Mathematics of Scattering

Group Invariant Scattering, Mallat S., Communications in Pure and Applied Mathematics, vol. 65, no. 10, pp. 1331-1398, Oct. 2012. (PDF)

This paper constructs translation invariant operators on \(\bf L^2 (R^d)\), which are Lipschitz continuous to the action of diffeomorphisms. A scattering propagator is a path ordered product of non-linear and non-commuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz continuous to the action of \(\bf C^2 \) diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform which is translation invariant.

Scattering coefficients also provide representations of stationary processes. Expected values depend upon high order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on \( {\bf L^2} (G)\), where \( G \) is a compact Lie group, and are invariant under the action of \( G \). Combining a scattering on \(\bf L^2 (R^d) \) and on \( {\bf L^2} (SO(d)) \) defines a translation and rotation invariant scattering on \( \bf L^2 (R^d) \).

Scattering Representations for Recognition, Bruna J., Ph.D. thesis, École Polytechnique, Palaiseau, France, Nov. 2012. (PDF)

The thesis develops further properties of scattering operators and its applications on pattern and texture recognition. The non-linear structure of the transform is characterized from the Lipschitz stability to additive noise and geometric deformations. The regularity of the integral scattering in the transformed domain is related to signal decay, similarly as in Fourier transforms. Scattering representations also define a new tool to characterize and identify Multifractal measures and functions.