A Borsuk-Ulam theorem for digital images

Abstract

The Borsuk-Ulam theorem states that a continuous function $f:S^n \to \R^n$ has a point $x\in S^n$ with $f(x)=f(-x)$. We give an analogue of this theorem for digital images, which are modeled as discrete spaces of adjacent pixels equipped with $\Z^n$-valued functions. In particular, for a concrete two-dimensional rectangular digital image whose pixels all have an assigned "brightness" function, we prove that there must exist a pair of opposite boundary points whose brightnesses are approximately equal. This theorem applies generally to any integer-valued function on an abstract simple graph. We also discuss generalizations to digital images of dimension 3 and higher. We give some partial results for higher dimensional images, and show a counter example which demonstrates that the full results obtained in lower dimensions cannot hold generally.