  We study the duality of reconstruction systems, which are $g$-frames in a
finite dimensional setting. These systems allow redundant linear
encoding-decoding schemes implemented by the so-called dual reconstruction
systems.
  We are particularly interested in the projective reconstruction systems that
are the analogue of fusion frames in this context. Thus, we focus on dual
systems of a fixed projective system that are optimal with respect to erasures
of the reconstruction system coefficients involved in the decoding process. We
consider two different measures of the reconstruction error in a blind
reconstruction algorithm.
  We also study the projective reconstruction system that best approximate an
arbitrary reconstruction system, based on some well known results in matrix
theory. Finally, we present a family of examples in which the problem of
existence of a dual projective system of a reconstruction system of this type
is considered.
