Uniqueness to Inverse Medium Scattering with One Incoming Wave Johannes Elschner and Guanghui Hu
update: 2019-01-22 09:28:25     

A challenging open problem in inverse time-harmonic scattering is to determine if the far field pattern (also called scattering amplitude) for scattering of one incident plane wave at a fixed energy completely determines the scatterer. It is widely believed that it is true for a large class of scatterers, but little progress has been made so far. In 1967, Lax and Phillips firstly mentioned Schiffer’s idea to show that the shape of a sound-soft obstacle can be uniquely determined by far-field patterns for an infinite number of incident plane waves with distinct directions. Colton and Slemman [1] proved that the data of a single incident wave are sufficient under an a priori smallness assumption on the size of the scatterer. Cheng and Yamamoto [2] verified a global uniqueness within the class of sound-soft polygonal scatterers. However, the proofs of these local and global uniqueness results do not apply to penetrable scatterers. Motivated by corner scattering problems [3], J. Elschner and G. Hu recently consider uniqueness in inverse medium scattering with a single plane wave. It was verified in [4, 5] that interfaces with corners or edges are capable of scattering every incident wave non-trivially (that is, the scattered field cannot vanish identically). Moreover, the shape of a convex penetrable obstacle of polygonal or polyhedral type can be uniquely determined by a single far-field pattern. A sharp local uniqueness result is also derived in [6] to show that interfaces even with a weakly singular point always scatter; see Fig. 1.


Fig.1 The two penetrable scatterers D1 and D2 on the left hand side cannot generate the same far-field pattern (or scattered field) due to the presence of the singular point O lying on the boundary of D2. This phenomenon was rigorously explained in [4-6] by analytical approach and singularity analysis of the inhomogeneous Laplace equation in a cone.

These uniqueness results provide insights into whether the measurement data are sufficient to determine the shape of an unknown object, playing important roles in inversion algorithms (e.g., optimization-based iterative schemes). Further, the arguments of Elschner and Hu [4-6] exclude non-scattering energies in a large class of singularly perturbed domains and attract particular interests in inverse scattering communities. Ref. [4] was selected as one of the highlight articles of the journal Inverse Problems and Ref. [5] was published on the prestigious journal ARMA on applied mathematics in 2018. 


[1]     D. Colton and B. D. Sleeman, IMA J. Appl. Math., 31 (1983): 253--259.

[2]     J. Cheng and M. Yamamoto, Inverse Problems, 19 (2003): 1361-1384.

[3]     E. Blasten, L. Paivarinta and J. Sylvester, Commun. Math. Phys., 331 (2014): 725-753.

[4]     J. Elschner and G. Hu, Inverse Problems, 31 (2015): 015003.

[5]     J. Elschner and G. Hu, Arch. Rational Mech. Anal., 228 (2018): 653-690.

[6]     L. Li, G. Hu and J. Yang, Inverse Problems, 34 (2018): 075002. 

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