Going back to Suresh's blog entry on NIPS2012, I noted his mention of Stephane Mallat's scattering operator as featured recently in Small-sample brain mapping: sparse recovery on spatially correlated designs with randomization and clustering. There Michael Eickenberg Alexandre Gramfort and Bertrand Thirion were showing a potential match-up between the scattering operation and what is actually happening in the brain (Multilayer Scattering Image Analysis Fits fMRI Activity in Visual Areas also here). Since I last featured the code, there seems to have been a larger edition I missed. Here it is, with a tutorial.

- Image scattering toolbox v3 : Released July 11 2012.

A supporting document of interest is the PhD thesis of Joan Bruna entitled Scattering Representations for Recognition which came out in Nov 2012. The abstract reads:

This thesis addresses the problem of pattern and texture recognition from a mathematical perspective. These high level tasks require signal representations enjoying speciﬁc invariance, stability and consistency properties, which are not satisﬁed by linear representations. Scattering operators cascade wavelet decompositions and complex modulus, followed by a lowpass ﬁltering. They deﬁne a non-linear representation which is locally translation invariant and Lipschitz continuous to the action of diffeomorphisms. They also deﬁne a texture representation capturing high order moments and which can be consistently estimated from few realizations.The thesis derives new mathematical properties of scattering representations and demonstrates its eﬃciency on pattern and texture recognition tasks. Thanks to its Lipschitz continuity to the action of diﬀeomorphisms, small deformations of the signal are linearized, which can be exploited in applications with a generative aﬃne classiﬁer yielding state-of-the-art results on handwritten digit classiﬁcation. Expected scattering representations are applied on image and auditory texture datasets, showing their capacity to capture high order moments information with consistent estimators. Scattering representations are particularly eﬃcient for the estimation and characterization of fractal parameters. A renormalization of scattering coeﬃcients is introduced, giving a new insight on fractal description, with the ability in particular to characterize multifractal intermittency using consistent estimators.

As of right now, this approach is orthogonal to what we do in vanilla compressive sensing.

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## 2 comments:

I listened to the talk and it was brilliant (but the papers are somewhat cumbersome). Why do you think it is orthogonal to compressed sensing? It is a nonlinear transform that is stable with respect to translation and transformations, and has rather surprising and unusual properties, but I do not feel that it is orthogonal :)

Ivan,

In "vanilla" compressed sensing, the transform is linear (most of the time people use a random projection but not all the time). Mallat's transform is nonlinear hence the comment that the two approaches are orthogonal.

In both case, after the transforms have been applied, one can compare the reduced versions using the Euclidian norm.

In Mallat's case, the transformed elements are unique if they are a translation or a rotation or... away from one element.

In generic compressive sensing, the transformed elements have nearly the same euclidian norm as their larger version.

what I like about compressive sensing is this element of randomness as it provides a clear path towards a substition to biological system. But that's just me.

Igor.

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