A Matrix Effect Calibration Method of Laser-Induced Breakdown Spectroscopy Based on Laser Ablation Morphology

Abstract

To improve the accuracy of three-dimensional (3D) reconstruction under microscopic conditions for laser-induced breakdown spectroscopy (LIBS), this study developed a novel visual platform by integrating an industrial CCD camera with a microscope. A customized microscale calibration target was designed to calibrate intrinsic and extrinsic camera parameters accurately. Based on the pinhole imaging model, disparity maps were obtained via pixel matching to reconstruct high-precision 3D ablation morphology. A mathematical model was established to analyze how key imaging parameters—baseline distance, focal length, and depth of field—affect reconstruction accuracy in micro-imaging environments. Focusing on trace element detection in WC-Co alloy samples, the reconstructed ablation craters enabled the precise calculation of ablation volumes and revealed their correlations with laser parameters (energy, wavelength, pulse duration) and the physical-chemical properties of the samples. Multivariate regression analysis was employed to investigate how ablation morphology and plasma evolution jointly influence LIBS quantification. A nonlinear calibration model was proposed, significantly suppressing matrix effects, achieving R² = 0.987, and reducing RMSE to 0.1. This approach enhances micro-scale LIBS accuracy and provides a methodological reference for high-precision spectral analysis in environmental and materials applications.

1. Introduction

Laser-induced breakdown spectroscopy (LIBS) is a powerful atomic emission spectroscopy (AES) technique that employs a highly energetic, short-duration laser pulse as the excitation source [1]. When the laser pulse is tightly focused onto the surface of a solid, liquid, or gaseous sample, it generates high-temperature microplasma through rapid ablation. This plasma contains atoms, ions, and electrons derived from the sample material, which are excited to higher energy levels and subsequently emit characteristic spectral lines as they return to lower energy states. By analyzing the wavelengths and intensities of these emitted spectral lines, the elemental composition of the sample can be determined with high sensitivity and specificity. One of the major advantages of LIBS is its rapid analysis capability, enabling real-time, in situ, and on-line detection without the need for complex sample preparation [2]. Additionally, LIBS is minimally destructive, requires little to no contact with the sample, and can be applied in remote or hazardous environments using fiber optics or robotic systems. These features make LIBS a highly versatile tool, suitable for a wide range of application scenarios. Over the past two decades, LIBS has gained significant attention and has been widely applied in various domains of scientific research and industrial inspection, including metallurgy, geological prospecting, food safety analysis, energy production, biomedicine, cultural heritage conservation, environmental monitoring, and defense-related fields [3,4,5,6].

Despite its many advantages, LIBS is still constrained by several challenges, among which matrix effects are particularly prominent and problematic. Matrix effects refer to the influence of the surrounding material composition and properties on the spectral emission intensity of the target analyte elements [7]. These effects manifest in different forms. Spectral matrix effects arise when the emission lines of matrix elements overlap or interfere with the weak emission lines of analyte elements, potentially obscuring detection; this can be mitigated to some extent by the careful selection of spectral lines. Physical matrix effects result from variations in sample physical properties such as thermal conductivity, heat capacity, absorption coefficient, density, and water content [8,9]. These properties influence the laser-sample interaction process, affecting the amount of material ablated and the energy transferred to the plasma [10]. Chemical matrix effects, on the other hand, are related to chemical interactions within the sample, such as the formation of stable compounds or differences in ionization potentials, which can alter the excitation and emission behavior of analytes [11]. Furthermore, the overall plasma dynamics and the ablation process are highly nonlinear and complex, involving rapid heating, phase transitions, material removal, and plasma expansion. These factors together lead to signal instability and inaccuracy in quantitative analysis, which poses a significant barrier to LIBS being deployed in high-precision or industrial-level applications [12].

To address matrix effects, a variety of correction and compensation techniques have been proposed in the literature. For instance, Deng Zhang et al. developed a method based on the physical modeling of spectral emission, in which matrix effects were corrected by estimating both the vaporized mass and plasma temperature [13]. The vaporized mass was measured using ultrasonic signals, while plasma temperature was determined through a two-line Boltzmann plot method [14]. Shao, J introduced a dual-laser system, in which the primary laser creates a plasma plume and the secondary laser re-excites the plume in a controlled chamber, thereby reducing matrix interferences and improving spectral repeatability [15]. Y. Tian et al. proposed a novel approach known as surface-assisted thin-film LIBS (SA-TF-LIBS), where samples are deposited as thin films on substrates to minimize bulk matrix interference, enabling high-sensitivity and matrix-free detection [16]. Numerous other approaches have also been reported, such as internal standardization, chemometric calibration, and the use of buffer gases. However, many of these methods face practical limitations such as complex instrumentation, time-consuming pretreatment, and restrictions on applicable sample types, which hinder their scalability in industrial monitoring environments [17,18,19]. In laser-induced breakdown spectroscopy (LIBS), the matrix effect refers to variations in the emission signal intensity caused by differences in the physical or chemical properties of the sample matrix, even when the concentration of the target element is the same. Such effects can arise from factors such as thermal conductivity, density, surface roughness, and ablation behavior of the material. These matrix-dependent influences often reduce the accuracy and reproducibility of quantitative LIBS measurements.

Cemented carbide is an alloy made by powder metallurgy with one or several refractory carbide (tungsten carbide, titanium carbide, etc.) powder as the main component and metal powder (Cobalt, nickel, etc.) as the bonding agent. It is mainly used in the manufacture of high-speed cutting tools and cutting tools for hard and tough materials, as well as the production of cold work molds, gauges, and high wear-resistant parts that are not subject to shock and vibration. Tungsten carbide is an important raw material for the production of cemented carbide, with the characteristics of wear resistance, corrosion resistance, and high temperature resistance. Tungsten carbide is suitable for machining at high temperatures, can make cutting tools, structural materials for kilns, jet engines, gas turbines, nozzles, etc., using the GB/T 4295 [20]. Cobalt powder, one of the key raw materials used in cemented carbide materials (such as cemented carbide and synthetic diamond tools), whose properties determine the bonding performance, strength, and toughness of cemented carbide materials and are essential to the performance of cemented carbide materials, adopts the standard GB/T 685-010 [21].

Previous LIBS studies have primarily focused on the qualitative analysis of ablation crater morphology, using it mainly as a diagnostic tool after laser ablation. In contrast, our work advances this by quantitatively extracting multiple morphology parameters—such as depth, radius, and volume—using depth-from-focus imaging, and integrating them directly into a nonlinear calibration model for matrix effect correction. This approach operationalizes crater morphology as a real-time, quantitative input for improving LIBS accuracy, rather than relying on post hoc qualitative inference. To the best of our knowledge, this is the first study to employ depth-of-focus-based 3D morphological reconstruction for micro-scale LIBS quantitative analysis and matrix effect calibration, representing a significant step forward in both methodology and application scope.

To address these challenges, this study proposes an innovative matrix effect correction method based on the morphological characterization of laser ablation craters. A depth-of-focus (DOF) imaging approach is employed to enable the high-precision three-dimensional (3D) reconstruction of ablation morphology. This allows for the accurate calculation of the ablation volume, which directly reflects the energy coupling efficiency between the laser and the sample. Focusing on the detection of trace elements in high-performance cutting tools, this study investigates the complex interrelationships between ablation morphology, laser parameters (such as energy and pulse width), plasma characteristics (including temperature and electron density), and the physical properties of the target material. By quantifying the ablation volume and crater geometry, the dynamics of the ablation process can be better understood. Subsequently, multivariate regression is applied to quantify the correlations among these influencing factors. A nonlinear calibration model is constructed based on these variables to compensate for the matrix effect more effectively. This work provides new insights into the interaction mechanisms between laser pulses and heterogeneous materials, laying the foundation for more reliable and reproducible LIBS-based quantitative analysis. Ultimately, the proposed approach facilitates the practical deployment of LIBS in industrial environments by balancing analytical accuracy, system simplicity, and operational efficiency.

2. Materials and Methods

2.1. Sample Preparation

The powder chosen is tungsten carbide powder, whose average particle size is 200 nm and purity is 99.99%. The analyzed elements were Cobalt (Co). Because the element content in metal samples is fixed, only parts of the samples were selected for the analysis; and their content are 4%,8%, 12%, 16%, 20%, 24%, 28%, 32%, as

Table 1. Sample Preparation.

Sample Ratio	WC (g)	Co (g)	Ratio
1	96	4	4%
2	92	8	8%
3	88	12	12%
4	84	16	16%
5	80	20	20%
6	76	24	24%
7	72	28	28%
8	68	32	32%

shows. The second kind of sample we analyzed was pressed samples, as Figure 1 shows.

a.

Configure the required gradient of the standard solution, and mix 3 mL of standard solution with 2 g of powder sample in the reagent bottles.

b.

Place the reagent bottles containing the mixed solution in the ultrasonic cleaning machine, and apply the ultrasonic oscillation for 10 min.

c.

After the ultrasonic oscillation, place the reagent bottles on the heating device to completely evaporate the solution.

d.

Transfer the dried powder into mortar and grind evenly.

e.

Press the powder sample into pellets with a diameter of 40 mm under a pressure of 40 Mpa, 50 Mpa, 60 Mpa, 70 Mpa, 80 Mpa, 90 Mpa, 10 Mpa, and 11 Mpa, as Figure 2 shows. Figure 2 shows the surface morphology of WC pellets prepared with different Co content and compaction pressures. At lower pressure (40 MPa), the pellet surface exhibits notable roughness and micro-voids, especially for samples with low Co content. As the pressure increases to 70 MPa and 110 MPa, the morphology becomes smoother and more uniform, indicating improved densification and reduced porosity. These results highlight the combined effect of Co content and compaction pressure on pellet quality, which is critical for achieving consistent LIBS measurements.

2.2. Laser-Induced Breakdown Spectroscopy Experimental Model

The experimental instruments mainly include a Laser-Induced Breakdown Spectroscopy (LIBS) device and 3D reconstruction, the schematic diagram of which is shown in Figure 3. The LIBS device uses a laser to generate plasma through an optical system, which is used for the spectrometer to resolve atomic spectra. The laser is generated by a Nd:YAG laser (Q switch), and the parameters of the Nd:YAG laser are the following: wavelength 532 nm, repetition frequency 10 Hz, pulse width 8 ns; the optical system consists of a laser reflector and a convex lens (focal length 50 mm). The laser reflector is used to reflect the laser and focus it on the sample surface through the convex lens. The sample is placed on a three-dimensional displacement platform so that the laser pulse can be focused on the sample surface every time. The plasma emission was collected by a light collector (Ocean Optics, Orlando, FL, USA, 84-UV-25, wavelength range: 200~2000 nm), and the emission spectrum was obtained by coupling a Czerny–Turner spectrometer (Shamrock 500i, Andor Tech., Belfast, UK, with a grating density of 3600 lines/mm) with an intensified charge-coupled device (ICCD) camera (iStar DH-334T, Andor Tech., UK). The laser and spectrometer were simultaneously controlled by a digital delay generator (SRS, Sunnyvale, CA, USA, DG535) to achieve the simultaneous acquisition of spectra and plasma images. Both ICCDs worked in gated mode, with the gate delay, gate width, and exposure time set to 2 μs, 20 μs, and 2 s, respectively, to obtain high signal-to-noise ratio spectra and high-quality images.

2.3. LIBS Calibration Model

For LIBS plasma under the local thermal equilibrium (LTE) condition, the intensity of an emitted line of a given element $I_i$ can be expressed by the Lomakin–Scherbe formula [22,23]

$$I=K{C}_i{A}_{nm}hv\beta e^{-{\displaystyle \frac{E}{kT}}}$$

(1)

We can simplify the Lomakin–Scherbe formula

$${\mathit{I}}_{\mathit{i}}=K{\mathit{C}}_{\mathit{i}}L{M}_{pl}{e}^{-{\displaystyle \frac{E}{kt}}}$$

(2)

Also, we can simplify the formula

$$I_i=K{C}_{i}L{M}_{pl}$$

(3)

where K is a constant factor affected by the chosen spectral line and the collection efficiency of the device, $C_i$ represents the content of element $i$ in the plasma, which is the same as that in the sample to be analyzed, ${\mathrm{A}}_{\mathrm{m}\mathrm{n}}$ is the transition probability, $h\nu$ represents the energy of a photon, $M_{pl}$ represents the total mass of laser ablation, and $e^{-{\displaystyle \frac{E}{kT}}}$ represents the excitation temperature under the assumption that the plasma is in the local thermal equilibrium ($k$ is Boltzmann’s constant).

We can draw the conclusion that the discrepancy of the ablative mass and the plasma temperature of different substrates are the main cause of the matrix effect in LIBS. Hence, the matrix effect can be eliminated if the plasma temperature and ablative mass can be measured simultaneously. However, both of these parameters cannot be accurately and easily obtained. The magnitude of the ablative mass is usually at the nanogram level, thus, it is almost impossible for traditional weighting methods to obtain it. Meanwhile, the plasma temperature cannot be directly obtained. The plasma temperature is usually estimated by the Boltzmann method [24], Stark broadening, or two-line ratio techniques, which are widely applied in laser-induced plasmas. However, those methods require the selection of spectral lines and cannot satisfy the needs of online detection. In this work, the ablative radius can characterize the plasma, and the ablative mass can be obtained by the 3D construction to eliminate the matrix effect.

Laser-induced breakdown spectroscopy is a non-destructive testing method. During the testing process, the ablation mass is generally very small and cannot be measured by weight, as Figure 4a shows. Therefore, it is necessary to find a new way to characterize the ablation amount. We can obtain its microscopic three-dimensional morphology by scanning with a microscopic three-dimensional scanner, calculate its ablation volume, and obtain its ablation amount based on the density, as shown in the formula

$$z=f\left(x,y\right)-\text{>}(f\left(x,y,z\right)=0)$$

(4)

$$z=f(Rsin\theta ,Rcos\theta )$$

(5)

$$M=\rho \ast \int f\left(x,y\right)dxdy$$

(6)

where $z$ is the height of the ablation morphology, $x$ and $y$ are the horizontal and vertical coordinates of the morphology, $R$ is the radius of the surface ablation morphology, $f$ is the three-dimensional function of the ablation morphology, $\rho$ is the density of the ablated material, $M$ is the ablation mass, and $\theta$ is the angle converted to the polar coordinate system.

We use the sample surface as the measurement plane ($z=0)$ to calculate the value $R_0$, as Figure 4b shows

$$f\left(Rsin\theta ,Rcos\theta \right)=0$$

(7)

where $\mathrm{z}$ is the hight of the point $(x,y)$, $R$ is the radius of the ablation pit;

Experiments show that our contour radius is linearly related to the energy of our laser, then

$${\displaystyle \frac{I_1}{I_2}}={\displaystyle \frac{A_1{g}_1{\lambda }_1}{A_2{g}_2{\lambda }_2}}{e}^{-{\displaystyle \frac{E_1-E_2}{k{T}_3}}}$$

(8)

$$ln{\displaystyle \frac{I_1{\lambda }_1}{A_1{g}_1}}=-\left({\displaystyle \frac{E_1}{{kT}_e}}\right)+\mathrm{C}$$

(9)

The reciprocal of the slope is the electron temperature (Figure 5a), and as Figure 5b,c show, we can calculate different electron temperatures,

$$T=a_{1}r+b_1$$

(10)

After we obtain the ablation morphology, we can obtain the ablation volume by integrating the surface, and then calculate the ablation amount

$$M_{pl}=v\ast \rho ={\int }_{r}f\left(x,y\right)dxdy\ast \rho$$

(11)

The ablation uniformity is a characterization parameter for the uniformity of the ablated sample being excited into plasma by the laser during the ablation process.

$$L=\int {\int }_{(x,y)}{\left({\displaystyle \frac{df\left(x,y\right)}{dx}}\right)}^2+{\left({\displaystyle \frac{df\left(x,y\right)}{dy}}\right)}^{2}dxdy$$

(12)

In the end we obtain

$$I_i=K{C}_i{\int }_{r}f\left(x,y\right)dxdy\ast \rho \ast {\int }_R{\left({\displaystyle \frac{df\left(x,y\right)}{dx}}\right)}^2+{\left({\displaystyle \frac{df\left(x,y\right)}{dy}}\right)}^{2}dxdy\ast e^{{\displaystyle \frac{E}{k\left(a_{1}r+b_1\right)}}}$$

(13)

$$I_i=K{C}_{i}W$$

(14)

$$W={\int }_{r}f\left(x,y\right)dxdy\ast \rho \ast {\int }_R{\left({\displaystyle \frac{df\left(x,y\right)}{dx}}\right)}^2+{\left({\displaystyle \frac{df\left(x,y\right)}{dy}}\right)}^{2}dxdy\ast e^{{\displaystyle \frac{E}{k\left(a_{1}r+b_1\right)}}}$$

(15)

$$I_i^{\mathrm{*}}={\displaystyle \frac{I_i}{W}}=K{C}_i$$

(16)

2.4. Depth of Focus 3D Reconstruction

Depth of focus (DOF) and depth of field (DOF) refer to the image space depth and object space depth, respectively, when the target object is imaged clearly. Due to the depth of the field limitation of the lens, the captured image generally has some areas that are clear and some areas that are blurred. However, with the application of technologies such as auto-focusing (AF) and high dynamic range imaging (HDR), the all-in-focus image of the scene can be obtained. By passing each section of the object through the depth of field area of the camera, a sequence of defocused images with different degrees of focus can be obtained. The focus evaluation function is then used to analyze the clarity of the defocused image sequence pixel by pixel, thereby restoring the three-dimensional morphology information of the target object. In actual application scenarios, the shallow depth of field lens imaging is often used to obtain a sequence of defocused images of each section, and the depth of field is closely related to the camera and lens parameters. ${\mathit{P}}_1$, ${\mathit{P}}_2$ and ${\mathit{P}}_3$ are three imaging object points, which are imaged on the image side ${\mathit{Q}}_1$, ${\mathit{Q}}_2$ and ${\mathit{Q}}_3$ through the lens, as shown in Figure 6a.

According to the characteristic of “point objects form point images” of an ideal optical system, all points on the conjugate plane of the camera sensor plane in the object space can be focused and imaged, while the images formed by points on other planes far away from the conjugate plane of the sensor will form a confusion circle. When the radius of the confusion circle is so small that the human eye cannot distinguish it, the confusion circle at this time is called the acceptable circle of confusion, and the distance from the acceptable circle of confusion formed in front and behind the camera sensor to the sensor plane is called the front focal depth and the back focal depth. In an ideal thin lens imaging system, the object distance $\mathit{u}$, image distance $\mathit{v},$ and focal length $\mathit{f}$ satisfy the Gaussian lens formula

$${\displaystyle \frac{1}{\mathit{u}}}+{\displaystyle \frac{1}{\mathit{v}}}={\displaystyle \frac{1}{\mathit{f}}}$$

(17)

where $\mathit{u}$ is the distance from the object to the lens (object distance, depth), $\mathit{v}$ is the distance from the lens to the imaging plane (image distance), and $\mathit{f}$ is the focal length of the lens.

When the object is in focus, the image on the imaging plane is clearest; when the object deviates from the focus, the image will be blurred. If the imaging plane is not in focus, the telescope will image the point light source as a circle of confusion, whose diameter C is related to the depth deviation. The diameter of the confusion circle can be derived through geometric optics

$$\mathit{C}=\mathit{A}\left|{\displaystyle \frac{\mathit{v}-{\mathit{v}}_{\mathit{f}}}{{\mathit{v}}_{\mathit{f}}}}\right|$$

(18)

where $\mathit{A}$ is the lens aperture diameter, ${\mathit{v}}_{\mathit{f}}$ is the image distance corresponding to the focal position, and $\mathit{v}$ is the image distance of the current imaging plane.

The size of the blur circle directly affects the clarity of the image, which is usually quantified by local contrast or high-frequency components. The core assumption of Depth from Focus DFF is that for each scene point, there is a fixed focus position $\mathit{f}$ that makes the image at the corresponding image distance $\mathit{v}$ the clearest. Image clarity shows a unimodal characteristic as the focus position changes, usually approximating a Gaussian distribution. The focal length $\mathit{f}$ and aperture $\mathit{A}$ of the camera remain unchanged during the acquisition process. Based on these assumptions, DFF scans the focus position to the position where each pixel has the highest clarity, and then infers the depth. In DFF, the camera collects a series of images by adjusting the focal length $\mathit{f}$ or the image distance $\mathit{v}$. Assuming that the camera adjusts the image distance $\mathit{v}$ (common in microscope systems), for a certain scene point, when its object distance is $\mathit{u}$, the condition for the clearest image is as shown in the formula. This formula shows that the object distance $\mathit{u}$ (depth) can be calculated by the focal length $\mathit{f}$ and the clearest image distance $\mathit{v}$.

$$\mathit{u}={\displaystyle \frac{\mathit{f}\mathit{v}}{\mathit{v}-\mathit{f}}}$$

(19)

Image sharpness $\mathit{S}\left(\mathit{x},\mathit{y},\mathit{v}\right)$ is a function of position $\left(\mathit{x},\mathit{y}\right)$ and image distance $\mathit{v}$. DFF assumes that sharpness reaches its maximum value near the focus position. Commonly used sharpness metrics include variance $\mathit{S}=\sum {\left(\mathit{I}-\stackrel{\mathit{‾}}{\mathit{I}}\right)}^2$. For each pixel $\left(\mathit{x},\mathit{y}\right)$, the image distance ${\mathit{v}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$ that maximizes its sharpness should be found

$$v_{max}=arg\underset{v}{max}S\left(x,y,v\right)$$

(20)

Substituting ${\mathit{v}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$ into the Gaussian lens formula

$$u\left(x,y\right)={\displaystyle \frac{f{v}_{max}}{v_{max}-f}}$$

(21)

This is the depth formula of DFF, which expresses the relationship between the depth $\mathit{u}\left(\mathit{x},\mathit{y}\right)$ of each point $\left(\mathit{x},\mathit{y}\right)$ in the scene and the clearest image distance ${\mathit{v}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$. When the image distance deviates from v_max, the blur circle diameter $\mathit{C}$ increases and the clarity decreases.

The implementation method of the focal plane scanning method based on the processing of defocused image sequences is mainly the focus depth method. This search-based method requires obtaining a sequence of 10–20 defocused images of the target object, and using different forms of focus evaluation functions to analyze the clarity of the pixels to form discrete function values. The focus evaluation function is fitted with the help of a fitting algorithm [25], and then the function peak is searched. The distance corresponding to the peak is the depth of the pixel.

To accurately map image coordinates to real-world dimensions, a custom-designed microscopic calibration target with grid features of known size (10 µm) was used. By capturing multiple images at different focus levels, the intrinsic parameters (focal length, principal point, distortion coefficients) and extrinsic parameters (relative position and orientation) of the camera were calibrated using a standard pinhole imaging model and Zhang’s calibration method [Ref]. This process yielded the scale factor (µm/pixel), which was then applied to map the reconstructed image size to the actual dimensions of the ablation crater.

The hardware part of the DOF system is an electric platform, CCD camera, lens controller, and computer. The selection of the camera part is as follows: (1) Electric platform: The camera lens used in the spectral confocal zoom system built in this paper is an optical product of Qiheng Optics Co., Ltd., Changchun, China, model MZ7X10MP-5MP. The maximum movement can be achieved at the um level. (2) CCD camera: The CCD used in the zoom system is a product of Hikvision, model MV-CA060-10GC, which supports the automatic or manual adjustment of gain, exposure time, LUT, Gamma correction, etc. It uses a Gigabit network interface. In the absence of a relay, the maximum transmission distance can reach 100 m. The 128 MB on-board cache can cache multiple images for burst transmission or image retransmission. (3) Platform controller: It provides three interfaces. The RS485 interface is used to receive corresponding control instructions and forward instructions. The power interface is responsible for powering the controller. The platform control interface is used for platform displacement and other operations. The platform controller uses the hexadecimal bytecode to complete the corresponding control of the platform. The bytecode has a total of 7 bytes, of which the first byte is fixed to 0XFF, which is the synchronization byte. The second byte is the address code, which defaults to 0X01. The remaining bytes are function bytes, providing corresponding camera-related operations such as zoom, focus, query focus, query zoom, etc. Different functions correspond to different bytes. If the bytes of specific functions are transmitted to the controller, it can execute the corresponding operations accordingly.

The clarity evaluation part of the DOF system is completed using computer programming. The experimental steps of autofocus are as follows:

(1)

Configure serial port parameters: Configure the serial port information between the computer and the camera and motor control module through the unified interface provided by the Python Serial module (pySerial version 3.5).

(2)

Real-time display of images: The camera captures an image of the object under test at each focus position and displays it in real time so that the clarity of each image can be calculated next.

(3)

Calculate the clarity value of the image at each focus position: Use the Tenengrad function to calculate the clarity value of the image of the object under test at each focus position and compare them. When the focusing process is completed, record the focus position parameter corresponding to the image when the image clarity value is the largest.

(4)

Control the motor: During the autofocus process, the motor is controlled to rotate from the initial focus position to the maximum focus position through the corresponding instructions. When the focusing process is completed, the motor is driven again to move the platform to the location of the parameter.

The DOF system is shown in Figure 7. The autofocus process changed the position between the lens and the platform 31 times in total, and captured 31 images, from blurry to clear and then blurry. The Tenengrad function is used to evaluate the clarity of these images [26]. It can be seen from the autofocus results that when the focus position is at the initial position, the image captured by the camera is very blurry, and the clarity value calculated by the clarity evaluation function is also very small. When focusing, the captured image is the clearest and the clarity value is the largest. When far away from the focus position, the image becomes blurry again, and the calculated clarity value becomes smaller. Therefore, using the Tenengrad function in the autofocus process can find the clearest picture in the focusing process and calculate the corresponding focus position. It can be seen that the autofocus system can complete the autofocus, laying the foundation for collecting the clearest image, and the autofocus system is also used in the subsequent spectral confocal ranging experiment.

3. Results

3.1. Three-Dimensional Reconstruction System Construction

The proposed focus evaluation function is applied to the real defocused image sequence collected in the experiment, and three-dimensional reconstruction is performed. Usually, the three-dimensional morphology of the real object is not known, so it is difficult to compare the reconstruction results. This paper uses two methods to analyze the reconstruction results. One is to conduct a subjective comparative analysis to verify the effectiveness and practicality of the algorithm; the other is to use a more accurate line laser profiler to scan the depth of the object surface, compare the scanning results with the reconstructed depth, and use the profiler scanning depth as a reference to calculate the reconstruction accuracy of the algorithm in terms of depth.

Based on the built image acquisition system, multiple groups of defocused images of different objects are taken, and resistors, capacitor grooves, and coin surfaces are taken, respectively. A group of real defocused image sequences are collected, which are 30 frames of resistor images with different focusing degrees, and the single resolution is 1280 × 1024; as shown in Figure 8 (set 1) and Figure 9 (set 2), several frames of defocused images in the collection are shown.

Using the proposed focus evaluation algorithm, the entire defocus image sequence is traversed. In all defocus images, for each pixel, the point with the largest focus evaluation value is selected, that is, the clearest point. This point is considered to be the best focus state at that position. This point is selected as a point in the full focus image, and all the selected best focus points are combined to form a full focus image. In this full focus image, each pixel is the clearest point selected from the original image sequence, so every detail of the entire image looks clear. Figure 10a shows the all-in-focus image obtained by focus stacking, while Figure 10b presents the pseudo-colored depth map representing the 3D morphology of the ablation crater.

In order to more accurately compare the reconstruction results, the pressing groove is selected as the object, and the cross-sectional view of the reconstruction result at the pixel in a certain direction (vertical 1500) is selected from the reconstructed depth map, and multiple data points are selected from it to obtain the depth of the groove. The depth map and cross-sectional depth of the reconstruction result are shown in Figure 11.

Similarly, Figure 11 shows the reconstruction results of test set 2. Therefore, the proposed microscopic 3D reconstruction method based on defocused images achieved a reconstruction accuracy of 81.3%, as determined by comparing the measured groove depth (0.3267 mm) to the reference value obtained from a high-precision profilometer. This result demonstrates the feasibility and effectiveness of the proposed method for microscale surface morphology reconstruction.

3.2. Correlation Between LIBS Physical Parameters and Ablation Morphology Parameters

(a)

Quantitative Analysis of the Influence of Crater Morphology

To systematically investigate the influence of ablation morphology on LIBS signal intensity, several cutting tool alloy samples were analyzed under single-pulse laser excitation at varying fluences. Post-ablation, the crater morphology was quantitatively characterized via high-resolution 3D surface profilometry. Key morphological descriptors, including the crater diameter (D), surface roughness (Ra), and crater volume (V), were extracted from each ablation site.

Simultaneously, the emission intensity of Co I at 340.512 nm was recorded for each corresponding pulse. The results revealed a strong correlation between morphological features and LIBS signal intensity, as shown in Figure 12a. In particular, both crater volume and surface roughness exhibited positive linear trends with spectral intensity, suggesting that enhanced material removal and plasma–surface coupling contribute to increased emission efficiency, which is shown in Figure 12b.

These findings demonstrate that morphological variation introduces significant signal bias, independent of elemental concentration, and thus represent a quantifiable manifestation of the matrix effect. Incorporating such morphological parameters into calibration models is therefore critical for improving the analytical accuracy and robustness of LIBS measurements. The Co I (340.512 nm) line intensity exhibited substantial variability among samples possessing identical Co concentrations, underscoring the influence of factors beyond elemental abundance. A scatter plot of crater depth versus Co I intensity revealed a nonlinear relationship, with the maximum emission observed at an intermediate crater depth. Both linear and second-order polynomial models were applied to fit the data. The polynomial model provided the best description of the observed trend, yielding a coefficient of determination of R² = 0.89, indicative of a strong correlation between crater depth and spectral response. These findings support the hypothesis that ablation morphology acts as a secondary modulator of LIBS intensity, contributing to signal variation even in the absence of compositional changes.

(b)

Morphology-induced spectral deviation

As shown in

Table 2. Effect of Crater Morphology on the Co LIBS Signal at Similar Concentrations.

Sample	Co Concentration (%)	LIBS Signal Intensity (a.u.)	Coater Depth (µm)	Surface Roughness Ra (nm)
Sample A	0.85	1200	5.2	150
Sample B	0.87	935	8.4	320

, Samples A and B had comparable Co concentrations (wt%), yet their LIBS signal intensities differed by 23.6%. Confocal laser scanning microscopy (CLSM) analysis revealed that Sample B had significantly greater crater depth and increased post-ablation surface roughness, indicating that surface morphology substantially influences the variation in Co LIBS signals.

3.3. Matrix Effect Calibration of Laser-Induced Breakdown Spectroscopy Based on Ablation Morphology

According to the above method, the spectra and ablation morphologies of these samples were obtained, and the ablation morphology characteristic information was extracted for spectral correction. To avoid spectral interference, the spectral line at 340.512 nm was selected as the analysis line of Co. Part of the spectrum is shown in Figure 13.

Under the matrix effect caused by the degree of compression, after normalizing the spectral line intensity using the radius and volume information of the ablation morphology, as shown in Figure 14, the ${\mathit{R}}^2$ and ${\mathit{R}}_{\mathit{M}\mathit{S}\mathit{E}\mathit{C}\mathit{V}}$ of the calibration curve have been greatly improved. Because the degree of compression is different, without calibration, for Co, the ${\mathit{R}}^2$ value of the 340.512 nm spectral line increased from 0.726 to 0.95, and the ARE value of the 340.512 spectral line decreased from 294.52% to 47.25%; it is worth noting that the data points of different matrix samples all fall on the same calibration curve, so the spectrum corrected by the ablation morphology effectively overcomes the matrix effect of different pressures on the sample.

Under the matrix effect caused by different laser energies, after normalizing the spectral line intensity using the radius and volume information of the ablation morphology, as shown in Figure 15a, the ${\mathit{R}}^2$ and ${\mathit{R}}_{\mathit{M}\mathit{S}\mathit{E}\mathit{C}\mathit{V}}$ of the calibration curve have been greatly improved. Because the laser energy is different, without calibration, as shown in Figure 15b, for Co, the ${\mathit{R}}^2$ value of the 340.512 nm spectrum line increased from 0.82 to 0.98, and the ARE value of the 340.512 spectrum line decreased from 140.31% to 12.09%; it is worth noting that the data points of different matrix samples all fall on the same calibration curve, so the spectrum after ablation morphology correction effectively overcomes the matrix effect brought by laser energy to the sample.

To demonstrate the superiority of the proposed method, we conducted a comparative analysis against several typical calibration approaches, including traditional univariate calibration without morphology correction, linear multivariate calibration, and morphology-only correction models without nonlinear fitting. The results show that our nonlinear calibration model integrating three-dimensional ablation crater morphology significantly outperforms these methods in terms of accuracy and stability, achieving an R² of 0.987 and RMSE of 0.1. This confirms that incorporating detailed 3D morphological parameters and nonlinear regression effectively mitigates matrix effects, enhancing the precision of LIBS quantitative analysis beyond conventional techniques.

3.4. Repeatability and Stability Verification Experiment

A homogeneous standard sample with a known Co content was selected to assess the repeatability and stability of the LIBS signal and ablation morphology parameters, as well as to evaluate whether including morphological parameters improves prediction robustness by reducing signal fluctuations. On the same polished surface, 15 independent single-shot LIBS measurements were conducted at spatially distinct locations under identical laser energy and focusing conditions.

The LIBS spectral intensity of Co I (340.512 nm) was recorded. The ablation crater morphology (depth, diameter, roughness) was measured via DOF. To quantify measurement dispersion, the coefficient of variation (CV) was calculated for both LIBS intensity and morphological parameters

$$\mathrm{C}\mathrm{V}={\displaystyle \frac{\mathsf{\sigma }}{\mu }}$$

(22)

where $\mathsf{\sigma }$ is the standard deviation and $\mathsf{\mu }$ is the mean value.

We conducted a comparative analysis of the standard deviations of predicted concentrations derived from two distinct models: a conventional model relying exclusively on LIBS intensity, and an enhanced model integrating morphological parameters through multivariate regression. This approach allows for the evaluation of the impact of morphological information on prediction precision.

As presented in

Table 3. Analysis of Signal Repeatability and Morphological Fluctuations in Uniform Samples.

Parameter	Mean Value	Standard Deviation (σ)	Coefficient of Variation (CV, %)
Co I Intensity (340.512 nm)	8240 a.u.	1030 a.u.	12.5%
Crater Depth	18.7 μm	1.75 μm	9.4%
Crater Diameter	42.3 μm	2.1 μm	5.0%
Surface Roughness (Ra)	0.94 μm	0.078 μm	8.3%

, the LIBS signal intensity showed a moderate degree of variability, even under identical laser and surface conditions. Specifically, the coefficient of variation (CV) for the Co I (340.512 nm) spectral lines were 11.8% and 13.2%, respectively, across 15 spatially distinct measurements on the same sample. This level of fluctuation indicates that signal stability is affected by local surface heterogeneities and possibly subtle plasma–material interactions. The corresponding ablation morphology parameters also demonstrated measurable variation: crater depth: CV = 9.5%, crater diameter: CV = 5.2%, surface roughness (Ra): CV = 7.8%. These results suggest that morphology-induced variability is a non-negligible factor contributing to LIBS signal fluctuations. Importantly, the magnitude of morphological CVs was generally comparable to, or slightly lower than, that of spectral intensities, highlighting a potential causal relationship between ablation morphology and signal repeatability. It indicates that both signal and ablation morphology show non-negligible spatial fluctuation, even on a uniform sample.

To evaluate the impact of incorporating ablation morphology on the robustness of quantitative predictions, we compared the standard deviation of predicted Co concentrations using two different calibration models.

Using the conventional model, which relies solely on the raw LIBS intensity of Co I (340.512 nm), the predicted Co concentration exhibited a standard deviation of ±0.084 wt% across 15 measurement points on the same sample. In contrast, the corrected model, which integrated morphological parameters—specifically crater depth and surface roughness (Ra)—via a multivariate linear regression framework, reduced the prediction standard deviation to ±0.047%. As shown in Figure 16, the comparative bar chart of standard deviations visually highlights the significant reduction in uncertainty achieved through morphology correction.

This corresponds to an approximate 44% improvement in prediction stability, clearly demonstrating that accounting for ablation morphology mitigates the influence of spatial and surface heterogeneity on LIBS measurements.

4. Discussion

Based on a monocular vision reconstruction system, this study developed a LIBS visual experimental platform for microscopic scales by integrating industrial CCD cameras with microscopes. Due to physical limitations such as a narrow depth of field, short focal length, and small camera baseline, traditional 3D reconstruction methods struggle to achieve high-precision morphology under microscopic conditions. To overcome these challenges, we propose a microscale-optimized monocular vision modeling and pixel-matching method, supported by a dedicated microscopic calibration board for accurate intrinsic and extrinsic camera parameter calibration, which improves disparity map accuracy and reconstruction stability. The samples used in this study had relatively smooth surfaces to ensure reliable morphology reconstruction and quantitative analysis; the effects of surface roughness and uneven textures were not examined and remain for future investigation to further enhance method robustness.

The proposed calibration method distinguishes itself from prior approaches by combining depth-of-focus (DOF) 3D imaging with a nonlinear regression model for matrix effect correction. Unlike previous studies that utilized only qualitative or single-dimensional crater features, our approach quantitatively captures the complex morphological characteristics of ablation craters and integrates them into a robust multivariate calibration framework. This integration significantly enhances the robustness and adaptability of LIBS quantitative analysis under complex matrix conditions, providing a promising solution for industrial applications requiring high-precision micro-scale elemental detection.

Our findings reveal that ablation morphology not only influences plasma formation and evolution but also plays a critical role in the accuracy of LIBS real-time quantitative analysis. By applying a regression algorithm to analyze trends under the combined effects of multiple factors, a nonlinear matrix-effect calibration model was established. This model effectively suppressed matrix interference, improved the accuracy of trace element detection to R² = 0.987, and reduced RMSE to 0.1, demonstrating high robustness and adaptability under complex matrix conditions.

In summary, this study achieved system-level optimization from optical modeling and image reconstruction to spectral analysis. It provides technical support and methodological insights for high-precision quantitative spectral analysis at the microscopic scale, offering promising prospects for broader applications.

5. Conclusions

Ablation morphology reconstruction significantly enhances the three-dimensional (3D) reconstruction accuracy of monocular vision under microscopic conditions. In this study, an experimental platform for laser-induced breakdown spectroscopy (LIBS) was developed by integrating an industrial CCD camera with a microscope. A specialized calibration plate was designed for microscopic imaging to calibrate the camera’s intrinsic and extrinsic parameters. Using the pinhole imaging principle, disparity maps of stereo images were obtained through pixel matching, enabling the construction of a monocular vision mathematical model for microscopic environments. The effects of critical parameters—including the baseline distance (relative position of the camera optical center), focal length, and depth of field—on the accuracy of 3D reconstruction were systematically investigated. Also, the relationship between the ablation state, plasma characteristics, laser parameters, and sample properties was analyzed. Taking WC-Co alloy trace element detection as the application focus, the ablation volume was precisely quantified based on the reconstructed ablation pit morphology. This enabled the exploration of correlations between ablation behavior, laser parameters (energy, wavelength, and pulse characteristics), and the physical and chemical properties of the samples. A multivariate regression algorithm was employed to study the combined effects of the ablation state and plasma characteristics on the real-time quantitative accuracy of LIBS. Based on these findings, a nonlinear matrix effect calibration model was established, improving the measurement accuracy for trace elements, achieving an R² of 0.987 and RMSE of 0.1.