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We develop a theory for the analysis of binary-valued digital pictures, particularly blobs and arcs, on a fairly large class of nonuniform as well as uniform mosaics. These so-called "acute mosaics" can represent a wide variety of schemes for digitizing images in machine vision systems. These mosaics can also represent a variety of retinas in human and animal visual systems. We prove the existence of minimum-perimeter-polygon (MPP) representations of digitized blobs on arbitrary mosaics and the uniqueness of MPP's in acute mosaics. We introduce the concept of "relative convexity," and show its relation to the MPP. We show that the MPP reflects the concavities of the digitized blob. We describe and prove the validity of simple tests for cellular convexity and cellular straightness. We present and prove the validity of an algorithm for computing the MPP of so-called "normal complexes" (a broad class of digitized blobs) on an acute mosaic.