# Pixelwise Phase Unwrapping Based on Ordered Periods Phase Shift

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## Abstract

**:**

## 1. Introduction

## 2. Related Methods

#### 2.1. Basic Method

#### 2.2. Phase Unwrapping

## 3. Proposed Method

#### 3.1. Ordered Periods Phase Shift (OPPS)

#### 3.1.1. Basic Method

#### 3.1.2. Realizing Unique Tuple Identification

#### 3.1.3. Spatial Continuity of Patterns

#### 3.1.4. Decoding Projected Patterns

#### 3.1.5. Measurement Interpolation

#### 3.1.6. Projection Period Limitations

#### 3.2. Four-Step OPPS

#### 3.2.1. Basic Method

#### 3.2.2. Realizing Unique Tuple Identification

#### 3.2.3. Spatial Continuity of Patterns

#### 3.2.4. Decoding Projection Patterns

## 4. Experiments

#### 4.1. Evaluation of Motion Error

#### 4.2. Evaluation of Global Illumination

#### 4.3. High-Speed OPPS

^{®}Xeon

^{®}CPU E5-2687W v4 and NVIDIA Quadro M5000 GPU. Using a frame rate of 500 fps, the projector with resolution set at 1024 × 768 and the camera with resolution set at 640 × 480 were synchronized. Figure 8 shows the system implemented and the experimental environment.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Examples of the projection pattern using the phase-shift method. (

**a**) Three-step phase shift and (

**b**) Two-plus-one phase shift.

**Figure 2.**Design of projection pattern for the three-step OPPS. (

**a**) Standard projection patterns ${I}_{1}^{p}$, ${I}_{2}^{p}$, and ${I}_{3}^{p}$. (

**b**) Incorporating ordered structure (Section 3.1.1). Three steps are reordered; thus, there are ${}_{3}{P}_{3}=6$ combinations. (

**c**) Constrained to uniquely identify tuple (Section 3.1.2). (

**d**) Spatial continuity (Section 3.1.3).

**Figure 3.**Design of projection pattern for the four-step OPPS. (

**a**) Standard projection patterns ${I}_{1}^{p}$, ${I}_{2}^{p}$, ${I}_{3}^{p}$, and ${I}_{4}^{p}$. (

**b**) Incorporating ordered structure (Section 3.2.1). Four steps are reordered; thus, there are ${}_{4}{P}_{4}=24$ combinations. (

**c**) Constrained to uniquely identifying tuple (Section 3.2.2). (

**d**) Spatial continuity (Section 3.2.3).

**Figure 5.**Motion errors. (

**a**) RMSE for noise $\sigma =0\%$, (

**b**) RMSE for noise $\sigma =1\%$, and (

**c**) RMSE for noise $\sigma =5\%$.

**Figure 8.**Experimental environment. The high-speed projector and high-speed camera were synchronized, and the camera was placed vertically above the projector to perform the measurements. The measurement results were obtained in real-time and displayed in the monitor screen.

**Figure 9.**Measurement results in real environment. Each row shows the 3D shape reproduced using ${I}_{1}^{c}$, ${I}_{2}^{c}$, ${I}_{3}^{c}$, and ${I}_{4}^{c}$ (starting from left). From the data measured at 500 fps, data at 200 ms intervals were extracted. Starting from the top, the measurement results are shown at time t = 0, 200, 400, and 600 ms.

$\mathit{\kappa}$ in Figure 2c | 1 | 2 | 3 | 4 | 5 | 6 |

hash ($\kappa $) | 1 | 4 | 2 | 3 | 5 | 6 |

${\mathit{I}}_{3}^{\mathit{c}}<{\mathit{I}}_{1}^{\mathit{c}}<{\mathit{I}}_{2}^{\mathit{c}}$ | ${\mathit{I}}_{2}^{\mathit{c}}<{\mathit{I}}_{1}^{\mathit{c}}<{\mathit{I}}_{3}^{\mathit{c}}$ | ${\mathit{I}}_{3}^{\mathit{c}}<{\mathit{I}}_{2}^{\mathit{c}}<{\mathit{I}}_{1}^{\mathit{c}}$ | ${\mathit{I}}_{1}^{\mathit{c}}<{\mathit{I}}_{2}^{\mathit{c}}<{\mathit{I}}_{3}^{\mathit{c}}$ | ${\mathit{I}}_{2}^{\mathit{c}}<{\mathit{I}}_{3}^{\mathit{c}}<{\mathit{I}}_{1}^{\mathit{c}}$ | ${\mathit{I}}_{1}^{\mathit{c}}<{\mathit{I}}_{3}^{\mathit{c}}<{\mathit{I}}_{2}^{\mathit{c}}$ | |

tuple | (1,2,3) | (1,3,2) | (2,1,3) | (2,3,1) | (3,1,2) | (3,2,1) |

$\kappa $ in Figure 2c | 1 | 2 | 3 | 4 | 5 | 6 |

$\mathit{\kappa}$ in Figure 3c | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

hash($\kappa $) in Figure 3d | 1 | 2 | 3 | 4 | 33 | 14 | 15 | 16 | 13 | 34 | 27 | 24 | 10 | 8 | 5 | 6 |

$\kappa $ in Figure 3c | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 |

hash($\kappa $) in Figure 3d | 7 | 11 | 12 | 9 | 21 | 22 | 23 | 28 | 25 | 46 | 47 | 20 | 17 | 30 | 31 | 44 |

$\kappa $ in Figure 3c | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 |

hash($\kappa $) in Figure 3d | 41 | 42 | 43 | 32 | 37 | 38 | 39 | 40 | 29 | 18 | 19 | 48 | 45 | 26 | 35 | 36 |

**Table 4.**Decoding in proposed method (four-step). Based on condition of ${I}_{max}^{c}$ and ${I}_{min}^{c}$, the corresponding tuple and ${I}_{sin}^{c}$ and ${I}_{cos}^{c}$ are determined. Moreover, the tuple index ${\kappa}_{\mathrm{tuple}}$ is also similarly determined.

${I}_{max}^{c}$ | ${I}_{3}^{c}$ | ${I}_{4}^{c}$ | ${I}_{4}^{c}$ | ${I}_{1}^{c}$ | ${I}_{1}^{c}$ | ${I}_{2}^{c}$ | ${I}_{2}^{c}$ | ${I}_{3}^{c}$ | ${I}_{2}^{c}$ | ${I}_{4}^{c}$ | ${I}_{3}^{c}$ | ${I}_{1}^{c}$ |

${I}_{min}^{c}$ | ${I}_{4}^{c}$ | ${I}_{3}^{c}$ | ${I}_{1}^{c}$ | ${I}_{4}^{c}$ | ${I}_{2}^{c}$ | ${I}_{1}^{c}$ | ${I}_{3}^{c}$ | ${I}_{2}^{c}$ | ${I}_{4}^{c}$ | ${I}_{2}^{c}$ | ${I}_{1}^{c}$ | ${I}_{3}^{c}$ |

tuple | (1,2,3,4) | (1,2,4,3) | (4,1,2,3) | (3,1,2,4) | (3,4,1,2) | (4,3,1,2) | (1,3,4,2) | (1,4,3,2) | (1,3,2,4) | (1,4,2,3) | (4,1,3,2) | (3,1,4,2) |

${I}_{sin}^{c}$ | ${I}_{1}^{c}$ | ${I}_{1}^{c}$ | ${I}_{2}^{c}$ | ${I}_{2}^{c}$ | ${I}_{3}^{c}$ | ${I}_{3}^{c}$ | ${I}_{1}^{c}$ | ${I}_{1}^{c}$ | ${I}_{1}^{c}$ | ${I}_{1}^{c}$ | ${I}_{2}^{c}$ | ${I}_{2}^{c}$ |

${I}_{cos}^{c}$ | ${I}_{2}^{c}$ | ${I}_{2}^{c}$ | ${I}_{3}^{c}$ | ${I}_{3}^{c}$ | ${I}_{4}^{c}$ | ${I}_{4}^{c}$ | ${I}_{4}^{c}$ | ${I}_{4}^{c}$ | ${I}_{3}^{c}$ | ${I}_{3}^{c}$ | ${I}_{4}^{c}$ | ${I}_{4}^{c}$ |

${\kappa}_{\mathrm{tuple}}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

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## Share and Cite

**MDPI and ACS Style**

Tabata, S.; Maruyama, M.; Watanabe, Y.; Ishikawa, M.
Pixelwise Phase Unwrapping Based on Ordered Periods Phase Shift. *Sensors* **2019**, *19*, 377.
https://doi.org/10.3390/s19020377

**AMA Style**

Tabata S, Maruyama M, Watanabe Y, Ishikawa M.
Pixelwise Phase Unwrapping Based on Ordered Periods Phase Shift. *Sensors*. 2019; 19(2):377.
https://doi.org/10.3390/s19020377

**Chicago/Turabian Style**

Tabata, Satoshi, Michika Maruyama, Yoshihiro Watanabe, and Masatoshi Ishikawa.
2019. "Pixelwise Phase Unwrapping Based on Ordered Periods Phase Shift" *Sensors* 19, no. 2: 377.
https://doi.org/10.3390/s19020377