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Super-resolution is the art of recovering spikes from their low-pass projections. Over the last decade specifically, several significant advancements linked with mathematical guarantees and recovery algorithms have been made. Most super-resolution algorithms rely on a two-step procedure: deconvolution followed by high-resolution frequency estimation. However, for this to work, exact bandwidth of low-pass filter must be known; an assumption that is central to the mathematical model of super-resolution.

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.