  The co-sparse analysis model for signals assumes that the signal of interest
can be multiplied by an analysis dictionary \Omega, leading to a sparse
outcome. This model stands as an interesting alternative to the more classical
synthesis based sparse representation model. In this work we propose a
theoretical study of the performance guarantee of the thresholding algorithm
for the pursuit problem in the presence of noise. Our analysis reveals two
significant properties of \Omega, which govern the pursuit performance: The
first is the degree of linear dependencies between sets of rows in \Omega,
depicted by the co-sparsity level. The second property, termed the Restricted
Orthogonal Projection Property (ROPP), is the level of independence between
such dependent sets and other rows in \Omega. We show how these dictionary
properties are meaningful and useful, both in the theoretical bounds derived,
and in a series of experiments that are shown to align well with the
theoretical prediction.
