POTENCIAL HIDRÓGENO

In this chapter we have briefly reviewed several traditional and state-of-the-art principles and related classical, diagnostic and robust methods from different domains with their advantages and disadvantages. Although these have been reviewed for general datasets, their relevance to 3D point clouds and large datasets is mentioned when needed, specifically because we are dealing with large datasets with low dimensionality. In the following chapters, we will adopt the relevant principles and methods in our proposed methods or will be used for comparison.

In the next chapter, we will investigate and propose algorithms for robust planar surface fitting in 3D point cloud data.

“Don’t judge each day by the harvest you reap but by the seeds that you plant.”

Robert Stevenson

“. . . the statistician knows. . . that in nature there never was a normal distribution, there never was a straight line, yet with normal and linear assumptions, known to be false, he can often derive results which match, to a useful approximation, those found in the real world.”

George Box

Chapter 3

Robust Planar Surface Fitting

3.1

Introduction

Surface reconstruction and recognition is an important task for extracting information from point cloud processing in subjects including computer vision, computer graphics, computational geometry, photogrammetry, reverse engineering, remote sensing and robotics (Hoppe et al., 1992; Huang and Menq,

2001; Vosselman and Maas, 2010; Weber et al., 2012; Heo et al., 2013). Fitting planes and estimating subsequent plane parameters are essential in point-based representations. Much research has been carried out on accurate local surface fitting and local point set property (e.g. normal) estimation. In surface reconstruction, commonly the quality of the approximation of the output surface depends on how well the estimated surface normals approximate the true normals of the sampled surface (Dey et al., 2005). Surface segmentation, reconstruction, object modelling and rendering are related to each other. They are closely related to local normal and curvature estimation and mostly depend on accurate plane fitting (Hoffman and Jain, 1987; Hoppe et al., 1992; Huang and Menq, 2001; Li et al., 2010). The accuracy of plane extraction and fitting is a cornerstone for later steps of the object modelling pipeline. The Least Squares

(LS) method is the most well-known classical method for parameter estimation, and Principal Component Analysis (PCA) is one of the most popular techniques mainly used for dimension reduction. Both of these methods and their variants are popular for plane fitting. It is known that the methods are influenced by outliers and can lead to inconsistent and misleading results (Mitra and Nguyen,

2003). Point cloud data is acquired mostly by various measurement processes using a number of instruments (sensors) and can easily be distorted by noise and/or outliers. Sotoodeh (2006) pointed out that the physical limitations of the sensors, boundaries between 3D features, occlusions, multiple reflectance and noise can produce off-surface points that appear to be outliers. Apart from noise and outlier contamination, fitting an accurate plane to point cloud data can be complicated by non-uniform point density, incomplete regions of scanned objects and the presence of multiple structures. Many people use RANdom SAmple Consensus (RANSAC) to reduce outlier/noise effects and for robust model parameter estimation (Schnabel et al., 2007; Gallo et al., 2011; Sanchez and Zakhor, 2012). LS, its equivalent PCA and RANSAC are the three most popular and classical techniques for fitting planes in 3D data (Hoppe et al.,

1992; Pauly et al., 2002; Schnabel et al., 2007; Klasing et al., 2009; Deschaud and Goulette,2010;Heo et al., 2013).

It is logical, if the uncertainties of the sampled points in the point cloud data are known, then the outliers can be tested against prior knowledge. However, this is not always possible, and if it is possible, it is non-trivial. It has been demonstrated that the uncertainty of a point is highly depended on the attributes of the scanner and the scanner geometry, such as distance and surface orientation (Bae et al.,

2005; Soudarissanane et al., 2011). This information is not always available, such as when a scene comprises multiple co-registered laser scans acquired from different positions. The properties only relate to a single point, not to the local sampled surface model. The surface properties will be based on pooled variance models based on each scan. It should also be pointed out that to find the scanner geometry, the local surface must be adequately defined, a process that will also be affected by the presence of outliers and could cause errors in the calculated uncertainties. In recognition of these factors, this chapter focuses on examining the points robustly, based on the local neighbourhood distribution.

of statistics have been proposed (Huber, 1981; Hampel et al., 1986; Stewart,

1999). Robust statistics produce statistical procedures, which are stable w.r.t. small changes or deviations in the data, and even large changes in the underlying data pattern cannot cause a complete failure of the procedures. Alternatively, diagnostic statistics condition on the fit using standard methods to attempt to diagnose incorrect assumptions, allowing the analyst to modify them and refit under the new set of assumptions (Huber, 1991; Stahel and Weisberg, 1991). It is known that the two branches are complementary and their combined use is argued to be more effective for producing highly robust estimators and to detect multiple outliers and/or high leverage points (Rousseeuw and Leroy, 2003).

We want fast fitting as well as accuracy of planar surfaces to be able to efficiently process point clouds consisting of large amounts of unorganized points. We propose six variants of Fast-MCD (Rousseeuw and Driessen, 1999) and Deterministic MCD (DetMCD; Hubert et al., 2012) based robust algorithms that use diagnostic, robust and the combination of diagnostic and robust (diagnostic-robust) statistical approaches for planar surface fitting in 3D point cloud data, which are able to find outliers and robust estimates of the parameters at the same time. The proposed robust plane fitting methods also produce robust normal and curvature values. The accuracy and robustness of the methods are compared w.r.t. the size of the data, outlier percentage, point density variation, speed of computation and for different applications in point cloud analysis. We compare the new methods with LS, PCA, RANSAC and MSAC (M-estimator SAmple Consensus), and show that the results from the proposed methods are significantly better than the existing methods.

The remaining sections are as follows: Section 3.2 presents a short literature review. Section 3.3 formulates the necessary calculations for the proposed algorithms. Section 3.4 implements the proposed algorithms for fitting planes and to get robust local normals and curvatures from the best-fit-plane. In Section 3.5, we experiment, evaluate and compare the results and the performance of the proposed techniques with the other methods using simulated and real mobile laser scanning (MLS) datasets. Section3.6 explores the computational speed and effort of the proposed algorithms followed by conclusions in Section 3.7.