# Multidimensional Rational Covariance Extension with Approximate Covariance Matching

## TL;DR

This paper extends the multidimensional rational covariance extension problem to handle approximate covariance matching, ensuring well-posedness and demonstrating applications in spectral estimation and texture generation.

## Contribution

It introduces a framework for approximate covariance matching in RCEP, connecting soft and hard constraints, and proves their well-posedness with practical examples.

## Key findings

- The problems are well-posed under approximate covariance matching.
- Soft and hard constraint problems are connected via a homeomorphism.
- Applications include spectral estimation and texture synthesis.

## Abstract

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.