We consider a general method of deprojecting two-dimensional images to
reconstruct the three-dimensional structure of the projected object,
assuming axial symmetry. The method consists of the application of the
Fourier slice theorem to the general case in which the axis of symmet
ry 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 deproje
ction of X-ray, Sunyaev-Zeldovich (SZ), and gravitational lensing maps
of rich clusters of galaxies. For known values of the Hubble constant
H-0 and inclination angle, the quality of the projection depends on h
ow exact the extrapolation in the cone of ignorance is. in the case in
which the axis of symmetry is perpendicular to the line of sight and
the image is noise-free, the deprojection is exact. Given an assumed v
alue of H-0, the inclination angle can be found by matching the deproj
ected structure out of two different images of a given cluster, e.g.,
SZ and X-ray maps. However, this solution is degenerate with respect t
o its dependence on the assumed H-0, and a third independent image of
the given cluster is needed to determine H-0 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 structur
e of rich clusters and the value of H-0 and to place constraints on th
e physics of the intracluster gas and its relation to the total mass d
istribution. The method is demonstrated using a simple analytic model
for cluster dark matter and gas distributions and is shown to provide
a stable and unique reconstruction of the cluster three-dimensional st
ructure.