Classification with Invariant Scattering Representations
A scattering transform defines a signal representation which is invariant to translations and Lipschitz continuous relatively to deformations. It is implemented with a non-linear convolution network that iterates over wavelet and modulus operators. L…
Authors: Joan Bruna, Stephane Mallat
CLASSIFICA TION WITH INV ARIANT SCA TTE RING REPRESENT A TIONS J oan Bruna and St ´ ephane Mallat CMAP , ´ Ecole Polytechnique Rue de Saclay , Palaiseau, F rance ABSTRA CT A scattering transform defines a signal representation which is in variant to translations an d Lipschitz co ntinuou s relatively to deform ations. It is implemented with a n on-linear con- volution network th at iterates over wav elet an d m odulus op - erators. Lipschitz contin uity locally linear izes defor mations. Complex classes of signals and textures can be modeled with low-dimensional affine space s, compu ted with a PCA in the scattering d omain. Classification is performe d with a penal- ized m odel selection. State of th e a rt results are obtained for handwritten digit recognitio n over small training sets , and for texture classification. 1 Index T erms — Imag e classification, Inv ariant repre sen- tations, local im age descriptor s, pattern re cognition, textur e classification. 1. INTR ODUCTION Affine space mo dels are simple to com pute with a Principal Compone nt Analysis (PCA) but ar e no t appro priate to ap- proxim ate signal classes that inclu de complex forms of vari- ability . Image classes are often inv ariant to rigid transform a- tions suc h as translations or rotatio ns, and in clude elastic de- formation s, which define highly non-lin ear manifolds. T ex- tures may also be r ealizations of stro ngly no n-Gaussian pro- cesses that cannot be discriminated with linear models either . Kernel m ethods define d istances d ( f , g ) = k Φ( f ) − Φ( g ) k , with op erators Φ wh ich ad dress the se issues by map - ping f and g into a space of much high er dim ension. How- ev er , in variance properties and learning requirements on small training sets, rath er suggest to imp lement a dimension ality reduction . Scattering op erators constructed in [ 9, 10], a re inv ariant to global translations and Lip schitz co ntinuou s rela ti vely to lo- cal deformatio ns, up to a log term, thus providing lo cal trans- lation inv ariance th rough the linearization of such deforma- tions. These scatterin g operator s crea te inv ariance by a ver - aging inter ference coefficients, which capture signal interac- tions at several scales and orientations. This paper mo dels complex signal class es with low-dimensional af fine spaces in 1 This work is funded by the ANR grant 0126 01. the scatter ing domain , which are compu ted with a PCA. The classification is perform ed by a penalized model selection. Scattering operators m ay also be inv ariant to any comp act Lie subgro up of GL ( R 2 ) , su ch as rotatio ns, but we conce n- trate on translation in variance, which c arries the main diffi- culties and already covers a wide range of classification ap- plications. Section 2 revie ws the construction of scattering operator s with a cascade of wav elet transforms and modu - lus operators, wh ich defines a non-linea r con v olution network [6]. Section 3 shows that learning a ffi ne scattering mod el spaces has a linear complexity in the nu mber of training sam- ples. Section 4 describes state of th e art classification re- sults obtained f rom limited numb er of training samples in the MNI ST hand-wr itten digit database, and for texture clas- sification in the CUREt database. Software is av ailable at www.cmap.pol ytechnique.fr /scattering . 2. SCA TTERING OP ERA TORS In order to build a r epresentation which is locally translation in variant, a scattering tra nsform begins from a w a velet rep- resentation. T ranslation inv ariance is obtain ed by prog res- si vely ma pping hig h f requency wa velet coefficients to lower frequen cies, with modulu s operators described in Section 2.1. Scattering operator s, defined in 2.2, iterate over wavelet mod- ulus o perators. Section 2.3 sh ows that it defines a tran slation in variant r epresentation , wh ich is Lipschitz contin uous to de- formation , up to a log term. 2.1. W avelet Modulus Propagato r A wav elet tran sform extracts in formation at different scales and orien tations by conv olving a signal f with dilated band- pass wa velets ψ γ having a spatial orientation angle γ ∈ Γ : W j,γ f ( x ) = f ⋆ ψ j,γ ( x ) wit h ψ j,γ ( x ) = 2 − 2 j ψ γ (2 j x ) . At the largest scale 2 J , low-frequ encies are carried by a low- pass scaling function φ : A J f = f ⋆ φ J , with φ J ( x ) = 2 − 2 J φ (2 − J x ) and R φ ( x ) dx = 1 . The resulting w a velet r ep- resentation is W J f = { A J f , W j,γ f } j
Original Paper
Loading high-quality paper...
Comments & Academic Discussion
Loading comments...
Leave a Comment