Elsevier

Computer-Aided Design

Volume 31, Issue 7, June 1999, Pages 449-457
Computer-Aided Design

Segmentation of measured point data using a parametric quadric surface approximation

https://doi.org/10.1016/S0010-4485(99)00042-1Get rights and content

Abstract

In reverse engineering, a shape containing multi-patched surfaces is digitized, the boundaries of these surfaces should be detected. The objective of this paper is to introduce a new and computationally efficient segmentation technique for extracting edges, and partitioning the 3D measured point data based on the location of the boundaries.

The procedure begins with the identification of edge points. An automatic edge-based approach is developed on the basis of local geometry. A parametric quadric surface approximation method is used to estimate the local surface curvature properties. The least-square approximation scheme minimizes the sum of the squares of the actual Euclidean distance between the neighborhood data points and the parametric quadric surface. The surface curvatures and the principal directions are computed from the locally approximated surfaces. Edge points are identified as the curvature extremes, and zero crossings, which are found from the estimated surface curvatures.

After edge points are identified, edge-neighborhood chain-coding algorithm is used for forming boundary curves. The original point set is then broken down into subsets, which meet along the boundaries, by scan line algorithm. All point data are applied to each boundary loops to partition the points to different regions. Experimental results are presented to verify the developed methods.

Introduction

Efficient manufacturing of curved objects is an important issue in modern industry as more industrial products are being designed with sculptured or free-formed surfaces. For designing the sculptured products with geometric modeling, a computer-aided design is already used extensively in the industry. Nevertheless, there remain objects which, for various reasons, are not originally described using a computer-aided design system [1], [2]. Thus it is increasingly necessary to create geometric models of existing objects. The industrial jargon for this process is reverse engineering.

Reverse engineering typically starts with measuring an existing object so that a geometric model can be deduced in order to exploit the advantages of CAD/CAM technologies. When a shape containing multi-patched surfaces is scanned, it is imperative to detect the shape changes along the boundaries of these surfaces. Scanned points should be divided between different regions on the basis of the location of the boundaries. Surfaces can then be reconstructed for the points belonging to each of these regions [3], [4]. While reverse engineering is a simple concept, interpreting a measured model is a difficult and complex problem. For a human carrying out reverse engineering, it is easy to recognize if a point is an edge point or not, or to which surface a point belongs to. For the computer, while it is not easy to do segmentation even for simple shapes, further difficulties arise when trying to segment free-form shapes. So, in the standard approach to reverse engineering, a manual or semi-automatic segmentation is performed to divide the point data into a suitable patchwork of smooth surface regions [5]. An ideal reverse engineering system would be able to process a set of point data automatically; however, there is a long way to go to accomplish this.

To divide the whole measurement data points into regions according to shape-change detection has been a long-standing research problem. Previous researches have utilized two basically different approaches to the partitioning of measurement point data, namely edge-based and face-based methods [6], [7], [8], [9], [10]. The first method is one popular approach of a two-stage process, edge detection and linking. This works by trying to find boundaries in the point data representing edges between surfaces. If edges are being sought, an edge-linking process follows, in which disjoint edge points are connected to form continuous edges. This technique thus infers the surfaces from the implicit segmentation provided by the edge curves.

The second technique or ‘face-based’ method goes in the opposite order, and tries to infer connected regions of points with similar properties as groups of points belonging to the same surface, with edges then being derived by other computations from the surfaces. The advantages of a face-based approach are obvious for objects composed of quadric primitives, because the primitives are inherently polynomial in nature. However, face-based approaches suffer from the difficulties in applying to free-form surfaces [5], [10], [11], [12]. In this paper, an edge-based approach has been selected to segment a set of point data including free-form surfaces for this reason.

This paper presents an automatic and computationally efficient segmentation technique, which extracts edges, and partitions the measured point data. Two types of surface boundaries are considered: surface discontinuities and curves of surface-curvature extrema. Surface discontinuities of type G0 and G1 indicate physical events on object surfaces. A G0 discontinuity corresponds to the occlusion of one surface by another (called a step). A G1 discontinuity occurs at a vivid edge of the object surface. The curves of surface-curvature extrema correspond to surface ridge or valley lines, which have been considered as important shape descriptors in extracting surface boundaries. The domain of available point data is usually a simple connected planar region. Any line parallel to the z-axis penetrates the surface at most once.

The procedure begins with the edge points detection. In the edge detection process, the local surface curvature properties are used to identify boundary present in the measured data. Surface curvature makes it much more suitable than global properties for handling free-form surfaces. For describing the local geometry, a parametric quadric surface approximation method is applied. A locally approximated surface minimizes the square sum of the actual Euclidean distances from the neighborhood point data. The surface curvatures and the principal directions are computed from locally least-square approximated surfaces. Edge points are identified as the curvature extremes, and zero-crossings among the estimated surface curvatures.

In the edge-linking process, the edge points are connected to form continuous edge loops. It is assumed that component surfaces are smooth internally and bounded by edge loops. These edges may be partly or entirely the boundaries of some underlying patch, or they may also be trimming curves cutting across the patch structure. In the partitioning process, the original point set can be broken down into subsets, which meet along the edges, by a scan line algorithm. The number of intersection between a scan line and a closed boundary loop is used to decide the interior or exterior of the boundary loops. Based on these techniques, experimental results are given to verify the developed approach.

Section snippets

Detection of edge points

Various approaches are used to detect edge points in the measured data. Edge-based methods can be subdivided into three groups by the edge detection schemes as edge operator method, edge region based segmentation, and surface curvature based segmentation. Edge operators may not be effective in noisy data and have not been popular for measured data [6], [10]. The edge region technique carries out segmentation by focusing on the neighborhoods of the edge regions only. In this technique, the range

Partitioning of measured point data

In partitioning process, the original point set is broken down into subsets, which meet along the boundaries. A method is needed to judge whether the location of a point is the interior or exterior of a boundary loop. Scan line algorithm is used for partitioning of the point data. The algorithm decides the interior or exterior based on the number of intersection between a scan line and a closed boundary loop [16]. When a measured point is given, a scan line is created parallel to x-axis from

Conclusions

The principal objective of this work was to develop a new and computationally efficient segmentation method based on local surface curvature properties. The method breaks down the original point set into subsets, which meet along the surface boundaries.

A parametric quadric surface approximation was applied to estimate local curvature and principal directions. In the parametric surface approximation, the actual Euclidean distances were computed between the point data and the approximated

Minyang Yang is a professor of Mechanical Engineering at the Korea Advanced Institute of Science and Technology. He received BS and MS degrees in Seoul National University, and PhD in Mechanical Engineering from MIT in 1977, 1982 and 1986, respectively. He was a visiting professor at the Technical University, Berlin in 1991. His current research interests include NC/CAM, precision machining and machine tools.

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Minyang Yang is a professor of Mechanical Engineering at the Korea Advanced Institute of Science and Technology. He received BS and MS degrees in Seoul National University, and PhD in Mechanical Engineering from MIT in 1977, 1982 and 1986, respectively. He was a visiting professor at the Technical University, Berlin in 1991. His current research interests include NC/CAM, precision machining and machine tools.

Eungki Lee is a PhD candidate in the Department of Mechanical Engineering at the Korea Advanced Institute of Science and Technology. He received his BS in mechanical engineering from the Yonsei University in 1992 and MS in Precision Engineering and Mechatronics from the Korea Advanced Institute of Science and Technology in 1994. His research interests include NC/CAM, reverse engineering and high speed machining.

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