Three-dimensional probability density functions via tomographic inversion

In many experimental observation systems where the goal is to record a three-dimensional observation of an object, or a set of objects, a lower-dimensional projection of the intended subject is obtained. In some situations only the statistical properties of such objects are desired: the three-dimensional probability density function. This article demonstrates that under special symmetries this function can be obtained from either a one- or two-dimensional probability density function which has been obtained from the observed, projected data. Standard tomographic theorems can be used to guarantee the uniqueness of this function, and a natural basis set can be used in computing the three-dimensional function from the one- or two-dimensional projection. The theory of this inversion is explored using theoretical and computational methods with examples of data taken from scientific experiments.