  The Fast Proximal Gradient Method (FPGM) and the Monotone FPGM (MFPGM) for
minimization of nonsmooth convex functions are introduced and applied to
tomographic image reconstruction. Convergence properties of the sequence of
objective function values are derived, including a $O\left(1/k^{2}\right)$
non-asymptotic bound. The presented theory broadens current knowledge and
explains the convergence behavior of certain methods that are known to present
good practical performance. Numerical experimentation involving computerized
tomography image reconstruction shows the methods to be competitive in
practical scenarios. Experimental comparison with Algebraic Reconstruction
Techniques are performed uncovering certain behaviors of accelerated Proximal
Gradient algorithms that apparently have not yet been noticed when these are
applied to tomographic image reconstruction.
