  We consider a general method of deprojecting 2D images to reconstruct the 3D
structure of the projected object, assuming axial symmetry. The method consists
of the application of the Fourier Slice Theorem to the general case where the
axis of symmetry is not necessarily perpendicular to the line of sight, and is
based on an extrapolation of the image Fourier transform into the so-called
cone of ignorance. The method is specifically designed for the deprojection of
X-ray, Sunyaev-Zeldovich (SZ) and gravitational lensing maps of rich clusters
of galaxies. For known values of the Hubble constant, H0, and inclination
angle, the quality of the projection depends on how exact is the extrapolation
in the cone of ignorance. In the case where the axis of symmetry is
perpendicular to the line of sight and the image is noise-free, the
deprojection is exact. Given an assumed value of H0, the inclination angle can
be found by matching the deprojected structure out of two different images of a
given cluster, e.g., SZ and X-ray maps. However, this solution is degenerate
with respect to its dependence on the assumed H0, and a third independent image
of the given cluster is needed to determine H0 as well. The application of the
deprojection algorithm to upcoming SZ, X-ray and weak lensing projected mass
images of clusters will serve to determine the structure of rich clusters, the
value of H0, and place constraints on the physics of the intra-cluster gas and
its relation to the total mass distribution.
