Accurate phase detection is crucial for optical measurement precision and sample thickness reconstruction, especially for weakly absorbing or transparent specimens. Traditional bright-field microscopy lacks phase sensitivity, leading to phase imaging techniques like wave-front sensing, iterative phase retrieval (IPR), and digital holography. IPR is widely used due to its simple setup and robustness. However, multi-plane phase retrieval (MPR) requires precise alignment, which is challenging. Existing solutions include optical path modifications, which increase system complexity and cost, and image registration techniques, which neglect tilt misalignment. These limitations affect phase reconstruction accuracy and require additional manual adjustments. Therefore, it is necessary to propose a digital automated strategy for precise calibration of MPR.

To address these issues, we propose an adaptive joint calibration-based multi-plane phase retrieval (ACMPR) technique to develop a low-cost, high-performance phase imaging system, eliminating the need for precise optical alignment or additional markers. The proposed method integrates auto-focusing, cross-correlation calibration, and tilted plane calibration to compensate for displacement errors and tilt errors digitally. This significantly reduces reliance on complex experimental setups and precision mechanical adjustments, improving the system’s robustness and flexibility. In the calibration stage, we use the cross-correction method to correct the displacement error. Then the Tamura of gradient (ToG) method is utilized to evaluate the tilt error and diffraction distance of the first image. In this way, a precise digital calibration of each parameter in the lensfree system can be achieved, since the optimal reconstruction plane has been determined. In the reconstruction stage, we employ a multi-plane phase retrieval (total variation-based adaptive phase retrieval, TV-APR) algorithm that incorporates spatial weighting and total variation (TV) regularization to accelerate the iterative convergence process. Additionally, we adopt Bluestein-based angular spectrum propagation, which enhances computational efficiency by enabling fast diffraction calculations. These techniques collectively ensure high-quality phase retrieval while maintaining computational efficiency.

The ACMPR method effectively compensates for displacement and tilt errors in multi-plane diffraction imaging, as confirmed by simulation and experimental results. Unlike traditional methods such as adaptive cascade calibrated (ACC) method, which only correct displacement errors, ACMPR simultaneously calibrates both displacement and tilt errors, ensuring precise image reconstruction. Cross-correlation-based registration eliminates displacement errors, while the ToG method accurately estimates diffraction distances, resulting in optimal system alignment. In resolution chart reconstruction experiments as shown in Fig. 7, ACMPR achieves an ultimate resolution of 3.2 μm, successfully resolving the seventh group’s second line pair. In contrast, the ACC method, affected by tilt-induced defocus, reduces resolution to approximately 4.4 μm. By correcting both errors, ACMPR enables the diffraction imaging system to reach its theoretical resolution limit, significantly enhancing image quality. Additionally, ACMPR proves robust under complex conditions, as demonstrated in biological sample reconstructions shown in Figs. 8 and 9. It successfully restores fine details in pancreatic cancer tissue and osteosarcoma cell slices, whereas the traditional methods suffer from uncorrected tilt errors, leading to image distortion. Another advantage of ACMPR is its effectiveness in large-error scenarios, maintaining high-quality reconstruction even with displacement errors up to 10 pixel and tilt errors up to 20°. The high structure similarity index measure (SSIM) and normalized cross-correlation (NCC) evaluation values can also indicate quantitatively the effectiveness of ACMPR as shown in Table 1. Hence ACMPR is well-suited for phase imaging, especially in the portable and miniaturized applications. ACMPR is also ideal for label-free imaging as it can successfully reconstruct the fine phase structures. It indicates that ACMPR outperforms conventional calibration techniques by precisely compensating for displacement and tilt errors. Its accuracy, robustness, and adaptability make it a powerful tool for computational imaging, biomedical imaging, and diffraction-based optical systems, achieving high-resolution, distortion-free reconstructions in challenging imaging environments.

ACMPR method can address the alignment and calibration challenges in multi-plane phase retrieval by providing a fully digital calibration and reconstruction approach. It utilizes cross-correlation calibration to correct displacement errors and an autofocusing algorithm to simultaneously calibrate tilt errors and diffraction distances. Once the system is accurately calibrated, the TV-APR algorithm is applied for iterative recovery of the complex amplitude of the object plane, achieving phase reconstruction at the theoretical resolution limit. To optimize ACMPR’s calibration performance, various autofocusing and image-matching algorithms are compared, ensuring the best calibration accuracy. The method is further benchmarked against the APR algorithm based on the in-line assumption and the ACC algorithm with displacement correction, demonstrating its superiority in handling systematic errors. Experimental results show that ACMPR effectively reconstructs object-plane information across various sample types and error conditions, whereas conventional methods suffer from defocus blurring and other inaccuracies. Unlike traditional multi-plane approaches, ACMPR maintains robust performance under different misalignment conditions, proving its effectiveness and adaptability. It provides a promising digital strategy for high-resolution phase retrieval, computational light-field imaging, and the miniaturization of optical microscopy systems. By offering an automated, precise calibration framework, ACMPR enables enhanced imaging performance and extends the applicability of phase retrieval techniques in next-generation optical systems.