Imaging Schemes for Perfectly Conducting Cracks

Abstract

Abstract. We consider the problem of locating perfectly conducting cracks and estimating their geometric features from multi-static response matrix measurements at a single or multiple frequencies. A main objective is to design specific crack detection rules and to analyze their receiver operating characteristics and the associated signal-to-noise ratios. In this paper we introduce an analytic framework that uses asymptotic expansions which are uniform with respect to the wavelength-to-crack size ratio in combination with a hypothesis test based formulation to construct specific procedures for detection of perfectly conducting cracks. A central ingredient in our approach is the use of random matrix theory to characterize the signal space associated with the multi-static response matrix measurements. We present numerical experiments to illustrate some of our main findings.

Non-Iterative Imaging of Thin Electromagnetic Inclusions From Multi-Frequency Response Matrix

“…. , M [3,17]. Since the first M columns of the matrix U(ω) and V(ω), {u 1 Thus, we can construct an image function at given frequency ω as:…”

“…, M , and of small magnitude at x ∈ R 2 \Γ. Unfortunately, image function (8) at single frequency offers an image with poor resolution [3,23] (also see Figure 4). In order to improve the imaging performance, we suggest a normalized image function at several frequencies {ω f : f = 1, 2, .…”

“…However, (9) does not contain the singular values, i.e., it will allow a more accurate result in the presence of random noise. Finally, based on the statistical hypothesis testing [3], multiple frequencies should guarantee the imaging performance via higher signal-to-noise ratio (SNR). For reader's convenience, we briefly state it in the appendix.…”

“…We suggest [3,16,19] for a more detailed description. Let us assume that measured MSR matrix in the presence of single thin inclusion Γ is polluted by Additive White Gaussian Noise (AWGN):…”

“…However, in the imaging of thin inclusion of both dielectric and magnetic contrast with respect to the embedding homogeneous space R 2 case, one obtains a poor result so that an improvement is required. Recently, an effective and robust multifrequency algorithm has been proposed so as to find the location of small perfectly conducting cracks [3] and dielectric thin inclusions within a homogeneous half-space [23], but a suitable algorithm for extended inclusions of both dielectric and magnetic contrast is still expected.…”