Reconstruction of acoustic source with high-curvature part

Tao Li; Youjun Deng; Xiaoping Fang

doi:10.3934/ipi.2022067

Article Contents

2023, Volume 17, Issue 3: 629-645. Doi: 10.3934/ipi.2022067

This issue Previous Article Determination of piecewise homogeneous sources for elastic and electromagnetic waves Next Article A distance function based cascaded neural network for accurate polyps segmentation and classification

Reconstruction of acoustic source with high-curvature part

1.
School of Mathematics and Statistics, Central South University, Changsha, Hunan, China
2.
College of Science, Hunan University of Technology and Business, Changsha 410205, China
3.
Key Laboratory of Hunan Province for Statistical Learning Intelligent Computation, Hunan University of Technology and Business, Changsha 410205, China

^*Corresponding author: Xiaoping Fang
^*Corresponding author: Xiaoping Fang

Received: August 2022

Revised: November 2022

Early access: December 2022

Published: June 2023

The work of Y. Deng was supported by NSFC-RGC Joint Research Grant No. 12161160314 and NSF grant of China No. 11971487. The work of X. Fang was supported by NSF grant of China No. 72001077, Humanities and Social Sciences Foundation of the Ministry of Education No. 20YJC910005, NSF grant of Hunan No. 2021JJ30192 and PSCF of Hunan No. 18YBQ077.

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Abstract

In this paper, we are concerned with the reconstruction of acoustic sources which contain high-curvature components. We consider the Fourier method developed by Wang X et. al. in [35], and take high-curvature boundary into consideration when choosing Fourier coefficients. We investigate the sophisticated relationship between high frequency Fourier coefficients and high-curvature boundary of the acoustic source, and propose a better cutoff procedure which can greatly enhance the effects in reconstruction of high-curvature boundary. The stability of the proposed method is also derived. Numerical experiments are presented to show the effectiveness of our method. The results may be a new perspective for super-resolution reconstruction of acoustic sources.

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Mathematics Subject Classification: Primary: 35C20; Secondary: 35A22.

Citation:

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Figure 1. Schematic illustration of perturbation $ D^a $ along $ x_2 $-axis

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Figure 2. Schematic illustration of perturbation $ D^a $ along $ x_1 $-axis

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Figure 3. The exact $ f $ and the reconstructions with noise level $ \delta $. (a), (d), (g), (j) and (m) true $ f $; (b), (e) and (h) reconstructions of $ f $ with $ |p|\le30,\ |q|\le30 $; (k) and (n) reconstructions of $ f $ with $ |p|\le40,\ |q|\le40 $; (c) and (f) reconstructions of $ f $ with $ |p|\le60,\ |q|\le15 $; (i) reconstruction of $ f $ with $ |p|\le15,\ |q|\le60 $; (l) reconstruction of $ f $ with $ |p|\le20,\ |q|\le80 $; (o) reconstruction of $ f $ with $ \{(p,q)|\ |p|\le60,\ |q|\le15 \ \text{or}\ |p|\le60,\ |p+q|\le15\} $

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Figure 4. The exact $ f $ and the reconstructions with noise level $ \delta $. (a), (d), (g), (j) and (m) true $ f $; (b), (e) and (h) reconstructions of $ f $ with $ |p|\le30,\ |q|\le30 $; (k) and (n) reconstructions of $ f $ with $ |p|\le40,\ |q|\le40 $; (c) and (f) reconstructions of $ f $ with $ |p|\le60,\ |q|\le15 $; (i) reconstruction of $ f $ with $ |p|\le15,\ |q|\le60 $; (l) reconstruction of $ f $ with $ |p|\le80, |p+q|\le20 $; (o) reconstruction of $ f $ with $ \{(p,q)|\ |p|\le60,\ |q|\le15 \ \text{or}\ |p|\le60,\ |p+q|\le15\} $

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Figure 5. The exact $ f $ and the reconstructions with noise level $ \delta $. (a), (d), (g), (j) and (m) true $ f $; (b), (e) and (h) reconstructions of $ f $ with $ |p|\le30,\ |q|\le30 $; (k) and (n) reconstructions of $ f $ with $ |p|\le40,\ |q|\le40 $; (c) and (f) reconstructions of $ f $ with $ |p|\le60,\ |q|\le15 $; (i) reconstruction of $ f $ with $ |p|\le15,\ |q|\le60 $; (l) reconstruction of $ f $ with $ |p|\le80, |p+q|\le20 $; (o) reconstruction of $ f $ with $ \{(p,q)|\ |p|\le60,\ |q|\le15 \ \text{or}\ |p|\le60,\ |p+q|\le15\} $

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Table 1. Relative errors of high-curvature part

	case 1	case 2	case 3	case 4	case 5
simple truncation	47.12%	45.53%	46.84%	31.60%	32.75%
curvature characteristic truncation	27.82%	25.51%	27.27%	20.21%	27.56%

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Table 2. Relative errors of source $ f $

	case 1	case 2	case 3	case 4	case 5
simple truncation	13.07%	13.15%	13.03%	10.74%	11.48%
curvature characteristic truncation	14.58%	15.03%	14.67%	12.47%	12.71%

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Table 3. Relative errors of high-curvature part

	case 1	case 2	case 3	case 4	case 5
simple truncation	47.45%	44.31%	47.25%	41.77%	27.10%
curvature characteristic truncation	27.90%	27.02%	28.99%	27.73%	23.10%

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Table 4. Relative errors of source $ f $

	case 1	case 2	case 3	case 4	case 5
simple truncation	12.95%	12.96%	12.53%	11.20%	11.48%
curvature characteristic truncation	14.46%	14.21%	14.42%	14.15%	12.68%

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Table 5. Relative errors of high-curvature part

	case 1	case 2	case 3	case 4	case 5
simple truncation	44.57%	50.45%	44.47%	35.24%	32.88%
curvature characteristic truncation	28.77%	28.80%	28.42%	25.05%	24.06%

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Table 6. Relative errors of source $ f $

	case 1	case 2	case 3	case 4	case 5
simple truncation	11.26%	11.19%	11.16%	9.71%	9.77%
curvature characteristic truncation	12.68%	12.62%	12.35%	12.90%	10.85%

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