1.1. Motiviations
The importance of automotive exteriors has grown as competition in the automotive industry has intensified. Automotive outer panels include narrow feature lines that exhibit long curved geometries [
1], and the shape and dimensions of feature lines are essential because they determine the appearance of a car. Examples of feature lines on an automotive door outer panel and a sectional view are illustrated in
Figure 1 and
Figure 2, respectively. Automotive outer panels are manufactured using the stamping process, which is a metal-forming process that combines drawing and bending deformations. Therefore, it is difficult to form the panel into a designed shape. Feature lines with larger angles and smaller radii are more difficult to create and are known as subtle feature lines [
2]. For example, an automaker defines a feature line with an angle greater than 155° and a radius less than 48 mm as a subtle feature line and manages it separately.
It is difficult to measure and inspect the shape of a subtle feature line because of the precision required and its small size. The subtle feature line section includes an arc with lines at either ends, as shown in
Figure 2, and the radius (R) of the arc and angle (θ) between the two lines specifies the shape. With an increase and decrease in the angle and radius of the subtle feature line section, respectively, the width of the subtle features decreases, making the measurement and inspection difficult. Although the typical chord length of the arc in the section measures up to 10 mm, precision, measured in micrometers, is required to estimate the arc’s radius. In addition, the roughness of the sheet metal panel and noise in the measured data make inspecting subtle feature lines difficult.
1.2. State of the Art and Related Work
Template matching helps inspect the geometry of subtle features in automakers; however, it is not accurate and does not produce numerical results. There are two types of measurement methods: noncontact and contact methods [
3]. A coordinate measuring machine (CMM) employs a contact method that moves a probe along the outer surface and measures its movement in three dimensions [
4]. The measurement speed of the CMM is extremely low in contrast to other methods. A non-contact 3D scanner can quickly and simultaneously measure coordinates over a large area; however, its measurement accuracy is lower than contact methods. Typical methods are not suitable for measuring subtle features regarding the measurement scale and precision.
A confocal microscope [
5] and surface-roughness testers can be used to measure the subtle feature line. Confocal microscopy, a non-contact mechanism, possesses high precision for small areas; however, it takes hours to measure a small region. A surface-roughness tester measures the surface’s roughness using a contact method and measures the surface profiles with sub-micrometer resolutions. However, surface-roughness testers are typically capable of measuring surface profiles up to a length of 10 mm. Lee et al. measured the sections of subtle feature lines using a surface-roughness tester and demonstrated that the tester is a good method for measuring sections of the subtle feature line [
2]. This study used data points of the surface profiles measured using a surface-roughness tester.
It is difficult to estimate the angle and radius based on the measured points on a subtle feature line. As shown in
Figure 2, the angle between the two lines and the radius of the central arc are the main parameters of the subtle feature line. The angles and radius can be inspected via the line-arc-line curve by approximating the measured points on a line-arc-line curve. In this study, the random sampling consensus (RANSAC) algorithm proposed by Fischler and Bolles [
6] was used to approximate the measured points on a line-arc-line curve. RANSAC is an iterative method for estimating the parameters of a mathematical model from a dataset containing noise and outliers. This process includes two iterative steps. First, the algorithm randomly selects a sample subset of data from the input dataset and estimates the parameters of the model based on the sample data. Next, the algorithm checks the elements of the entire dataset that are consistent with the model obtained in the first step. The number of sample subsets was sufficiently small enough to determine the model parameters. If a data element does not fit the model within the tolerance limit, it is considered an outlier; otherwise, it is an inlier.
Chida and Masuda used the RANSAC algorithm to estimate the shapes from a measured point cloud [
7]. They subdivided a point cloud into small regions to extract surfaces in practical time. Kawashima et al. used an algorithm to fit the line of a pipe using laser-scanned point clouds [
8]. Robertson et al. applied the algorithm to classify discontinuities and extract parametric models from low-quality three-dimensional data [
9]. This study used the RANSAC algorithm to estimate an arc based on a measured dataset. In this study, computation times were sufficiently quick because the number of data points was minor in contrast to the number of point clouds from the surface.
Fitting geometric shapes to point data is a common geometric problem. Many researchers studied fitting a geometric model to a point sequence or point cloud. Numerous techniques, such as least-squares, principal component analysis, optimal transport, graph theory, and machine learning, have been used for fitting geometric shapes to point data. Currently, circle fitting has become a trivial problem, as most algorithms are public and easily accessible over the internet. Bo et al. proposed a graph-based method to fit B-spline curves to a planar point cloud, in which multiple curves with intersections were extracted [
10]. Suzuki et al. proposed a method for matching computer-aided design (CAD) data and scan data [
11]. The scan data in their study were largely deformed simulation models, and the deformed CAD data were obtained using the matching method. He et al. [
12] fitted B-spline curves to numerical control (NC) milling paths using the modified progressive and iterative approximation method proposed by Qi et al. [
13]. They incorporated the energy term into a progressive iterative approximation method to avoid numerical instability and they lowered chord errors by stretching the energy term.
Various fitting methods have been proposed for point data with noise and outliers [
14,
15,
16,
17]. Most of these methods first remove noise through clustering, thinning, or averaging; however, they often lose noteworthy features. Song proposed an algorithm to reconstruct curves from unorganized points using clustering, confining, and thinning. In this study, a minimal spanning tree is used to remove noise from the data [
17]. Ghorbani and Khameneifar proposed an automatic reconstruction method for sectional airfoil profiles obtained from inspection data points [
18]. They used a recursive weighted local least-squares technique to fit curves within the uncertainty measurement of data. The measured data in this study include wave and random noise, and the geometric model is known as a line-arc-line curve.
1.3. Contributions
This paper proposes a curve-fitting algorithm and analyzes its stability in practical applications. The algorithm fits a line-arc-line curve to the data points obtained by measuring the cross-section of the subtle feature lines using a surface-roughness tester. Three types of data points were used to analyze the stability of the algorithm: simulated data, measured data from laboratory specimens, and measured data from real panels. The main contributions of this paper are outlined as follows:
A curve-fitting algorithm for evaluating the accuracy of the subtle feature line is presented.
Numerical, experimental, and actual specimens are tested to verify the proposed algorithm.
Moreover some applications of the proposed algorithm are stated as follows:
The rest of this paper is organized as follows:
Section 2 provides details of the proposed algorithm.
Section 3 describes the results and analysis of the algorithm. Finally, the last section concludes this paper.