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We formulate and investigate a statistical inverse problem of a randomtomographic nature, where a probability density function on $\mathbb{R}^3$ isto be recovered from observation of finitely many of its two-dimensionalprojections in random and unobservable directions. Such a problem is distinctfrom the classic problem of tomography where both the projections and the unitvectors normal to the projection plane are observable. The problem arises insingle particle electron microscopy, a powerful method that biophysicistsemploy to learn the structure of biological macromolecules. Strictly speaking,the problem is unidentifiable and an appropriate reformulation is suggestedhinging on ideas from Kendall's theory of shape. Within this setup, wedemonstrate that a consistent solution to the problem may be derived, withoutattempting to estimate the unknown angles, if the density is assumed to admit amixture representation.

On random tomography with unobservable projection angles