M2-spectral estimation: A relative entropy approach

A Well-Posed Multidimensional Rational Covariance and Generalized Cepstral Extension Problem

“…In the case that the random field is periodic, we are still able to find a solution matching the covariances and approximately the cepstral coefficients for any d ≥ 1, see [33]. Such a result is not surprising because it corresponds to discretizing the power spectral density in the frequency domain [18,19], and indeed the existence of such a solution for d > 1 is also guaranteed using the classic definitions of logarithmic moments and entropy [21,32]. However, there is a fundamental difference between our problem and the classic one in the case of d > 2.…”

“…Such a discretization in the frequency domain has an interesting interpretation in the time domain (or more precisely, the "space" domain). Indeed, it corresponds to considering a periodic stationary random field as explained in [32].…”

A fast robust numerical continuation solver to a two‐dimensional spectral estimation problem

“…The aforementioned theories are established for random processes and one-dimensional systems, that is, random fields and dynamical systems that depend on one index, in most cases the time. Motivated by many practical applications involving multidimensional systems and random fields such as image processing [24] and parameter estimation in automotive radar systems [25][26][27], the research in rational covariance extension has also been extended to the multidimensional case, see [28][29][30][31][32][33][34][35]. Discrete versions of the theory that facilitate numerical computation have also been developed in [36,37], following the idea in [38,39] for the 1-d case.…”

“…As a first step towards efficient algorithms in the multidimensional case, in this paper we propose a fast implementation of the classical Newton's method to solve a 2-d spectral estimation problem formulated as a moment-constrained optimization problem where the Itakura-Saito pseudo-distance is used as the objective function. Such a choice of the objective function first appeared in [46] and later was further developed in [19,32] where it was shown that the solution is indeed rational and of bounded complexity (in terms of the McMillan degree). For the convenience of numerical computation, we mainly deal with the dual optimization problem which typically has much fewer number of variables than the primal problem.…”

Optimal Transport between Gaussian random fields

“…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].…”