A Measure-Theoretic Sheaf Framework for Shape Analysis

We develop a sheaf-theoretic framework for shape analysis of covered shapes, where measure is introduced only after the underlying local-to-global topological structure has been established. Starting from a finite cubical complex together with a finite admissible cover, we build a finite topological model from the overlap structure of the cover and define a component sheaf on this model. The local data of the sheaf records the connected pieces visible on individual patches, while the maps between them describe how these pieces fit together across overlaps. The resulting degree-zero sheaf cohomology recovers the connected components of the full shape by the standard descent of locally constant functions. We show that the isomorphism class of this sheaf is an invariant of covered shapes, that it strictly refines β0 for a fixed labeled cover and can distinguish some shapes with identical full Betti vectors, although it does not determine higher Betti numbers in general, and that it behaves functorially under symmetries preserving the cover. For dyadic covers, we distinguish the overlapping closed cover used by the sheaf from a paired half-open measurable partition, investigate measurable refinements, introduce monotone notions of local complexity under corrected refinement hypotheses, construct canonical sheaf-induced measures on both the index set and the ambient domain, and establish convergence and localization results for regular closed sets under pixel refinement. The sheaf-theoretic axiomatization therefore offers two complementary benefits: topologically, it provides a finite-space and cohomological invariant of covered shapes; measure-theoretically, it gives a principled hierarchy of quantitative summaries built only after the local gluing structure has been preserved.