  We consider estimating a random vector from its noisy projections onto low
dimensional subspaces constituting a fusion frame. A fusion frame is a
collection of subspaces, for which the sum of the projection operators onto the
subspaces is bounded below and above by constant multiples of the identity
operator. We first determine the minimum mean-squared error (MSE) in linearly
estimating the random vector of interest from its fusion frame projections, in
the presence of white noise. We show that MSE assumes its minimum value when
the fusion frame is tight. We then analyze the robustness of the constructed
linear minimum MSE (LMMSE) estimator to erasures of the fusion frame subspaces.
We prove that tight fusion frames consisting of equi-dimensional subspaces have
maximum robustness (in the MSE sense) with respect to erasures of one subspace,
and that the optimal subspace dimension depends on signal-to-noise ratio (SNR).
We also prove that tight fusion frames consisting of equi-dimensional subspaces
with equal pairwise chordal distances are most robust with respect to two and
more subspace erasures. We call such fusion frames equi-distance tight fusion
frames, and prove that the chordal distance between subspaces in such fusion
frames meets the so-called simplex bound, and thereby establish connections
between equi-distance tight fusion frames and optimal Grassmannian packings.
Finally, we present several examples for construction of equi-distance tight
fusion frames.
