Geometric segmentation of 3D scanned surfaces

Highlights

• A new method to segment geometric features in discrete geometric models is proposed.

• Sharp edges, defective zones and 10 different types of regular points are recognized.

• The method requires just a few setting parameters that are not critical.

• It works with real scanned geometries, highly noised and not well-sampled models.

• The point type association is not affected by the singular properties of the point.

Abstract

The geometric segmentation of a discrete geometric model obtained by the scanning of real objects is affected by various problems that make the segmentation difficult to perform without uncertainties. Certain factors, such as point location noise (coming from the acquisition process) and the coarse representation of continuous surfaces due to triangular approximations, introduce ambiguity into the recognition process of the geometric shape. To overcome these problems, a new method for geometric point identification and surface segmentation is proposed.

The point classification is based on a fuzzy parameterization using three shape indexes: the smoothness indicator, shape index and flatness index. A total of 11 fuzzy domain intervals have been identified and comprise sharp edges, defective zones and 10 different types of regular points. For each point of the discrete surface, the related membership functions are dynamically evaluated to be adapted to consider, point by point, those properties of the geometric model that affects uncertainty in point type attribution.

The methodology has been verified in many test cases designed to represent critical conditions for any method in geometric recognition and has been compared with one of the most robust methods described in the related literature.

Introduction

The process of organizing the nodes of a triangular mesh into groups, which correspond to surface features, is called data segmentation. Segmentation is an important and preliminary stage in many applications involving geometric models described by point clouds or triangular meshes. There are two criteria for surface segmentation  [1]: semantic segmentation and geometric segmentation. Semantic segmentation methods attempt to segment the surface (typically “natural objects”) into meaningful pieces as expected from a human observer. Geometric segmentation methods, in contrast, aim at dividing the mesh into clusters that have similar geometric properties. These methods are mainly used to recognize the nominal geometry of discrete geometric models obtained by reverse engineering. These two criteria give rise to completely different categories of segmentation methods. This paper is concerned with geometric segmentation.

The geometric segmentation of a surface utilizes the identification of the peculiar characteristics of the geometric shape and, subsequently, assigns the identification a linguistic attribute of a portion of the surface: spherical, ruled, toroidal, planar and generic. Geometric segmentation is typically based on point classification (Umbilical, Hyperbolic, Parabolic, Elliptical and Planar), which is performed through the analysis of the differential geometric properties (DGP) that locally characterize the surface shape. The classification of a point on an analytical surface is unambiguous and uniquely defined because it is performed by a deterministic process that is based on a curvature analysis at that point. A precondition necessary to evaluate the DGP is that the point on the surface must be regular. On an analytical surface, a point is regular if the two following conditions are satisfied  [2]:

• There exist derivatives of all orders; the surface at the point is a differentiable surface $\left(C^{\infty }\right)$;

• There exists a tangent plane.

The vertices of a tessellated surface are naturally non-regular points because only $C^0$ continuity is verified. In this case, regularity must be assumed as an attribute of the vertex whose neighborhood can be considered as similar to a smooth surface: the same surface from which the tessellation is derived. The differential geometric properties at a mesh’s point that are recognized to be regular are not properties intrinsic to the point; rather, it is at the location of the points in the neighborhood that information concerning regularity and DGP is retained. Because discrete surfaces are not regular by nature, the recognition of regularity and the estimation of the DGP are uncertain. Although the studies described in the literature for geometric segmentation are based on the DGP analysis, no study considers the uncertainties in the attribution of regularity. In this work, the uncertainties in regularity recognition are taken into consideration and are used in a single process aimed at detecting non-regular zones and 10 different types of geometric surfaces.

Very often, discrete surfaces come from the measure of a real object. In these cases, the source of uncertainty is due both to the measurement errors and to the non-ideality of the acquired object geometry, which introduces noise in the tessellated model. Uncertainties in DGP estimation also depend on the acquired point density and on the way the sampling is performed. This is highlighted in cases A, B and C in Fig. 1. The three tessellated models are obtained from an analytical cylindrical surface that has been sampled in three different ways. When estimated on the tessellated surface, the Gaussian curvature $\left(K\right)$, which is theoretically null, gives rise to different values depending on the method used to evaluate the DGP (paraboloid approximation  [3], discrete differential-geometry-operator-based method  [4]) and on the way the surface is tessellated. A parabolic point is recognized for $K=0$, but this value is never found except in case C, where the points of the tessellated model are aligned with the generatrix and the angles at a vertex are less than $\pi /2$. The previous case demonstrates the uncertainty in the attribution of parabolic point types in a tessellated surface when the value of $K$ is different from 0. A point, for which the Gaussian curvature is not zero, but even if it is not very far from zero, can be parabolic, although it can be elliptic or hyperbolic.

To solve for the uncertainty, in this work, the geometric type attribution is performed in a possibilistic way so that all of the cases compatible with the estimated curvatures are considered, each case having a proper level of possibility. If it is to be selective, the criterion used to state that a point is of a specific type has to be restrictive, but it also needs to be sufficiently broad to consider the intrinsic uncertainties when evaluating the curvatures. In this paper, the possibility of the point type is assigned by a dynamic criterion that, point by point, considers the specific state of the tessellated surface (noise and surface quality sampling). This approach exceeds the limitations of the methods reported in the related literature that use a rigid recognition criterion based on a comparison of the geometric identifier parameter with a unique threshold value assigned for the whole geometric model. To address indecision in assigning a type to a point, which may have different possible attributions, the nature of the other points in the neighborhood zone is investigated and considered. Thus, the point type is identified as a property of a finite compact area by a region-growing algorithm driven by a similarity criterion.

The most appropriate way to implement the methodology proposed here is using the theory of possibility, or fuzzy set theory. For assigning possibilities, the typical parameters used to classify the point type (Mean curvature $H$, Gaussian curvature $K$ and Curvedness index $R$   [5]) have been replaced by three new parameters that are more suited to implement a geometric segmentation method based on fuzzy logic.