  In our companion paper "Multidimensional rational covariance extension with
applications to spectral estimation and image compression" we discussed the
multidimensional rational covariance extension problem (RCEP), which has
important applications in image processing, and spectral estimation in radar,
sonar, and medical imaging. This is an inverse problem where a power spectrum
with a rational absolutely continuous part is reconstructed from a finite set
of moments. However, in most applications these moments are determined from
observed data and are therefore only approximate, and RCEP may not have a
solution. In this paper we extend the results to handle approximate covariance
matching. We consider two problems, one with a soft constraint and the other
one with a hard constraint, and show that they are connected via a
homeomorphism. We also demonstrate that the problems are well-posed and
illustrate the theory by examples in spectral estimation and texture
generation.
