Obtaining three dimensional probability density functions from projected data

In many experimental observation systems where the goal is to record a 3-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 is desired: the three dimensional probablility density function. This article demonstrates that under special symmetries this function can be obtained from a 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 two dimensional projection. Here, the theory of this inversion is explored from a theoretical and numerical point of view with some examples of data functions taken from scientific experiments