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Abstract
This paper introduces a novel phase unwrapping algorithm based on the investigation of the arctangent function. Acting as a differential amplifier associated with nonlinear response, the arctangent operation facilitates distinguishing various wrap types. Phase jumps from two shifted-wrapped maps aligned on the same grids are combined as a pair to function as differential mode inputs. This discerns wraps from continuous and discontinuous surfaces through distinct outputs (threshold phase values). Simulations and experiments validate the feasibility of the proposed wrap-type identification. A complete mathematical form is derived to explain the effects of shifted wraps due to various causes. This paper significantly contributes to the foundational understanding of the 2π ambiguity problem in phase unwrapping.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The measurement of discontinuous surfaces poses a notable obstacle to phase unwrapping techniques. Various solutions involve the utilization of object-based information as auxiliary data. These aids assist in the removal of inherent discontinuities in unwrapping paths. However, this task becomes more intricate due to interferometric factors. Hence, an ongoing need persists for a straightforward and accurate unwrapping solution.
Raw phase data extracted from inverse trigonometric functions is confined within the range $-\pi$ to $\pi$. This allows the phase unwrapping process to be easily accomplished by integrating neighboring pixel phase differences and adjusting by integral multiples of 2$\pi$ for phase wraps. However, the presence of physical discontinuities complicates this integration. When depth variations exceed one optical wavelength, the phase difference between neighboring pixels surpasses $|\pi |$, leading to a 2$\pi$ phase jump (wrap), according to Nyquist theory. Unlike real wraps, which arise purely from arctangent operations, these jumps are fake wraps that should not be unwrapped. This obscures real wraps, resulting in the 2$\pi$ ambiguity phenomenon. This ambiguity introduces unwrapping errors and leads to task failure. [1].
The straightforward solution to the unwrapping issue is to eliminate these ambiguities during the unwrapping process. Once resolved, the unwrapping paths are guaranteed, simplifying the task to match ideal scenarios. However, managing the 2$\pi$ ambiguities is challenging. With only object-based information, no solution exists. The determination of wrap types appeared impossible until the introduction of the Wrap-type phase unwrapping (WTPU) algorithm proposed in this paper.
To circumvent the difficulty of identifying wrap types within a phase map and in accordance with the Nyquist limit, unwrapping schemes in the spatial domain focus on detecting residues or utilizing various quality maps to identify inconsistent phase areas caused by noise or physical discontinuities [2–6]. On the other hand, temporal unwrapping strategies involve recording sequences of intermediate phase maps for dynamic objects or employing fringe patterns with multiple frequencies for measuring static objects. In the latter case, the high-resolution map can be magnified proportionally to the ratio of the low-frequency image [7–11].
Unfortunately, both of these unwrapping strategies rely on a fixed cutting threshold at $\pm \pi$ (due to the Nyquist limit), and the arctangent operation remains unnoticed as it is often treated as a linear system, whether in the spatial or temporal domain. As a result, the implicit effects of the arctangent function are overlooked, and valuable information is considered only in terms of object-based [12] or coded fringe pattern-based input data. Examples of such coded fringe patterns include circular [13,14], colored [15], gray coded [16], or staggered patterns [17]. This limitation is the reason why these methods encounter complications imposed by the Nyquist limit, whether in terms of measurement operations, as temporal unwrapping methods do, or computational requirements, as spatial unwrapping methods do.
The proposed algorithm aims to explicitly identify a specific wrap type associated with a $2\pi$ jump within a wrapped phase distribution by utilizing shifted phases (Fig. 1). It utilizes the arctangent operation as a differential amplifier associated with nonlinear amplification, aiding in the discernment of different wrap types. To accomplish this, the object is shifted by a relative phase in relation to a reference plane. This establishes differential phase jumps, situated at identical grid points before and after the shift, which serve as a pair of differential inputs for the arctangent amplifier. An inherent $2\pi$ cycle invariance, along with branch thresholds, becomes evident among the shifted wraps.
Fig. 1. Schematic of wrap-type identification based on the arctangent differential amplifier. The nonlinear response of the tile slides along the arctangent curve, where $dh$ remains unchanged while the phase angle of $d\theta$ changes, which associated with branch thresholds $\mathcal {T}_{HRc}$ and $\mathcal {T}_{HRd}$,
By leveraging these distinctive wrap types, the algorithm directly identifies unique unwrapping paths on a wrapped phase map. It effectively unwraps real wraps without ambiguity, retains physical discontinuities, and can even rectify noisy pixels. Contrasting conventional phase unwrapping methods, the proposed algorithm overcomes the challenge of insufficient information for wrap type differentiation and adheres to the Nyquist limit without necessitating redundant steps.
This paper introduces the Wrap-Type Phase Unwrapping (WTPU) algorithm. Section 2 provides an in-depth exploration of the algorithm’s details. Subsection 2.1 outlines wrap type identification principles, and Subsection 2.2 addresses shifted wrapped map generation and related parameter considerations. Experimental outcomes are shared in Sections 3 and 4. Section 3 covers results from a computer-simulated surface, while Section 4 discusses practical experiments using a fringe projection system. Finally, the paper concludes with Section 5, providing a summary and overall research conclusion.
2. Principle
2.1 Wrap-type identification
Figure 1 illustrates the fundamental concept of determining wrap types, divided into two parts: preparing the phase-jump pair as differential inputs and identifying specific wrap types using the nonlinear response of the arctangent operation.
In Fig. 1, consider a tile of an object under test with a depth measurement $dh$ and a phase difference $d\theta$. Upon moving along the optical axis by $\Delta l$ relative to a reference plane, a corresponding $\delta$ phase shift occurs before and after the movement. Representing the initial and shifted phases of a discontinuous object as $\Phi (\boldsymbol {r})$ and $\Theta (\boldsymbol {r})$ respectively, the resulting original and second wrapped phase maps, $\varphi (\boldsymbol {r})$ and $\eta (\boldsymbol {r})$, are derived as follows:
Here, $k(\boldsymbol {r})$ is the unknown integer array that we aim to determine through the unwrapping procedure. The symbol $\mathcal {W} [\cdot ]$ signifies the wrapping operator, and $\boldsymbol {r}$ indicates the pixel position in a 2D-discretized space.
Application of the difference operator to Eqs. (1–2) yields:
Here $i$ and $j$ represent the pixel coordinates of adjacent points, i.e., $i=(x,y)$ and $j=(x,y+1)$. The relationship between $\delta$ and $\Delta l$ is such that $\delta$ is solely used to redistribute the positions of fringes and does not affect the absolute phase differences ($d\theta$) between neighboring points. As a result, the relationship $\Delta \Phi (\boldsymbol {r}) = \Delta \Theta (\boldsymbol {r})$ holds true. This equivalence can also be expressed as:
By combining the equivalent relationships described in Eqs. (3–4) with the definition of the $\mathcal {W}$ operation:
For instance, $\mathcal {W}[-\pi /4] \boldsymbol {==} \mathcal {W}[7\pi /4]$ as demonstrated in Fig. 1.
Considering this essential condition for treating $\Delta \varphi (\boldsymbol {r})$ and $\Delta \eta (\boldsymbol {r})$ as differential inputs for the arctangent operation, the identification of random noise takes precedence. This stems from the fact that the two wrapped phases originate from separate and independent experiments, causing noise to manifest differently in both phases. If one of the wrapped phases exhibits noise within a specific area, the other phase may remain noise-free at those corresponding coordinates. Mathematically, this translates to:
This violation contradicts the condition stated in Eq. (7), prompting the need for further modification to adhere to this condition. This can be easily achieved by incorporating other suitable nearby values in the subsequent unwrapping process.
On the contrary, when Eq. (7) holds true and either $\Delta \varphi (\boldsymbol {r}) > \pi$ or $\Delta \eta (\boldsymbol {r}) > \pi$, Eq. (7) can be simplified to $|\Delta \varphi (\boldsymbol {r})|+|\Delta \eta (\boldsymbol {r})|= 2\pi$. This simplification establishes a robust criterion for differential inputs. Consequently, this $2\pi$ cycle invariance can be further extended to distinguish real or physical discontinuous wraps, all due to the nonlinearity inherent in the arctangent curve, as
The symbol $\varepsilon$ denotes the error tolerance for continuous phase, influenced by measurement accuracy and object roughness. On the other hand, $\tau$ represents the threshold for physical discontinuities, reflecting the intrinsic properties of the object ($dh$). In essence, both values should correspond to the phase value of the surface shape divided by the $2\pi$ mode, expressed as:
Remarkably, this method of distinguishing wrap types hinges on the nonlinearity of the arctangent function. In a linear scenario, the physical parameter measured at each object point would be confined to the same phase range due to a constant magnification factor. If, for instance, the arctangent curve were a straight line, the phase difference $d\theta$ would exhibit uniformity across various ranges, thus preventing the differentiation of wrap types. Consequently, the piecewise function (Eq. (10)) cannot be deduced from Eq. (9).
The invalidation of the linear model highlights the significance of the criteria from an opposing perspective. Additionally, the details of the $\mathcal {W}$ operator utilized in Eqs. (3)–(4) are elaborated below:
The relationships established in Eqs. (12) and (13) serve as critical preconditions for determining the validity of the criteria in Eqs. (7) and (8), even though they exhibit a similar mathematical form to $\varphi (\boldsymbol {r}) = \mathcal {W}[\Phi (\boldsymbol {r})]$ and $\eta (\boldsymbol {r}) = \mathcal {W}[\Theta (\boldsymbol {r})]$.
Furthermore, considering the expression for $k\boldsymbol (r)$ in Eq. (1a): if a real wrap is detected, the phase value should be incremented or decremented by $\pm 2\pi$ accordingly, depending on its sign. On the other hand, if a physical discontinuity is detected, indicating a sudden step-like phase change, the phase value should remain wrapped. Consequently, the unwrapping issue in both cases has been successfully addressed:
In conclusion, the proposed wrap-type phase unwrapping algorithm uncovers the inherent behavior of the arctangent function, which is typically obscured within the intricate process of wrapping, using the concise notation: $\mathcal {W} [\cdot ]$.
2.2 Practical generation of two shifted wrapped phase maps and related parameter considerations
In practical experiments, the second wrapped phase distribution outlined above can be obtained by directly adding $\delta$ to $\varphi (\boldsymbol {r})$ and applying the $\mathcal {W} [\cdot ]$ operator, as shown in Eq. (15):
However, an obvious shortcoming in this case is that Eq. (8) would be invalid.
To simplify the procedure of capturing measurements at varying depths from an object to a reference plane without removing the object or translating the reference plane, a phase-stepping technique can be used, while keeping Eqs. (7)–(8) valid. For example, a general fringe projection system is depicted in Fig. 2, where a four-step phase algorithm is applied.
Let $I_i(\boldsymbol {r})$ and $\hat {I}_i(\boldsymbol {r}), i = 1,2,3,4$ represent two sets of four intensity values independently captured in the same measurement environment, both associated with a projection of the same set of sinusoidal fringe patterns with a one-fourth period phase shift [20]. By rearranging the start step of $\hat {I}_i(\boldsymbol {r})$, two shifted phase maps with a $\delta = \pi$ are generated and utilized in the experiments presented in this paper, as expressed by the following equations:
This approach enables the creation of wrapped phase maps with specific $\delta$ values, such as $\pi /2$ and $\pi$. This feasibility arises from $\delta$ not impacting the fundamental characteristics of the measured object; it is subsequently eliminated during the derivative process. In practical applications, $\delta$ can be assigned any arbitrary value that is not a multiple of $2\pi$.
The threshold value $\tau$ employed for identifying phase discontinuities is contingent upon the specific object under investigation and can be chosen empirically. To accurately trace discontinuity edges in phase maps, it is recommended to use a threshold value within the range of 0.5$\pi$ to 1.3 $\pi$. On the other hand, the continuous threshold value $\epsilon$ is determined by the precision of the experimental system and is theoretically expected to be 0. Thus, $\epsilon$ should be smaller than the discontinuity threshold value ($\epsilon < \tau )$. In general, a range of 0.05$\pi$ to 0.15$\pi$ is advised for the value of $\epsilon$.
While this unwrapping algorithm proves effective, it is not immune to problems. The occurrence of random noise at the same locations in two phase maps or high noise levels surpassing the discontinuity threshold ($\epsilon > \tau$) can introduce errors and compromise the reliability of the approach. To alleviate the impact of these noise effects, it becomes necessary to either restrict noise levels or implement pre-filtering measures.
The procedure of this proposed method is concluded in Fig. 2. It is important to clarify that although some phase unwrapping algorithms may have similar names or measurement steps, their underlying principles differ significantly. For example, Souza et al. [21] introduced an algorithm that balances residue charges by utilizing shifted wrapped phase maps to achieve fast processing times, while Stetson [22] and Khmaladze et al. [23] rely on shifting the phase to find continuous regions. However, these methods still only use the phase data or related information without employing the nonlinear information from the arctangent function. This limits their applicability to specific interferometric fields, making them less effective in other applications.
Consequently, the criteria established in Eqs. (7)–(10) are the main contribution of this paper, which reveals a complete framework to describe the implied nonlinear connection between different wrap types and the arctangent function in a rigorous mathematical form.
3. Simulation results
A classical spiral surface was selected as it poses a significant challenge for most path-following unwrapping algorithms due to its sharp discontinuities at intertwined spirals, as seen in the wrapped phase map in Fig. 3(a). Consequently, there are points on this surface that have a high potential for $2\pi$ ambiguity phase errors during the unwrapping process, as demonstrated by Goldstein’s branch-cuts algorithm in Figs. 3(b)-(d). Reference [1] provides further examples of failed results.
Fig. 3. Results of a spiral surface. (a) The wrapped phase map; (b) branch cuts established by Goldstein’s algorhitm [2] and the corresponding unwrapped results shown in 2D and 3D forms in (c) and (d), respectively.
The second phase map is shown in Fig. 4(b) by adding a $\pi$ phase shift to the original underlying phase and rewrapping, according to Eqs. (1)–(2). Figures 4(c) and (d) show cross sections along the center line in both the vertical and horizontal directions. Two sets of points are labeled C/C’ and D/D’ to compare the different wrap types in $2\pi$ jumps. The $2\pi$ jumps resulting from physical discontinuities can be seen more clearly in Fig. 4(d) since no real wraps exist along that direction. It is evident that $2\pi$ errors can easily occur if the physical discontinuous wraps shown in Fig. 4(c) or (d) are incorrectly unwrapped. Figure 4(e) shows the positions of the physical discontinuous wraps and real wraps in different colors. As seen in Fig. 4(f), these assist WTPU in accurately unwrapping the maps.
Fig. 4. Results of a spiral surface using WTPU. (a)-(b) The original and shifted wrapped phase maps; (c)-(d) horizontal and vertical cross sections along the center lines of the wrapped phase of (a)-(b), respectively; (e) identified real wraps (red) and physical discontinuities (white); (f) corresponding unwrapped result. Points C/C’ and D/D’ represent marked positions where real and physical discontinuous wraps occur.
4. Experimental verification
To demonstrate the capabilities of WTPU in practical applications, a machine part with steep discontinuities (Fig. 5(a)) and an ear plaster model with multiple layers of varying depths (some exceeding one fringe period) were chosen to be measured using a general fringe projection setup, as shown in Fig. 2. A Sony VPL-EX 175 3LCD projector, a Basler ace 2040-180 CCD camera, and a laptop computer comprise the setup. The two wrapped phase maps are obtained using the aforementioned four-step phase algorithm, as illustrated in Eqs. (16) and (17). For simplicity, the following experimental results demonstrate only the initial step of the eight fringe patterns from the first set.
Fig. 5. Experimental results of a machine part. (a) Intensity image, (b) corresponding recorded fringe pattern (416 $\times$ 444 pixels), (c)-(d) the original and shifted wrapped phase maps; (e)-(f) unwrapped results of (a) by using Goldstein’s algorithm [2] and Servin’s method [9], respectively.
Due to their surface complexity, both the machine part with steep discontinuities and the ear plaster model with multiple layers and varying depths pose major obstacles for conventional spatial unwrapping approaches, as evidenced by the errors obtained using the Goldstein’s method (Figs. 5(e), 8(e)) [2]. These errors tend to occur at locations with physical discontinuities (indicated by red arrows) and propagate throughout the rest of the image. This type of failure is typical, as unwrapping approaches misinterpret the geometry boundaries as actual wraps.
By combining the second wrapped phase maps depicted in Fig. 5(d) and Fig. 8(d), the proposed algorithm successfully unwrapped the wrapped phase. The identified wrap types are shown in Fig. 6(a) and Fig. 9(a), respectively. Figure 10 displays the cross sections taken along the horizontal center line of the two wrapped images of Fig. 8(c) and Fig. 8(d) using the ear surface profile as a test object that is more general and features an irregular shape and lacks symmetry, in contrast to the machine part. Jump A/A’ denotes a detected physical discontinuous wrap that is easily misinterpreted by conventional unwrapping algorithms, but its discontinuity can be preserved to maintain a real phase feature by using WTPU. Both the identified wraps (Figs. 6(a), 9(a)), and the cross section demonstrate the efficiency of the proposed algorithm in distinguishing wrap types caused by distinct sources, which has a significant impact on discontinuous-phase unwrapping cases. The results of the unwrapping process are displayed in Figs. 6(b) and 9(b).
Fig. 6. Results of WTPU. (a) The distinguished physical discontinuities (white), real wraps (red) and the noise (green), $\epsilon = 0.05\pi, \; \tau = 0.7\pi$ are used; (b) the unwrapped result of WTPU.
Figures 5(f) and 8(f) depict the intermediate unwrapped results obtained by using the Servin’s method [9] respectively, during iteration. Their final results are then used as a benchmark to compare with the proposed algorithm. During the measurement, two sets of projection patterns with different fringe spacings were used. One of these patterns had a fringe spacing of 10 mm, the same as that used in WTPU, while the other had a frequency that was five times larger [9].
Since the final unwrapping results of Servin’s and WTPU methods are almost identical in their 2D forms, as shown in Figs. 6(b) and 9(b), only the mathematical differences between WTPU and Servin’s method are presented (Figs. 7(b) and 11(b)). From the two Figs. 7(b) and 11(b), the same level of agreement (better than absolute 0.2 radians) was observed, except at the positions with noise where the temporal unwrapping strategy is not very reliable. Compared to the Goldstein’s method, WTPU provides a more comprehensive unwrapping result under inconsistent circumstances while also being easier to implement than the Servin’s method thanks to its utilization of the nonlinearity of the arctangent function.
Fig. 7. Quantitative comparison of unwrapped results. (a) The 3D rendering of the unwrapped result from Fig. 6(b), and the mathematical differences between the unwrapped results obtained by WTPU and Servin’s method [9].
Fig. 8. Experimental results of an ear plaster model. (a) Intensity image, (b) corresponding recorded fringe pattern (768 $\times$ 512 pixels), (c) original wrapped phase map, (d) the shifted additional wrapped phase map, (e-f) unwrapped results from the Goldstein’s [2] and Servin’s [9] methods. The red arrows mark the positions of physical discontinuities.
Fig. 9. Results of WTPU. (a) The distinguished physical discontinuities (white), real wraps (red) and the noise (green), $\epsilon = 0.05\pi, \; \tau = 0.5\pi$ are used; (b) the unwrapped result of WTPU.
Fig. 10. Wrap-type identification performed along row 256 of the two shifted wrapped phase maps (Figs. 8(c-d)). Phase-jump pair A/A’ is a physical discontinuous wrap, distinguished from the real wraps (J0-J9) along the same line.
Fig. 11. Quantitative comparison of unwrapped results. (a) The 3D rendering of the unwrapped result from Fig. 9(b), and the mathematical differences between the unwrapped results obtained by WTPU and Servin’s method [9].
The execution time of WTPU is comparable to that of Servin’s temporal unwrapping [9], since both methods use only additive and subtractive operations.
5. Conclusion
In summary, the proposed algorithm allows for the direct and accurate differentiation of the 2$\pi$ ambiguity without requiring prior knowledge. This achievement is made possible by harnessing the inherent nonlinear nature of the arctangent function, as indicated by the recognizable fringe shifting effects [24] described in Eqs. (7)–(10). The branch thresholds utilized in Eqs. (7)–(10) are linked to the intrinsic frequencies of an object in the temporal domain, effectively bridging the gap between spatial and temporal unwrapping strategies. By treating each phase jump pair independently, the algorithm prevents error propagation and eliminates the need for constructing branch cuts or employing diverse fringe patterns with distinct frequencies. As long as a second wrapped phase map with shifted fringes can be obtained, this unwrapping algorithm, with its ability to determine wrap types, holds promise for various extended interferometric applications.
Funding
National Natural Science Foundation of China (61975161).
Acknowledgement
The author expresses sincere thanks to Dr. Hong Zhao from State Key Laboratory for Manufacturing Systems Engineering of Xi’an Jiaotong University who carefully and thoroughly read the manuscript and provided meaningful criticisms.
Disclosures
The author declares no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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