Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes

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

Scattered Data Approximation by LR B-Spline Surfaces: A Study on Refinement Strategies for Efficient Approximation

Locally refined B-spline (LRB) surfaces provide a representation that is well suited to scattered data approximation. When a data set has local details in some areas and is largely smooth elsewhere, 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 the resulting data volume and accuracy when applied to geospatial data sets with different structural behaviour. The relative performance of the refinement strategies is reasonably coherent for the different data sets and this paper concludes with some recommendations. An overall evaluation indicates that bi-quadratic LRB are preferable for the use cases tested, and that the strategies we denote as “full span" have the overall best performance.publishedVersio

Optimal Surface Fitting of Point Clouds Using Local Refinement

This open access book provides insights into the novel Locally Refined B-spline (LR B-spline) surface format, which 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 behavior and allows the modeling 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 iterative method can be modified to improve the accuracy in particular domains of a point cloud. The use of statistical information criterion can help determining the optimal threshold, the number of iterations to perform as well as some parameters of the underlying mathematical functions (degree of the splines, parameter representation). The resulting surfaces are well suited for analysis and computing secondary information such as contour curves and minimum and maximum points. Also deformation analysis are potential applications of fitting point clouds with LR B-splines.publishedVersio

LR B-splines to approximate bathymetry datasets: An improved statistical criterion to judge the goodness of fit

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.publishedVersio

Trivariate Spline Representations for Computer Aided Design and Additive Manufacturing

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.Comment: 30 pages, 14 figures. This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 68044