[Paper Review] Rigid-Motion Scattering for Texture Classification

Laurent Sifre, Stéphane Mallat|arXiv (Cornell University)|Mar 7, 2014

Image Retrieval and Classification TechniquesComputer Science17 references278 citations

TL;DR

This paper proposes rigid-motion scattering, a deep convolutional network that computes joint invariants to translations and rotations using wavelets on the rigid-motion group (SE(2)), preserving spatial-orientation relationships. It achieves state-of-the-art texture classification performance on multiple benchmarks, especially under large rotation and scaling variations, by combining wavelet transforms with logarithmic scaling invariance and efficient filter bank implementation.

ABSTRACT

A rigid-motion scattering computes adaptive invariants along translations and rotations, with a deep convolutional network. Convolutions are calculated on the rigid-motion group, with wavelets defined on the translation and rotation variables. It preserves joint rotation and translation information, while providing global invariants at any desired scale. Texture classification is studied, through the characterization of stationary processes from a single realization. State-of-the-art results are obtained on multiple texture data bases, with important rotation and scaling variabilities.

Motivation & Objective

• To develop a stable, non-learning-based representation that captures joint invariance to translations and rotations in texture images.
• To address the limitation of separable translation and rotation invariance, which discards critical joint spatial-orientation information.
• To extend scattering transforms—originally for translation invariance—to the non-commutative rigid-motion group (SE(2)) for improved discriminative power.
• To incorporate scale invariance through logarithmic non-linearity and data augmentation, enhancing robustness to geometric variability.
• To demonstrate state-of-the-art performance on texture classification under high geometric variability using a fixed, non-learned architecture.

Proposed method

• Constructs a scattering transform on the rigid-motion group SE(2), where convolutions are performed jointly over spatial position and rotation angle.
• Uses wavelets defined on SE(2) to compute multi-scale, multi-orientation representations that preserve joint translation-rotation information.
• Applies successive wavelet transforms followed by modulus operations to generate invariant features, forming a deep, non-learned hierarchical network.
• Implements the transform efficiently via a cascade of separable spatial and angular convolutions with downsampling, enabling fast computation.
• Introduces logarithmic non-linearity and scale augmentation to achieve partial invariance to scaling, improving robustness to image rescaling.
• Employs a PCA classifier on scattering coefficients for small training sets and a linear SVM for larger, more variable datasets like FMD.

Experimental results

Research questions

• RQ1 Can joint invariance to translations and rotations be achieved more effectively than by separating the two transformations?
• RQ2 How does preserving the joint spatial-orientation structure in wavelet transforms affect texture classification performance under geometric variability?
• RQ3 To what extent can a non-learning-based scattering network achieve state-of-the-art results on texture classification with large rotation and scaling variations?
• RQ4 What is the impact of incorporating scale invariance via logarithmic non-linearity and data augmentation on classification accuracy?
• RQ5 Can rigid-motion scattering generalize across diverse texture datasets with varying levels of deformation and material variability?

Key findings

• Rigid-motion scattering achieves 53.28% classification accuracy on the KTH-TIPS dataset using YUV color channels, outperforming prior single-feature methods.
• On UIUC and UMD datasets, rigid-motion scattering with scale invariance improves accuracy by over 10% compared to standard translation scattering.
• The method achieves 52.2% accuracy on the FMD dataset using a linear SVM, which is, to the authors' knowledge, the best result for a single feature.
• The addition of logarithmic non-linearity and scale invariance significantly boosts performance on datasets with high scaling variability.
• The approach maintains strong performance even with small training sets, demonstrating low-variance, ergodic representations suitable for single-realization texture analysis.
• The scattering framework is reproducible using the ScatNet MATLAB toolbox, enabling consistent benchmarking across datasets.