The steerable graph Laplacian and its application to filtering image data-sets

TL;DR

This paper introduces the steerable graph Laplacian, an adaptive, rotation-invariant operator for processing image datasets on low-dimensional manifolds, improving convergence and enabling efficient noise filtering using Fourier modes.

Contribution

The paper proposes the steerable graph Laplacian, extending the standard graph Laplacian to account for all rotations, with improved convergence and efficient eigenfunction computation for image data processing.

Findings

Steerable GL converges to the Laplace-Beltrami operator with improved rate.

Eigenfunctions are Fourier modes times eigenvectors, computed via FFT.

Effective noise filtering demonstrated on cryo-EM datasets.

Abstract

In recent years, improvements in various image acquisition techniques gave rise to the need for adaptive processing methods, aimed particularly for large datasets corrupted by noise and deformations. In this work, we consider datasets of images sampled from a low-dimensional manifold (i.e. an image-valued manifold), where the images can assume arbitrary planar rotations. To derive an adaptive and rotation-invariant framework for processing such datasets, we introduce a graph Laplacian (GL)-like operator over the dataset, termed ${\textit{steerable graph Laplacian}}$. Essentially, the steerable GL extends the standard GL by accounting for all (infinitely-many) planar rotations of all images. As it turns out, similarly to the standard GL, a properly normalized steerable GL converges to the Laplace-Beltrami operator on the low-dimensional manifold. However, the steerable GL admits an improved convergence rate compared to the GL, where the improved convergence behaves as if the intrinsic dimension of the underlying manifold is lower by one. Moreover, it is shown that the steerable GL admits eigenfunctions of the form of Fourier modes (along the orbits of the images' rotations) multiplied by eigenvectors of certain matrices, which can be computed efficiently by the FFT. For image datasets corrupted by noise, we employ a subset of these eigenfunctions to "filter" the dataset via a Fourier-like filtering scheme, essentially using all images and their rotations simultaneously. We demonstrate our filtering framework by de-noising simulated single-particle cryo-EM image datasets.