Non-convex Relaxation of Optimal Transport for Color Transfer Between Images

Abstract

In this paper we study two forms of blurring effects that may appear in the reconstruction of 3D Electron Microscopy (EM), specifically in single particle reconstruction from random orientations of large multi-unit biomolecular complexes. We model the blurring effects as being due to independent contributions from: (1) variations in the conformation of the biomolecular complex; and (2) errors accumulated in the reconstruction process. Under the assumption that these effects can be separated and treated independently, we show that the overall blurring effect can be expressed as a special form of a convolution oper-ation of the 3D density with a kernel defined on SE(3), the Lie group of rigid body motions in 3D. We call this form of convolution mixed spatial-motional convolution. We discuss the ill-conditioned nature of the deconvolution needed to deblur the reconstructed 3D density in terms of parameters associated with the unknown probability in SE(3). We pro-vide an algorithm for recovering the conformational information of large multi-unit biomolecular complexes (essentially deblurring) under certain biologically plausible prior structural knowledge about the subunits of the complex in the case the blurring kernel has a special form.