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Modern high-resolution microscopes, such as the scanning tunnelingmicroscope, are commonly used to study specimens that have dense and aperiodicspatial structure. Extracting meaningful information from images obtained fromsuch microscopes remains a formidable challenge. Fourier analysis is commonlyused to analyze the underlying structure of fundamental motifs present in animage. However, the Fourier transform fundamentally suffers from severe phasenoise when applied to aperiodic images. Here, we report the development of anew algorithm based on nonconvex optimization, applicable to any microscopymodality, that directly uncovers the fundamental motifs present in a real-spaceimage. Apart from being quantitatively superior to traditional Fourieranalysis, we show that this novel algorithm also uncovers phase sensitiveinformation about the underlying motif structure. We demonstrate its usefulnessby studying scanning tunneling microscopy images of a Co-doped iron arsenidesuperconductor and prove that the application of the algorithm allows for thecomplete recovery of quasiparticle interference in this material. Our phasesensitive quasiparticle interference imaging results indicate that the pairingsymmetry in optimally doped NaFeAs is consistent with a sign-changing s+- orderparameter.

Dictionary Learning in Fourier Transform Scanning Tunneling Spectroscopy