On random tomography with unobservable projection angles

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.Comment: Published in at http://dx.doi.org/10.1214/08-AOS673 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sparse approximations of protein structure from noisy random projections

“…In order to illustrate the details (and effectiveness) of this discrete regularization approach, we revisit an artificial example presented in Panaretos (2009), where a three-dimensional mixture of four Gaussian kernels was to be recovered given its projections at randomly chosen directions. The pseudoparticle potential density in three dimensions was given by…”

“…The method employed in Panaretos (2009) to perform the deconvolutions required for the construction of the estimator was a direct spectral approach based on results on Toeplitz forms [Grenander and Szegö (1958), Pisarenko (1973)]. The approach performed well on noiseless projections, but would fail completely even with very small amounts of noise.…”

“…Their construction requires the solution of further inverse problems, with severe instabilities due to the presence of noise, and the approximate nature of the modeling framework. In this paper we propose a framework for implementing estimators such as those proposed in Panaretos (2009) under sparsity constraints. Our approach combines L 1 -regularization using Least Angles Regression with the special geometry of the sample space to yield a procedure applicable to actual electron microscope data.…”

“…Although this is conceptually obvious, it can be a serious hurdle to statistical estimation: for example, if one wishes to parametrize the unknown density using a Fourier expansion, the Fourier coefficients will not be invariant to changes of the coordinate system. Nevertheless, the shape of the density ρ can potentially be recovered [Panaretos (2009)]. The shape of ρ, denoted [ρ], encodes the totality of characteristics of ρ that are invariant with respect to the coordinate system…”

A new ab initio reconstruction method from unknown-direction projections of 2D binary set

“…This approach is shared by all current methods [3] except [4]. In [4], Panaretos shows that the object density can be estimated without direction assignment under assumption on it. Nevertheless, the proposed method is noise sensitive and has not been tested on real data.…”