Residue calibrated least-squares unwrapping algorithm for noisy and steep phase maps

Abstract

This work proposes a robust unwrapping algorithm for noisy and steep phase maps based on the residue calibrated least-squares method. The proposed algorithm calculates and calibrates the residues in the derivative maps to get a noise-free Poisson equation. Moreover, it compensates for the residuals between the wrapped and unwrapped phase maps iteratively to eliminate approximation errors and the smoothing effect of the least-squares method. The robustness and efficiency of the proposed algorithm are validated by unwrapping simulated and experimentally wrapped phase maps. Compared with the other three typical algorithms, the proposed algorithm has the most effective performance in noisy and steep phase unwrapping, providing a reliable alternative for practical applications.

Residue-guided phase unwrapping in fringe projection measurements using second differences

“…This is particularly noticeable in regions with noise-induced discontinuities. execution for unwrapping two phase maps in figures 6 9 using RGPU was approximately half second, significantly faster than the time required by RCLS [20]. This execution time is also comparable to Servin's method [22], without the need for two different frequency patterns.…”

“…Figures 6 and 9 also depict masks generated by Bone's algorithm [26] and the proposed algorithm. Figure 7 presents a 3D view of the unwrapped result obtained using the RCLS method [20] and compares its mathematical differences with the proposed method. Similarly, figure 10 illustrates the 3D perspective of the unwrapped results using Bone's method [26] and Servin's method [22], along with the mathematical differences when compared to the proposed algorithm.…”

“…Figures 8 and 11 present cross-sections along the horizontal centerlines of their respective phase maps for comparison. In figure 8, the RCLS technique [20], though robust and utilizing residues, still employs the minimum norm solution unwrapping strategy, affecting the phase map's fidelity, especially for non-smooth furnace plate materials.…”

“…Current methods for phase unwrapping can be categorized into three main groups: path-dependent methods [12][13][14][15][16][17], minimum norm methods [18][19][20][21], and path-independent (temporal) unwrapping approaches [22][23][24][25]. Temporal unwrapping algorithms excel at unwrapping objects with physical discontinuities, but they face challenges in noisy conditions due to their reliance on the ratio between multiple wavelengths (low and high frequencies), which amplifies noise during phase unwrapping.…”

“…The latter category, on the other hand, is robust but tends to smooth out desired information. For instance, the residue calibrated least-squares (RCLS) unwrapping algorithm [20] aims to remove residues in phase derivatives iteratively but can lead to over-smoothing.…”