Atomic force microscopy (AFM) is one of the most advanced tools for high-resolution imaging and manipulation of nanoscale matter. Unfortunately, standard AFM imaging requires a timescale on the order of seconds to minutes to acquire an image which makes it complicated to observe dynamic processes. Moreover, it is often required to take several images before a relevant observation region is identified. In this paper we show how to significantly reduce the image acquisition time by undersampling. The reconstruction of an undersampled AFM image can be viewed as an inpainting, interpolating problem, or a special case of compressed sensing. We argue that the preferred approach depends upon the type of image. Of the methods proposed for AFM, images containing high frequencies should be reconstructed using basis pursuit from data collected in a spiral pattern. Images without too much high frequency content should be reconstructed using interpolation.