Benchmarking compressed sensing, super-resolution, and filter diagonalization

Abstract

Signal processing techniques have been developed that use different strategies to bypass the Nyquist sampling theorem in order to recover more information than a traditional discrete Fourier transform. Here we examine three such methods: filter diagonalization, compressed sensing, and super-resolution. We apply them to a broad range of signal forms commonly found in science and engineering in order to discover when and how each method can be used most profitably. We find that filter diagonalization provides the best results for Lorentzian signals, while compressed sensing and super-resolution perform better for arbitrary signals.

Comment on “Benchmarking compressed sensing, super‐resolution, and filter diagonalization” (Int. J. Quantum Chem. 2016, 116, 1097)

“…The recent publication by Markovich et al, [ ] which from now on we refer to as the Paper , reports a study comparing three methods, super‐resolution (SR), compressed sensing (CS), and FDM, and based on five numerical examples, concludes that both SR and CS outperform FDM. In what follows we explain why this comparison is done incorrectly and, in addition, identify a number of technical flaws that we found in the Paper .…”

“…4, the line list determined from solving HIP (1) can be used to estimate IðxÞ directly. Consequently, the FDM spectral estimate may result in high (or even infinite) resolution [depending on how well condition (3) is satisfied], while the resolution of the finite FT spectrum is still limited by the uncertainty principle (6).…”

“…erence 5, in which FDM performance was critically assessed by applying it to noisy data, both synthesized and experimental.The recent publication by Markovich et al,[6] which from now on we refer to as the Paper, reports a study comparing three methods,…”

Response to: “Comment on benchmarking compressed sensing, super‐resolution, and filter diagonalization”

“…During the course of peer review, we were asked to compare FDM to our methods. As we performed a more thorough comparison, and wanted to explore the data‐fitting features and observed instability further, we ultimately performed the published comparisons …”

“…As we performed a more thorough comparison, and wanted to explore the datafitting features and observed instability further, we ultimately performed the published comparisons. [4] While we did evaluate the various methods on their performance for data fitting, we have observed that the reported L 2 errors provide a lower bound for the parameter errors. This can be seen by first starting with the following definition of a Lorentzian:…”