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

Harmening, Corinna; Neuner, Hans

doi:10.1515/jag-2016-0036

Cited by 12 publications

(10 citation statements)

References 14 publications

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“…Since previous geodetic literatures [21,22,28] has solved the model selection problem through well-known penalization information criteria: the AIC and BIC, it is necessary in this section to compare Vuong's non-nested hypothesis test with this widely used approach. It is noticeable that there are close connections between AIC, BIC, and Vuong's test.…”

Section: Discussionmentioning

confidence: 99%

“…This limitation serves as the motivation for discriminating between estimated surface models in order to select the most appropriate one. In the context of model selection, Harmening and Neuner [21,22] investigated statistical methods based on information criteria and statistical learning theory for selecting the optimal number of control points within B-spline surface estimation. Another possibility is to compare the (log-)likelihoods of competing models directly by means of the general testing principle by D.R.…”

Section: Introductionmentioning

confidence: 99%

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Model Selection for Parametric Surfaces Approximating 3D Point Clouds for Deformation Analysis

et al. 2018

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Deformation monitoring of structures is a common application and one of the major tasks of engineering surveying. Terrestrial laser scanning (TLS) has become a popular method for detecting deformations due to high precision and spatial resolution in capturing a number of three-dimensional point clouds. Surface-based methodology plays a prominent role in rigorous deformation analysis. Consequently, it is of great importance to select an appropriate regression model that reflects the geometrical features of each state or epoch. This paper aims at providing the practitioner some guidance in this regard. Different from standard model selection procedures for surface models based on information criteria, we adopted the hypothesis tests from D.R. Cox and Q.H. Vuong to discriminate statistically between parametric models. The methodology was instantiated in two numerical examples by discriminating between widely used polynomial and B-spline surfaces as models of given TLS point clouds. According to the test decisions, the B-spline surface model showed a slight advantage when both surface types had few parameters in the first example, while it performed significantly better for larger numbers of parameters. Within B-spline surface models, the optimal one for the specific segment was fixed by Vuong's test whose result was quite consistent with the judgment of widely used Bayesian information criterion. The numerical instabilities of B-spline models due to data gap were clearly reflected by the model selection tests, which rejected inadequate B-spline models in another numerical example.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Model Selection for Parametric Surfaces Approximating 3D Point Clouds for Deformation Analysis

et al. 2018

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“…The optimal number of control points to be estimated can be interpreted as a model selection problem and can either be solved by classical model selection strategies or by structural risk minimisation (cf. [34,35]). The latter strategy has the advantage that the degrees p and q can also be taken into account during model selection, rather than choosing them arbitrarily [36].…”

Section: Non-distorted Regions and Final Transformation Parametersmentioning

confidence: 99%

Laser Scanner–Based Deformation Analysis Using Approximating B-Spline Surfaces

2021

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Due to the increased use of areal measurement techniques, such as laser scanning in geodetic monitoring tasks, areal analysis strategies have considerably gained in importance over the last decade. Although a variety of approaches that quasi-continuously model deformations are already proposed in the literature, there are still a multitude of challenges to solve. One of the major interests of engineering geodesy within monitoring tasks is the detection of absolute distortions with respect to a stable reference frame. Determining distortions and simultaneously establishing the joint geodetic datum can be realised by modelling the differences between point clouds acquired in different measuring epochs by means of a rigid body movement that is superimposed by distortions. In a previous study, we discussed the possibility of estimating these rigid body movements from the control points of B-spline surfaces approximating the acquired point clouds. Alternatively, we focus on estimating them by means of constructed points on B-spline surfaces in this study. This strategy has the advantage of larger redundancy compared to the control point–based strategy. Furthermore, the strategy introduced allows for the detection of rigid body movements between point clouds of different epochs and for the simultaneous localisation of areas in which the rigid body movement is superimposed by distortions. The developed approach is based on B-spline models of epoch-wise acquired point clouds, the surface parameters of which define point correspondences on different B-spline surfaces. Using these point correspondences, a RANSAC-approach is used to robustly estimate the parameters of the rigid body movement. The resulting consensus set initially defines the non-distorted areas of the object under investigation, which are extended and statistically verified in a second step. The developed approach is applied to simulated data sets, revealing that distorted areas can be reliably detected and that the parameters of the rigid body movement can be precisely and accurately determined by means of the strategy.

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“…However, the proposed approach of a stochastic modelling of the deformation offers two advantages: on the one hand, the challenge of a consistent surface parametrization, which is necessary to compare different B-spline surfaces (cf. Harmening and Neuner 2017), is circumvented. On the other hand, only in a unified B-spline-based framework for handling rigid body movements and distortions, the former can be a-priori eliminated according to Harmening and Neuner (2016b).…”

Section: Basic Ideas Of the Developed Approachmentioning

confidence: 99%

A spatio-temporal deformation model for laser scanning point clouds

Harmening

Neuner

2020

J Geod

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The establishment of the terrestrial laser scanner changed the analysis strategies in engineering geodesy from point-wise approaches to areal ones. During recent years, a multitude of developments regarding a laser scanner-based geometric state description were made. However, the areal deformation analysis still represents a challenge. In this paper, a spatio-temporal deformation model is developed, combining the estimation of B-spline surfaces with the stochastic modelling of deformations. The approach's main idea is to model the acquired measuring object by means of three parts, similar to a least squares collocation: a deterministic trend, representing the undistorted object, a stochastic signal, describing a locally homogeneous deformation process, and the measuring noise, accounting for uncertainties caused by the measuring process. Due to the stochastic modelling of the deformations in the form of distance-depending variograms, the challenge of defining identical points within two measuring epochs is overcome. Based on the geodetic datum defined by the initial trend surface, a pointto-surface-and a point-to-point-comparison of the acquired data sets is possible, resulting in interpretable and meaningful deformation metrics. Furthermore, following the basic ideas of a least squares collocation, the deformation model allows a time-related space-continuous description as well as a space-and time-continuous prediction of the deformation. The developed approach is validated using simulated data sets, and the respective results are analysed and compared with respect to nominal surfaces.

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

Cited by 12 publications

References 14 publications

Model Selection for Parametric Surfaces Approximating 3D Point Clouds for Deformation Analysis

Model Selection for Parametric Surfaces Approximating 3D Point Clouds for Deformation Analysis

Laser Scanner–Based Deformation Analysis Using Approximating B-Spline Surfaces

A spatio-temporal deformation model for laser scanning point clouds

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