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We introduce a novel approach for the polygonization of Multi-view Stereo (MVS) meshes of buildings, which results in compact and topologically valid models. The main characteristic of our method is structure awareness, i.e., the recovery and preservation of the initial mesh primitives and their adjacencies. Our proposed methodology consists of three main stages: (a) primitive detection via mesh segmentation, (b) encoding of primitive adjacencies into a graph, and (c) polygonization. Polygonization is based on the approximation of the original mesh with a candidate set of planar polygonal faces. On this candidate set, we apply a binary labelling formulation to select and assemble an optimal set of faces under hard constraints that ensure that the final model is both manifold and watertight. Experiments on various building models demonstrate that our simplification method can produce simpler representations for both closed and open building meshes. Furthermore, these representations highly conform to the initial structure and are ready to be used for spatial analysis. The source code of this work is freely available at https://github.com/VasileiosBouzas/MeshPolygonization.

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In 1999, Dar conjectured if there is a stronger version of the celebrated Brunn-Minkowski inequality. However, as pointed out by Campi, Gardner, and Gronchi in 2011, this problem seems to be open even for planar o-symmetric convex bodies. In this paper, we give a positive answer to Dar’s conjecture for all planar convex bodies. We also give the equality condition of this stronger inequality. For planar o-symmetric convex bodies, the log-Brunn-Minkowski inequality was established by B¨or¨oczky, Lutwak, Yang, and Zhang in 2012. It is stronger than the classical Brunn-Minkowski inequality, for planar o-symmetric convex bodies. Gaoyong Zhang asked if there is a general version of this inequality. Fortunately, the solution of Dar’s conjecture, especially, the definition of “dilation position”, inspires us to obtain a general version of the log-Brunn-Minkowski inequality. As expected, this inequality implies the classical Brunn-Minkowski inequality for all planar convex bodies.