Abstract
An important source of a priori information is the geometric structure of the data indexing space, be it space for images, or time for audio signals. This parameterization defines groups of transformations such as translations. Linear and translation-covariant operators are called convolutions. An important class of translation invariants is obtained by diagonalizing convolutions with the Fourier transform and removing the phase with a modulus.
A diffeomorphism is an operator that deforms physical space with a regular, invertible function. In one dimension, a diffeomorphism can be locally approximated by translation and dilation. The action of a small diffeomorphism is not linearized by the modulus of the Fourier transform, as it can produce instabilities at high frequencies.
Time-frequency representations localize Fourier information, within the limit of the uncertainty theorem, by projecting data onto time-frequency atoms. Window Fourier transforms enable us to obtain parsimonious representations that highlight the temporal or spatial evolution of a signal's frequencies. However, this localization is not sufficient to avoid the instabilities induced by high-frequency diffeomorphisms.
