  Compressed sensing (CS) is a valuable technique for reconstructing
measurements in numerous domains. CS has not yet gained widespread adoption in
scanning tunneling microscopy (STM), despite potentially offering the
advantages of lower acquisition time and enhanced tolerance to noise. Here we
applied a simple CS framework, using a weighted iterative thresholding
algorithm for CS reconstruction, to representative high-resolution STM images
of superconducting surfaces and adsorbed molecules. We calculated
reconstruction diagrams for a range of scanning patterns, sampling densities,
and noise intensities, evaluating reconstruction quality for the whole image
and chosen defects. Overall we find that typical STM images can be
satisfactorily reconstructed down to 30\% sampling - already a strong
improvement. We furthermore outline limitations of this method, such as
sampling pattern artifacts, which become particularly pronounced for images
with intrinsic long-range disorder, and propose ways to mitigate some of them.
Finally we investigate compressibility of STM images as a measure of intrinsic
noise in the image and a precursor to CS reconstruction, enabling a priori
estimation of the effectiveness of CS reconstruction with minimal computational
cost.
