Geometric segmentation of 3D scanned surfaces☆
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]:
- (i)
There exist derivatives of all orders; the surface at the point is a differentiable surface ;
- (ii)
There exists a tangent plane.
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 , 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 , 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 . The previous case demonstrates the uncertainty in the attribution of parabolic point types in a tessellated surface when the value of 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 , Gaussian curvature and Curvedness index [5]) have been replaced by three new parameters that are more suited to implement a geometric segmentation method based on fuzzy logic.
Section snippets
Related works
Three are the main approaches used in the literature for the geometric segmentationof surfaces [6], [7]: the edge-based, region-based and hybrid methods.
The edge-based approach [8], [9], [10] aims to identify the boundary zones between the geometric features shaping the model. This identification is performed by evaluating the discontinuities in the smoothness of the original surfaces, which in turn is performed by analyzing the data retained at the location of the points of the mesh or cloud.
The new segmentation procedure
The surface segmentation method presented in this paper is based on a region-growing approach and also performs, in a unique growing process, the edge identification. This method determines the uncertainty in feature identification by classifying the points based on the theory of possibility and by driving the growing process by a similarity criterion. The method consists of the following main phases:
- (1)
Association of the possibility of the point type. A total of 11 different point types are
Case studies
The proposed procedure for shape recognition and surface segmentation has been implemented in an original software package coded in Visual C++.
The method has been tested in many different cases. The three most significant cases are reported here and are discussed. The first test case (Fig. 13(a)) refers to an acquired real object where five different types of points and seven geometric features are found. In this object, there are sharp and smooth transitions that identify critical situations
Conclusion and discussion
This paper has described a new method based on fuzzy set theory to segment geometric features in discrete geometric models. The proposed method is shown to properly address some of the problems that typically affect other geometric feature recognition methods presented in the related literature and in commercial software.
The method is new in many aspects:
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the region growing strategy takes advantage of the simultaneous analysis of the geometrical differential properties and the surface
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This paper has been recommended for acceptance by Dr. Vadim Shapiro.