In this SpringerBrief, we went through the mathematical concepts of LR B-splines in all its facets, explaining in details how the spline space can be refined and the different strategies for adaptive approximation. Through detailed example with seabed and terrain data sets, we have highlighted how adaptive surface approximation of various noisy and scattered point clouds is performed concretely. We showed how to deal with challenges raised when working with real data set such as voids or outliers. We presented numerous applications that can be derived from ”transforming data to information”. More specifically, we reviewed:

  1. 1.

    How adaptive local refinement can be performed by combining multi-level B-spline approximation and least squares with a low computational burden,

  2. 2.

    How parameters such as tolerance, number of iterations of the algorithm, and refinement strategies affect the surface approximation,

  3. 3.

    How statistical concepts can be used to judge the goodness of fit and determine parameters of the surface approximation, such as the tolerance, or the bidegree of the splines space,

  4. 4.

    How outliers can be removed efficiently and how to fuse data from different sources to perform efficient surface approximation,

  5. 5.

    How voids can be handled by applying trimming,

  6. 6.

    The potential of LR volumes for spatio-temporal analysis of point clouds,

  7. 7.

    The computation of contour lines from the mathematical approximations as an additional application.

We have illustrated the principle of surface approximation with various examples, using data from terrestrial laser scanner, sonar, and terrain or seabed data set. We have proposed the LR B-spline surface and volume as a new and promising format for representing noisy and scattered point clouds in a compact form. This approximation method provides a middle road between the rigid, but effective regularity of the raster format and the large flexibility of triangulated surfaces. LR B-spline surfaces are smooth and can, due to their adaptive potential, represent local detail without a drastic increase in data size. For point clouds coming from sensors having a very high data rate and containing millions of points, the computational time of the surface approximation stays manageable: This is a strong argument for a wide acceptance of the local iterative fitting to represent scattered and noisy data mathematically. The surfaces can be exported as rasters in various resolutions as well as collections of tensor product spline surfaces. All software are freely available to promote usage.

7.1 A Promising Application: The LR B-Spline Volume for Spatio-temporal Analysis of Geomorphological Changes

The analysis of spatio-temporal deformation is one of the most promising applications of LR B-spline surfaces and volumes. These latter will provide a framework for analysing geomorphological changes without having to work on noisy and scattered point clouds of different quality, coming from different sensors. Detecting and analysing the movements of, e.g., sand dunes and snow masses is a potential use of LR B-spline volumes to gain a better understanding in the underlying geophysical processes. Similar applications are conceivable within the context of geodetic deformation analysis of structures, such as tunnels or dams. Advanced methods could take features such as extremal points, ridges and valleys into account. Outliers should be efficiently identified during the computation of the approximating surface and removed during the iterative algorithm presented. The outlier problematic is an important topic for further work and has been addressed in the present SpringerBrief.

7.2 Ongoing Research

Ongoing research focuses on identifying the most optimal surface. New criteria could be investigated to judge the quality of the approximation by accounting for the sensor noise, setting a tolerance and a stop criterion for the number of iterations in the approximation algorithm. We have introduced this challenge within the context of model selection, but alternative strategies should be developed. They could depend on the application at hand, as a balance between computational time, needed accuracy of the surface and overfitting avoidance. The detection and analysis of deformation or changes between two or more surface approximations or inside an LR B-spline volume remain an open topic, for which the definition of distance between mathematical surfaces should be further investigated. This latter should account for both the uncertainty of the measurements and the surface/volume approximation.

Last but not least, we expect that the capacity of LIDAR and sonar type data acquisition technology will continue to grow in the next decades. This makes in even more urgent to turn the enormous more or less structured point clouds into mathematically structured information that can be efficient interrogated and analysed. We believe that representations that allow the granularity of the representation to adapt to the local behaviour of the underlying information intrinsic in point clouds are essential in this respect.