Super-resolution Results for a 1D Inverse Scattering Problem

Abstract

In this work we consider the one-dimensional (1D) inverse scattering problem of super-resolving the location of discrete point scatters satisfying the 1D Helmholtz equation. This inverse problem has important applications in the detection of shunt faults in electrical transmission lines and leaks in water pipelines where usually only low frequency spectral information is available from measurements. We formulate the inverse scattering problem as a sparse reconstruction problem and apply convex optimization to super-resolve the location of point scatters. We extend previous results and prove that we can super-resolve up to 4 points and 5-18 points with infinite precision if the points are separated by 1/(2f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ) and 1/f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> respectively (f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> is the maximum frequency we can measure). This is over 4 times closer than previous results. Simulation results are used to demonstrate the effectiveness of the approach.