1. Introduction
In precision measurement, many basic applications with optical sensing technologies are applied, such as the laser interferometer principle, piezoelectric actuator principle, and micro-encoding principle [
1,
2,
3,
4]. One of their common characteristics is that they have exacting requirements on the environment, such as vacuum and equipment components, so it is necessary to control the correlative uncertainty sources [
5]. For example, errors of air wavelengths and refractive index are the essential uncertainty sources [
6] and would lead to Abbe and accumulation errors for multiple degree-of-freedom (DOF) tasks, due to such principles only enabling the measurement of a single DOF [
7]. To avoid these limitations, vision-based techniques are used for precision measurement [
8,
9,
10,
11]. Currently, the vision-based technique is an appealing method for precision measurement because of its technological advantages, including the visualization result, multiple DOF measurement, easy operation and installation, etc. A novel method that attempts to integrate computer vision and ultra-precision machining technologies [
11,
12,
13] has been presented, which is feasible and promising for precision measurement due to its simplicity, space-saving, low-cost, and high-robust features, etc. The template matching algorithm has been employed for image processing and determined the absolute position of the selected image in the global map [
11]. Moreover, a unique surface topography named polar microstructure has been developed for the abovementioned measurement method [
12,
13]. Polar microstructures aim to serve as a unique global map used for the subsequent matching measurement.
Inspired by the polar coordinate system, a polar microstructure is presented [
11], as shown in
Figure 1. To generate the polar microstructure surface topography with straight lines and concentric circle trails, a process chain introduced in the literature [
14] is designed and fabricated by a combination of single-point diamond turning (SPDT) and single-point diamond grooving (SPDG) processes. It is worth noting that the polar microstructure with form accuracy within the micrometer range still possesses a high discrimination rate. Taking into account the specific characteristics on the surface of the polar microstructure, both the geometric pattern and the arrangement of pixel intensity values are unique. This is due to the unique gray-scale intensity distribution of the polar microstructure that ensures the high reliability and robustness for the optical precision measurement with this polar microstructure.
However, the determination of machining parameters is necessary to fabricate polar microstructures. There are many factors to consider in order to meet the functional performance requirements, which include the parameters for microscopic observation, etc. Currently, there are only a few examples of research concerning the optical resolution of micro-structured surface influenced by cutting strategy [
15,
16,
17,
18]. In this paper, an investigation into the influence of machining parameters of polar microstructures to satisfy the extraction of the feature point matching is presented. First, the chosen field of view was investigated. After that, the modeling of polar microstructures is described, and the algorithm for detecting the feature points is explained. In the results and discussion section, three important parameters for the cutting strategy for polar microstructures are identified, which are tool geometry, depth of cut, and groove spacing. The optimized parameters were obtained according to the simulation and experimental results.
3. Modeling of Surface Generation for Polar Microstructures
According to the process chain combining SPDG and SPDT, a polar microstructure is a workpiece with a specific surface microstructure.
Figure 6 shows the processing principle of SPDT and SPDG. The workpiece is mounted and rotates on the spindle, and the diamond tool mounted on the machine slides on the machine tool.
SPDT(Pre) is an end processing operation that generates the initial workpiece surface, so the initial surface topography model (STM) of the workpiece needs to be firstly established. In this processing operation, the spindle rotational speed for the SPDT is 2000 r/min and the feed rate is 2 mm/min. The geometrical relationship of the initial STM between the tool nose (arc tip) and the initial ST is shown in
Figure 7.
Theoretically, the formation of the initial surface topography is related to the tool and cutting parameters, including the arc tip radius, spindle rotational speed, and feed rate. As shown in
Figure 7,
can be represented by Equation (1):
where
is the tool feed rate,
is the spindle rotational speed of the workpiece, and
is deduced as Equation (2):
where
is the period number from point
to point
. The position relationship between point
and point
is more clearly referred to in
Figure 8b.
and
are the projecting position coordinates of point
on the diamond tool path plane.
According to the movement of the turning face in SPDT(Pre), the cutting tool path formed is a spiral of Archimedes or uniform speed spiral. The cutting tool path of SPDT(Pre) relative to the surface of the workpiece is shown as
Figure 8a. As shown in
Figure 8a, point
is the center point of the uniform speed spiral; the five-x magnification views of the red circle in
Figure 8a is shown in
Figure 8b, which depicts the position relationship between the projecting position of point
and point
; the projecting position of point
is a point on the line
.
Based on the geometrical relationship, Equations (3)–(5) can be derived as follows:
where
is the feeding time of the tool nose between point
and point
and
is the polar radius of
.
and
are the projecting position coordinates of point
on the diamond tool path plane.
is the counterclockwise angle from line
to line
.
According to the Pythagorean theorem, it is easy to obtain the height
of point
, which is derived by Equation (6):
where
is the cutting depth.
The initial STM establishing the workpiece in SPDT(Pre) is accomplished, and the derivation is similar to the STM established in SPDT(Circle) and SPDG(Line). The difference between the model established in SPDT and SPDG is the cutting tool path of the tool nose relative to the workpiece. The tool path of the SPDT(Circle) model is a group of concentric circles. For the SPDG model, the tool path is a series of parallel grooves. To distinguish the parameters, , , and represent the workpiece surface height in the SPDT(Pre) model, the SPDT(Circle) model, and the SPDG model, respectively.
Combining the process chain model, the surface topography of the polar microstructure is generated as a consequence of the processing models of SPDT and SPDG. For an arbitrary point on the surface of the workpiece, the surface height
results from the minimum height between
,
, and
, which can be defined as Equation (7):
where
is the workpiece machined area. Hence, the process chain model has been established.
According to the models of SPDT and SPDG, simulation experiments under different machining parameters were conducted.
Figure 9 shows one of the comparison groups between simulated and experimental results; a high degree of similarity between the two surface texture images is found. The result demonstrates that the proposed model is capable of representing the actual machining conditions of the polar microstructure surface. In other words, the simulated surface topography is able to be used in the further image processing as a substitute of the measured surface images. Furthermore, additional attention should be paid to the simulation model, which is able to reduce the cost and improve the efficiency.
4. Feature Point Detection
The modeling of polar microstructures aims to provide their surface topography, which can be shown in the form of images for the feature point detection, which is very significant for the further computer vision-based measurements. The measurement is realized by image matching, and the matching is accomplished by a series of corresponding feature points distributed in different images. As a result, it is necessary to develop the algorithm for feature point detection. There have been many studies providing feature point detection methods, such as canny edge [
22], Difference of Gaussians (DoG) [
23], and the principal curvature-based region (PCBR) [
24]. In this paper, a method named fast and robust feature-based positioning (FRFP) [
13] is presented for feature point detection.
Feature point extraction aims to construct a Hessian Matrix (HM) to generate points of interest for feature extraction, named ‘Feature point.’ Constructing the HM aims to find image stable edge points and blob points and provides a basis for the next step of feature extraction. The HM of an image expressed as
is given in Equation (8):
The filtered HM by Gaussian filtering is expressed in Equation (9):
To increase the speed of the algorithm, this comes up with one box filter (BF) to replace the Gaussian filter (GF). A schematic diagram of a GF and a BF is shown in
Figure 10.
,
, and
are used as the approximation for
,
, and
. The upper two figures, as shown in
Figure 10, are the values of the second derivative of the 9 × 9 GB template in the vertical direction on the image, which are
and
, and the lower two images are approximated by using a BF, which are
and
. As shown in
Figure 10, the pixel values of the white, black, and gray parts are 1, −2, and 0.
Since the integral image method is used for image convolution, BF increases the computational speed of the algorithm. The integral image method is a fast algorithm that only needs to traverse an image to get the sum of all the pixels in the image, which improves the efficiency of image eigenvalue calculation. The concept of the integral image is shown in
Figure 11. For any point
in an integral image, its value is the sum of the gray values of the rectangular region from the upper-left point of the original image to the point
, which can be expressed as Equation (10):
The filtering calculated by BF of an image is equal to calculating the pixel sums among the regions of the image, that is, the strength of the integral graph, and can be simply realized by inquiring an integral graph. Since the integral of a region is determined for a given image, the only work that needs to be done is to calculate the values of the four vertices of this region in the integral image, which can be obtained by two-step addition and two-step subtraction. The mathematical formula is expressed in Equation (11):
When a point is brighter or dimmer than other points surrounding it, the discriminant of the HM yields a local extremum, which determines the position of such a key point. The discriminant of the HM can be derived by Equation (12):
In Equation (12), constant C has no effect on the comparison of key points. In this way, when Gaussian second-order differential filtering is used with
and the template size is 9 × 9 as the smallest scale space value to filter the image,
in Equation (12) can be expressed in Equation (13):
Equation (12) is then simplified to Equation (14):
where a weighting factor
is used to remedy the error caused by BF approximation. Additionally, a response value is standardized based on the filter size to ensure that the Frobenius norm of the filter of any size is uniform. At a certain image point, its blob response value is represented by the approximate Hessian matrix determinant. A response image of all detected points on a certain scale is formed after all pixel points are traversed. Moreover, taking diverse template sizes obtains a multi-scale blob response map, which is applied to feature point localization.
Figure 12 shows a sample of detection results by using FRFP algorithms. It is shown that the main DFP are usually distributed around the designed intersection points, as shown in
Figure 3. This demonstrates that the detection algorithm is suitable for accurate detection of the feature points of polar microstructures. However, in order to reduce the localization errors of feature points, the algorithm increases the threshold of detection to ensure that the DFP are accurate and stable enough. On the basis of the above principles, the greater the number of feature points to be detected, the greater the number of subsequent matching points, and the more accurate the final measurement result.
The following section presents an investigation of three machining factors, which are tool nose radius, depth of cut, and groove spacing. For each factor, different machining values of the parameters were simulated to output the surface image of polar microstructures. Feature point detection was then conducted, and the number of detected points was found as the most important criterion for performance evaluation of polar microstructures.
6. Conclusions
In this study, the influence of the cutting strategy on polar microstructures was investigated. Considering the FOV, the rough size range of grooving spacing was determined. The offset spacing was also designed for a 360-degree measurement. After that, the modeling results of the surface topography were compared with the measured result, which indicated that it is capable of using the simulated surface images for further processing, which greatly reduces the cost. A FRFP-based algorithm was introduced to detect the feature points, and the results show that the polar microstructure was well designed and the algorithm is suitable for detection. Lastly, a series of simulations and experiments was conducted to investigate the influence of machining parameters on the performance of polar microstructures. The three main parameters focused on were tool geometry, depth of cut, and groove spacing. Some experiments were also conducted to demonstrate the accuracy of the simulated results. The optimized parameters were finally chosen for further machining. Other machining parameters, such as cutting speed effect, should be further investigated in future work. This work contributes to the parameter optimization of optic-functional microstructure surface through both theoretical and experimental study.