We explore the application of optimal inversion techniques to astronomical data with a goal of developing a set of procedures for the determination of the three dimensional structure of astronomical sources. Astronomical data present a particularly difficult problem in inversion because: In any observation, 3 of 6 spatial and velocity dimensions are lost in projection onto the plane of the sky and the line of sight velocity. In any inversion, we would like to solve for a number of physical parameters. Generally, these parameters are closely related in their effect on the single observable, the sky brightness.

The dimensional deficiency leaves us with an unavoidably large degree of ambiguity (non-uniqueness) in any solution, while the inter-related parameters lead to a high probability of correlated errors and hence instability in the presence of to noise.

We show how constraints of symmetry and smoothness source allow us to handle an inversion with an insufficiently sampled data base and mutually dependent solution parameters (mathematically ill-posed and ill-conditioned). The constraints represent a priori information incorporated into the solution; thus very highly constrained inversions are similar to model fitting. In any case the inversion procedure provides us with quantitative statistics on the goodness of fit which may be used to assess the degree of ambiguity in a particular model, and the expected errors and cross-correlated errors on the parameters defining the source structure.

We briefly discuss the background and motivation, and outline the procedure in general terms. We refer to papers published in the Ap. J. where different aspects of the inversion are applied to observational data bases collected at the VLA.