A Methodological Proposal for the Comparison of 3D Photogrammetric Models

The field of simplification of geometric surfaces still lacks a formal and universally recognized definition of the error, which should involve both the approximation of the shape and the conservation of the other attributes of the mesh (starting from the colour). In order to solve this problem, we propose a hypothesis of methodological comparison that allow the evaluation of differences between two homologous surfaces, quantified employing the Hausdorff distance. The main advantage of this method is the independence from sampling techniques used to produce the mesh, without losing its characteristics of objectivity and generality. The Hausdorff distance geometrically represents the distance between two sets A and B in a suitable metric space, and it is defined as the maximum between the excess of A over B and the excess of B over A. This value is then compared with the average length of the diagonals of the “bounding boxes” of the homologous models, i.e. the parallelepipeds corresponding to the minimum volume that completely envelops each set; this results in an effective representation of error in relative terms.