  In applications of scanning probe microscopy, images are acquired by raster
scanning a point probe across a sample. Viewed from the perspective of
compressed sensing (CS), this pointwise sampling scheme is inefficient,
especially when the target image is structured. While replacing point
measurements with delocalized, incoherent measurements has the potential to
yield order-of-magnitude improvements in scan time, implementing the
delocalized measurements of CS theory is challenging. In this paper we study a
partially delocalized probe construction, in which the point probe is replaced
with a continuous line, creating a sensor which essentially acquires line
integrals of the target image. We show through simulations, rudimentary
theoretical analysis, and experiments, that these line measurements can image
sparse samples far more efficiently than traditional point measurements,
provided the local features in the sample are enough separated. Despite this
promise, practical reconstruction from line measurements poses additional
difficulties: the measurements are partially coherent, and real measurements
exhibit nonidealities. We show how to overcome these limitations using natural
strategies (reweighting to cope with coherence, blind calibration for
nonidealities), culminating in an end-to-end demonstration.
