Flexible building primitives for 3D building modeling

Abstract

3D building models, being the main part of a digital city scene, are essential to all applications related to human activities in urban environments. The development of range sensors and Multi-View Stereo (MVS) technology facilitates our ability to automatically reconstruct level of details 2 (LoD2) models of buildings. However, because of the high complexity of building structures, no fully automatic system is currently available for producing building models. In order to simplify the problem, a lot of research focuses only on particular buildings shapes, and relatively simple ones. In this paper, we analyze the property of topology graphs of object surfaces, and find that roof topology graphs have three basic elements: loose nodes, loose edges, and minimum cycles. These elements have interesting physical meanings: a loose node is a building with one roof face; a loose edge is a ridge line between two roof faces whose end points are not defined by a third roof face; and a minimum cycle represents a roof corner of a building. Building primitives, which introduce building shape knowledge, are defined according to these three basic elements. Then all buildings can be represented by combining such building primitives. The building parts are searched according to the predefined building primitives, reconstructed independently, and grouped into a complete building model in a CSG-style. The shape knowledge is inferred via the building primitives and used as constraints to improve the building models, in which all roof parameters are simultaneously adjusted. Experiments show the flexibility of building primitives in both lidar point cloud and stereo point cloud.

Introduction

Buildings form the dominant type of man-made objects in urban scenes. The building models are necessary to visualize and analyze the urban environment (Brook et al., 2011). Possible applications include video games, augmented reality, 3D navigation, solar potential analysis, micro-scale environment modeling and 3D Geographic Information Systems. During past decades methodologies have been developed to automatically reconstruct building models from digital surface models or airborne laser scanning data (Haala and Kada, 2010, Vanegas et al., 2010, Wang, 2013). Although research has progressed significantly, the reconstruction problem still presents many challenges. An urban scene can be represented as a Digital Surface Model (DSM) or a TIN mesh which is derived from laser scanning systems (Frueh and Zakhor, 2003) or stereo imagery (Hirschmuller, 2008). Such data contain outliers and have spatially heterogeneous point distributions. Meanwhile, this representation demands a lot of storage space. Researchers propose mesh simplification procedures to reduce the amount of data while preserving sharp features and inhibiting noise (Fleishman et al., 2005, Zhou and Neumann, 2010). However, sharp features are detected by local points and are therefore inaccurate. Buildings are reconstructed without applying shape constraints and are therefore imprecise. Furthermore, semantic information is lost as a roof patch can be split into several triangles with different normal.

The above-mentioned methods are usually called data-driven; other methods are model-driven methods (Dorninger and Pfeifer, 2008, Huang et al., 2011, Karantzalos and Paragios, 2010, Oude Elberink and Vosselman, 2009, Verma et al., 2006, Zebedin et al., 2008). Model-driven methods reconstruct buildings with incomplete data and reconstruct all roofs at the same time. However, the usage of model-driven methods is limited because they require a library of prior defined models. Furthermore, many buildings still cannot be represented because of the high complexity of building structures. Verma et al. (2006) describe how a building can be represented as a group of assembled building primitives by analyzing the roof topology graphs. They define primitive shapes as I-shaped, L-shaped and U-shaped, and search for these using a roof topology graph. Primitive shapes are limited so only relative simple buildings can be described. Oude Elberink and Vosselman (2009) expand the topological primitive library to include gable, hip, gambrel, etc. The primitive shapes defined by Oude Elberink are finer in granularity than those defined by Verma and are, therefore, more flexible. However there are still many building shapes are not described in the library. A lot of other studies also use building primitives to integrate building structures (Hammoudi and Dornaika, 2010, Huang et al., 2011, Kada and McKinley, 2009, Lafarge et al., 2010).

This paper goes further and thoroughly divides buildings into basic elements of the topology graph: loose nodes, loose edges and minimum cycles. As any roof topology graph can be decomposed into such basic elements, buildings with a planar structure can be assembled according to the primitive shapes defined. Ameri and Fritsch (2000) also found that roof structures can be modeled in a generic manner based solely on the 3D intersection of adjacent roof segments. Perera et al. (2012) noted that graph cycles represent 3D roof corners. We precisely define the 3D corners as minimum cycles in the roof topology graph, and provide detailed proof of its formation mechanism. On the other hand, minimizing the size of building primitives decreases their robustness. In cases in which little information can be inferred from the data, it remains difficult to determine new primitives properly. However, nowadays increasing point densities allow the robust identification of smaller primitives.

Another weak point of model-driven methods is that these models demand strict regularity, which cannot always be satisfied by all buildings. The strict constraints prevent the reconstructed roof patch from exactly meeting the regularity requirements. Inspired by Brenner (2004), constraints used in this paper are much looser in comparison with typical model-driven methods. We propose a method for automatically checking data to determine whether constraints should be enforced. The constraints are subsequently used to simultaneously adjust all roof parameters. Several methods globally find surface constraints which are iteratively learned and enforced through constraint optimization (Li et al., 2011, Rottensteiner, 2006, Zhou and Neumann, 2012). Angle constraints and alignment constraints are enforced separately. As constraints are not independent, the iteration strategy will achieve one constraint while losing another, especially when considering concurrent relationships between several planes. We use a constraint least-squares fitting algorithm to simultaneously enforce all constraints while minimizing the residual of fitting points to planes.

Outer boundaries of building models are usually constructed based on boundary points. However, because points are discrete, the boundary directly derived from them zigzags. Many researchers try to improve the outer boundaries by introducing constraints, such as parallel and orthogonal ones, or building principal directions (Matei et al., 2008, Sampath and Shan, 2010, Vosselman et al., 2005, Zhou and Neumann, 2008). Typically, they obtain all outer boundaries for one building at the same time. Boundary points are detected by α-shape or convex hull algorithms. They subsequently detect and refine boundary lines of all roofs simultaneously. This solution leaves two unanswered questions. Firstly, many roofs are inside the building footprint polygon, so their boundaries cannot be detected. Secondly, not all roof polygons follow the principal direction. In this paper, we detect and refine outer boundaries facet by facet, so we detect all boundaries and achieve a greater precision.

We propose an algorithm that provides solutions that address the above-mentioned problems. The workflow of our method is shown in Fig. 1. Building points are detected and segmented, followed by intersection line detection. A roof topology graph is constructed by grouping roof faces (graph nodes) and intersection lines (graph edges). A building is decomposed into a group of building part, whose structures are predefined in the new library of building primitives. Structure knowledge is inferred and employed to determine constraints in order to simultaneously adjust all roofs. Inner boundaries and vertexes are found and refined according to the building primitives searched. Outer boundaries are hypothesized by inferring the available context boundaries. A complete building model is subsequently achieved by grouping all building parts.

The contributions of this paper can be summarized as follows:

• Flexible building primitives. The building primitives are defined according to basic elements in roof topology graphs. This means that all buildings with arbitrary complex structures can be assembled by using such building primitives. This definition facilitates the use of model-driven methods for all kinds of buildings.

• Adaptive constraints. The buildings are decomposed into building primitives, which introduced constraints for guiding reconstruction. We do not make all constraints obligatory. A decision is taken as to which constraints are necessary and which constraints could be left aside based on both inferred building primitives and input data.

• Minimum enclosing polygon. Outer boundaries are created by intersecting roof planes with the hypothesized vertical or horizontal planes. Roof polygons are subsequently closed by inferring all detected boundary lines.

The remainder of this paper is organized as follows: in Section 2 we discuss basic elements of the topology graph and their physical meaning in representing surface structures. Based on Section 2, in Section 3 we give a definition of a building primitive library and introduce an algorithm that searches for building primitives. The building primitives supply geometry regularities, which are applied to improve the quality of the model. In Section 4 we give a definition of geometry regularities, the inferring method, and the algorithm for applying constraints. In Section 5 we demonstrate how to obtain enclosed roof polygons by grouping the inner corners and inner boundaries, and the inferred outer corners and outer boundaries. As the whole algorithm is dependent upon the roof topology graph, any error in the roof topology graph would seriously affect the quality of the model. In Section 6 we give an automatic method and a manual method of correcting any errors in the roof topology graphs. In Section 7 we discuss the results of our method as applied to a dataset of Vaihingen (Germany) and Enschede (The Netherlands), including lidar point cloud and photogrammetric point cloud. The article ends in Section 8 with concluding remarks, a discussion of open questions and suggestions for further work.