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Figure 1.

Wavelet transform of an image (), computed with a cascade of convolutions with filters over =4 scales and =4 orientations. The low-pass and =4 band-pass filters are shown on the first arrows. (Online version in colour.)

Filter bank. Wavelet transforms can be computed with a fast multiscale cascade of filters, which is at the core of deep network architectures. At each scale 2 j , we define a low-pass filter w j ,0 which increases the averaging scale from 2 j −1 to 2 j , and band-pass filters w j , k which compute each wavelet: 4.3 Let us write x j ( u ,0)= x ϕ j ( u ) and x j ( u , k )= x ψ j , k ( u ) for k ≠0. It results from ( 4.3 ) that for 0< j J and all 1≤ k K : 4.4 These convolutions may be subsampled by 2 along u ; in that case x j ( u , k ) is sampled at intervals 2 j along u .

Phase removal contraction. Wavelet coefficients x j ( u , k )= x ψ j , k ( u ) oscillate at a scale 2 j . Translations of x smaller than 2 j modify the complex phase of x j ( u , k ) if the wavelet is complex or its sign if it is real. Because of these oscillations, averaging x j with ϕ J outputs a zero signal. It is necessary to apply a nonlinearity which removes oscillations. A modulus ρ ( α )=| α | computes such a positive envelope. Averaging ρ ( x ψ j , k ( u )) by ϕ J outputs non-zero coefficients which are locally invariant at a scale 2 J : 4.5 Replacing the modulus by a rectifier gives nearly the same result, up to a factor 2. One can prove [ 9 ] that this representation is Lipschitz continuous to actions of diffeomorphisms over , and thus satisfies ( 3.2 ) for the metric ( 3.4 ). Indeed, the wavelet coefficients of x deformed by g can be written as the wavelet coefficients of x with deformed wavelets. Small deformations produce small modifications of wavelets in , because they are localized and regular. The resulting modifications of wavelet coefficients are of the order of the diffeomorphism metric | g | Diff .

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William "Bill" Bray; b. 22 Mar 1829, Clairborn Co., Tn; d. 13 Aug 1883; Milan, Sullivan Co., Mo.; 51y 11m 22d Civil War: Co. H, 4th Ind. Cav.
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