An SVD Approach for Estimating the Dimension of Phaseless Data on Multiple Arcs in Fresnel Zone

Abstract

In this article, we tackle the question of evaluating the dimension of the data space in the phase retrieval problem. With the aim to achieve this task, we first exploit the lifting technique to recast the quadratic model as a linear one. After that, we evaluate analytically the singular values of the lifting operator, and we quantify the dimension of the data space by counting the number of “significant” singular values. In the last part of the article, we show some numerical results in order to corroborate our analytical prediction on the singular values’ behavior of the lifting operator and on the dimension of the data space. The analysis is performed for a 2D scalar geometry consisting of an electric current strip whose square magnitude of the radiated field is observed on multiple arcs of circumference in Fresnel zone.

The Dimension of Phaseless Near-Field Data by Asymptotic Investigation of the Lifting Operator

“…It is worth highlighting that Equation (67) is very similar to Equation (46) shown in [35], i.e., M ≈ 4P π βa u max . The latter represents the number of relevant singular values of the lifting operator when the observation domain is an ensemble of P observation arcs in Fresnel zone subtending an angular sector [−θ max , θ max ].…”

“…The latter represents the number of relevant singular values of the lifting operator when the observation domain is an ensemble of P observation arcs in Fresnel zone subtending an angular sector [−θ max , θ max ]. As can be seen, the relevant difference between Equation (67) of this article and Equation (46) in [35] is that η max is replaced by sin θ max . This is perfectly consistent with the fact that the variable η in Fresnel zone can be approximated by sin θ.…”

“…Let us remark that in the case of data in amplitude and phase, an analytical estimation of the dimension of data space has been provided for several configurations [32][33][34]; instead, in the case of phaseless measurements, it has been evaluated only for a strip source observed in Fresnel region [35]. Hence, this work represents an extension of [35] to the case of near-field data.…”

On the Sampling of the Fresnel Field Intensity over a Full Angular Sector

“…In this section, an analytical evaluation of the dimension of data space is provided. Different methods can be exploited to achieve this task, here, an SVD-based approach that allows estimating the dimension of data space by counting the number of significant singular values of the lifting operator A is exploited [52].…”

Numerical Study on Multiple Arcs in a Pyro-Breaker Based on the Black-Box Arc Model

“…Arc modeling is considered to be a practicable way to study the arc characteristics and analyze experiment results. Plenty of works have been devoted to the utilization of arc models in circuit breakers [4][5][6]. If an arc model has a perfect correspondence with the experiment results, it can be used as the theoretical basis for the structural design and optimization.…”