[Paper Review] How to quantify fields or textures? A guide to the scattering transform

Sihao Cheng, Brice Ménard|arXiv (Cornell University)|Nov 30, 2021

Machine Learning in Materials Science1 references24 citations

TL;DR

The paper promotes the scattering transform as a training-free, interpretable statistics framework that captures beyond-power-spectrum information in fields/textures, with stable, compact descriptors.

ABSTRACT

Extracting information from stochastic fields or textures is a ubiquitous task in science, from exploratory data analysis to classification and parameter estimation. From physics to biology, it tends to be done either through a power spectrum analysis, which is often too limited, or the use of convolutional neural networks (CNNs), which require large training sets and lack interpretability. In this paper, we advocate for the use of the scattering transform (Mallat 2012), a powerful statistic which borrows mathematical ideas from CNNs but does not require any training, and is interpretable. We show that it provides a relatively compact set of summary statistics with visual interpretation and which carries most of the relevant information in a wide range of scientific applications. We present a non-technical introduction to this estimator and we argue that it can benefit data analysis, comparison to models and parameter inference in many fields of science. Interestingly, understanding the core operations of the scattering transform allows one to decipher many key aspects of the inner workings of CNNs.

Motivation & Objective

• Motivate the need for informative descriptors beyond the power spectrum for fields and textures.
• Introduce the scattering transform as an interpretable, training-free estimator with good stability properties.
• Explain the key operations (wavelet convolutions, modulus, averaging) and how they yield translation- and rotation-invariant descriptors.
• Discuss how scattering coefficients relate to CNNs and provide practical, non-technical guidance for data analysis and inference.

Proposed method

• Describe the scattering transform as a hierarchy of wavelet convolutions, modulus, and average to produce translation-invariant descriptors.
• Define 0th-, 1st-, and 2nd-order scattering coefficients S0, S1, S2 and their reduced forms when averaging over orientations.
• Explain energy partition, scale interactions, and how normalization yields dimensionless statistics.
• Show how wavelets provide localized, scale-aware analysis that captures non-Gaussian, morphological information beyond the power spectrum.
• Highlight the connection to simplified CNNs with pre-defined kernels, modulus nonlinearities, and pooling via averaging.

Experimental results

Research questions

• RQ1 Can scattering coefficients capture non-Gaussian morphology and interactions across scales beyond the power spectrum?
• RQ2 How do first- and second-order scattering coefficients relate to scale interactions and energy distribution in fields/textures?
• RQ3 What are the stability, convergence, and interpretability properties of scattering descriptors for scientific data analysis?
• RQ4 How does the scattering framework compare conceptually to CNNs and traditional spectral methods for parameter inference and model comparison?

Key findings

• Scattering coefficients provide a compact, interpretable set of descriptors that capture information beyond the power spectrum, including morphology and scale interactions.
• The energy of the input field can be partitioned across scattering orders, enabling a stable, multi-layer description with rapid convergence in many practical cases.
• Modulus operations move information toward lower frequencies, enhancing robustness to deformations and noise while revealing sparsity.
• Normalization of scattering coefficients yields dimensionless statistics that facilitate comparison and inference.
• The approach bridges between traditional spectral analysis and CNN-based methods by using predetermined wavelets and avoiding training.
• Second-order scattering, even with a small number of scales, can yield informative summaries for a wide range of fields.