  Crystallographic image processing (CIP) techniques may be utilized in
scanning probe microscopy (SPM) to glean information that has been obscured by
signals from multiple probe tips. This may be of particular importance for
scanning tunneling microscopy (STM) and requires images from a sample that is
periodic in two dimensions. The image-forming current for multiple tips in STM
is derived in a more straightforward manner than prior approaches. The Fourier
spectrum of the current for p4mm Bloch surface wave functions and a pair of
delta function tips reveals the tip-separation dependence of various types of
image obscurations. In particular our analyses predict that quantum
interference should be visible on a macroscopic scale in the form of bands
quite distinct from the basket-weave patterns a purely classical model would
create at the same periodic double STM tip separations. A surface wave function
that models the essential character of highly (0001) oriented pyrolytic
graphite (technically known as HOPG) is introduced and used for a similar
tip-separation analysis. Using a bonding H_2 tip wave function with significant
spatial extent instead of this pair of infinitesimal Dirac delta function tips
does not affect these outcomes in any observable way. This is explained by
Pierre Curie's well known symmetry principle. Classical simulations of multiple
tip effects in STM images may be understood as modeling multiple tip effects in
images that were recorded with other types of SPMs). Our analysis clarifies why
CIP and crystallographic averaging work well in removing the effects of a blunt
SPM tip (that consist of multiple mini-tips) from the recorded 2D periodic
images and also outlines the limitations of this image processing techniques
for certain spatial separations of STM mini-tips.
