This open access book provides insights into the novel LR B-spline surface format to represent terrain and seabed data in a compact way

AbstractThe surface approximation obtained with adaptive strategies using locally refined (LR) B-splines depends on the degrees of freedom of the spline space, the tolerance from which the refinement is performed, the noise level of the scattered observations, the refinement strategy and the bidegree of the spline space. The choice of the best model is a challenging task that can be partially answered with statistical criteria, such as the Akaike Information Criterion (AIC). Here we relax the assumption that the approximation error should be normally distributed and with equal variance and propose the use of the student distribution to compute the AIC. We apply the AIC to decide which tolerance, refinement level, or polynomial bidegree are the most adequate for an optimal fitting. We highlight how the resulting AIC can be combined with more usual criteria to judge the goodness of fit of the surface approximation.

Locally refined spline surfaces (LRB) is a representation well suited for scattered data approximation. When a data set has local details in some areas and is largely smooth in other, LR B-splines allow the spatial distribution of degrees of freedom to follow the variations of the data set. An LRB surface approximating a data set is refined in areas where the accuracy does not meet a required tolerance. In this paper we address, in a systematic study, different LRB refinement strategies and polynomial degrees for surface approximation. We study their influence on data volume and accuracy when applied to geospatial data sets with different structural behaviour. The performance of the refinement strategies is to some degree coherent and the article concludes with some recommendations. An overall evaluation indicates that bi-quadratic LRB are preferable for the uses cases tested, and that the strategies we denote as 'full span' have the overall best performance.

AbstractGeospatial data acquisition of terrains with contact-free sensors such as Terrestrial or Airbone Laser Scanners generates scattered and noisy point clouds. Performing a surface approximation is an efficient way to reduce and structure the recorded point clouds. To that end, LR B-splines are attractive as they allow a local refinement, on the contrary to the tensor product B-spline and raster surfaces. By comparing the approximation error with a given tolerance, a local refinement is performed. We apply this adaptive refinement strategy to landslides data sets from Alpine terrain in Austria. We show how different epochs of the point clouds can be analyzed with LR B-spline volumes for spatio-temporal visualisation of deformation. We highlight the potential of a time-differenced LR B-splines volume for analysing geomorphological changes. A further application of this method is the drawing of contour lines.

AbstractA locally refined (LR) B-spline surface is a piecewise polynomial surface for which the distribution of the surface coefficients can be locally adapted. Such a mathematical representation is interesting for fitting scattered and noisy data, as the local behaviour of a real point cloud may require more degrees of freedom only locally. The number of redundant surface coefficients is minimized, which avoids the fitting of the point cloud’s noise. The surface approximation is performed iteratively either by solving a least squares system or by a local approximation method. This procedure allows for mesh refinement in domains where the distance between a current surface and the point cloud exceeds a prescribed tolerance. In this way, parts of the LR B-spline surface obtained at previous steps may be kept unchanged. This chapter aims at explaining the adaptive fitting using local refinement with LR B-splines. We present two examples with simulated point clouds to illustrate the methodology.

In recent years there has been an explosion of interest in digital twinning in many disciplines, including the manufacturing, geospatial, and medical domains. A core topic of importance in modelling digital twins, is reconstruction of geometric models from raw data. Despite the diversity of requirements in the vast space of digital twin applications, methods for geometric reconstruction can often be transferred between disciplines with only minor modifications. In this paper we present some recent results related to how advances in machine learning over the last decade can be used for data-driven geometric reconstruction in the medical, geospatial and manufacturing domains.

Geospatial data acquisition of terrains produces huge, noisy and scattered point clouds. An efficient use of the acquired data requires structured and compact data representations. Working directly in a point cloud is often not appealing. To face this challenge, approximation with tensor product B-spline surfaces is attractive. It reduces the point cloud description to relatively few coefficients compared to the volume of the original point cloud. However, this representation lacks the ability to adapt the resolution of the shape to local variations in the point cloud. The result is frequently that noise is approximated and that surfaces have unwanted oscillations. Locally Refined (LR) B-spline surfaces were introduced to face this challenge and provide a tool for approximating Geographic Information System point clouds. In our LR B-spline based approximation algorithm, iterative least-squares approximation is combined with a Multilevel B-spline Approximation to reduce memory consumption. We apply the approach to data sets from coastal regions in Norway and the Netherlands, and compare the obtained approximation with a raster method. We further highlight the potential of LR B-spline volumes for spatio-temporal visualisation of deformation patterns.

The task of representing remotely sensed scattered point clouds with mathematical surfaces is ubiquitous to reduce a high number of observations to a compact description with as few coefficients as possible. To reach that goal, locally refined B-splines provide a simple framework to perform surface approximation by allowing an iterative local refinement. Different setups exist (bidegree of the splines, tolerance, refinement strategies) and the choice is often made heuristically, depending on the applications and observations at hand. In this article, we introduce a statistical information criterion based on the t-distribution to judge the goodness of fit of the surface approximation for remote sensing data with outliers. We use a real bathymetry dataset and illustrate how concepts from model selection can be used to select the most adequate refinement strategy of the LR B-splines.

AbstractPerforming surface approximation of geospatial point clouds with locally refined (LR) B-splines comes with several challenges: (i) Point clouds have varying data density, (ii) outliers should be eliminated without deleting features, (iii) voids, also called holes, or data gaps should be treated specifically to avoid the drop of the approximated surface in domains without points. These factors tend to be even more challenging when point clouds acquired from different sensors having different noise characteristics are fused together. The data set becomes non-uniform and the fusing process itself involves a risk of an increased noise level. In this chapter, we provide some tools to answer those specific challenges. We will use terrain and seabed data and show didactically how to perform adaptive surface approximation with local refinement and to select customized parameters. We will further address the problem of choosing an appropriate tolerance for performing an adaptive fitting, and discuss the refinement strategies within the context of LR B-splines. The latter is shown to provide a promising framework for surface fitting of heterogeneous point clouds from various sources.

AbstractThe univariate minimal support B-spline basis (UMB) has been used in Computer Aided Design (CAD) since the 1970s. Freeform curves use UMB, while sculptured surfaces are represented using a tensor product of two UMBs. The coefficients of a B-spline curve and surface are respectively represented in a vector and a rectangular grid. In CAD-intersection algorithms for UMB represented objects, a divide-and-conquer strategy is often used. Refinement by knot insertion is used to split the objects intersected into objects of the same type with a smaller geometric extent. In many cases the intersection of the resulting sub-objects has simpler topology than the original problem. The sub-objects created are represented using their parents’ UMB format and deleted when the sub-problem is solved. Consequently, no global representations of the locally refined bases are needed. This is contrary to when locally refined splines are used for approximation of large point sets. As soon as a B-spline is locally refined, the regular structure of UMB objects in CAD is no longer valid. In this chapter we discuss how Locally Refined B-splines (LR B-splines) address this challenge and present the properties of LR B-splines.

This report documents the program and the outcomes of Dagstuhl Seminar 21471 “Geometric Modeling: Interoperability and New Challenges”. This seminar was initially planned on May 2021, and was delayed due to the pandemic. The seminar took place as a hybrid version with on site and remote participants. It provided a great opportunity for exchanges which, as pointed out by participants, were very appreciated in this period where international scientific interactions have been diminished. This report summarizes the seminar communications, first by providing the abstracts of the talks which present recent results in geometric modeling. Moreover, the scientific exchanges during the seminar provided a great basis for scientific discussions that resulted to the included five reports which highlight the new and future challenges in Geometric Modeling.

The novel Locally Refined B-spline (LR B-spline) surface format is suited for representing terrain and seabed data in a compact way. It provides an alternative to the well know raster and triangulated surface representations. An LR B-spline surface has an overall smooth behaviour and allows the modelling of local details with only a limited growth in data volume. In regions where many data points belong to the same smooth area LR B-splines allow a very lean representation of the shape by locally adapting the resolution of the spline space to the size and local shape variations of the region. The surfaces generated approximate the smooth component of a cloud of data points within user specified tolerances. The method can be modified to improve the accuracy in particular domains and selected data points. The resulting surfaces are well suited for analysis and computing secondary information such as contour curves and minimum and maximum points. The surfaces can be translated into a raster or a tessellated surface of desired quality or exported as collections of tensor product spline surfaces. Data transfer can also be performed using Part 42 of ISO 10303 (the STEP standard) where LR B-splines were published in 2018.

The focus on locally refined spline spaces has grown rapidly in recent years due to the need in Isogeometric Analysis (IgA) of spline spaces with local adaptivity: a property not offered by the strict regular structure of tensor product B-spline spaces. However, this flexibility sometimes results in collections of B-splines spanning the space that are not linearly independent. In this paper we address the minimal number of Minimal Support B-splines (MS B-splines) and of Locally Refined B-splines (LR B-splines) that can form a linear dependence relation. We show that such minimal numbers are six for MS B-splines and eight for LR B-splines. Further results are established to help detecting collections of B-splines that are linearly independent.

Given a spline space spanned by Truncated Hierarchical B-splines (THB), it is always possible to construct a spline space spanned by Locally Refined B-splines (LRB) that contains the THB-space. Starting from configurations where the two spline spaces are equal, we adress what happens to the properties of the LRB-space when it is modified by local one-directional refinement at convex corners of, and along edges between dyadic refinement regions. We show that such local modifications can reduce the number of B-splines over each element to the minimum prescribed by the polynomial bi-degree, and that such local refinements can be used for improving the condition numbers of mass and stiffness matrices.

Digital representations targeting design and simulation for Additive Manufacturing (AM) are addressed from the perspective of Computer Aided Geometric Design. We discuss the feasibility for multi-material AM for B-rep based CAD, STL, sculptured triangles as well as trimmed and block-structured trivariate locally refined spline representations. The trivariate spline representations support Isogeometric Analysis (IGA), and topology structures supporting these for CAD, IGA and AM are outlined. The ideas of (Truncated) Hierarchical B-splines, T-splines and LR B-splines are outlined and the approaches are compared. An example from the EC H2020 Factories of the Future Research and Innovation Actions CAxMan illustrates both trimmed and block-structured spline representations for IGA and AM.

The focus on locally refined spline spaces has grown rapidly in recent years due to the need in Isogeoemtric analysis (IgA) of spline spaces with local adaptivity: a property not offered by the strict regular structure of tensor product B-spline spaces. However, this flexibility sometimes results in collections of B-splines spanning the space that are not linearly independent. In this paper we address the minimal number of B-splines that can form a linear dependence relation for Minimal Support B-splines (MS B-splines) and for Locally Refinable B-splines (LR B-splines) on LR-meshes. We show that the minimal number is six for MS B-splines, and eight for LR B-splines. The risk of linear dependency is consequently significantly higher for MS B-splines than for LR B-splines. Further results are established to help detecting collections of B-splines that are linearly independent.

This report documents the program and the outcomes of Dagstuhl Seminar 17221 Geometric Modelling, Interoperability and New Challenges. While previous Dagstuhl seminars on geometric modeling were focused on basic research, this seminar was focused on applications of geometric modeling to four topic areas: big data and cloud computing, multi-material additive manufacturing, isogeometric analysis, and design optimization. For this purpose we brought together participants from industry urgently in need of better solutions, researchers in the above application areas, and researchers in the geometric modeling community.

Digital environmental data are becoming commonplace and the amount of information they provide is complex to process, due to the size, variety, and dynamic nature of the data captured by sensing devices. The paper discusses an evaluation framework for comparing methods to approximate observed rain data, in real conditions of sparsity of the observations. The novelty brought by this experimental study stands in the geographical area and heterogeneity of the data used for evaluation, aspects which challenge all approximation methods. The Liguria region, located in the north-west of Italy, is a complex area for the orography and the closeness to the sea, which cause complex hydro-meteorological events. The observed rain data are highly heterogeneous: two data sets come from measured rain gathered from two different rain gauge networks, with different characteristics and spatial distributions over the Liguria region; the third data set come from weather radar, with a more regular coverage of the same region but a different veracity. Finally, another novelty of the paper is brought by the proposal of an application-oriented perspective on the comparison. The approximation models the rain field, whose maxima and their evolution is essential for an effective monitoring of meteorological events. Therefore, we adapt a storm tracking technique to the analysis of the displacement of maxima computed by the different methods, used as a dissimilarity measure among the approximation methods analyzed.