Optimal scan planning with enforced network connectivity for the acquisition of three-dimensional indoor models

https://doi.org/10.1016/j.isprsjprs.2021.07.013Get rights and content

Abstract

The positioning of laser scanners for indoor surveying is still a time and cost expensive process. This article proposes an optimization approach for computing an admissible sensor placement with the minimal number of sensor view point positions. The approach facilitates both wall and floor surveying based on a floorplan of the study object. Optimal solutions are calculated by solving an Integer Linear Program that respects manufacturer specifications incorporating constraints such as full coverage. To enable a subsequent co-registration of the scans, a flow-based constraint formulation ensuring the connectivity of the selected positions in an appropriately defined geometric intersection graph is introduced. The method has been evaluated on real-world objects and compared to heuristic methods that have frequently been used for related problems. Our solutions outperform heuristic approaches regarding both running time and the number of TLS stations. In a case study with a larger floorplan of an institute building and with different parameter settings, our method resulted in a solution with at least two stations less compared to a solution generated by an expert.

Introduction

Sensor technology has been rapidly developed over the last decade which opened up many opportunities for applications such as the fast and complete 3D-mapping and wide range surveying of complex sites. Terrestrial Laser Scanners (TLS) in particular are used in many disciplines such as geodesy for the change detection and deformation monitoring of man-made structures (Mukupa et al., 2017, Nothnagel et al., 2019), Architectural Engineering and Construction (AEC) for construction site monitoring (Zhang and Arditi, 2020, Wang and Kim, 2019, Riveiro and Lindenbergh, 2019) or for the documentation of the as-built state of existing buildings based on scan-to-BIM techniques (Brumana et al., 2020). Despite the wide range of applications and usefulness of TLS, the process of data acquisition and the related tasks pipeline is, however, not yet optimal and it is still both time and cost expensive. In this context, the optimal positioning of laser scan instruments, particularly in complex sites, is of great importance to find a satisfying trade-off between economic efficiency and reliable results. To this end, the laser scanner configurations have to fulfill a range of requirements ensuring an efficient laser scanning process that leads to point clouds of high quality. In particular, a full coverage of the scene of interest is mandatory. In this context, several laser scanning positions are often needed to satisfy the coverage of a whole scene. Consequently, the overlap between the laser scans which stem from different positions should be sufficient for subsequent registration tasks. Besides, scan instrument related restrictions such as range limits should be respected. The problem which deals with the optimal satisfaction of this range of constraints is called the View Planning Problem (VPP). In the more general context of determining an optimum design of a geodetic network, the problem is also known as first-order design problem. This is up to now still an open subject of research (Wujanz et al., 2016). In practice, however, the positioning of TLS instruments is mostly performed ad-hoc based on the operator’s experience.

In this article, we propose a novel approach for the optimal planning of laser scan positions for the acquisition of three-dimensional indoor models based on a building floorplan without requiring a BIM model in advance. In this context, the problem to find the minimal number of station point positions for capturing a whole 3D-indoor environment can be safely reduced to searching such positions in a 2D-space. Herewith, an input polygon is required which corresponds to a horizontal projection on the 3D-scene at the level of the laser scanner centroid. The resulting station point positions, however, correspond to 3D-points relative to the laser scan horizon line. Based on a 2D-map of the indoor object, our method guarantees an optimal placement of the sensors satisfying a full coverage of both floor and wall surfaces. The method is guarantying a full coverage as long as the proposed TLS positions are free to use and all occluding objects are known and represented in the floorplan. Obviously, due to occlusions caused by furniture and similar objects, the full coverage is not possible in such situations. The completeness of the captured scene is assured respecting all mentioned constraints, i.e., concerning the range and incidence angle. In this context, we formulated the VPP as an Integer Linear Program (ILP) which aims to minimize the number of TLS stations satisfying these constraints. Compared to most related approaches where the overlap between the view points is fully disregarded or not considered assuming the availability of target points for a subsequent registration process, we address this key issue in an explicit way. To this end, we present a flow-based formulation of the problem using Mixed Integer Linear Programming (MILP) ensuring not only a sufficient bilateral overlap between neighbored stations but also the global connectivity of the station network. Fig. 1 illustrates an optimal station network as result of our method for surveying the whole wall surfaces of the cathedral of Cologne in Germany. The connectivity graph reflects the connectivity property between the determined stations (red stars). Beyond simple polygons, our method can deal with polygons with holes. This allows obstacles to be taken into account such as pillars as can be seen in the same figure. According to the CityGML specification (Gröger et al., 2012), our approach paves the way for a transition from available LOD-0 models, i.e. footprint of a building, towards needed LOD-4 models which include 3D-interior building information. Our method can be further applied for the change detection and deformation monitoring of a-priori surveyed building interiors. In this context, our method can be applied to perform multiple and repeated measurements for a subsequent quality and change detection analysis. In the context of the saving of cultural heritage, the usage of TLS is more accurate in order to satisfy this task where a high data quality is a prerequisite. Since a mobile mapping system is in general more expensive than a TLS system, the economical factor is also playing an important role for the choice of TLS. Nevertheless, in mobile laser scanning (MLS) applications it is favorable to acquire all scans from a short trajectory. This could be achieved by combining our ILP formulation with an ILP formulation for the traveling salesperson problem (Miller et al., 1960) taking MLS-specific aspects into account by incorporating additional constraints adequately. In this context, Kuželka and Surovỳ (2021) demonstrated recently the suitability of mathematical optimization for the planning of trajectories for 3D scene reconstruction.

VPP is strongly related to the Art Gallery Problem (AGP) where the task is to place a minimum number of guards sufficient to cover a gallery. The latter is usually represented as a polygon P. Even for a simply connected polygonal region P, the AGP problem is known to be an NP-hard problem (Lee and Lin, 1986). Such problems are extraordinarily complex in the sense that efficient exact algorithms are unlikely to exist. This is usually considered a good justification for choosing a heuristic or, alternatively, an exact approach via (Mixed) Integer Linear Programming. Although the existing MILP solvers have an exponential worst-case running time, MILP-based approaches often show a good performance for problem instances occurring in practice. In many situations one can consider the solver as a black box that, after being provided with a list of variables, an objective function, and a set of constraints as input, returns an optimal variable assignment as output. Especially when dealing with large sets of constraints, however, one needs to intervene in the optimization process. This option is provided with callback utilities of state-of-the-art MILP solvers, which we employ in this article to initialize the model with a small set of constraints and to add so-called lazy constraints on demand, only if they turn out to be violated during the optimization.

To summarize, the main contribution of this article is the design and implementation of a Mixed Integer Linear Program for the optimal planning of a geodetic network which consists of several laser scanner positions. Beyond enforcing the constraints related to sensor properties and scanning geometry, our method is taking particular care of ensuring further connectivity constraints for the registration of the separately captured point clouds subsequently leading to a complete three-dimensional indoor model of the scene of interest. To demonstrate the feasibility of our approach, we applied it on several real-world instances and compared it with examples stemming from existing approaches for a further evaluation.

The remainder of this article is structured as follows. Section 2 gives a review of the most relevant related research. Section 3 introduces the addressed problem and our formulation accordingly where its Sections 3.1 Floor coverage problem, 3.2 Wall coverage problem, 3.3 Network connectivity based on a flow model address the floor coverage, wall coverage, and network connectivity, respectively. After demonstrating our conducted experiments and discussing their outcomes in Section 4, we summarize and conclude the article in Section 5.

Section snippets

Art gallery problem.

As mentioned, the problem at hand is closely related to the Art Gallery Problem, a well-studied problem in computer science and computational geometry in particular. The problem to solve consists in the minimization of the number of watchmen or guards for the monitoring of an art gallery bounded by a simple polygon P of n vertices. Herewith, a guard is a point in the polygon. Depending on the type of the guards, different versions of AGP can be encountered in the literature. If guards coincide

Laser scanning network design

As mentioned, we aim at minimizing the size of a set S of watchmen or station points sufficient to cover an indoor environment represented by a polygon P of n vertices. Fig. 2(a) depicts an example of such a polygon. Generally, we allow P to contain holes. Our target is to guarantee a full coverage of the walls and the ground surfaces of P separately or both of them simultaneously. In any case, we assume that the points that need to be covered are given as a continuous set RP. In this context,

Experimental results

This section discusses the experiments that we performed. We compare the results achieved with our approach with those of existing heuristic methods of Soudarissanane and Lindenbergh (2011) and Jia and Lichti (2019). To this aim, we take the polygons which have been used in those works as illustrative examples. We implemented our method in Java and performed the experiments on an Intel Xeon W-2125. The machine is clocked at 4.0GHz and has 128 GB RAM. We solved the ILP and MILP problems using

Conclusion

This article presented an automatic approach for the optimal scan planning of laser scanner stations for the acquisition of three-dimensional indoor models. From an outline polygon of the scene to scan, a minimum and sufficient number of scanner positions is determined with regard to a set of constraints including manufacturer range and recommended incidence angle values. This has been formulated and addressed as an Integer Linear Program (ILP). Compared to related methods, our approach

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors gratefully thank Mr. Martin Blome for providing the expert solution.

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