Problem Formulation

It is a fundamental idea to segmentation that points in a segment have similar characteristics. First, we investigate the problems of identifying the underlying pattern of surface points, and then formulate the expected characteristics for the points to be in the same object surface.

It is logical to assume that every point in a sufficiently-small (reasonably sized) local area (neighbourhood) is on a planar surface. This assumption of points on a plane is useful, and leads to using the information about local saliency features to check the behaviour of a point on a smooth surface. Figure 5.1(shown in one dimension for clarity) illustrates that under certain conditions points on different local planar surfaces (Figure 5.1a) may be on the same smooth surface (Figure 5.1b), where three different planes of different orientations appeared as a single smooth surface. We see three planar surfaces (Figure 5.1a) that appear to have discontinuities (gaps) between them. The first two planes from the left appear to have a crease edge and the last two planes appear to have a step edge at their boundaries. If the gaps between the two boundary points of different planar surfaces are not enough to consider them separate then they may be co-surface points under certain coherence criteria, otherwise discontinuity appears in the gaps.

(a) (b) Euclidean Distance (ED)

Orthogonal Distance (OD) _{Coplanar/smooth surface point} Outlier Normal Best-fit-plane Coplanarity distance threshold

Figure 5.1 (a) Three different planar surfaces, and (b) co-planar smooth surface.

5.3.1.1 Edges, Gaps and Outliers

In this Section, we explain why edges, gaps and outliers should be considered carefully at the time of segmentation. Figure 5.1 (in the previous section) shows

that points near gaps between two boundary points and edges/corners need more attention when determining which points belong to which specific surface. Edges and gaps are the two situations where properties or attributes of the surface point may be falsely estimated. The presence of outliers may intensify the problems of edges and gaps. Therefore, exploring the properties of edges, gaps and outliers in the data needs to be investigated for a proper understanding about an object’s surfaces, and will help when estimating reliable surface point attributes and for formulating appropriate test criteria for robust and accurate segmentation.

Edges: Many authors use edge/corner information to separate different surfaces.

Near edges and corners, known as geometric singularities, normals are usually differently oriented and discontinuous. Hoffman and Jain(1987) stated that edge points may delineate surface patches and therefore be useful for modelling. A common effect is rounded or smoothed normal estimates along edges (Castillo et al., 2013). Three most common types of edges in point cloud data shown in Figures 5.2(a, b and c) are: (i) step/jump edges that occur where a surface undergoes a discontinuity and the boundary points on the two parallel planes close to the discontinuity have the same orientation, (ii) crease/corner edges e.g. where two sides of a roof meet, and (iii) smooth or virtual edges that can be characterized by continuity of the orientation of normals i.e. smoothly changing across the surface, but discontinuities of curvature e.g. where curvature goes from +ve to −ve suddenly. Usually, a step edge appears when an object obstructs another object, and for the crease edges, the normals of the surface points are influenced by different planes.

(a) (b) (c)

Figure 5.2 Three types of edges; normals (red arrows): (a) step edge, (b) crease edge, and (c) smooth edge.

Gaps: Improper sensor alignment, error in data acquisition because of faulty sensors, unexpected interruption in data collection, surface point density variation and/or obstacles that may obstruct the laser pulse may cause gaps

such as discontinuities and holes in the data. Figure 5.3 shows some types of gaps that are common in point cloud data. There is a possibility to wrongly join the two different surfaces together into one segment if the gaps between two individual surface points cannot be identified properly. In addition, real gaps can be filled by faulty boundary extension in the presence of outliers/noise. So, a thorough analysis of the neighbouring surface points based on their proximity criteria is useful for a proper understanding about the gaps between the relevant neighbouring points, and helps to avoid the problems of misleading gaps.

(a) (b) Gaps (c)

Figure 5.3 Gaps in different surface positions, red and green arrows show normal orientation and directions of gaps, respectively: (a) gap between two horizontally distant planes, (b) gap between two horizontally as well as vertically distant planes, and (c) gap between two vertically distant planes.

Outliers: Usually outliers are classified as the points that are far from the

majority of points in the data, and/or do not follow the same pattern as the majority of points (Barnett and Lewis, 1995; Rousseeuw and Leroy, 2003). Moreover, the existence of multiple structures in a dataset may create psuedo-outliers. In a general sense, noise can appear as off-surface points and behave like outliers in many cases (Sotoodeh, 2006). Covariance statistics based on an outlier contaminated local neighbourhood may produce inaccurate normals and curvatures. The presence of outliers in different positions (especially around or on the edges and boundaries) on a surface causes errors in the estimates of local saliency features such as normals. For example, the effect may cause continuous/smoothed normals along the edges and corners. Outliers between two points in a neighbourhood can produce erroneous discontinuities in a homogeneous surface. Points in a local neighbourhood in the presence of outliers results in the tangent plane being biased to the direction of the outliers. The inclusion of outliers between the gaps of two neighbouring surface points

can erroneously join two surfaces. Figure 5.4 illustrates the influence of outliers in different positions in a local neighbourhood that causes the change of real orientation of the local plane. Figure 5.4a shows how the presence of an outlier between two vertically distant parallel planes changes the orientations of the two, and wrongly joins them together, Figure 5.4b shows how an off-surface point may appear as an outlier and changes the orientation of a plane to its own direction, and Figure 5.4c shows how the presence of an outlier between a pair of horizontally distant surfaces joins them erroneously.

Figure 5.4 Influence of the presence of an outlier (red point) in different positions: (a) outlier between two vertically distant parallel planes, (b) outlier as an off-surface point, and (c) outlier between two horizontally distant co-planar surfaces.