Mesh-connected computers are almost exclusively used for low-level, local image processing, yet are also suitable for higher level tasks. The authors illustrate this by giving optimal (in the O-notational sense) algorithms for computing several geometric properties of figures. For example, given a black/white picture stored one pixel per processing element of an nxn mesh-connected computer, the authors give theta (n) time algorithms for determining the extreme points of the convex hull of each component, for determining the distance to the nearest neighboring component of each component for determining internal distances in each component, and for computing the diameter of each component. The present results show that the mesh-connected computer provides significantly better solutions.