# Fast Fourier-Based Phase Unwrapping on the Graphics Processing Unit in Real-Time Imaging Applications

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## Abstract

**:**

## 1. Introduction

## 2. Method and Implementation

#### 2.1. Schofield, Volkov and Zhu Phase Unwrapping

#### 2.2. Preconditioned Conjugate Gradient Phase Unwrapping

#### 2.3. Parallel Implementation of the Discrete Cosine Transform

## 3. Numeric and Experimental Results

#### 3.1. Schofield, Volkov and Zhu Phase Unwrapping

**Table 1.**Processing times of central processing unit (CPU)-and graphics processing unit (GPU)-based phase unwrapping algorithms, executed on phase maps of varying pixel sizes. The last two columns include the corresponding processing times of optimized GPU-accelerated discrete cosine transforms (DCTs) and inverse DCTs (IDCTs).

Image Pixel Size | CPU (FFT) (ms) | GPU (MM) incl. mem. trans. (ms) | GPU (MM) excl. mem. trans. (ms) | Acceleration Factor (×) | DCT/IDCT incl. mem trans. (ms) | DCT/IDCT excl. mem trans. (ms) |
---|---|---|---|---|---|---|

256 × 256 | 9.6 | 1.6 | 1.2 | 6.0 | 0.5 | 0.1 |

512 × 512 | 40.2 | 4.1 | 3.2 | 9.8 | 1.3 | 0.3 |

640 × 480 | 48.5 | 4.9 | 4.0 | 9.9 | 1.5 | 0.5 |

1024 × 1024 | 182.9 | 20.1 | 16.7 | 9.1 | 5.9 | 2.5 |

2048 × 2048 | 735.9 | 136.4 | 128.4 | 5.4 | 27.9 | 19.8 |

**Figure 1.**Phase unwrapping procedure for two phase maps obtained with different imaging modalities (first row (

**a**–

**c**): electron holography, second row (

**d**–

**f**): phase-sensitive magnetic resonance imaging (MRI)). The first column (a,d) shows the input wrapped phase map; the second column (b,e) depicts the integer number of 2π that is added to the respective wrapped phase map to end up with the unwrapped phase map, included in the third column (c,f). Scale bars are included in the third column of subfigures only but apply to the other images in their respective row, also.

#### 3.2. Preconditioned Conjugate Gradient Phase Unwrapping

**Figure 2.**The preconditioned conjugate gradient algorithm is able to correctly unwrap a randomly generated phase map containing regions where phase data is set to zero in three iterations (IT 1-3) by weighting the intermediate preconditioned conjugate gradient (PCG) solutions according to a user-defined input quality mask. Using a simple binary mask, the phase misses (circled in red) are reduced from seven to none in three iterations.

#### 3.3. Real-Time Optical Profilometry

**Figure 3.**Single-frame excerpt from a multi-image video recording of real-time optical profilometry measurements of moving pieces of fabric (Media 1). (

**a**) One of three phase-shifted input images. (

**b**) Wrapped phase map. (

**c**) Unwrapped phase map. (

**d**) 3D perspective-view of (c).

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Jeught, S.V.d.; Sijbers, J.; Dirckx, J.J.J.
Fast Fourier-Based Phase Unwrapping on the Graphics Processing Unit in Real-Time Imaging Applications. *J. Imaging* **2015**, *1*, 31-44.
https://doi.org/10.3390/jimaging1010031

**AMA Style**

Jeught SVd, Sijbers J, Dirckx JJJ.
Fast Fourier-Based Phase Unwrapping on the Graphics Processing Unit in Real-Time Imaging Applications. *Journal of Imaging*. 2015; 1(1):31-44.
https://doi.org/10.3390/jimaging1010031

**Chicago/Turabian Style**

Jeught, Sam Van der, Jan Sijbers, and Joris J. J. Dirckx.
2015. "Fast Fourier-Based Phase Unwrapping on the Graphics Processing Unit in Real-Time Imaging Applications" *Journal of Imaging* 1, no. 1: 31-44.
https://doi.org/10.3390/jimaging1010031