This paper is an essay in axiomatic foundations for discrete geometry intended, in principle, to be suitable for digital image processing and (more speculatively) for spatial reasoning and description as in AI and GIS. Only the geometry of convexity and linearity is treated here. A digital image is considered as a finite collection of regions, regions are primitive entities (they are not sets of points). The main result (Theorem 20) shows that finite spaces are sufficient. The theory draws on both region-based topology (also known as mereotopology) and abstract convexity theory.