Point Cloud Vibration Compensation Algorithm Based on an Improved Gaussian–Laplacian Filter

Abstract

In industrial environments, steel plate surface inspection plays a crucial role in quality control. However, vibrations during laser scanning can significantly impact measurement accuracy. While traditional vibration compensation methods rely on complex dynamic modeling, they often face challenges in practical implementation and generalization. This paper introduces a novel point cloud vibration compensation algorithm that combines an improved Gaussian–Laplacian filter with adaptive local feature analysis. The key innovations include (1) an FFT-based vibration factor extraction method that effectively identifies vibration trends, (2) an adaptive windowing strategy that automatically adjusts based on local geometric features, and (3) a weighted compensation mechanism that preserves surface details while reducing vibration noise. The algorithm demonstrated significant improvements in signal-to-noise ratio: 15.78% for simulated data, 6.81% for precision standard parts, and 12.24% for actual industrial measurements. Experimental validation confirms the algorithm’s effectiveness across different conditions. This approach achieved a practical, implementable solution for surface inspection in steel plate surface inspection.

1. Introduction

The rapid expansion of the global shipbuilding industry has intensified the demand for stringent quality control protocols in large-scale metal plate inspection. These plates, characterized by diverse specifications and dimensions, are prone to various surface defects including corrosion pitting, mechanical abrasion, and impact deformation, all of which can significantly compromise structural integrity. The fundamental limitations of traditional contact-based metrology methods, particularly their temporal inefficiency and potential for surface damage during measurement, have accelerated the development and adoption of advanced non-contact inspection technologies [1,2,3].

Line-scanning laser metrology systems represent a state-of-the-art solution that combines high-precision measurement capability, rapid inspection speeds, and non-destructive evaluation methodology, making them particularly well-suited for high-throughput shipyard quality control processes [4,5]. By incorporating high-resolution motion encoders for precise spatial position tracking, these systems enable accurate three-dimensional surface topography reconstruction and quantitative defect characterization, achieving exceptional spatial resolution (≤0.1 mm) in controlled environments [6,7,8]. The compelling advantages of this technology have driven its widespread implementation across diverse industrial applications [9].

However, significant challenges arise from the complex dynamic behavior of metal plates during inspection, particularly the multi-modal vibration phenomena induced by roller conveyor systems, which introduce systematic measurement uncertainties that compromise both the geometric accuracy and spatial completeness of three-dimensional reconstructions [10,11,12]. These measurement deviations are further amplified by environmental disturbances prevalent in industrial settings, such as mechanical vibrations, thermal fluctuations, and electromagnetic interference, underscoring the critical need for robust vibration compensation strategies [13,14].

In precision ball screw-driven laser metrology systems, as the line-scanning laser slides along the encoder track, the encoder, composed of rotating components, may cause gaps and misalignment in transmission mechanisms (such as slides and ball screws) due to its mechanical structure’s high-speed operation, determining that vibration primarily occurs in the radial direction. Extensive experimental studies have revealed that key system parameters—notably the ball screw preload configuration [15], joint interface stiffness [16], and mechanical boundary conditions [17]—fundamentally govern the system’s dynamic behavior. The encoder subsystem, which forms the cornerstone of precise position measurement, demonstrates particularly high susceptibility to vibrational disturbances. Recent research has revealed that encoder measurement uncertainties exhibit complex spectral characteristics comprising both deterministic components (arising from mechanical periodicity) and stochastic elements (stemming from environmental perturbations) [18,19]. While active vibration suppression through mechanical design optimization has shown promise, its implementation often requires substantial system modifications, imposing significant economic barriers for existing industrial installations [20].

Point cloud vibration compensation methodologies have emerged as a pivotal research direction in modern metrology, offering novel solutions for measurement accuracy enhancement. While conventional approaches rely predominantly on statistical filtering [21] and frequency-domain decomposition [22], these methods often struggle to maintain the integrity of localized surface features while suppressing vibration-induced deviations. Recent advances in deep learning architectures present promising alternatives [23]; however, their practical implementation is constrained by training data quality requirements and limited generalization capabilities across different measurement scenarios [24,25].

Recent research initiatives have validated the effectiveness of integrating high-precision laser scanning systems with sophisticated computational algorithms, demonstrating significant improvements in noise suppression and system performance optimization under real-world industrial conditions [26,27,28]. Successfully addressing these metrological challenges in shipyard environments is crucial not only for enhancing inspection reliability but also for optimizing manufacturing throughput and quality assurance processes. These advancements establish metrology-based quality control as a critical domain requiring continued scientific investigation and technological innovation [29]. However, current algorithms typically require a large amount of computing resources [30,31], resulting in a reduced possibility of large-scale promotion. These inherent limitations highlight the pressing need for highly efficient and adaptive compensation strategies that can effectively handle diverse vibration while preserving measurement accuracy and local feature.

Traditional vibration compensation approaches have primarily relied on complex dynamic system models for vibration component extraction, which present substantial challenges in terms of required expertise, computational resources, and generalization capabilities across different applications. To address these limitations, this paper introduces a novel point cloud vibration compensation algorithm that combines an improved Gaussian–Laplacian filter with adaptive local feature analysis. The key innovations include (1) an FFT-based vibration factor extraction method that effectively identifies vibration trends, (2) an adaptive windowing strategy that automatically adjusts based on local geometric features, and (3) a weighted compensation mechanism that preserves surface details while reducing vibration noise.

2. Vibration Compensation Algorithm

Dynamic solution approaches have high professional requirements, complexity, and low universality. Additionally, point clouds contain large amounts of data, resulting in low computational efficiency, and different system interference factors are intricately intertwined, making it difficult to eliminate them individually. This paper focuses on compensating the acquired data.

The actual acquired point cloud data containing vibration information can be denoted as ${\mathrm{p}\mathrm{c}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}$ (the line laser data with respect to time $t$ can be written as ${\mathrm{f}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}\left(\mathrm{t}\right)$), and the ideal point cloud data containing actual object surface features can be denoted as ${\mathrm{p}\mathrm{c}}_{\mathrm{f}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}}$ (the line laser data with respect to time t can be written as ${\mathrm{f}}_{\mathrm{f}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}}\left(\mathrm{t}\right)$). Due to various system errors and external disturbances, there exists a difference between ${\mathrm{f}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}\left(\mathrm{t}\right)$ and ${\mathrm{f}}_{\mathrm{f}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}}\left(\mathrm{t}\right)$, defined as $∆f\left(t\right)$, yielding the following equation:

$${\mathrm{f}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}\left(\mathrm{t}\right)={\mathrm{f}}_{\mathrm{f}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}}\left(\mathrm{t}\right)+∆\mathrm{f}\left(\mathrm{t}\right),$$

(1)

The vibration component of ${\mathrm{f}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}\left(\mathrm{t}\right)$ compared to the corresponding ${\mathrm{f}}_{\mathrm{f}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}}\left(\mathrm{t}\right)$ is denoted as vibration(t). Therefore,

$${\mathrm{f}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}\left(\mathrm{t}\right)={\mathrm{f}}_{\mathrm{f}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}}\left(\mathrm{t}\right)+\mathrm{v}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(\mathrm{t}\right)$$

(2)

The overall processing approach algorithm flowchart is shown in Figure 1:

2.1. Overall Trend Extraction

The Fourier transform of the signal is computed as

$$Y\left(k\right)={\sum }_{n=0}^{L-1}x(n)e^{-{\displaystyle \frac{\mathrm{j}2\pi \mathrm{kn}}{L}}},$$

(3)

where $L$ is the signal length, and $x\left(n\right)$ is the input signal.

The single-sided amplitude spectrum is then calculated as

$$P\left(k\right)=\left\{\begin{array}{c}{\displaystyle \frac{\left|Y\left(k\right)\right|}{L}}, k=0 or k=L/2\\ {\displaystyle \frac{\left|Y\left(k\right)\right|}{L}}, 0<k<L/2 \hfill \end{array}\right.,$$

(4)

The frequency values are given by

$$f_k={\displaystyle \frac{k}{L}},k=\mathrm{0,1},\dots ,\lfloor L/2\rfloor ,$$

(5)

And the phase information is extracted as

$${\varphi }_k=\mathrm{\angle }Y\left(k\right),k=\mathrm{0,1},\dots ,\lfloor L/2\rfloor ,$$

(6)

The overall variation trend of the data can be represented as

$$feature\left(t\right)={\sum }_{k=1}^K{A}_k\cos \left({\displaystyle \frac{2\pi f_{k}t}{L}}+{\varphi }_k\right),$$

(7)

where $K$ is the number of peak values, $A_k$ is the corresponding amplitude, $f_k$ is the frequency value $f$ corresponding to the maximum value of the power spectrum $S\left(f\right)$, and ${\varphi }_k$ is the corresponding phase.

2.2. Signal Similarity Evaluation Metrics

To evaluate the effectiveness of extracting data variation trends, drawing from the similarity calculations of LIU N., Zhang B., and Baba T. et al. [32,33,34], this paper proposes a multi-dimensional similarity evaluation metric, as shown below:

$$\mathrm{S}\mathrm{i}\mathrm{m}\left(D,\widehat{ D}\right)={\alpha }_1\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}\left(D, \widehat{D}\right)+{\alpha }_2\left(1-NAE\right)+{\alpha }_3\mathrm{S}\mathrm{S},$$

(8)

where $D$ is the original signal data sequence, $\widehat{D}$ is the reconstructed signal data sequence, $Corr(D,\widehat{D})$ is the Pearson correlation coefficient, $NAE$ is the normalized average error, $SS$ is the spectral similarity score and α₁, α₂, α₃ are weighting coefficients (∈[0,1]) with α₁ + α₂ + α₃ = 1.

The overall temporal similarity is calculated using the Pearson correlation coefficient:

$$\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}\left(D,\widehat{ D}\right)={\displaystyle \frac{\sum _{t=1}^L\left(D\left(t\right)-\overline{D}\right)\left(\widehat{D}\left(t\right)-\overline{\widehat{D}}\right) }{\sqrt{\sum _{t=1}^L{\left(D\left(t\right)-\overline{D}\right)}^2 \sum _{t=1}^L{\left(\widehat{D}\left(t\right)-\overline{\widehat{D}}\right)}^2 }}},$$

(9)

where $D\left(t\right)$ is the original signal based on time $t$, $\widehat{D}\left(t\right)$ is the reconstructed signal based on time $t$, and $L$ is the length of the signal sequence.

Amplitude similarity is calculated using the normalized average absolute error:

$$\mathrm{N}\mathrm{A}\mathrm{E}={\displaystyle \frac{\mathrm{M}\mathrm{A}\mathrm{E}}{\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}}}={\displaystyle \frac{\frac{1}{L}\sum _{t=1}^L \left|D\left(t\right)-\widehat{D}\left(t\right)\right|}{\max \left(D\right)-\min \left(D\right)}},$$

(10)

Spectral similarity is calculated using the normalized spectral Euclidean distance:

$$S\mathrm{S}=1-\sqrt{{\displaystyle \frac{{\sum }_f{\left(\left|Y\left(f\right)\right|-\left|\widehat{Y}\left(f\right)\right|\right)}^2}{{\sum }_f{\left|Y\left(f\right)\right|}^2}}},$$

(11)

where $f$ is frequency, $Y\left(f\right)$ is the Fourier transform of the original signal, and $Ŷ\left(f\right)$ is the Fourier transform of the reconstructed signal.

2.3. Adaptive Gaussian Smoothing

The window size is adaptively adjusted based on local data features. The calculation formula is as follows:

$${WindowSize}_{score}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\left|x-x_{ref}\right|}{{\mu }_{diff}}}+{\displaystyle \frac{\left|\nabla x\right|}{{\mu }_{grad}}}\right),$$

(12)

where $x$ is the original signal,$x_{ref}$ is the reference simulated trend, $\nabla x$ is the gradient, ${\mu }_{diff}$ and ${\mu }_{grad}$ are two normalization factors.

${\mu }_{diff}$ is the reference deviation normalization factor. The calculation formula is as follows:

$$u_{diff}=median\left(\left|x-x_{ref}\right|\right)+k\times std\left(\left|x-x_{ref}\right|\right),$$

(13)

where $median\left(\left|x-x_{ref}\right|\right)$ is the median absolute deviation from the reference, $std\left(\left|x-x_{ref}\right|\right)$ is the standard deviation of the deviations and $k$ is a scaling factor to account for data variability.

${\mu }_{grad}$ is the gradient normalization factor. The calculation formula is as follows:

$$u_{grad}=median\left(\left|\nabla x\right|\right)+k\times std\left(\left|\nabla x\right|\right),$$

(14)

where $median\left(\right|\nabla x\left|\right)$ is the median absolute gradient, $std\left(\right|\nabla x\left|\right)$ is the standard deviation of the gradient magnitudes and $k$ is the same scaling factor as above.

The algorithm considers both spatial distance and signal similarity weights (maintaining local adjustment features by referencing overall variation trends). The calculation formula is as follows:

$$w\left(x\right)=w_{dist}\left(x\right)\times w_{dev}\left(x\right)=e^{-{\displaystyle \frac{{\left(x-c\right)}^2}{2{\sigma }_x^2}}}\cdot e^{-{\displaystyle \frac{{\left(x-x_{ref}\right)}^2}{2{\sigma }_x^2}}}=e^{-\left({\displaystyle \frac{{\left(x-c\right)}^2+{\left(x-x_{ref}\right)}^2}{2{\sigma }_x^2}}\right)},$$

(15)

where $x$ is the current point being processed, $c$ is the center point of the current window position, $x_{ref}$ is the reference value from the overall trend, ${\sigma }_x$ is the standard deviation of local data variation, $w_{dist}\left(x\right)$ is the spatial distance weight component, and $w_{dev}\left(x\right)$ is the deviation weight component.

2.4. Laplacian Operator Feature Enhancement

The discrete Laplacian operator for one-dimensional signals is defined as

$${\nabla }^{2}D\left(t\right)=D\left(t+1\right)-2D\left(t\right)+D\left(t-1\right),$$

(16)

where $D\left(t\right)$ is the signal value at position $t$, $D(t-1)$ is the signal value at position $t-1$, $D(t+1)$ is the signal value at position $t+1$, and ${\nabla }^2$ is the Laplacian operator.

Based on this, iterative Laplacian smoothing can be expressed as

$$D_{k+1}\left(t\right)=D_k\left(t\right)+\lambda {\nabla }^2{D}_k\left(t\right),$$

(17)

where $D_k\left(t\right)$ is the signal value at position $t$ in the $k$th iteration, $\lambda$ is the smoothing factor where $\lambda \in \left(\mathrm{0,1}\right)$, and $k$ is the number of iterations.

To better preserve local features, an adaptive smoothing factor is introduced, as shown below:

$$\lambda \left(t\right)={\lambda }_{base}\times e^{-{\displaystyle \frac{\beta \left|\nabla D\left(t\right)\right|}{\sigma }}},$$

(18)

where ${\lambda }_{base}$ is the base Laplacian smoothing factor, $\beta$ is the sensitivity parameter, $\nabla D\left(t\right)$ is the local gradient, and $\sigma$ is the standard deviation of local gradients.

The adaptive smoothing formula becomes

$$D_{k+1}\left(t\right)=D_k\left(t\right)+\lambda \left(t\right)\left[D_{k+1}\left(t\right)-2D_k\left(t\right)+D_k\left(t-1\right)\right],$$

(19)

The window size is adjusted based on local signal characteristics according to

$$w\left(t\right)=w_{base}+⌊\alpha \times \left(1-{\displaystyle \frac{\left|\nabla D\left(t\right)\right|}{max\left|\nabla D\right|}}\right)⌋,$$

(20)

where $w_{base}$ is the basic window size, and $\alpha$ is the window scaling factor.

2.5. Algorithm Code

The pseudocode for the processing Algorithm 1 is as follows:

Algorithm 1. PlaneDenoiseViaGaussLap.

Input: P = {P1, P2,  , Pₙ}   //set of point cloud by line-scanning laser Output: Q = {Q1, Q2, …, Qₙ}  //denoised point cloud set 1: Π ← RANSAC_FitPlane(P)   //fit to establish reference plane Π 2: Q ← ∅     //initialize empty output point cloud 3: for each line Pi∈P do: 4:  if |Pi| > threshold do: 5:    D ← Dist(Pi, Π)      //compute signed distances to Π as original data 6:  end if 7:  Ŝ ← ∅   //initialize smoothed distance set 8:  //adaptive Gaussian smoothing phase 9:   for each d∈D do: 10:  σ ← LocalVar(D, d, ω)  //estimate local noise characteristics $11:{\omega }_{\hspace{1em}adapt}$ ← ω(1 + ασ) //adapt window size based on local variance $12:ŝ \leftarrow \mathrm{GaussSmooth}(\mathrm{d}, {\omega }_{adapt}$) //apply adaptive Gaussian kernel to smooth 13:   Ŝ ← Ŝ ∪ {ŝ} 14:   end for 15:    //Laplacian enhancement phase to preserve sharp features 16:    L ← Ŝ 17:    for iteration = 1 to K do: 18:   for j = 2 to |L|−1 do: 19:     ∇2$L_j$ ← L[j + 1] − 2L[j] + L[j − 1] //Compute discrete Laplacian operator 20:       L[j] ← L[j] + β∇2$L_j$  //Update using Laplacian enhancement factor β 21:     end for 22:   end for 23:    Qi ← Reconstruct(Pi, L, Π) //Reconstruct denoised points 24: end for 25: return Q   //Return point cloud

Tips: RANSAC plane fitting may encounter challenges or potential failures in several scenarios, including highly curved or non-planar surfaces; multi-plane structures and sparse or unevenly distributed point clouds. However, in our specific application involving steel plates, these limitations have minimal impact on the results for the following reasons: the steel plates in our study exhibit only slight angles (typically less than 5° deviation across the surface), making the planar approximation valid for our purposes. And the experiments employ a high-precision industrial laser scanning device that ensures uniform point cloud distribution.

2.6. Evaluation Criteria

Signal-to-noise ratio (SNR) is the ratio between the signal and noise in a system, calculated as

$$SNR=10 lg{\displaystyle \frac{P_S}{P_N}},$$

(21)

where $P_S$ is the effective power of the signal, and $P_N$ is the effective power of the noise.

The SNR serves as the data measurement standard. Visual comparisons are made between pre- and post-vibration compensation laser data randomly selected from one line, alongside four types of visualization graphs: adjacent point differences, local variance, local curvature, and local features (absolute value of local curvature).

Frequency analysis employs the Welch method, dividing the signal into $L$ overlapping segments, each of length $M$. The period estimate for the i-th segment is as follows:

$${\widehat{P}}_i\left(f\right)={\displaystyle \frac{1}{MU}}{\left|{\sum }_{n=0}^{M-1} z_i\left[n\right]w\left[n\right]e^{-j2\pi fn}\right|}^2,$$

(22)

where $U={\displaystyle \frac{1}{M}}\sum _{n=0}^{M-1}{w}^2\left[n\right]$ is the normalization factor for the window function, and $z_i\left[n\right]$ represents the data from the i-th segment.

The final Welch Power Spectral Density (PSD) estimate is the average of all segment periods:

$${\widehat{P}}_{xx}\left(f\right)={\displaystyle \frac{1}{L}}{{\sum }_{n=0}^{M-1} \widehat{P}}_i\left(f\right),$$

(23)

The horizontal axis is frequency, while the vertical axis shows the logarithm of power spectral density:

$$PS{D}_{dB}\left(f\right)=10{\log }_{10}\left({\widehat{P}}_{xx}\left(f\right)\right),$$

(24)

3. Experiments and Results Analysis

The entire experimental framework diagram is shown in Figure 2:

3.1. Experimental Equipment and Environment

The main equipment used in the experiments includes a line-scanning laser, incremental encoder, computer, and related auxiliary equipment. Specific equipment configurations are as follows:

Line-scanning laser: Model SENGO sG56N 040 × 060 (Shanghai Sengo Advanced Technology Co.,Ltd., Shanghai, China), with a resolution of 0.43 μm, profile data interval of 10.5 μm, and capable of achieving full-field scanning speeds up to 1000 frames/second.

Encoder: Using an incremental encoder, model SICK DBS50E-S4EK01000 (1000 lines) (SICK AG, Shenzhen, China), transmitting values through digital signal rising edges via an information output channel (A).

Computer: Data acquisition and processing performed on a computer equipped with an AMD Ryzen7 7840HS (Advanced Micro Devices, Inc. (AMD), California, United States) processor and 32 GB memory. MATLAB R2023a installed for data processing and analysis.

Auxiliary equipment: Includes rigid mounting brackets and calibration plates for fixing and calibrating the line-scanning laser and encoder, ensuring stability and accuracy during measurement.

For precision standard parts, the first priority at the hardware structure level was to adopt a design principle of separation from the main structure. They were installed on independent platforms while minimizing vibration amplification due to structural deformation, reducing interference from other system factors as much as possible. This ensures their complete independence from the system and freedom from interference.

Based on the parameters of the experimental equipment and numerous experimental attempts, the key parameters used in this article are shown as

Table 1. The values of key parameters.

Key Parameters	Value
$w_{base}$	50
$\alpha$	20
${\lambda }_{base}$	0.5
$\beta$	2

:

3.2. Simulated Data Verification

The base plane is an inclined plane with certain local features added (simulated here using multiple superimposed sine waves), plus some random white noise, generated using code to create simulated data.

The algorithm was verified on the simulated plane point cloud, repeated 10 times and averaged. The data obtained are shown in

Table 2. SNR comparison before and after vibration compensation for three data sets.

Data Type	Original SNR (dB)	Post-Compensation SNR (dB)
Simulated Data	61.66	71.39
Standard Bearing	57.56	61.48
Actual Plane	61.91	69.49

. The original data, trend plot and comparison before and after vibration compensation for a random laser line of simulated data are shown in Figure 3:

It can be observed from the image that the main trend of the original data can be extracted through the simple superposition of sine waves; the data before vibration compensation exhibit severe jitter near adjacent points, while the compensated data address this issue and retain the overall trend (such as the overall curvature of changes) as well as local features (such as depressions in certain areas).

This paper compares four types of visualizations: the difference between adjacent points, local variance, local curvature, and local features (the absolute value of local curvature). The visual images are in Figure 4:

Comparing the four basic parameter visualization plots, the rebuild curve lying below the original curve indicates that the algorithm has changed data; the amplitudes of adjacent point differences, local variance, local curvature, and local features are all significantly reduced, proving that the algorithm effectively reduces local dramatic changes and improves local smoothness. In terms of overall trends, the positions of the peaks are essentially overlapping, demonstrating that the local features have been preserved.

The results obtained by analyzing the frequency domain using the Fourier transform are shown in Figure 5:

Comparing before and after compensation, the low-frequency data are essentially overlapping, which proves that the algorithm retains local features; the energy of the mid to high frequencies of the compensated signal is significantly lower than that before compensation, which proves that the compensation indeed reduces high-frequency vibrations.

The verification through the aforementioned experiment demonstrated that using Fourier transform analysis can quickly extract the overall trend of the data by identifying the main frequency components and their corresponding amplitudes and phases. Based on local data features, adaptively dividing windows and using the improved Gaussian-Laplacian filter for vibration compensation can effectively preserve local features.

3.3. Standard Parts Experimental Verification

A line-scanning laser was used to scan a precision standard bearing with a diameter of 129.991 ± 0.001 mm. Due to the relatively small number of points collected in the point cloud, direct cylinder fitting produced poor results.

The photographs taken at the site of the experimental system are shown in Figure 6:

Based on these considerations, the experimental scheme was designed, where independent precision standard parts were used for auxiliary detection, with the line laser simultaneously scanning both the precision standard parts and the steel plate under test, ensuring identical external interference for both. First, vibration compensation was performed and verified on the point cloud of the precision standard parts’ surface. Once the algorithm proved effective, it was similarly applied to the steel plate’s surface point cloud. The overall algorithm approach is shown in Figure 7:

After setting up the experimental apparatus and collecting the required data, preprocessing was performed (including point cloud filtering, noise filtering, and outlier removal), followed by data reading, format parsing, format conversion, mapping data points to three-dimensional space, and precise region of interest (ROI) location. Each line laser scan was segmented based on data characteristics, and surface point cloud data for each object were segmented according to object features.

The Iterative Closest Point (ICP) algorithm was first proposed by P.J. Besl et al. [35] in their 1992 paper published in IEEE Transactions on Pattern Analysis and Machine Intelligence. Registration can be achieved by iteratively computing to minimize an objective function, finding the transformation matrix between the same point clouds in different poses. For point clouds with different numbers of points, in each iteration, the algorithm finds the closest point in the second point cloud for each point in the first point cloud and calculates the transformation based on these closest point pairs.

For generating ideal state standard point clouds without external interference based on strict mathematical logic, a series of arc points are generated in three-dimensional space, and the arcs are copied along the axial direction at fixed intervals to assemble the ideal point cloud. The specific steps are as follows:

We can assume that the line-scanning laser collects data with a lateral spacing of $\mathsf{\delta }y$, forward direction spacing of $\mathsf{\delta }x$, N data points distributed over the range $\mathsf{\Delta }y$, and point cloud data obtained from the precision standard part surface over time t as $p{c}_{meas}$. Based on its X-axis spacing $\mathsf{\delta }x$, Y-axis spacing $\mathsf{\delta }y$, and the dimensions of the precision standard part used, calculations can be performed to construct $p{c}_{bearing}$, which is the point cloud data of a precision standard part in an ideal state.

When evaluating the ICP algorithm, the Root Mean Square Error (RMSE) of matching point pairs is typically used to quantitatively assess the accuracy of registration results. The calculation formula is

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\left(y_i-f_i\right)}^2},$$

(25)

where $yᵢ$ represents the transformed coordinates of points from the original data, $fᵢ$ represents the coordinates of corresponding points in the target point cloud, and $n$ is the number of valid matching point pairs.

The vibration compensation algorithm processing steps for the standard bearing are as follows:

1.

The registration algorithm is sensitive to point cloud density and initial position. Based on the sampling density, data scale, and other characteristic information of the acquired surface point cloud data (denoted as $p{c}_{meas}$), simulated point clouds are generated by computing the ideal standard point cloud model (denoted as $p{c}_{cylinder}$) to ensure consistent sampling density;

2.

Based on the dimensions of $p{c}_{meas}$, $p{c}_{cylinder}$ is shape-sized trimmed, iterating until $p{c}_{cylinder}$ is slightly larger than the surface-collected point cloud;

3.

Given that the analytic surface equation of $p{c}_{cylinder}$ is ${\Phi }_{org}$, ICP is performed between $p{c}_{cylinder}$ and $p{c}_{meas}$. The transform obtained from point cloud registration is applied to $p{c}_{feature}$ to obtain ${\Phi }_{refer}$;

4.

Since $p{c}_{meas}$ exhibits certain spatial irregularities, the distance from points in each laser data line to the reference surface ${\Phi }_{refer}$ is calculated to obtain the $dist$ data;

5.

For the $dist$ portion of the bearing point cloud data corresponding to each line laser data of $p{c}_{meas}$, the vibration interference $vibration\left(t\right)$ is removed.

Here, since the reference surface ${\Phi }_{refer}$ also has relatively few points, directly calculating its mathematical analytical expression would result in large errors. Therefore, this paper uses the transform matrix obtained from ICP with the standard point cloud that has a known analytical expression. The two point cloud’s registration results are shown in Figure 8:

The RMSE values were recorded for each registration between the two point clouds, which were repeated 10 times and averaged, resulting in an RMSE value of approximately 0.02 mm.

For the standard bearing, the algorithm was validated, repeated 10 times, and averaged. The data obtained are shown in Table 2. The comparisons before and after vibration compensation for a random laser line of the standard bearing data are shown in Figure 9, Figure 10 and Figure 11:

The effectiveness of the algorithm is further verified by the standard data of precision parts.

The analysis results similar to those obtained from simulated data, with the standard data from precision bearing, demonstrate the method’s effectiveness. However, there are wave peaks in some areas (around point 1800) at Figure 10, which, combined with actual surface observation, can be attributed to corrosion.

3.4. Actual Plane Data Verification

The algorithm was verified on the collected actual plane point cloud, repeated 10 times and averaged. The data obtained are shown in Table 2. The comparisons before and after vibration compensation for a random laser line of actual plane data are shown in Figure 12, Figure 13 and Figure 14:

The analysis is similar to that of the simulated data, with the algorithm showing clear effectiveness on actual data. Frequency analysis revealed that the dominant vibration components in this experimental setup predominantly occurred within the 0–0.3 Hz range, which is characteristic of this paper’s specific mechanical system configuration (incorporating the line-scanning laser and ball screw drive). However, it is crucial to emphasize that this frequency range is an equipment-specific constraint rather than an inherent limitation of the algorithm itself.

The SNR data of the above three data before and after vibration compensation are as follows.

3.5. Comparison Between Gaussian Smoothing and Adaptive Algorithm on Actual Plane

When directly applying Gaussian smoothing to single laser line data, the results are shown in Figure 15:

Simple Gaussian smoothing can only eliminate vibrations globally but cannot distinguish local features.

When directly applying the Laplacian operator to single laser line data, the results are shown in Figure 16:

The data become smoother near local extrema among adjacent points, making features more prominent, but vibrations cannot be eliminated.

The results of using the improved Gaussian–Laplacian filter based on local data features are shown in Figure 17:

While compensating for vibration, it effectively preserved the local data features of the depression on the right side (around point 1800) (with profile intervals of 10.5 μm, 150 points approximately equals 1.6 mm).

3.6. Comparison Before and After Compensation on Actual Planes

The comparisons before and after vibration compensation for a random laser line of the standard bearing data are shown in Figure 18:

The alternating color patterns effectively illustrate the transformation before and after compensation. In the original point cloud, pronounced fluctuations are readily observable, indicating the presence of vibration. The magnified regions reveal that the unprocessed data points oscillate around the compensated data profile. After compensation, the point cloud exhibits significantly reduced local vibration, characterized by smoother point-to-point transitions while preserving the intrinsic surface features.

4. Discussion

The development and implementation of high-precision surface defect detection systems for large steel plates represent a critical challenge in modern industrial manufacturing. This research addresses a fundamental aspect of this challenge by focusing on vibration compensation in point cloud data acquisition, particularly for line-scanning laser systems mounted on ball screw-driven platforms.

This paper‘s experimental results demonstrate the effectiveness of the proposed algorithm across different scenarios, with significant improvements in signal-to-noise ratios. However, several important considerations emerge from this research that warrant further investigation, as outlined below.

4.1. Current Achievements and Limitations

The algorithm’s performance shows strong capabilities in handling localized vibration compensation, particularly in the frequency range typical of this experiment systems (0.1–0.3 Hz). The adaptive window sizing and improved Gaussian–Laplacian filtering effectively preserve local features while reducing vibration-induced noise. However, the current validation focuses primarily on localized measurements, representing only a subset of the challenges present in full-scale industrial applications.

4.2. Future Research Directions

The transition from laboratory validation to industrial implementation presents several key challenges. Based on this paper‘s findings, several critical research directions emerge, as outlined below:

1.

Developing synchronized measurement systems for large-scale applications;

2.

Integrating compensation for geometric deformations;

3.

Enhancing robustness against diverse industrial environmental conditions;

4.

Extending the algorithm’s capability to handle broader frequency ranges.

5. Conclusions

This research establishes a foundation for point cloud vibration compensation in precision measurement systems through the development of an improved Gaussian–Laplacian filter algorithm. The method demonstrates significant improvements in signal-to-noise ratios across different test scenarios: 15.78% for simulated data, 6.81% for standard parts, and 12.24% for actual plane measurements.

While these results validate the effectiveness of this approach, they represent a solution for local issues in comprehensive surface quality inspection for large steel plates. The algorithm’s current implementation focuses on specific experimental conditions, providing a foundation for future expansions.