Special Section on SMI 2020Data-driven quasi-interpolant spline surfaces for point cloud approximation
Graphical abstract
Introduction
Due to the recent progress in the acquisition by laser scanners, photogrammetry and diagnostic devices and the popularity of remote sensing technologies, there has been an exponential growth in the availability of data and the need of efficient representations. A good representation model must be, at the same time, efficient, that is based on the minimum amount of data, yet effective, that is able to support conservative conclusions and keep as much information as possible. To handle large data volumes it is often necessary to adopt approximation strategies so that a single point becomes representative of a region or a set of properties [1], [2]. Given a set of measurements (and their spatial locations), its model approximation has to permit deducing information about the process that generated those data, even at locations different from those at which the measurements were obtained. Surface reconstruction, terrain elevation estimation and the approximation of rainfall and pollution fields over Digital Elevation Models (DEMs) are all examples of spatial data approximation. Moreover, an efficient approximation allows the recovery of the digital representation of a physical shape that commonly contains a variety of properties and defects, such as: geometric features; noise and outliers; perturbations introduced when the acquisition conditions are not optimal (low resolution of instruments, motions, etc.) or different acquisition techniques collect data sets of different resolution (laser scans or photogrammetry); incomplete data (e.g., in the acquisition of broken artefacts or scans of objects partially occluded [3]).
Quasi-interpolation schemes [4], [5] are popular for data approximation because, unlike traditional least squares approximations, they do not require solving a linear system. We use B-splines as basis functions because they are computationally convenient, for instance with respect to the common radial basis functions, as (piecewise) polynomial bases require low-order integration schemes to be exactly computed. The use of piecewise algebraic approximations makes our method suitable also to CAD applications, where B-splines and NURBS are de facto the standard tools. Moreover, the treatment of essential boundary conditions is more natural for structured approaches like the spline-based ones than in meshless methods, as it relies on the number of repetition of each knot value.
The wQISA implementation proposed in this paper is specifically designed to define powerful prediction methods from low quality points. Starting from the theoretical wQISA definition given in [6] for a single tensor-product mesh, in this work we derive a multi-level approximation algorithm and provide an extended comparative analysis with other methods in the literature. The main contributions of the paper include: a detailed description of the wQISA method when used for surface approximation; a multi-level approximation algorithm based on a data-driven definition of the weight functions; an extensive comparison of the wQISA outcome with other well-known continuous approximation methods.
The remainder of the paper is organized as follows. Section 2 overviews the literature on data approximation focusing on methods related to continuous surface approximation. Section 3 overviews the wQISA method and its properties, together with a multi-level implementation of the method based on a data-driven mesh refinement strategy. Section 5 extensively compares the wQISA outcome on different real use cases, with respect to a number of well-known approximation methods and a set of performance indicators that are detailed in Section 4. Discussions and concluding remarks are provided in Section 6.
Section snippets
Previous work
The literature on data approximation is vast and we cannot do justice to all contributions. We limit our review to methods whose output satisfies some smoothness requirement, either local or global. For a more complete list of methods related to our use cases, we refer to [7] for a recent survey on surface reconstruction methods, to [8] for an overview on methods for modelling terrain data and to [9] for a comparative analysis of methods for approximating rainfall data.
Meshless methods. Kriging
Weighted Quasi Interpolant Spline Approximations
Quasi-interpolation is a low-cost and accurate procedure in function approximation theory. The term quasi-interpolation has been interpreted differently according to the context. We follow the Cheney [33] definition of a quasi interpolant as any linear operator L of the formwhere is a function being approximated, (xi, yj) are given nodes and are functions at our disposal. Differently from classic Quasi Interpolation methods that
Experimental settings
Despite the popularity of data approximation, we could not find a benchmark able to address all the use cases we are targeting (surface reconstruction, terrain modelling and spatial data measurement approximation); thus we propose a new comparative analysis. In this section we detail the approximation methods and the performance metrics we adopt in our work.
Experimental simulations
In our experiments we consider real data coming from different use cases. We classify the data as affected by different levels of noise: low (e.g., 3D point clouds from high-quality laser scans), average (e.g., terrains from Lidar or sonar acquisition) and high (e.g., air pollution and rainfall measurements from sensors and radars). The data size varies from few dozens to hundreds of thousands of points. To provide a fair comparative analysis – and avoid any method to overfit – a data-driven
Concluding remarks
The weighted quasi interpolant spline approximation (wQISA) is a simple and robust procedure to obtain a spline approximation of point clouds. This paper presents a data-driven implementation of wQISA [6], inspired by the supervised learning paradigm. The method has been tested on several real-world data from different application domains and different levels of noise. Experiments have shown that wQISA is able to accurately generalize to data different from those used to define it, i.e., it can
CRediT authorship contribution statement
Andrea Raffo: Methodology, Investigation, Software, Formal analysis, Validation, Visualization, Writing - original draft, Writing - review & editing. Silvia Biasotti: Conceptualization, Supervision, Methodology, Resources, Visualization, Writing - original draft, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 675789. The work has been partially developed in the CNR-IMATI activities DIT.AD021.080.001 and DIT.AD009.091.001. The authors also thanks: Dr. Bianca Falcidieno and Dr. Michela Spagnuolo for the fruitful discussions; Dr. Oliver J. D. Barrowclough, Dr. Tor Dokken and Dr. Georg Muntingh for their concern as supervisors; Dr. Vibeke Skytt
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