Abstract We address the problem of simplifying two‐dimensional polygonal partitions that exhibit strong regularities. Such partitions are relevant for reconstructing urban scenes in a concise way. Preserving long linear structures spanning several partition cells motivates a point‐line projective duality approach in which points represent line intersections, and lines possibly carry multiple points. We propose a simplification algorithm that seeks a balance between the fidelity to the input partition, the enforcement of canonical relationships between lines (orthogonality or parallelism) and a low complexity output. Our methodology alternates continuous optimization by Riemannian gradient descent with combinatorial reduction, resulting in a progressive simplification scheme. Our experiments show that preserving canonical relationships helps gracefully degrade partitions of urban scenes, and yields more concise and regularity‐preserving meshes than common mesh‐based simplification approaches.

This paper considers a particular case of the Optimal Homologous Chain Problem (OHCP) for integer modulo 2 coefficients, where optimality is meant as a minimal lexicographic order on chains induced by a total order on simplices. The matrix reduction algorithm used for persistent homology is used to derive polynomial algorithms solving this problem instance, whereas OHCP is NP-hard in the general case. The complexity is further improved to a quasilinear algorithm by leveraging a dual graph minimum cut formulation when the simplicial complex is a pseudomanifold. We then show how this particular instance of the problem is relevant, by providing an application in the context of point cloud triangulation.