Objective  Objective Phase Measurement Deflectometry (PMD) is widely employed in free-form surface transmission wavefront detection due to its simplicity, high accuracy, and broad detection range. Achieving high-precision phase acquisition is a critical step in the measurement and detection process. The phase unwrapping task, crucial in optics, plays a pivotal role in optical interferometry, magnetic resonance imaging, fringe projection profilometry (FPP), and other fields ^1-4. The challenge lies in recovering a continuously varying true phase signal from the observed wrapped phase signal within the range of −π, π). While the ideal phase unrolling involves adding or subtracting 2π at each pixel based on the phase difference between adjacent pixels, practical applications face challenges such as noise and phase discontinuity, leading to poles in the wrapped phase ⁵. These poles result in accumulated computational errors during the unwrapping process, causing phase unwrapping failures. Various methods are employed to unwrap and obtain the real phase distribution. To address these challenges, this paper proposes a phase unwrapping algorithm based on an improved U-Net network.

  Methods  During the model training process, a composite loss function is defined to train the network based on the specific problem of spatial phase unwrapping. To address these challenges, this paper proposes a phase unwrapping algorithm based on an improved U-Net network. This algorithm utilizes U-Net as the basic network, integrates the CBiLSTM module for modeling time series, introduces an attention mechanism for enhanced generalization, and explores optimized loss functions. The proposed network model is validated through simulated and real datasets, showcasing its outstanding performance under noise, discontinuity, and aliasing conditions.The introduction of the attention mechanism enables better capture of global spatial relationships, while CBiLSTM effectively captures and stores long-term dependencies through memory unit structures. Memory units selectively remember and forget parts of the input signal information, enhancing their ability to handle long sequence data modeling tasks. The paper defines a composite loss function tailored to the spatial phase unwrapping problem during the model training process.Comparative experiments between the proposed network and classic models, such as U-Net ²⁰, Res-UNet ²¹, and methods by Wang ¹³ and Perera et al. ¹⁹, demonstrate the robustness of the proposed network under severe noise and discontinuities. Additionally, it showcases computational efficiency in performing spatial phase unwrapping tasks.

  Results and Discussions  Fig.10 shows the comparison between the predicted absolute phase and the real phase output by the wrapped phase after training the network model proposed in this article. Through the construction of the encoder-decoder model, the introduction of the CBiLSTM module and the attention mechanism module, and the composite The definition of the loss function, after comparing with other models, verifies the improvement in accuracy and reduction in training of the network model proposed in this article in the three situations mentioned above. Through simulation experiments and verification, by enhancing the deep learning model's ability to pay attention to key phase information, the network model proposed in this article can improve the accuracy and robustness of phase unwrapping, and promote further development in fields such as optical measurement and phase imaging.

  Conclusions  This paper addresses the challenge of wrapped phase unwrapping by introducing a novel convolutional architecture framed as a regression problem. The proposed network incorporates several enhancements within the encoder-decoder framework, notably featuring a CBiLSTM module and a soft attention mechanism. Comparative analyses with existing phase unwrapping methods demonstrate the network's remarkable performance in achieving precise phase unwrapping, even in severe noise, discontinuities, and aliasing. Notably, the network showcases exceptional unwrapping capabilities without necessitating extensive training on large datasets. Moreover, it exhibits significantly reduced computational time, rendering it well-suited for tasks requiring accuracy and expeditious phase unwrapping.Validation experiments conducted on real laboratory datasets further affirm the outstanding performance of the proposed network. The introduced model empowers phase unwrapping tasks under challenging conditions, such as severe noise, discontinuities, and aliasing, surpassing the limitations of traditional methods. Comparative assessments with other deep learning models reveal a normalized root mean square error (NRMSE) as low as 0.75%. The advancement in unwrapped phase technology holds substantial significance for optical free-form surface detection, contributing to enhanced measurement accuracy, precise control of optical parameters, optimization of optical design, and quality assurance in optical manufacturing and detection processes.