The laser imaging system is a type of noncontact and active 3D measurement system that projects a collimated stripe laser light of distinct frequency on a target object and measures the deformation of the laser light according to the shape of the object through a camera [1–6]. The 3D shape data can be obtained based on triangulation [7]. The laser imaging methods are used in a myriad of fields, such as cultural relic and scene 3D reconstruction [8, 9], welding [10–13], underwater surveying and exploration [14–16], railway inspection [17], and biometrics [18–20].

Much of the current research in stripe laser imaging has concerned with noise reduction [1, 2, 10, 15] and laser camera calibration [7, 21]. Extracting the center of the laser beam is also a crucial task for shape measurement. Because the laser light appears across several pixels in the image, it is difficult to accurately determine the center position of the laser light. Conventional methods simply select the pixel with maximum intensity among the laser pixels, calculate the average position of the two peaks of the laser pixels [22], or fit a parabolic curve to the laser pixels to find the maximum position of the curve [23–25] in the laser image. Convolutional neural networks (CNN) and feature clustering were combined for an automatic laser stripe detection algorithm [26]. These methods are based on the empirical observation of laser images, not on the physical phenomena of laser light.

The intensity of a laser beam is generally known as physically based on a Gaussian distribution. In [27], a Gaussian model of laser intensity on a target object is presented as a function of distance to the laser source. This model can be used to determine the center position of the laser beam on the object. However, because the object to be measured has an irregularly curved surface, not only the distance but also the slant angle of the surface influences the laser light. On the slant surface of the object, each end of the laser beam is at a different distance from the laser source causing the Gaussian beam to skew. Based on that observation, this study aims to present a modified Gaussian model for a laser light on a slant surface to improve accuracy and incorporate a mathematical interpretation of the physical cause of the skewed laser distribution. The experiments will verify the proposed model and show that it is able to extract the accurate center of a laser light on the object surface at a slant angle.

The paper is organized as follows: Related works are presented in Section II. Section III presents the modified Gaussian model for laser beam intensity to include the slant angle of a target object. The experiment and results are discussed in Section IV and concluded in Section V.

Various ever-evolving methods exist to detect the center of a laser beam in an image. A common method is to detect the edges of the laser using a Sobel operator [12] and compute their geometric center. The gray gravity method is widely used [22, 28], though this method is sensitive to laser saturation [4]. When derivative filters are convolved with the image, the zero-crossing of the result gives the peak location [29, 30]. The Otsu image segmentation method can be used to isolate the laser stripe from the background, so that center detection algorithms can more accurately calculate the center of laser beam [31, 32]. Automatic algorithms utilizing neural networks are useful for industrial applications [26, 33]. The optical properties of the experiment surface can be used to select a derivative filter for increased accuracy [3]. These methods have various demerits, such as complexity, processing time, and costly memory requirements. Figure 1 shows the common issues in the laser imaging.

Figure 1. Common issues in laser imaging. (a) Noise, (b) saturation, and (c) asymmetry.

Because the physical nature of laser intensity follows a Gaussian distribution, the Gaussian curve fitting is the most appropriate for modeling the laser beam in an image. The sum of multiple Gaussian curves of varying variances and standard deviations is adopted for the curve fitting to produce more accurate results than a single Gaussian curve [4]. In [34], the optimal laser center was estimated using weighted Gaussian signals of varying expectations and variances. While the effects of asymmetry due to object curvature on the gray distribution model were studied in [35], the reasoning behind it was not discussed.

This study proposes a modified mathematical model for Gaussian curve fitting of a laser beam in an image. This modification incorporates the slant angle of a target object to enhance accuracy and explain for the physical meaning behind the asymmetry of the beam distribution.

3.1. Gaussian Distribution of Laser Intensity

The intensity of a Gaussian laser beam in the image plane can be expressed as a Gaussian distribution of the form:

where I₀ is the laser intensity at the center of the beam, i is the pixel index in the image, µ is the mean, and w(y) represents the beam width as a function of the distance from the laser source [27]. The beam width can be expressed as

where w_o is the beam waist, y_R is the Rayleigh length, D is the distance from the laser source to the focal point, and y represents the distance from the laser source to the object. The Rayleigh length, y_R, is given by πw_o²/λ where λ denotes the wavelength of the laser. For example, if a red laser with 650 nm wavelength is adopted and the beam waist is 0.1 mm, the value of y_R is 48.3 mm.

In the region far outside the Rayleigh length, (D − y) >> y_R between the focal point and the laser source, the beam width has an approximately linear relationship with the distance to the laser, represented by

where w_s is defined as w_o/y_R. It is assumed in this study that a target object is placed at the position where Eq. (3) is satisfied.

3.2. Laser Beam Width as a Function of Slant Angle

A simple stripe laser imaging system is shown in Fig. 2, where the laser is illuminated on a curved object. Figure 3 shows the slant Gaussian model where θ is the slant angle at the laser spot on the object. The distance to the object at i = 0 is denoted by d₀, and the point on the object surface at i = 0 is represented by P₀ = [0 d₀]^t. The tangential vector on the object surface is $\overrightarrow{A}$, which is expressed as [cos θ sin θ]^t. Thus, a point on the object surface along i can be expressed as

Figure 2. Stripe laser imaging on curved object and image example.

Figure 3. Slant Gaussian model for laser light on object.

Eq. (4) is expanded as

Because the beam width depends on the distance (y), Eq. (1) can be rewritten as

Computer simulations of this Gaussian model are shown in Fig. 4 for ±50° slant angles. As expected, the slope for the +10° curve is demonstrably steeper on the right side as this side is closer to the focal point. On the contrary, the −10° curve exhibits a contrasting slope: It is steeper on the left side due to its closer distance to the focal point. Eq. (6) explains the skew symmetry of the laser beam distribution in the image. Using this model, extracting the more accurate center of the laser beam is possible.

Figure 4. Simulation graphs of slant Gaussian model. (a) 50°, (b) −50°.

Figures 5 and 6 are the simulation results presenting the differences in the extracted center points of the laser image when using the geometric mean and the proposed Gaussian model according to the slant angle of the target object. The center points of the laser detected using the geometric mean method are denoted with colored circles in Fig. 5. As the slant angle increases, the skew also increasingly shifts to the left and the geometric mean also shifts away from the center point of the slant Gaussian curve.

Figure 5. Intensity distribution skew as a result of slant angle.

Figure 6. Difference in center positions of laser image obtained from geometric mean and the proposed Gaussian model. (a) Positive angles, (b) negative angles.

4.1. Experiment Setup

A monocular camera and a stripe laser source were used in the experiment to verify the proposed model. Table 1 shows the specification of the experimental setup of the laser imaging system.

TABLE 1. Specification of experimental setup.

Camera Resolution	2,048 × 1,536
Image Sensor Size	1/1.8″
Pixel Size (μm)	3.45
Laser-object Distance (mm)	200
Imaging Area (mm2)	49.3 × 37
Laser Wavelength (nm)	650

The wavelength of the infrared laser source is longer than 780 nm and that of the red laser is 650 nm. The longer wavelength of the infrared laser source leads to a shorter Rayleigh length. However, because the infrared laser is invisible and requires an additional optical filter in camera, this study adopts the red laser for the convenience of experiments.

Instead of the target object, the laser source was installed at a slant angle and the slant angle is adjustable. A flat object covered with smooth, nonreflective white paper was placed beneath the camera. The experiment setup and a sample laser light image are shown in Fig. 7.

Figure 7. Experiment setup and experiment image.

Laser images were obtained for angles from −70° to 70° at 5° steps. The well-known Gaussian image filtering algorithm is employed to reduce the speckle noise at an early stage of image processing before applying the curve fitting to find the center of the laser image in this study. The parameters I₀, w₀, D, d₀, µ, and θ of the proposed model in Eq. (6) were obtained from the curve fitting algorithm using the gradient descent method.

4.2. Results

Figure 8 shows the curve fitting of the proposed model for ±10° and ±70°. For positive angles, the left side of the Gaussian distribution has a less steep slope than the right side because the left side of the target object is closer to the laser source as shown in the figures. The opposite is true for negative angles.

Figure 8. Curve fitting of the proposed Gaussian model. (a) ±10°, (b) ±70°.

The extracted center position of the laser light in the image using the proposed method and the conventional geometric mean of the two peak positions of the laser pixels [22] were compared. Figure 9 shows the difference in the extracted center position at each slant angle obtained by the two methods. As the slant angle increases, the difference in pixels of the two center extraction methods increases. An increasing slant angle causes an increasing difference in the beam width at either end of the object. This results in a Gaussian distribution that is increasingly skewed towards one side. The increasing skew is expressed as an increasing difference between the simple geometric mean of the two laser peaks and the peak of the slant Gaussian curve. This result accords with the simulation results explained in Fig. 5.

Figure 9. Comparison of center extraction methods. (a) Positive angles, (b) negative angles.

Table 2 shows the value of the slant angle obtained from the proposed slant Gaussian model with respect to the ground truth value of the slant angles of the target object. The average error between the ground truth values and those of the proposed Gaussian model is 0.3487° for the positive angles and −0.2803° for the negative angles.

TABLE 2. Ground truth vs. experimental results of slant angle.

Positive Angles (deg)
Ground Truth	10	15	20	25	30	35	40	45	50	55	60	65	70
Result of Experiments	10.0073	14.9970	19.8064	24.9512	29.6117	34.9572	39.3906	44.9498	49.4236	54.9543	59.1255	64.9804	68.3118

Negative Angles (deg)
Ground Truth	−10	−15	−20	−25	−30	−35	−40	−45	−50	−55	−60	−65	−70
Result of Experiments	−10.0599	−14.9817	−19.7952	−24.9643	−29.7594	−34.9519	−39.7336	−44.9516	−49.5814	−54.9581	−59.1232	−64.9686	−68.5273

This study proposed an imaging model of a laser light for extracting the accurate center of laser image. The width of a laser beam is a function of distance from the laser source. When a laser source illuminates a slant surface of a target object, the distance from the laser source to the surface varies from one side of the beam to the other, resulting in a change in the width of the laser image and a skewness in the intensity distribution.

Based on the physical phenomena of the laser light, the proposed model accounts for the difference in slope on each side of the intensity distribution by incorporating the slant angle of an object as a variable of the beam width. Thus, a more accurate Gaussian model is achieved. It was shown via experiments that there is a difference in the center pixel extracted using curve fitting of the proposed model and the geometric center method. In addition, the proposed model was able to accurately estimate the slant angle of a surface.

Research program funded by Seoul National University of Science and Technology.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.