Data-driven quasi-interpolant spline surfaces for point cloud approximation

Highlights

• A detailed description of the wQISA method when dealing with surface approximation.

• A multi-level approximation algorithm based on a data-driven weight definition.

• Comparative analysis of the wQISA outcome with other continuous approximations.

Abstract

In this paper we investigate a local surface approximation, the Weighted Quasi Interpolant Spline Approximation (wQISA), specifically designed for large and noisy point clouds. We briefly describe the properties of the wQISA representation and introduce a novel data-driven implementation, which combines prediction capability and complexity efficiency. We provide an extended comparative analysis with other continuous approximations on real data, including different types of surfaces and levels of noise, such as 3D models, terrain data and digital environmental data.

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.