On random tomography with unobservable projection angles

TL;DR

This paper addresses the challenge of reconstructing a 3D density from limited, randomly oriented 2D projections in a setting relevant to electron microscopy, proposing a mixture model approach to achieve consistent solutions despite unobservable angles.

Contribution

It introduces a novel formulation for a random tomography problem with unobservable angles and demonstrates a method for consistent density estimation using mixture models.

Findings

A reformulation based on shape theory makes the problem identifiable.

Consistent reconstruction is possible without estimating the unknown projection angles.

The approach is applicable to single particle electron microscopy data.

Abstract

We formulate and investigate a statistical inverse problem of a random tomographic nature, where a probability density function on $\mathbb{R}^3$ is to be recovered from observation of finitely many of its two-dimensional projections in random and unobservable directions. Such a problem is distinct from the classic problem of tomography where both the projections and the unit vectors normal to the projection plane are observable. The problem arises in single particle electron microscopy, a powerful method that biophysicists employ to learn the structure of biological macromolecules. Strictly speaking, the problem is unidentifiable and an appropriate reformulation is suggested hinging on ideas from Kendall's theory of shape. Within this setup, we demonstrate that a consistent solution to the problem may be derived, without attempting to estimate the unknown angles, if the density is assumed to admit a mixture representation.