  Quasiparticle interference imaging (QPI) offers insight into the band
structure of quantum materials from the Fourier transform of local density of
states (LDOS) maps. Their acquisition with a scanning tunneling microscope is
traditionally tedious due to the large number of required measurements that may
take several days to complete. The recent demonstration of sparse sampling for
QPI imaging showed how the effective measurement time could be fundamentally
reduced by only sampling a small and random subset of the total LDOS. However,
the amount of required sub-sampling to faithfully recover the QPI image
remained a recurring question. Here we introduce an adaptive sparse sampling
(ASS) approach in which we gradually accumulate sparsely sampled LDOS
measurements until a desired quality level is achieved via compressive sensing
recovery. The iteratively measured random subset of the LDOS can be interleaved
with regular topographic images that are used for image registry and drift
correction. These reference topographies also allow to resume interrupted
measurements to further improve the QPI quality. Our ASS approach is a
convenient extension to quasiparticle interference imaging that should remove
further hesitation in the implementation of sparse sampling mapping schemes.
