Multidimensional Rational Covariance Extension with Approximate Covariance Matching

Abstract

In our companion paper [54] 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 [54] 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.

Optimal Transport between Gaussian random fields

“…and, with some abuse of notation, ᾱ α α, 1 shows the corresponding geodesic defined in (23) with the constant weight function Ω(e jϑ ϑ ϑ ) = 1 −0.99 −0.99 1…”

“…These methods have been extended to: 1) stationary (i.e. homogeneous) random fields which are characterized by multidimensional power spectral densities [21], [22], [23], [24]; 2) stationary periodic random fields which are characterized by multidimensional power spectral densities whose domain is constituted by a finite number of points [25], [26]. It is worth noting that in the unidimensional case, the latter case boils down to the so called reciprocal processes, [27], [28], [29], [30], [31].…”

Graphical model selection for a particular class of continuous-time processes

“…We exploited problem (35) to show the existence of a unique solution of problem (27). A remarkable difference between (27) and (35) is that in the former the objective function is not differentiable while in the latter it is.…”

“…We exploited problem (35) to show the existence of a unique solution of problem (27). A remarkable difference between (27) and (35) is that in the former the objective function is not differentiable while in the latter it is. Accordingly, the importance of problem (35) is also that it allows to compute the optimal solution of (27) by applying gradient descent methods, [5].…”

“…Accordingly, such a covariance selection problem provides a covariance matrix corresponding to a graphical model with topology defined by Ω. Furthermore, such a paradigm represents a maximum entropy problem and thus it maximizes the information that can be encoded in the random vector x. Covariance selection problems received a great deal of attention, [8,9,10,15], as well as their dynamic generalizations which are limited to discrete-time models [1,2,3,6,7,16,27,31,33,36,37]. Interestingly, the optimal solution of (2) is such that Σ • = S −1 where S is given by solving the dual problem: min S 0 − log det(S) + tr(SΣ) subject to (S) ij = 0 ∀ (i, j) /…”