  Persistent homology, an algebraic method for discerning structure in abstract
data, relies on the construction of a sequence of nested topological spaces
known as a filtration. Two-parameter persistent homology allows the analysis of
data simultaneously filtered by two parameters, but requires a bifiltration --
a sequence of topological spaces simultaneously indexed by two parameters. To
apply two-parameter persistence to digital images, we first must consider
bifiltrations constructed from digital images, which have scarcely been
studied. We introduce the value-offset bifiltration for grayscale digital image
data. We present efficient algorithms for computing this bifiltration with
respect to the taxicab distance and for approximating it with respect to the
Euclidean distance. We analyze the runtime complexity of our algorithms,
demonstrate the results on sample images, and contrast the bifiltrations
obtained from real images with those obtained from random noise.
