Aspects of topology and geometry are used in analyzing continuous and discrete binary images in two dimensions. Several numerical properties of these images are derived which are " locally countable." These include the metric properties area and perimeter, and the topological invariant, Euler number. "Differentials" are defined for these properties, and algorithms are given. The Euler differential enables precise examination of connectivity relations on the square and hexagonal lattices. Easily computable binary image characterizations are introduced, with reference to a serial binary image processor (BIP) now being built. A precise definition of "localness" is given, and some implications for image computation theory are examined.