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Uniqueness theorems for the exponential X-ray transform

We consider the exponential X-ray transform with absorption which does not depend on the point and arbitrarily depends on the direction. The problem of the smallest set of projections sufficient for determining a function is investigated. We prove, for a finite distribution J, that any countable family of projections is sufficient if the projection centers do not accumulate to the convex hull of the support of J. A similar result is established for incomplete projections.