  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.
