Abstract
We consider uniqueness of two-dimensional parallel beam tomography in whichboth the object being imaged and the projection directions are unknown. This problem occursin certain practical applications. For example, in magnetic resonance imaging there may beuncertainty in the projection directions due to the involuntary motion of the patient. Thethree-dimensional version of this problem occurs in cryo electron microscopy of viral particles,where the projection directions may be completely unknown due to the random orientations ofthe particles being imaged. We show that the problem is related to some algebraic geometricproperties of a certain system of homogeneous polynomials. We also show that for sufficientlyasymmetric objects, the object is uniquely determined up to an orthogonal transformation bythe projection data from unknown directions.
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