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# A direct imaging method for the exterior and interior inverse scattering problems

• *Corresponding author: Yukun Guo

The work of D. Zhang, Y. Wu and Y. Wang are supported by NSFC grant 12171200. The work of Y. Guo is supported by NSFC grant 11971133 and the Fundamental Research Funds for the Central Universities

• This paper is concerned with the inverse acoustic scattering problems by an obstacle or a cavity with a sound-soft or a sound-hard boundary. A direct imaging method relying on the boundary conditions is proposed for reconstructing the shape of the obstacle or cavity. First, the scattered fields are approximated by the Fourier-Bessel functions with the measurements on a closed curve. Then, the indicator functions are established by the superposition of the total fields or their derivatives to the incident point sources. We prove that the indicator functions vanish only on the boundary of the obstacle or cavity. Numerical examples are also included to demonstrate the effectiveness of the method.

Mathematics Subject Classification: Primary: 78A46, 65N21; Secondary: 35R30.

 Citation: • • Figure 1.  An illustration of the inverse obstacle scattering problem

Figure 2.  An illustration of the interior inverse scattering problem

Figure 3.  The model scatterers for testing the reconstructions. (a) circle: $(\cos t, \sin t)$; (b) kite: $(\cos t+0.6\cos 2t-0.3, 1.3\sin t)$; (c) starfish: $(1+0.2\cos 5t)(\cos t, \sin t)$

Figure 4.  Reconstruction of sound-soft obstacles with 5% noise. Top row: $k = 3$; Bottom row: $k = 6$

Figure 5.  Reconstruction of sound-hard obstacles with 2% noise. Top row: $k = 4$; Middle row: $k = 5$; Bottom row: superposition with multiple frequencies $k = 3, 3.5, \cdots, 6$

Figure 6.  Reconstruction of sound-soft cavities with 5% noise. Top row: $k = 3$; Bottom row: $k = 5$

Figure 7.  Reconstruction of sound-hard cavities with 2% noise. Top row: $k = 4$; Middle row: $k = 5$; Bottom row: superposition with multiple frequencies $k = 3, 3.5, \cdots, 6$

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