb{R}^d$, with a wavelet transform."

I know what the wavelet packet decomposition is and I know what stability to deformations is (in the case of $C^2$ diffeomorphisms), but I don't understand why the quoted statement is true.

Perhaps I missed the explanation in the paper, but could someone provide a reference, an explanation or both? I assume it has something to do with the fact that a wavelet is localized.

1 Stéphane Mallat: Group Invariant Scattering, Communications on Pure and Applied Mathematics, Vol. LXV, 1331–1398 (2012), DOI: 10.1002/cpa.21413

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  • $\begingroup$ What paper is that? Knowing the paper probably makes answering easier. $\endgroup$
    – Joonas Ilmavirta
    Sep 26, 2014 at 19:42
  • $\begingroup$ I've added the link to my question. $\endgroup$
    – MRicci
    Sep 26, 2014 at 19:47

1 Answer 1

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Stability along log-frequency is attained via the constant Q transform, and along time via temporal averaging; this is proven in Theorem 2.12. Explained visually here, also see "Properties summary".

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