# 4D Building Reconstruction with Machine Learning and Historical Maps
## 1 Introduction  
Historical maps are the most powerful source of information for understanding urban phenomena and changes that contributed to defining our cities' actual shape. The growth and transformation of the urban patterns and landscapes can be analysed through these differently accurate, symbolised, and generalised representations of reality. Historical maps represent a graphically coded reduction of the three-dimensional (3D) world in the 2D space, which summarises the urban environment's main features. The cities' growth and changes--conditioned by preferred directions of expansion, natural constraints, and particular historical events--are impressed in these documents with several informative levels. Nevertheless, the 2D space-reduction of the maps entails an unavoidable loss of information, and particularly on the height of the built and natural environment. With the advent of digital technologies, a more realistic and complete representation of the world has become possible in its three and four dimensions (3D/4D). Several geomatic and modelling techniques have been developed in the last years to generate 3D/4D city models, derived with different levels of automation and input data [1; 2; 3; 4]. 3D/4D models can be digital copies of our cities when enriched with textural information or semantically enhanced when the geometry is linked to other attributes. Depending on their nature, they can be used for visualisation, simulations, geospatial analyses, planning activities, and many other applications [5]. The undisputable advantage offered by the three dimensions is the broader comprehension of the built spaces and relations in the urban pattern, as well as their interaction with the natural elements. Modelling in 3D multi-temporal versions of the same city (4D) can broaden how these relations and interactions are changed over time. Multi-temporal analyses and modelling of urban environments is typically performed usingimage or Light Detection and Ranging (LiDAR) data, deriving building height information from these data [6]. On the other hand, when only building footprints digitised from historical maps are available, the main issue is to derive the missing height information. Aims and Innovative AspectsThis work aims to explore existing machine learning solutions for inferring building heights from historical maps. Multi-temporal versions (4D) of the same city are generated with predicted height values, expanding the space information essential for more comprehensive geospatial analyses and 3D modelling applications. This work is part of the TOTEM project (4D Trento Time Machine) ([[https://totem.fbk.eu/](https://totem.fbk.eu/)]([https://totem.fbk.eu/](https://totem.fbk.eu/))), focused on the development of ICT (Information and Communications Technology) and AI (Artificial Intelligence) solutions for the valorisation of historical data (maps and photos) preserved in the archives of Trento (Italy). The developed methodology was firstly tested on four different historical maps of Trento (1851, 1887, 1908, and 1936). These maps (Section 3.1) depict many changes of the urban structure in the last 150 years, which involved both the historical city centre and the area outside the medieval defensive walls of Trento. The historical maps were georeferenced and building block shapes (\"footprints\"), including their partitioning, manually digitised in a GIS environment. A set of diverse polygons was thus generated and several attributes describing geometrical and neighbourhood features were computed for each digitised polygon (Section 4.1). Finally, machine learning algorithms were tested for predicting the block heights, using actual height values (derived from modern topographic data) as training data. The method has been verified (Section 4.4) adopting common machine learning quality metrics, examining in-depth the prediction performance on buildings still existing and with the same shape, or not existing anymore but documented in the historical photo.