Pixel Club: The Steerable Graph Laplacian and its Application to Filtering Image Datasets

Speaker:
Boris Landa (Tel-Aviv University)
Date:
Monday, 7.5.2018, 11:30
Place:
Room 337 Taub Bld.

In recent years, improvements in various scientific image acquisition techniques gave rise to the need for adaptive processing methods, particularly aimed for large data-sets corrupted by noise and deformations. Motivated by challenges in cryo-electron microscopy (cryo-EM), we consider the problem of reducing noise in a dataset of images admitting a certain unified structure. In particular, we consider datasets 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 exploit the geometry of the manifold in such datasets, 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 lowdimensional 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.

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