Mathematics – Statistics Theory
Scientific paper
2009-09-02
Annals of Statistics 2009, Vol. 37, No. 6A, 3272-3306
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/08-AOS673 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Scientific paper
10.1214/08-AOS673
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.
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