A study on the topological derivative-based imaging of thin electromagnetic inhomogeneities in limited-aperture problems

Abstract

The topological derivative-based non-iterative imaging algorithm has demonstrated its applicability in limited-aperture inverse scattering problems. However, this has been confirmed through many experimental simulation results, and the reason behind this applicability has not been satisfactorily explained. In this paper, we identify the mathematical structure and certain properties of topological derivatives for the imaging of two-dimensional crack-like thin penetrable electromagnetic inhomogeneities that are completely embedded in a homogeneous material. To this end, we establish a relationship with an infinite series of Bessel functions of integer order of the first kind. Based on the derived structure, we discover a necessary condition for applying topological derivatives in limited-aperture inverse scattering problems, and thus confirm why topological derivatives can be applied. Furthermore, we analyze the structure of multi-frequency topological derivative, and identify why this improves the single-frequency topological derivative in limited-aperture inverse scattering problems. Various numerical simulations are conducted with noisy data, and the results support the derived structure and exhibit certain properties of single- and multi-frequency topological derivatives.

When topological derivatives met regularized Gauss-Newton iterations in holographic 3D imaging

“…2 n−1 (n−1)! , as well as the relations h (1) n = j n + ıy n , and z n (kε) = −z n+1 (kε) + n(kε) −1 z n (kε) for z n = j n , h (1) n , we see that the coefficients a n (ε) and b n (ε) given by (63) behave like ε 2n+1 . Therefore, the source term (k 2 e − k 2 i )W ε,sc (z) = O(ε 3 ) andẼ ε ∼ E ε at zero order in ε.…”

“…These spherical functions constitute an orthonormal basis in L 2 (S 2 ). Denoting by j n the spherical Bessel functions of the first kind and by h (1) n the spherical Hankel functions, the sets of functions M…”

“…with coefficients a n,m = α n,m a n , b n,m = β n,m b n , c n,m = α n,m c n , d n,m = β n,m c n , a n = − , c n = keR[(h (1) n ) (keR))jn(keR)−h (1) n (keR)j n (keR))] jn(kiR)[h (1) n (keR)+keR(h…”

“…We then impose the transmission conditions (57) on |x| = R computing the cross product with the normal vectors. Notice that on the surface of a sphere |x| = R, M (1) n,m (k, Rχ) = j n (kR)curl S 2 Y m n (χ), N (1) n,m (k, Rχ) = χ ıkR n(n+1)j n (kR)Y m n (χ)+ 1 ıkR [j n (kR)+kRj n (kR)]∇ S 2 Y m n (χ),…”

Linear sampling method for imaging an extended crack using limited-range far-field data

“…Moreover, iteration-based schemes need the calculation of Fréchet derivative, appropriate regularization terms, and a priori information about the unknown crack. To avoid these difficulties, alternative methods have been developed, for example, MUltiple SIgnal Classification [12], [13], [14], [15], [16], [17], [18], [19], [9], topological derivatives [20], [21], [22], [23], [24], [25], Kirchhoff and subspace migration [26], [27], [28], [29], [30], [31], [32], [33], [34], and linear sampling methods [35], [36], [37], [38], [39], [40]. Among them, the linear sampling methods have been successfully applied for reconstructing shapes of various inhomogeneities.…”