Rigid-Motion Scattering For Image Classification

1 Ecole Polytechnique, CMAPPhD thesisRigid-Motion Scattering For ImageClassificationAuthor:LaurentSifreSu pervisor:Prof. St ephaneMallatDefended October 6th, 20141 AbstractImage classification is the problem of assigning a label thatbest describes the content ofunknown images, given a set of training images with known labels. This thesis introducesimage classification algorithms based on the Scattering transform, studies their propertiesand describes extensive classification experiments on challenging texture and object are high dimensional signals for which generic machine learning algorithms failwhen applied directly on the raw pixel space. Therefore, most successful approaches involvebuilding a specific low dimensional representation on whichthe classification is , the representation was engineered to reduce the dimensionality of images bybuilding invariance to geometric transformations while retaining discriminative recently, deep convolutional networks have achieved state-of-the-art results on mostimage classification tasks.

2 Such networks progressively build more invariant representationsthrough a hierarchy of convolutional layers where all the weights are thesis proposes several Scattering Scattering representa-tions have a structure similar to convolutional networks, but the weights of Scattering aredesigned to provide mathematical guaranty of invariance togeometric transformations, sta-bility to deformations and energy preservation. In this thesis, we focus on affine and morespecifically on Rigid-Motion transformations, which consist in translations and rotations,and which are common in real world Scattering is a cascade of two dimensional wavelet modulus operators whichbuilds translation invariance. We propose a first separablerigid-motion separable scatter-ing, which applies a first Scattering along the position variable to build translation in-variance, followed by a second Scattering transform along the rotational orbits of the firstscattering, to build invariance to any separable representation, separable Scattering hasthe advantage of simplicitybut also loses some information about the joint distribution of positions and orientationsin the intermediate layers of the representation.

3 We define ajoint Rigid-Motion scatteringwhich does retain this information. The joint Scattering consists in a cascade of waveletmodulus applied directly on the joint Rigid-Motion group. We introduce convolutions,wavelets, a wavelet transform and Scattering on the Rigid-Motion group and propose fastimplementations. Both separable and joint Scattering are applied to texture Image classi-fication with state-of-the-art results on most available texture , we demonstrate the applicability of joint Scattering and group convolutionson generic object Image datasets. It is shown that convolutional networks performancesare enhanced through the use of separable convolutions, similar to the Rigid-Motion con-volutions. Also, a non-invariant version of the Rigid-Motion Scattering is demonstrated toachieve results similar to those obtained by the first layersof convolutional esum eLa classification d Image consiste `a assigner un label `a une Image inconnue, etant donn eun ensemble d images d entra nement avec des labels connus.

4 Cette th`ese introduit desalgorithmes de classification bas es sur la transform ee enscattering, etudie leurs propri et eset d ecrit des exp eriences de classification sur des bases de donn ees de texture de d images sont des signaux de haute dimension pour lesquelsles algorithmes d appren-tissage echouent lorsque appliqu e directement sur l espace des pixels. La plupart des ap-proches qui fonctionnent construisent une repr esentation de basse dimension sur laquelle laclassification est effectu ee. Traditionnellement, cette repr esentation est con cue pour con-struire de l invariance aux transformations g eom etriques tout en retenant le plus d infor-mation discriminante. Plus r ecemment, les r eseaux convolutionnels ont supplant es cesrepr esentations sur la plupart des taches de classification d Image . Les r eseaux convo-lutionnels construisent des repr esentations progressivement de plus en plus invariantes `atravers une hi erarchie de couches o`u les poids sont th`ese propose plusieurs transform ees en Scattering .

5 Ces repr esentations ont unestructure similaire `a celles des r eseaux convolutionnels, mais les poids sont con cus pourfournir une garantie math ematique d invariance aux transformations g eom etriques, unestabilit e aux d eformations, et une conservation d energie. Cette th`ese se concentre surles transformations affines et plus particuli`erement sur les transformations rigides, tr`escourantes dans les Scattering en translation est une cascade de transform ee en ondelette et module, quiconstruit de l invariance par translation. Nous proposonsun premier Scattering s eparable,qui applique un premier Scattering en translation suivi d un second Scattering le long desorbites de rotation du premier Scattering , pour construirel invariance par toute repr esentation s eparable, le Scattering s eparable a l avantage de la sim-plicit e mais perd aussi l information de la distribution jointe de positions et d orientationsdans les couches interm ediaires de la repr esentation.

6 Nous proposons donc un scatteringjoint qui consiste en une cascade de transform ees en ondelettes module appliqu ees directe-ment sur le groupe joint des transformations rigides. Nous pr esentons les convolutions,ondelettes, transform ees en ondelettes et Scattering surce groupe joint de transformationsrigides. Les deux Scattering , s eparable et joint, sont appliqu es `a la reconnaissance de tex-tures et fournissent des r esultats comparables voire sup erieurs `a ceux de l etat de l art surla plupart des bases de donn ees de textures , nous d emontrons l applicabilit e du Scattering joint et des convolutions de groupesaux probl`emes de classification d objet g en eriques. Il est montr e que les performancesdes r eseaux convolutionnels sont am elior ees par l utilisation de convolutions s eparables,similaires `a celles que nous utilisons dans le Scattering joint.

7 Par ailleurs, une version non-invariante du Scattering joint permet d attendre des r esultats comparables `a ceux obtenusavec les premi`eres couches de r eseaux ephane, travailler avec toi m a apport e enorm ement,merci pour le temps et l energieque tu m as consacr e. Tes qualit es de scientifique continueront de m inspirer bien au-del`ade ces ann ees de th` remercie le DI de l ENS, et notamment les membres de l equipe data, Joan, Joakim,Ir`ene, Edouard, Vincent, Mia, Xiu-Yuan, Guy, Matthew et Gilles. Merci aussi aux visiteursnotamment Remco, Charles, Y-lan, Mike pour les discussionsint eressantes. Merci aussiaux membres du CMAP et aux autres doctorants et post-docs. Jeremercie aussi BertrandThirion, son equipe de Neurospin et toute l equipe de Google Brain, notamment Vincent,Mathieu, Beno t, Rajat, `a Claire pour m avoir soutenue.

8 Merci aussi a mes parents, ma famille, tous mesproches, amis et colocataires qui ont positivement ou n egativement impact e la probabilit ede l existence de ce Image Classification .. Filters Response Representations .. Fourier and Registration Invariants .. Averaged Filter Responses for Global Translation Invariance .. Local Descriptors and Transformation Invariance .. Hierarchical Invariant Representations .. Spatial Pyramid .. Higher Order Statistics .. Deep Convolutional Networks .. Translation Scattering .. Invariance to Geometric Transformation .. Elastic Deformation and the Affine Group .. Separable Invariants .. Separable Rigid-Motion Scattering .. Joint Invariants and Joint Rigid-Motion Scattering .. Joint versus Separable Invariants .. Rigid-Motion Wavelet Transform.

9 Joint Rigid-Motion Scattering .. Classification experiments .. Texture Classification .. Object Classification .. 252 Translation Scattering and Introduction .. Stability to Deformation .. Wavelet Transform .. Wavelets, Window and the Wavelet Transform Operator .. Oriented Gabor and Morlet Wavelets .. Fast Implementations .. Translation Scattering .. Wavelet Modulus Operator .. Cascading Wavelet Transform .. Scattering Properties .. Scattering Implementation .. Deep Convolutional Neural Networks .. 463 Separable Introduction .. Transformation Groups .. The Affine Group .. Subgroups of the Affine Group .. Separable Representations .. Separable Rigid-Motion Scattering .. Covariance Property of the Translation Scattering .. Wavelets on the Rotation Parameter.

10 Periodic Scattering on the Rotation Parameter .. Separable Scattering .. 604 Joint Introduction .. Joint versus Separable Invariants .. Separable Translation Invariance .. Separable Rigid-Motion Invariance .. Multiresolution Analysis on the Rigid Motion Group .. Rigid-Motion Convolutions .. Rigid-Motion Wavelet Transform .. Fast Rigid-Motion Wavelet Transform .. Joint Rigid-Motion Scattering .. Covariance of the Spatial Wavelet Transform .. Rigid-Motion Orbit and Rigid-Motion Scattering .. Covariance and Invariance Properties of Rigid-Motion Scattering .. 875 Texture Introduction .. Scattering of Stationary Processes .. Translation Expected and Windowed Scattering .. Separable Expected and Windowed Scattering .. Joint Expected and Windowed Scattering .. Classification with Separable Scattering .