A Voronoi poset

Given a set S of n points in general position, we consider all k-th order Voronoi diagrams on S, for k=1,...,n, simultaneously. We deduce symmetry relations for the number of faces, number of vertices and number of circles of certain orders. These symmetry relations are independent of the position of the sites in S. As a consequence we show that the reduced Euler characteristic of the poset of faces equals zero whenever n odd.Comment: 14 pages 4 figure

Limits of Voronoi Diagrams

In this thesis we study sets of points in the plane and their Voronoi diagrams, in particular when the points coincide. We bring together two ways of studying point sets that have received a lot of attention in recent years: Voronoi diagrams and compactifications of configuration spaces. We study moving and colliding points and this enables us to introduce `limit Voronoi diagrams'. We define several compactifications by considering geometric properties of pairs and triples of points. In this way we are able to define a smooth, real version of the Fulton-MacPherson compactification. We show how to define Voronoi diagrams on elements of these compactifications and describe the connection with the limit Voronoi diagrams.Comment: PhD thesis, 132 pages, lots of figure

Automatic tree parameter extraction by a Mobile LiDAR System in an urban context

e0196004In an urban context, tree data are used in city planning, in locating hazardous trees and in environmental monitoring. This study focuses on developing an innovative methodology to automatically estimate the most relevant individual structural parameters of urban trees sampled by a Mobile LiDAR System at city level. These parameters include the Diameter at Breast Height (DBH), which was estimated by circle fitting of the points belonging to different height bins using RANSAC. In the case of non-circular trees, DBH is calculated by the maximum distance between extreme points. Tree sizes were extracted through a connectivity analysis. Crown Base Height, defined as the length until the bottom of the live crown, was calculated by voxelization techniques. For estimating Canopy Volume, procedures of mesh generation and α-shape methods were implemented. Also, tree location coordinates were obtained by means of Principal Component Analysis. The workflow has been validated on 29 trees of different species sampling a stretch of road 750 m long in Delft (The Netherlands) and tested on a larger dataset containing 58 individual trees. The validation was done against field measurements. DBH parameter had a correlation R2 value of 0.92 for the height bin of 20 cm which provided the best results. Moreover, the influence of the number of points used for DBH estimation, considering different height bins, was investigated. The assessment of the other inventory parameters yield correlation coefficients higher than 0.91. The quality of the results confirms the feasibility of the proposed methodology, providing scalability to a comprehensive analysis of urban treesS

Robust Approach for Urban Road Surface Extraction Using Mobile Laser Scanning Data

Road surface extraction is crucial for 3D city analysis. Mobile laser scanning (MLS) is the most appropriate data acquisition system for the road environment because of its efficient vehicle-based on-road scanning opportunity. Many methods are available for road pavement, curb and roadside way extraction. Most of them use classical approaches that do not mitigate problems caused by the presence of noise and outliers. In practice, however, laser scanning point clouds are not free from noise and outliers, and it is apparent that the presence of a very small portion of outliers and noise can produce unreliable and non-robust results. A road surface usually consists of three key parts: road pavement, curb and roadside way. This paper investigates the problem of road surface extraction in the presence of noise and outliers, and proposes a robust algorithm for road pavement, curb, road divider/islands, and roadside way extraction using MLS point clouds. The proposed algorithm employs robust statistical approaches to remove the consequences of the presence of noise and outliers. It consists of five sequential steps for road ground and non-ground surface separation, and road related components determination. Demonstration on two different MLS data sets shows that the new algorithm is efficient for road surface extraction and for classifying road pavement, curb, road divider/island and roadside way. The success can be rated in one experiment in this paper, where we extract curb points; the results achieve 97.28%, 100% and 0.986 of precision, recall and Matthews correlation coefficient, respectively

Metrological intercomparison of six terrestrial laser scanning systems

6 p.Intercomparison among six terrestrial laser scanner systems focused on the measurement of small elements ( <; 0.5 m) is performed. Phase shift (PS) and time of flight (ToF) scanners are considered. Two standard artefacts containing three-dimensional printing spheres and steps of variable height are used for the experiment. Results show errors between -4.5 and 3.5 mm in the measurement of distances between step planes. The most stable systems for measuring small elements seem the Leica C10, Faro Photon and Riegl LMS Z390i. The quality of the results is linked to the overall quality of the system rather than the specific technology used for range measurement (PS or ToF) which does not appear to be a determining factor.S

A TWO-STEP FEATURE EXTRACTION ALGORITHM: APPLICATION TO DEEP LEARNING FOR POINT CLOUD CLASSIFICATION

Most deep learning (DL) methods that are not end-to-end use several multi-scale and multi-type hand-crafted features that make the network challenging, more computationally intensive and vulnerable to overfitting. Furthermore, reliance on empirically-based feature dimensionality reduction may lead to misclassification. In contrast, efficient feature management can reduce storage and computational complexities, builds better classifiers, and improves overall performance. Principal Component Analysis (PCA) is a well-known dimension reduction technique that has been used for feature extraction. This paper presents a two-step PCA based feature extraction algorithm that employs a variant of feature-based PointNet (Qi et al., 2017a) for point cloud classification. This paper extends the PointNet framework for use on large-scale aerial LiDAR data, and contributes by (i) developing a new feature extraction algorithm, (ii) exploring the impact of dimensionality reduction in feature extraction, and (iii) introducing a non-end-to-end PointNet variant for per point classification in point clouds. This is demonstrated on aerial laser scanning (ALS) point clouds. The algorithm successfully reduces the dimension of the feature space without sacrificing performance, as benchmarked against the original PointNet algorithm. When tested on the well-known Vaihingen data set, the proposed algorithm achieves an Overall Accuracy (OA) of 74.64% by using 9 input vectors and 14 shape features, whereas with the same 9 input vectors and only 5PCs (principal components built by the 14 shape features) it actually achieves a higher OA of 75.36% which demonstrates the effect of efficient dimensionality reduction

Distributed processing of Dutch AHN laser altimetry changes of the built-up area

The evolution and spreading of data capturing methods ranging from simple GPS devices like smart-phones to large scale imaging equipment – including very high resolution and hyperspectral cameras and LiDAR – resulted in an exponential growth in the amount of spatial data maintained by companies and organizations. At the same time methods for extracting information from such data are often behind in efficiency. In this paper we analyse the possibilities for nation-wide change detection of massive airborne laser altimetry point clouds, based on digital elevation models generated from them. The proposed workflow distinguishes modifications in the built-up area from other changes and noise. Our methodology combines different area processing spatial algorithms: object detection, noise filtering, morphological operations and clustering. Our proposed method is designed to scale dynamically on extensive datasets by processing a spatially partitioned input dataset in an easily parallelized manner. Favourable visualizations and aggregated representations of the results are examined, followed by a discussion of feasible validation methods. As a demonstration we showcase the implemented distributed evaluation of our workflow on the full Dutch altimetry archive – a dataset exceeding several terabytes of storage space – using a high-performance computing environment. While the average execution time was 47 h on a desktop computer, our solution only took less than 2.4 h to complete. The output was validated against the building layer of the TOP10NL topographic dataset, proving a 70% accuracy nation-wide and over 90% for urban areas. As a result our analysis shows that The Netherlands experienced an aggregated building volume change of 912.33 km3 between the acquisition of AHN-2 and AHN-3