  Ill-posed linear inverse problems appear in many image processing
applications, such as deblurring, super-resolution and compressed sensing. Many
restoration strategies involve minimizing a cost function, which is composed of
fidelity and prior terms, balanced by a regularization parameter. While a vast
amount of research has been focused on different prior models, the fidelity
term is almost always chosen to be the least squares (LS) objective, that
encourages fitting the linearly transformed optimization variable to the
observations. In this paper, we examine a different fidelity term, which has
been implicitly used by the recently proposed iterative denoising and backward
projections (IDBP) framework. This term encourages agreement between the
projection of the optimization variable onto the row space of the linear
operator and the pseudo-inverse of the linear operator ("back-projection")
applied on the observations. We analytically examine the difference between the
two fidelity terms for Tikhonov regularization and identify cases (such as a
badly conditioned linear operator) where the new term has an advantage over the
standard LS one. Moreover, we demonstrate empirically that the behavior of the
two induced cost functions for sophisticated convex and non-convex priors, such
as total-variation, BM3D, and deep generative models, correlates with the
obtained theoretical analysis.
