  To compute the persistent homology of a grayscale digital image one needs to
build a simplicial or cubical complex from it. For cubical complexes, the two
commonly used constructions (corresponding to direct and indirect digital
adjacencies) can give different results for the same image. The two
constructions are almost dual to each other, and we use this relationship to
extend and modify the cubical complexes to become dual filtered cell complexes.
We derive a general relationship between the persistent homology of two dual
filtered cell complexes, and also establish how various modifications to a
filtered complex change the persistence diagram. Applying these results to
images, we derive a method to transform the persistence diagram computed using
one type of cubical complex into a persistence diagram for the other
construction. This means software for computing persistent homology from images
can now be easily adapted to produce results for either of the two cubical
complex constructions without additional low-level code implementation.
