.png){kind=icon}
.png){kind=icon}
Return
An Image Distortion Correction Method Based on Sequence Recurrence
.png){kind=icon}
PRE
AI
DOI：10.3788/gzxb20265504.0410006.png)
.png){kind=icon}
Original request for help.png){kind=icon}
Share.png){kind=icon}
Save
Abstract
En 中文
In the realm of high-precision computer vision and photogrammetry, the accuracy of inverse distortion correction directly dictates the reliability of downstream tasks such as 3D reconstruction and visual measurement. Existing correction methods predominantly rely on an approximation strategy wherein the actual distorted image coordinates are utilized as substitutes for the unknown ideal coordinates to compute distortion terms. While computationally convenient, this approximation introduces systematic theoretical errors that are significantly amplified by the camera's focal length, resulting in non-negligible pixel-level deviations. The primary objective of this study is to eliminate this inherent systematic bias by proposing a novel, rigorous numerical framework based on sequence recurrence. By transforming the inverse solution of the non-linear distortion model into a fixed-point iteration problem, this research aims to achieve theoretically exact recovery of ideal image coordinates, thereby satisfying the stringent accuracy requirements of industrial metrology without resorting to computationally prohibitive global optimization techniques. To address the limitations of traditional approximation methods, this paper develops a mathematical model that treats distortion correction as a sequence recurrence problem, divided into two distinct strategies: a radial-only model and a coupled radial-tangential model. For scenarios dominated by radial distortion, the relationship between distorted and ideal radii is formulated as a seventh-degree algebraic equation involving the distortion coefficients. The study constructs a recursive sequence for the scaling factor, where the initial term is derived from the pixel's initial distortion scale. A recurrence formula is derived to iteratively update the ideal radial distance estimate based on the scaling factor from the previous iteration. The convergence of this sequence is mathematically proven using the contraction mapping principle, demonstrating that the derivative of the mapping function satisfies the contraction condition within the valid parameter range of real-world imaging systems, thus guaranteeing convergence to a unique fixed point. For complex scenarios involving both radial and tangential distortions, the inverse problem presents a high-dimensional non-linear coupling. The study proposes a joint recurrence model where the coordinate approximations from standard methods serve as initial values. A coordinate recurrence formula is established based on the inverse transformation of the complete distortion model. In each iteration step n, the coordinates are updated by subtracting the distortion offsets calculated from the current estimates, progressively refining the solution. This iterative process effectively decouples the non-linear terms numerically, driving the coordinate sequence toward the true ideal values. The algorithm is designed with a dynamic termination criterion based on the error threshold between successive iterations to balance precision and computational efficiency. Experiments were conducted using a D330M industrial structured-light camera system to validate the proposed method in terms of both 2D pixel-level accuracy and 3D reconstruction precision. In the 2D accuracy evaluation, an inverse-forward closed-loop consistency check was employed to quantify errors. The experimental data revealed that traditional approximation methods suffer from significant systematic errors, particularly at the image periphery, exhibiting a distinct center-to-edge error distribution gradient. In contrast, the proposed recurrence method demonstrated a dramatic improvement in accuracy. After merely a single iteration, the reprojection error was reduced by approximately two orders of magnitude compared to the traditional baseline. Specifically, for the radial-only recurrence model, the error exhibited exponential convergence, decaying from the sub-pixel level to the machine precision limit within 3 to 4 iterations. For the coupled distortion model, the error rapidly stabilized within 0.01 pixels after the first iteration. Heatmap analysis of the error distribution across 72 image blocks confirmed that the proposed method eliminates the spatial non-uniformity characteristic of traditional methods, yielding a spatially homogeneous error field. A further comparison of the two recursive strategies proposed in this paper shows that the reprojection error of method 1 is smaller than that of method 2. This is mainly because method 1 is not affected by the cross terms of tangential distortion and can iterate towards the optimal value at a faster rate. In the 3D reconstruction validation, the method was applied to measure standard matte ceramic spheres with a theoretical diameter of 50.801 9 mm and a center distance of 299.8513 mm. The traditional method yielded a diameter measurement error of 0.28 mm and a center distance error of 0.29 mm. Conversely, the proposed method achieved a significantly higher precision. The coupled recurrence model reduced the diameter measurement error to 0.038 mm and the center distance error to 0.15 mm. These results indicate a fundamental enhancement in geometric fidelity, validating the method's capability to suppress error propagation from the 2D image plane to 3D spatial coordinates. The proposed sequence recurrence-based distortion correction method offers a mathematically rigorous and computationally efficient solution to the inverse distortion problem. By abandoning the coordinate approximation assumption inherent in classical methods, it successfully bridges the gap between the distorted observation and the ideal projective geometry. The study concludes that the recurrence models possess robust convergence properties and can be seamlessly integrated into existing camera calibration pipelines. The method demonstrates particular superiority in high-precision measurement tasks where tangential and high-order radial distortions are non-negligible. While the coupled model is susceptible to local minima inherent to the non-convex nature of the full distortion polynomial, the achieved residual error is negligible for practical applications. Consequently, this approach provides a vital algorithmic foundation for advancing the precision of industrial visual inspection, robot navigation, and structured-light 3D reconstruction systems.
Keywords:
Distortion correction
Inverse distortion mapping
Sequence recursion
Radial-tangential coupled distortion
Reprojection error