scholarly journals LOCALLY REFINED SPLINES REPRESENTATION FOR GEOSPATIAL BIG DATA

2015 ◽  
Vol XL-3/W3 ◽  
pp. 565-570
Author(s):  
T. Dokken ◽  
V. Skytt ◽  
O. Barrowclough
Keyword(s):  
Degrees Of Freedom ◽  
Spline Approximation ◽  
Point Clouds ◽  
Data Representation ◽  
Ocean Floor ◽  
Sensor Data ◽  
B Spline ◽  
B Splines ◽  
Model Combining ◽  
Elevation Model

When viewed from distance, large parts of the topography of landmasses and the bathymetry of the sea and ocean floor can be regarded as a smooth background with local features. Consequently a digital elevation model combining a compact smooth representation of the background with locally added features has the potential of providing a compact and accurate representation for topography and bathymetry. The recent introduction of Locally Refined B-Splines (LR B-splines) allows the granularity of spline representations to be locally adapted to the complexity of the smooth shape approximated. This allows few degrees of freedom to be used in areas with little variation, while adding extra degrees of freedom in areas in need of more modelling flexibility. In the EU fp7 Integrating Project IQmulus we exploit LR B-splines for approximating large point clouds representing bathymetry of the smooth sea and ocean floor. A drastic reduction is demonstrated in the bulk of the data representation compared to the size of input point clouds. The representation is very well suited for exploiting the power of GPUs for visualization as the spline format is transferred to the GPU and the triangulation needed for the visualization is generated on the GPU according to the viewing parameters. The LR B-splines are interoperable with other elevation model representations such as LIDAR data, raster representations and triangulated irregular networks as these can be used as input to the LR B-spline approximation algorithms. Output to these formats can be generated from the LR B-spline applications according to the resolution criteria required. The spline models are well suited for change detection as new sensor data can efficiently be compared to the compact LR B-spline representation.

Generation and Modification of Non-Uniform B-Spline Surface Approximations to PDE Surfaces Using the Finite Element Method

1990 ◽  
Author(s):  
Joanna M. Brown ◽  
Malcolm I. G. Bloor ◽  
M. Susan Bloor ◽  
Michael J. Wilson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Spline Approximation ◽  
The Finite Element Method ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Spline Basis ◽  
Element Method

Abstract A PDE surface is generated by solving partial differential equations subject to boundary conditions. To obtain an approximation of the PDE surface in the form of a B-spline surface the finite element method, with the basis formed from B-spline basis functions, can be used to solve the equations. The procedure is simplest when uniform B-splines are used, but it is also feasible, and in some cases desirable, to use non-uniform B-splines. It will also be shown that it is possible, if required, to modify the non-uniform B-spline approximation in a variety of ways, using the properties of B-spline surfaces.

Choosing the optimal number of B-spline control points (Part 2: Approximation of surfaces and applications)

2017 ◽  
Vol 11 (1) ◽  
Author(s):  
Corinna Harmening ◽  
Hans Neuner
Keyword(s):  
Data Analysis ◽  
Laser Scanner ◽  
Point Clouds ◽  
Optimal Number ◽  
Deformation Analysis ◽  
Control Points ◽  
Trial And Error ◽  
B Spline ◽  
B Splines ◽  
Spline Surfaces

AbstractFreeform surfaces like B-splines have proven to be a suitable tool to model laser scanner point clouds and to form the basis for an areal data analysis, for example an areal deformation analysis.A variety of parameters determine the B-spline's appearance, the B-spline's complexity being mostly determined by the number of control points. Usually, this parameter type is chosen by intuitive trial-and-error-procedures.In [The present paper continues these investigations. If necessary, the methods proposed in [The application of those methods to B-spline surfaces reveals the datum problem of those surfaces, meaning that location and number of control points of two B-splines surfaces are only comparable if they are based on the same parameterization. First investigations to solve this problem are presented.

scholarly journals Tunnel Deformation Inspection via Global Spatial Axis Extraction from 3D Raw Point Cloud

Sensors ◽  
2020 ◽  
Vol 20 (23) ◽  
pp. 6815
Author(s):  
Cheng Yi ◽  
Dening Lu ◽  
Qian Xie ◽  
Jinxuan Xu ◽  
Jun Wang
Keyword(s):  
Cross Sections ◽  
Point Cloud ◽  
Large Scale ◽  
Spline Approximation ◽  
Point Clouds ◽  
Driving Safety ◽  
Inspection Method ◽  
B Spline ◽  
3D Point Clouds ◽  
Spatial Axis

Global inspection of large-scale tunnels is a fundamental yet challenging task to ensure the structural stability of tunnels and driving safety. Advanced LiDAR scanners, which sample tunnels into 3D point clouds, are making their debut in the Tunnel Deformation Inspection (TDI). However, the acquired raw point clouds inevitably possess noticeable occlusions, missing areas, and noise/outliers. Considering the tunnel as a geometrical sweeping feature, we propose an effective tunnel deformation inspection algorithm by extracting the global spatial axis from the poor-quality raw point cloud. Essentially, we convert tunnel axis extraction into an iterative fitting optimization problem. Specifically, given the scanned raw point cloud of a tunnel, the initial design axis is sampled to generate a series of normal planes within the corresponding Frenet frame, followed by intersecting those planes with the tunnel point cloud to yield a sequence of cross sections. By fitting cross sections with circles, the fitted circle centers are approximated with a B-Spline curve, which is considered as an updated axis. The procedure of “circle fitting and B-SPline approximation” repeats iteratively until convergency, that is, the distance of each fitted circle center to the current axis is smaller than a given threshold. By this means, the spatial axis of the tunnel can be accurately obtained. Subsequently, according to the practical mechanism of tunnel deformation, we design a segmentation approach to partition cross sections into meaningful pieces, based on which various inspection parameters can be automatically computed regarding to tunnel deformation. A variety of practical experiments have demonstrated the feasibility and effectiveness of our inspection method.

scholarly journals Spline Approximation, Part 2: From Polynomials in the Monomial Basis to B-splines—A Derivation

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2198
Author(s):  
Nikolaj Ezhov ◽  
Frank Neitzel ◽  
Svetozar Petrovic
Keyword(s):  
Numerical Stability ◽  
Spline Approximation ◽  
Point Of View ◽  
Direct Relation ◽  
Monomial Basis ◽  
B Spline ◽  
Numerical Examples ◽  
B Splines ◽  
Spline Representation ◽  
Lagrangian Form

In a series of three articles, spline approximation is presented from a geodetic point of view. In part 1, an introduction to spline approximation of 2D curves was given and the basic methodology of spline approximation was demonstrated using splines constructed from ordinary polynomials. In this article (part 2), the notion of B-spline is explained by means of the transition from a representation of a polynomial in the monomial basis (ordinary polynomial) to the Lagrangian form, and from it to the Bernstein form, which finally yields the B-spline representation. Moreover, the direct relation between the B-spline parameters and the parameters of a polynomial in the monomial basis is derived. The numerical stability of the spline approximation approaches discussed in part 1 and in this paper, as well as the potential of splines in deformation detection, will be investigated on numerical examples in the forthcoming part 3.

Dynamic Trajectory Planning for a Three Degrees-of-Freedom Cable-Driven Parallel Robot Using Quintic B-Splines

2020 ◽  
Vol 142 (7) ◽  
Author(s):  
Sen Qian ◽  
Kunlong Bao ◽  
Bin Zi ◽  
W. D. Zhu
Keyword(s):  
Trajectory Planning ◽  
Degrees Of Freedom ◽  
Parallel Robot ◽  
Planning Method ◽  
End Effector ◽  
B Spline ◽  
B Splines ◽  
Pick And Place ◽  
Two Factors ◽  
Three Degrees Of Freedom

Abstract This paper presents a new trajectory planning method based on the improved quintic B-splines curves for a three degrees-of-freedom (3-DOF) cable-driven parallel robot (CDPR). First, the conditions of positive cables’ tension are expressed in terms of the position and acceleration constraints of the end-effector. Then, an improved B-spline curve is introduced, which is employed for generating a pick-and-place path by interpolating a set of given via-points. Meanwhile, by expressing the position and acceleration of the end-effector in terms of the first and second derivatives of the improved B-spline, the cable tension constraints are described in the form of B-spline parameters. According to the properties of the defined pick-and-place path, the proposed motion profile is dominated by two factors: the time taken for the end-effector to pass through all the via-points and the ratio between the nodes of B-spline. The two factors are determined through multi-objective optimization based on the efficiency coefficient method. Finally, experimental results on a 3-DOF CDPR show that the improved B-spline exhibits overall superior behavior in terms of velocity, acceleration, and cables force compared with the traditional B-spline. The validity of the proposed trajectory planning method is proved through the experiments.

scholarly journals On the impact of correlations on the congruence test: a bootstrap approach

2020 ◽  
Vol 55 (3) ◽  
pp. 495-513 ◽  
Author(s):  
Kermarrec Gaël ◽  
Kargoll Boris ◽  
Alkhatib Hamza
Keyword(s):  
Statistical Significance ◽  
Spline Approximation ◽  
Point Clouds ◽  
Test Statistics ◽  
B Spline ◽  
Mode Decomposition ◽  
Spline Surfaces ◽  
Bootstrap Approach ◽  
The Impact ◽  
Congruence Test

AbstractThe detection of deformation is one of the major tasks in surveying engineering. It is meaningful only if the statistical significance of the distortions is correctly investigated, which often underlies a parametric modelization of the object under consideration. So-called regression B-spline approximation can be performed for point clouds of terrestrial laser scanners, allowing the setting of a specific congruence test based on the B-spline surfaces. Such tests are known to be strongly influenced by the underlying stochastic model chosen for the observation errors. The latter has to be correctly specified, which includes accounting for heteroscedasticity and correlations. In this contribution, we justify and make use of a parametric correlation model called the Matérn model to approximate the variance covariance matrix (VCM) of the residuals by performing their empirical mode decomposition. The VCM obtained is integrated into the computation of the congruence test statistics for a more trustworthy test decision. Using a real case study, we estimate the distribution of the test statistics with a bootstrap approach, where no parametric assumptions are made about the underlying population that generated the random sample. This procedure allows us to assess the impact of neglecting correlations on the critical value of the congruence test, highlighting their importance.

Modeling connected regions in arbitrary planar point clouds by robust B-spline approximation

2016 ◽  
Vol 76 ◽  
pp. 141-151 ◽  
Author(s):  
Thomas Mörwald ◽  
Jonathan Balzer ◽  
Markus Vincze
Keyword(s):  
Spline Approximation ◽  
Point Clouds ◽  
B Spline ◽  
Planar Point

scholarly journals Robust Spatial Approximation of Laser Scanner Point Clouds by Means of Free-form Curve Approaches in Deformation Analysis

2016 ◽  
Vol 10 (1) ◽  
Author(s):  
Johannes Bureick ◽  
Hamza Alkhatib ◽  
Ingo Neumann
Keyword(s):  
Laser Scanner ◽  
Point Clouds ◽  
Measured Data ◽  
Free Form ◽  
Deformation Analysis ◽  
B Spline ◽  
B Splines ◽  
Spline Fitting ◽  
Data Gaps ◽  
Spatial Approximation

AbstractIn many geodetic engineering applications it is necessary to solve the problem of describing a measured data point cloud, measured, e. g. by laser scanner, by means of free-form curves or surfaces, e. g., with B-Splines as basis functions. The state of the art approaches to determine B-Splines yields results which are seriously manipulated by the occurrence of data gaps and outliers.Optimal and robust B-Spline fitting depend, however, on optimal selection of the knot vector. Hence we combine in our approach Monte-Carlo methods and the location and curvature of the measured data in order to determine the knot vector of the B-Spline in such a way that no oscillating effects at the edges of data gaps occur. We introduce an optimized approach based on computed weights by means of resampling techniques. In order to minimize the effect of outliers, we apply robust M-estimators for the estimation of control points.The above mentioned approach will be applied to a multi-sensor system based on kinematic terrestrial laserscanning in the field of rail track inspection.

scholarly journals Parametric Surface Modelling for Tea Leaf Point Cloud Based on Non-Uniform Rational Basis Spline Technique

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1304
Author(s):  
Wenchao Wu ◽  
Yongguang Hu ◽  
Yongzong Lu
Keyword(s):  
Point Cloud ◽  
Principal Component ◽  
Point Clouds ◽  
Parametric Surface ◽  
B Spline ◽  
Surface Modelling ◽  
Modelling Method ◽  
3D Architecture ◽  
Tea Leaf ◽  
Fitting Error

Plant leaf 3D architecture changes during growth and shows sensitive response to environmental stresses. In recent years, acquisition and segmentation methods of leaf point cloud developed rapidly, but 3D modelling leaf point clouds has not gained much attention. In this study, a parametric surface modelling method was proposed for accurately fitting tea leaf point cloud. Firstly, principal component analysis was utilized to adjust posture and position of the point cloud. Then, the point cloud was sliced into multiple sections, and some sections were selected to generate a point set to be fitted (PSF). Finally, the PSF was fitted into non-uniform rational B-spline (NURBS) surface. Two methods were developed to generate the ordered PSF and the unordered PSF, respectively. The PSF was firstly fitted as B-spline surface and then was transformed to NURBS form by minimizing fitting error, which was solved by particle swarm optimization (PSO). The fitting error was specified as weighted sum of the root-mean-square error (RMSE) and the maximum value (MV) of Euclidean distances between fitted surface and a subset of the point cloud. The results showed that the proposed modelling method could be used even if the point cloud is largely simplified (RMSE < 1 mm, MV < 2 mm, without performing PSO). Future studies will model wider range of leaves as well as incomplete point cloud.

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