NDF of Scattered Fields for Strip Geometries

Sekehravani, Ehsan Akbari; Leone, Giovanni; Pierri, Rocco

doi:10.3390/electronics10020202

Cited by 12 publications

(21 citation statements)

References 27 publications

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“…The problems of the NDF and PSF of the linear inverse scattering have been addressed to estimate the achievable resolution for the 2D full rectangular geometry when the impinging plane waves and angular observation domains are limited. Therefore, the novelty of the manuscript relies on the extension of the validity of the approach introduced in [ 20 , 22 ] for curve geometries to a more general case. This also provides a larger scope for the applications, such as the localization problem considered in Section 3 .…”

Section: Discussionmentioning

confidence: 99%

“…The number of independent pieces of information that can be reliably reconstructed by an imaging algorithm when noise is present in the data is provided by the number of degrees of freedom (NDF). The NDF has been used in [ 18 ] for optical imaging applications; furthermore, the NDF has been considered in inverse source problems for square [ 19 ] and circumference geometries [ 20 ] and inverse scattering problems [ 20 , 21 , 22 ]. The behavior of the PSF is likewise connected to the NDF of the problem.…”

Section: Introductionmentioning

confidence: 99%

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Performance of the Linear Model Scattering of 2D Full Object with Limited Data

Sekehravani

Leone

Pierri

2022

Sensors

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Inverse scattering problems stand at the center of many important imaging applications, such as geophysical explorations, radar imaging, and synthetic-aperture radar (SAR). Several methods have been proposed to solve them when the full data are available, usually providing satisfactory reconstructions. However, it is impossible to acquire the full data in many practical circumstances, such as target detection and ground penetrating radar (GPR); consequently, only limited data are available. Thus, this paper focuses on the mathematical analysis and some numerical simulations to estimate the achievable resolution in reconstructing an object from the knowledge of the scattered far-field when only limited data are available, with multi-view excitations at a single frequency. We focus on 2D full rectangular geometry as the investigation domain (ID). We also examine the number of degrees of freedom (NDF) and evaluate the point spread function (PSF). In particular, the NDF of the considered geometry can be estimated analytically. An approximated closed-form evaluation of the PSF is recalled, discussed, and compared with the exact one. Moreover, receiving, transmission, and angle sensing modes are considered to apply the analysis to more realistic scenarios to highlight the difference between the corresponding NDF and the resulting resolution performances. Finally, interesting numerical applications of the resolution analysis for the localization of a collection of point-like scatterers are presented to illustrate how it matches the expectations.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Performance of the Linear Model Scattering of 2D Full Object with Limited Data

Sekehravani

Leone

Pierri

2022

Sensors

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“…The geometry of the problem is depicted in Figure 7. By following the strategy of [9,13], the whole NDF of the square can be provided approximately by summing the NDF of each strip, and the NDF of full view is equal to 96. The numerical results for the aspect limited case show that the NDF of the three modes is approximately equal to 38, and the only difference between them concerns the behavior of the singular values A comparison between the actual PSF of the bottom side of the three modes for three points located at 𝑥 = −1.5𝜆, 0𝜆, 2𝜆 are plotted in Figure 9.…”

Section: Square Geometrymentioning

confidence: 99%

“…The PSF behavior analysis is also connected to the Number of Degrees of Freedom (NDF) of the problem, i.e., the number of independent pieces of information that can be reconstructed reliably by an imaging algorithm in the presence of noise on data [5]. The NDF concept has been considered in [6][7][8] to be used for optical imaging applications and in solving inverse source [9][10][11] and inverse scattering problems [10,12,13].…”

Section: Introductionmentioning

confidence: 99%

Resolution of Born Scattering in Curve Geometries: Aspect-Limited Observations and Excitations

2021

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In inverse scattering problems, the most accurate possible imaging results require plane waves impinging from all directions and scattered fields observed in all observation directions around the object. Since this full information is infrequently available in actual applications, this paper is concerned with the mathematical analysis and numerical simulations to estimate the achievable resolution in object reconstruction from the knowledge of the scattered far-field when limited data are available at a single frequency. The investigation focuses on evaluating the Number of Degrees of Freedom (NDF) and the Point Spread Function (PSF), which accounts for reconstructing a point-like unknown and depends on the NDF. The discussion concerns objects belonging to curve geometries, in this case, circumference and square scatterers. In addition, since the exact evaluation of the PSF can only be accomplished numerically, an approximated closed-form evaluation is introduced and compared with the exact one. The approximation accuracy of the PSF is verified by numerical results, at least within its main lobe region, which is the most critical as far as the resolution discussion is concerned. The main result of the analysis is the space variance of the PSF for the considered geometries, showing that the resolution is different over the investigation domain. Finally, two numerical applications of the PSF concept are shown, and their relevance in the presence of noisy data is outlined.

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“…In particular, the far-field Green function, i.e., the kernel of the scattering operator, behaves similarly to an entire function of exponential type. This results in an abrupt decay of the singular values beyond a certain critical index, the so-called number of degrees of freedom (NDF) [ 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 ] of the scattered field. This singular value behavior, on one hand, is the result of the ill-posedness of the problem [ 31 , 32 ], which limits the achievable performance in the reconstructions.…”

Section: Introductionmentioning

confidence: 99%

Scattered Far-Field Sampling in Multi-Static Multi-Frequency Configuration

Maisto

Masoodi

Leone

et al. 2021

Sensors

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This paper deals with an inverse scattering problem under a linearized scattering model for a multi-static/multi-frequency configuration. The focus is on the determination of a sampling strategy that allows the reduction of the number of measurement points and frequencies and at the same time keeping the same achievable performance in the reconstructions as for full data acquisition. For the sake of simplicity, a 2D scalar geometry is addressed, and the scattered far-field data are collected. The relevant scattering operator exhibits a singular value spectrum that abruptly decays (i.e., a step-like behavior) beyond a certain index, which identifies the so-called number of degrees of freedom (NDF) of the problem. Accordingly, the sampling strategy is derived by looking for a discrete finite set of data points for which the arising semi-discrete scattering operator approximation can reproduce the most significant part of the singular spectrum, i.e., the singular values preceding the abrupt decay. To this end, the observation variables are suitably transformed so that Fourier-based arguments can be used. The arising sampling grid returns several data that is close to the NDF. Unfortunately, the resulting data points (in the angle-frequency domain) leading to a complicated measurement configuration which requires collecting the data at different spatial positions for each different frequency. To simplify the measurement configuration, a suboptimal sampling strategy is then proposed which, by an iterative procedure, enforces the sampling points to belong to a rectangular grid in the angle-frequency domain. As a result of this procedure, the overall data points (i.e., the couples angle-frequency) actually increase but the number of different angles and frequencies reduce and lead to a measurement configuration that is more practical to implement. A few numerical examples are included to check the proposed sampling scheme.

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NDF of Scattered Fields for Strip Geometries

Cited by 12 publications

References 27 publications

Performance of the Linear Model Scattering of 2D Full Object with Limited Data

Performance of the Linear Model Scattering of 2D Full Object with Limited Data

Resolution of Born Scattering in Curve Geometries: Aspect-Limited Observations and Excitations

Scattered Far-Field Sampling in Multi-Static Multi-Frequency Configuration

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