  A geometric form of information theory allows for reasonable, i.e.
probabilistic, evidence-ranking based, and generalized noise-level dependent,
classifications of the crystallographic and quasicrystallographic symmetries in
noisy digital images. Such classifications are based solely on the image pixel
intensity values, justifiable assumptions about the aggregate distribution of
generalized noise in the images, asymptotic extrapolations to zero-noise
images, and rational symmetry model selections with maximized predictive
accuracy in the presence of both symmetry-inclusion relations and
pseudo-symmetries. Preferring a well developed geometric form of information
theory over a theoretically possible geometric-Bayesian approach for these
classifications is the only subjective choice made. Using digital data planes
and assuming approximately Gaussian distributed generalized noise, reasonable
crystallographic and quasicrystallographic symmetry classifications can be made
for noisy images from both scanning probe and transmission electron
microscopes. A binary type classification of structurally very similar
mate-rials into either a quasicrystal or one of its rational/crystalline
approximants based on the approximate point symmetries in their noisy digital
images is proposed here for the first time.
