TY - JOUR

T1 - How Significant Are Differences Obtained by Neglecting Correlations When Testing for Deformation: A Real Case Study Using Bootstrapping with Terrestrial Laser Scanner Observations Approximated by B-Spline Surfaces

AU - Kermarrec, Gaël

AU - Paffenholz, Jens André

AU - Alkhatib, Hamza

N1 - Funding information: The publication of this article was funded by the Open Access Fund of the Leibniz Universität Hannover.

PY - 2019/9/1

Y1 - 2019/9/1

N2 - B-spline surfaces possess attractive properties such as a high degree of continuity or the local support of their basis functions. One of the major applications of B-spline surfaces in engineering geodesy is the least-square (LS) fitting of surfaces from, e.g., 3D point clouds obtained from terrestrial laser scanners (TLS). Such mathematical approximations allow one to test rigorously with a given significance level the deformation magnitude between point clouds taken at different epochs. Indeed, statistical tests cannot be applied when point clouds are processed in commonly used software such as CloudCompare, which restrict the analysis of deformation to simple deformation maps based on distance computation. For a trustworthy test decision and a resulting risk management, the stochastic model of the underlying observations needs, however, to be optimally specified. Since B-spline surface approximations necessitate Cartesian coordinates of the TLS observations, the diagonal variance covariance matrix (VCM) of the raw TLS measurements has to be transformed by means of the error propagation law. Unfortunately, this procedure induces mathematical correlations, which can strongly affect the chosen test statistics to analyse deformation, if neglected. This may lead potentially to rejecting wrongly the null hypothesis of no-deformation, with risky and expensive consequences. In this contribution, we propose to investigate the impact of mathematical correlations on test statistics, using real TLS observations from a bridge under load. As besides TLS, a highly precise laser tracker (LT) was used, the significance of the difference of the test statistics when the stochastic model is misspecified can be assessed. However, the underlying test distribution is hardly tractable so that only an adapted bootstrapping allows the computation of trustworthy p-values. Consecutively, the extent to which heteroscedasticity and mathematical correlations can be neglected or simplified without impacting the test decision is shown in a rigorous way, paving the way for a simplification based on the intensity model.

AB - B-spline surfaces possess attractive properties such as a high degree of continuity or the local support of their basis functions. One of the major applications of B-spline surfaces in engineering geodesy is the least-square (LS) fitting of surfaces from, e.g., 3D point clouds obtained from terrestrial laser scanners (TLS). Such mathematical approximations allow one to test rigorously with a given significance level the deformation magnitude between point clouds taken at different epochs. Indeed, statistical tests cannot be applied when point clouds are processed in commonly used software such as CloudCompare, which restrict the analysis of deformation to simple deformation maps based on distance computation. For a trustworthy test decision and a resulting risk management, the stochastic model of the underlying observations needs, however, to be optimally specified. Since B-spline surface approximations necessitate Cartesian coordinates of the TLS observations, the diagonal variance covariance matrix (VCM) of the raw TLS measurements has to be transformed by means of the error propagation law. Unfortunately, this procedure induces mathematical correlations, which can strongly affect the chosen test statistics to analyse deformation, if neglected. This may lead potentially to rejecting wrongly the null hypothesis of no-deformation, with risky and expensive consequences. In this contribution, we propose to investigate the impact of mathematical correlations on test statistics, using real TLS observations from a bridge under load. As besides TLS, a highly precise laser tracker (LT) was used, the significance of the difference of the test statistics when the stochastic model is misspecified can be assessed. However, the underlying test distribution is hardly tractable so that only an adapted bootstrapping allows the computation of trustworthy p-values. Consecutively, the extent to which heteroscedasticity and mathematical correlations can be neglected or simplified without impacting the test decision is shown in a rigorous way, paving the way for a simplification based on the intensity model.

KW - 3D point clouds

KW - B-spline surface approximation

KW - Bootstrap

KW - Deformation analysis

KW - Intensity model

KW - Laser tracker

KW - Mathematical correlation

KW - Terrestrial laser scanner

KW - Test statistics

UR - http://www.scopus.com/inward/record.url?scp=85071515807&partnerID=8YFLogxK

U2 - 10.3390/s19173640

DO - 10.3390/s19173640

M3 - Article

C2 - 31438564

AN - SCOPUS:85071515807

VL - 19

JO - Sensors

JF - Sensors

SN - 1424-8220

IS - 17

M1 - 3640

ER -