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We consider the problem of compressed sensing and propose new deterministic low-storage constructions of compressive sampling matrices based on classical finite-geometry generalized polygons. For the noiseless measurements case, we develop a novel exact-recovery algorithm for strictly sparse signals that utilizes the geometry properties of generalized polygons and exhibits complexity that depends on the sparsity value only. In the presence of measurement noise, recovery of the generalized-polygon sampled signals can be carried out effectively using a belief propagation algorithm. Experimental studies included in this paper illustrate our theoretical developments.