Calibrating unknown transformation relationships is an essential task for multi-robot cooperative systems. Traditional linear methods are inadequate to decouple and simultaneously solve the unknown matrices due to their intercoupling. This paper proposes a novel dual-robot accurate calibration method that uses convex optimization and Lie derivative to solve the dual-robot calibration problem simultaneously. The key idea is that a convex optimization model based on dual-robot transformation chain is established using Lie representation of SE(3). The Jacobian matrix of the established optimization model is explicitly derived using the corresponding Lie derivative of SE(3). To balance the influence of the magnitudes of the rotational and translational optimization variables, a weight coefficient is defined. Due to the closure and smoothness of Lie group, the optimization model can be solved simultaneously using Newton-like iterative methods without additional orthogonalization processing. The performance of the proposed method is verified through simulation and actual calibration experiments. The results show that the proposed method outperforms the previous calibration methods in terms of accuracy and stability. The actual experiments are used to compare the proposed method with two existing calibration methods, and the mean measurement error of a certified ceramic sphere is reduced from 0.9205mm and 0.5363mm to 0.4381mm, respectively.

In 3D microscopic imaging, the extremely shallow depth of field presents a challenge for accurate 3D reconstruction in cases of significant defocus. Traditional calibration methods rely on the spatial extraction of feature points to establish spatial 3D information as the optimization objective. However, these methods suffer from reduced extraction accuracy under defocus conditions, which causes degradation of calibration performance. To extend calibration volume without compromising accuracy in defocused scenarios, we propose a per-pixel calibration based on multi-view 3D reconstruction errors. It utilizes 3D reconstruction errors among different binocular setups as an optimization objective. We first analyze multi-view 3D reconstruction error distributions under the poor-accuracy optical model by employing a multi-view microscopic 3D measurement system using telecentric lenses. Subsequently, the 3D proportion model is proposed for implementing our error-based per-pixel calibration, derived as a spatial linear expression directly correlated with the 3D reconstruction error distribution. The experimental results confirm the robust convergence of our method with multiple binocular setups. Near the focus volume, the multi-view 3D reconstruction error remains approximately 8 μm (less than 0.5 camera pixel pitch), with absolute accuracy maintained within 0.5% of the measurement range. Beyond tenfold depth of field, the multi-view 3D reconstruction error increases to around 30 μm (still less than 2 camera pixel pitches), while absolute accuracy remains within 1% of the measurement range. These high-precision measurement results validate the feasibility and accuracy of our proposed calibration.

This article proposes a novel calibration framework for a multiview microscopic 3-D measurement system comprising multiple tilted-cameras with the Scheimpflug condition and one vertical projector. Conventionally, the accurate and independent point correspondences between images and planar targets achieve successful calibration. However, affected by our system's large imaging magnification and anisotropic perspectivity, targets’ poses must be limited in the shallow common focus volume to ensure correspondences’ extracting accuracy of all views. This imaging constraint will lead to insufficiently independent correspondences in calibration computation, where constraints will be insufficient in parameter estimation (under-constraint problems) and bundle adjustment (falling in local optima caused by ambiguous models). To this end, our method first relies on insufficiently independent correspondences to tackle the above problems. In calibrating multiple tilted-cameras, the proposed initialization method incorporates the lens equation as an additional constraint to solve the under-constraint problem. The subsequent multicamera bundle adjustment disambiguates by stereo constraints. Moreover, for the vertical projector, another novel unconventional calibration method utilizes geometric constraints to determine its projection matrix. The experiments prove that the estimation and optimization of parameters for multiple tilted-cameras are significantly enhanced with the smallest reprojection error near 0.1 pixel. In addition, the 3-D measurement result of the standard concentric cylinders has also been applied to demonstrate the accuracy of the vertical projector calibration and the whole calibration framework.

This article introduces a multiview 3-D microscopic profilometry system equipped with tilted cameras adhering to the Scheimpflug condition and a vertically aligned projector. It utilizes a novel 3-D cross-ratio invariant (3D-CRI) model that offers inherent self-correction, enhanced global consistency, and computational efficiency. The inherent self-correction mitigates optical contaminations like multiple reflections by using deviations from the epipolar line for optimal candidate selection. The model is also designed as spatial lines to associate the approximate linear error distribution across depths, thereby facilitating its further correction (e.g., proposed embedded linear compensation method to address data inconsistency among various binocular projector-camera setups). Additionally, the model simplifies computational demands by directly correlating phase differences to spatial coordinates. Furthermore, a conversion method from the triangular stereo to the 3D-CRI model is developed, avoiding the complex per-pixel calibration. The experiments are conducted to verify robustness, globally consistent accuracy, and speed performance. The experimental results validate that the system is robust to multiple reflections by testing on the printed circuit boards with dense components. The global consistency is also verified by the experiments with better quantitative metrics within the whole measurement range over ten times the depth of field. In particular, the quantitative metrics of the Z -axis are halved. The speed performance shows that our method is the fastest among competing techniques.