Open Access
December 2009 On random tomography with unobservable projection angles
Victor M. Panaretos
Ann. Statist. 37(6A): 3272-3306 (December 2009). DOI: 10.1214/08-AOS673


We formulate and investigate a statistical inverse problem of a random tomographic nature, where a probability density function on ℝ3 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.


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Victor M. Panaretos. "On random tomography with unobservable projection angles." Ann. Statist. 37 (6A) 3272 - 3306, December 2009.


Published: December 2009
First available in Project Euclid: 17 August 2009

zbMATH: 1193.60017
MathSciNet: MR2549560
Digital Object Identifier: 10.1214/08-AOS673

Primary: 60D05 , 62H35
Secondary: 44A12 , 65R32

Keywords: Deconvolution , mixture density , modular inference , Radon transform , Shape theory , single particle electron microscopy , Statistical inverse problem , trigonometric moment problem

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 6A • December 2009
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