Abstract
The wavelet transform is extended in two dimensions to images by defining several wavelets that are rotated, expanded and translated. The wavelet coefficients are calculated using convolutions. This representation is again shown to be stable and invertible. By suppressing the phase of these coefficients with a modulus, then performing spatial averaging, we obtain a representation that is locally invariant to translations. We study its stability to deformations, showing that the action of a small diffeomorphism can be approximated by a translation operator in space and along scales. The wavelet transform thus defines a deformation-stable representation, unlike the Fourier or window Fourier transform.
Hubel and Wiesel discovered that the " simple " neurons in the visual cortical area V1 behave like dilated linear filters. This representation is similar to a two-dimensional wavelet transform, with Gabor wavelets.
