Determining the shape of defects in non-absorbing inhomogeneous media from far-field measurements

Abstract

Abstract. We consider non-absorbing inhomogeneous media represented by some refraction index. We have developed a method to reconstruct, from far-field measurements, the shape of the areas where the actual index differs from a reference index. Following the principle of the Factorization Method, we present a fast reconstruction algorithm relying on far field measurements and near field values, easily computed from the reference index. Our reconstruction result is illustrated by several numerical test cases.

Nondestructive Testing of the Delaminated Interface between Two Materials

“…We start by proving a mixed reciprocity result in order to deal with the non-homogeneous background. This generalizes similar results in [23], [9] and [16] (see also [4] for a similar type of calculations).…”

“…Although in applications to nondestructive testing it is possible to have measurements all around, we remark that the inversion algorithm that we shall develop next can also be justified and implemented for limited aperture data (see Section 4.5 in [13]) as well as for near field data. However, the quality of the reconstruction is likely to be poor for small apertures which is usually the case for qualitative methods [23]. We also remark that for many problems in nondestructive testing, it is reasonable to assume that the background medium is known as we do here, since the background corresponds to the healthy object to be tested.…”

Adaptive refinement and selection process through defect localization for reconstructing an inhomogeneous refraction index

“…The main benefit is the possibility of considering Γ m = Γ e = S d−1 and complex valued indices. Furthermore, numerical examples in [12] show that this localization is effective for defects bigger than (approximately) one over six of the wavelength. Besides, in order to get satisfactory results in the successive resolutions of the Helmholtz equation, we have set the reconstruction mesh size to be about one over twenty of the wavelength.…”

An optimization approach for the localization of defects in an inhomogeneous medium from acoustic far-field measurements at a fixed frequency

“…Furthermore, it is easy to see that C Ψ satisfies the Helmholtz equation (1) on R d with n = n 0 . The unique continuation principle then yields C Ψ = 0, and thus Ψ = 0 by injectivity of C , which has been shown in the proof of [8,Proposition 5.4]. As a consequence, Ψ can not satisfy Ψ , u n0 (·, z) = 1.…”

“…This is the principle of the linear sampling method [5,7,3] or of the factorization method [9,12,1]. Yet, when looking for perturbations in nonhomogeneous background media, it is only recently that a factorization method has been proposed to reconstruct the shape of defects [14,8].…”