Identification and Modelling of Translational and Axial Symmetries and their Hierarchical Structures in Building Footprints by Formal Grammars

Buildings and other man‐made objects, for many reasons such as economical or aesthetic, are often characterized by their symmetry. The latter predominates in the design of building footprints and building parts such as façades. Thus the identification and modeling of this valuable information facilitates the reconstruction of these buildings and their parts. This article presents a novel approach for the automatic identification and modelling of symmetries and their hierarchical structures in building footprints, providing an important prior for façade and roof reconstruction. The uncertainty of symmetries is explicitly addressed using supervised machine learning methods, in particular Support Vector Machines (SVMs). Unlike classical statistical methods, for SVMs assumptions on the a priori distribution of the data are not required. Both axial and translational symmetries are detected. The quality of the identified major and minor symmetry axes is assessed by a least squares based adjustment. Context‐free formal grammar rules are used to model the hierarchical and repetitive structure of the underlying footprints. We present an algorithm which derives grammar rules based on the previously acquired symmetry information and using lexical analysis describing regular patterns and palindrome‐like structures. This offers insights into the latent structures of building footprints and therefore describes the associated façade in a relational and compact way.