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

In recent years, improvements in various scientific image acquisition techniques gave rise to the need for adaptive processing methods aimed for large data-sets corrupted by noise and deformations. In this work, we consider data-sets of images sampled from an underlying low-dimensional manifold (i.e. an image-valued manifold), where the images are obtained through arbitrary planar rotations. To derive the mathematical framework for processing such data-sets, we introduce a graph Laplacian-like operator, termed steerable graph Laplacian (sGL), which extends the standard graph Laplacian (GL) by accounting for all (infinitely-many) planar rotations of all images. As it turns out, a properly normalized sGL converges to the Laplace-Beltrami operator on the low-dimensional manifold, with an improved convergence rate compared to the GL. Moreover, the sGL admits eigenfunctions of the form of Fourier modes multiplied by eigenvectors of certain matrices. For image data-sets corrupted by noise, we employ a subset of these eigenfunctions to"filter"the data-set, essentially using all images and their rotations simultaneously. We demonstrate our filtering framework by de-noising simulated single-particle cryo-EM image data-sets.