MUSIC Algorithms for Grid Diagnostics

Abstract

The problem of detecting and localizing missing scatterers (faults) inside a known grid of small cross-sectional perfect-electric-conducting cylinders is dealt with. The case of a TM scalar 2-D geometry is considered, and the Multiple Signal Classification (MUSIC) spectral estimation technique is employed. Two different scattering models are employed and compared. The first one aims at localizing present objects within a free-space background medium. The second one aims at detecting the faults and exploits the Green's function of the full grid. The limitations of the first approach are pointed out and connected to the maximum dimension of the data space. Then, the second approach performs successfully when the fault number is lower than scattering objects.

Microwave Imaging by Means of Lebesgue-Space Inversion: An Overview

“…x n : Y → X acts rightly on the residual F (x n ) − y ∈ Y. The same holds true for (13), since F x n h n,k − (y − F (x n )) ∈ Y. Moreover, the subtraction in (10) is performed between the two operands x n and τ n F *…”

“…The same arguments of (18) and (19) apply to the inexact Newton approach too, leading to a new version of the inner iterations (13) in Banach spaces, involving the duality maps J Y r and J X * r * . Specifically, Step III of Algorithm 1 now reads: (III) INNER STEP: Find a (regularized) solution of the linear Equation (11) by means of an iterative minimization, with respect to h, of the n-th residual 1 2…”

“…Let us come back to the Landweber iterations (10) or (13), which are therein defined onto Hilbert spaces X and Y, so that F *…”

“…x n F x n h n,k − (y − F (x n )) belong to X. However, to extend the iterations (10) and (13) to Banach space setting, their forms need to be modified, since the terms F *…”

“…Consequently, proper inversion procedures need to be devised, to consider both these theoretical problems. To this end, several approaches have been proposed in the last years [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. They have been investigated in the context of Hilbert spaces, in most cases.…”