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.
Highlights
Inverse problems cover a wide range of applications in the electromagnetic and optical literature [1,2,3,4,5,6]
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 first one is that of exploiting the lifting technique to obtain a linear representation of the data. Once such a task has been achieved, the dimension of the data space can be computed by employing different tools like Singular-Value Decomposition (SVD) [1], the Gram–Schmidt orthogonalization (GSO) [25], or Principal Component Analysis (PCA) [33,34]
Summary
Inverse problems cover a wide range of applications in the electromagnetic and optical literature [1,2,3,4,5,6]. To overcome the traps issue, the lifting technique was introduced [26] The latter exploits a redefinition of the unknown space, which allows recasting the phase retrieval problem as a linear one. The first one is that of exploiting the lifting technique to obtain a linear representation of the data Once such a task has been achieved, the dimension of the data space can be computed by employing different tools like Singular-Value Decomposition (SVD) [1], the Gram–Schmidt orthogonalization (GSO) [25], or Principal Component Analysis (PCA) [33,34]. We choose to evaluate the dimension of the data space by exploiting the approach introduced in [35]; we estimate such a parameter by counting the number of “significant” singular values of the lifting operator.
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