Using Structural Risk Minimization to Determine the Optimal Complexity of B-Spline Surfaces for Modelling Correlated Poi
The increased use of areal measurement techniques in engineering geodesy requires the development of adequate areal analysis strategies. Usually, such analysis strategies include a modelling of the data in order to reduce the amount of data while preservi
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Abstract
The increased use of areal measurement techniques in engineering geodesy requires the development of adequate areal analysis strategies. Usually, such analysis strategies include a modelling of the data in order to reduce the amount of data while preserving as much information as possible. Free form surfaces like B-splines have been proven to be an appropriate tool to model point clouds. The complexity of those surfaces is among other model parameters determined by the number of control points. The selection of the appropriate number of control points constitutes a model selection task, which is typically solved under consideration of parsimony by trial-and-error procedures. In Harmening and Neuner (J Appl Geod 10(3):139–157, 2016; 11(1):43–52, 2017) a model selection approach based on structural risk minimization was developed for this specific problem. However, neither this strategy, nor standard model selection methods take correlations into account. For this reason, the performance of the developed model selection approach on correlated data sets is investigated and the respective results are compared to those provided by a standard model selection method, the Bayesian Information Criterion. Keywords
B-spline surfaces Correlated point clouds Model selection Point cloud modelling Structural risk minimization VC-dimension
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Introduction
With the development of the terrestrial laser scanner (TLS) a measuring instrument which allows a fast, efficient and contactless data acquisition moved into focus of engineering geodesy (Heunecke et al. 2013). The acquired data is of high spatial and temporal resolution and, therefore, forms an excellent basis to solve engineering geodetic tasks like geometric state descriptions or spatio-temporal deformation analyses. However, despite of its many advantages, the use of laser scanners also holds new challenges (see for example Mukupa et al. 2016 or Holst and Kuhlmann 2016).
C. Harmening () · H. Neuner Department of Geodesy and Geoinformation, TU Wien, Vienna, Austria e-mail: [email protected]
One of the major challenges is the development of appropriate deformation analysis strategies which are able to deal with the huge amount of data. When developing areal analysis strategies, the choice is between five possible ways to handle point clouds with respect to a subsequent deformation analysis (Ohlmann-Lauber and Schäfer 2011). Among them, a frequently used strategy is geometrybased. It includes a geometric modelling of the point clouds in order to reduce the amount of data while preserving as much information as possible. Applications of the geometrybased approach using geometric primitives like planes or cylinders can be found in Erdélyi et al. (2017), Lindenbergh and Pfeifer (2005) or Vezoˇcnik et al. (2009). However, when applying this approach to complex structures like domes, arch bridges or even natural objects, flexible mathematical functions are required. Free form surfaces like Bsplines have been proven to be particularly suitable to
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