# Fast γ Photon Imaging for Inner Surface Defects Detecting

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Tomographic Images Reconstructed in FPGA

#### 2.1. Image Reconstruction Algorithms

#### 2.1.1. Obtaining the Initial Image by Using FBP

#### 2.1.2. Optimizing Image Reconstruction Algorithms of BPML

_{n}is the nuclide concentration in pixel n, and Y

_{m}is the count value of coincidence events received by the m-th group of detectors. The coincidence events generated by the positrons annihilated at x

_{n}are received by the corresponding detector pairs of Y

_{m}. The number of received coincidence events obeys Poisson distribution by using the parameter of P

_{mn}x

_{n}, where P

_{mn}is the probability that the coincidence events generated at x

_{n}are counted on the basis of the corresponding detector pairs of Y

_{m}, i.e., the system matrix. The pixels are assumed to be independent of each other and to obey a Poisson distribution.

_{i}is the noise. According to the Bayesian theorem, the conditional probability of the count value of a coincidence event received under different pixel concentrations can be expressed as a likelihood function.

_{i}!) is a constant term of the reconstructed image. If this element is ignored, then Equation (14) can be transformed into:

#### 2.2. BPML Algorithms Built in FPGA

_{1}, we input data 1 to the data streaming to module 1, and then we input it to module 2 at t

_{2}; simultaneously, we input data 2 in the data stream to module 1. With this method, all modules are working at t

_{4}., when pipeline is working at maximum efficiency.

_{k}. Each group of data can be processed in parallel, and the three module circuits can process the different groups of data in parallel. After the first group of data completes the processing in the FFT_din and FFT_dout modules, the results are inputted to the iFFT_dout module. Simultaneously, after the second group of data complete the processing in the FFT_din module, the results are inputted to the FFT_dout module. The third group of data are inputted to the FFT_din module. In this method, these three filtering modules can be running in parallel. The parallel array structure of the FBP sub-circuits is shown in Figure 6.

_{k}, representing a row of sinogram data. The data are initially filtered in the frequency domain by the filter module and then buffered in PROJ_DATA_b BRAM (block random access memory), which is writable. Simultaneously, the index calculation unit calculates and obtains the storage address of the data from angle θ

_{k−}

_{1}. Furthermore, the data from θ

_{k−}

_{1}are input into PROJ_DATA_a BRAM, which executes a reading function. After the projection data under a certain angle are processed, the two sets of BRAM exchange their reading or writing functions and iterates to the next angle projection data without any delay. The division by sub-circuit ensures the continuity of the circuit’s sequence with high efficiency. A parallel array structure is designed by copying several sub-circuits (Figure 7).

_{i}

^{3q-3}is the i-th reconstruction pixel in the image obtained from the projection data of θ

_{3(q-1)}. Then, the reconstruction values of the same pixel from all of the data with different projection angles of θ are added, and a pixel of the reconstructed image is finally obtained.

## 3. Implementation of Edge Detection in FPGA

## 4. Experiments

#### 4.1. Preparation for Experiments

#### 4.2. A Experiment for a Model

#### 4.2.1. Completing the Experiment

#### 4.2.2. Algorithm Execution Time Analysis

#### 4.2.3. Internal Imaging Experiment of Hydraulic Parts

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 11.**Detection results based on two-directional detection and four-directional edge detection. (

**a**) Four sectors of a Derenzo image. (

**b**) Result of two-direction Sobel algorthim (connectivity ratio 0.6355). (

**c**) Result of four-direction Sobel algorthim (connectivity ratio 0.4844).

**Figure 13.**The experimental model. (

**a**)Physical structure of model 1. (

**b**) Solid work design of model 1.

**Figure 15.**The reconstructed images by BPML algorithm with one to four iterations in FPGA. (

**a**) One iteration. (

**b**)Two iterations. (

**c**) Three iterations. (

**d**) Four iterations.

**Figure 18.**The reconstructed images by the BPML algorithm with one to four iterations in the CPU. (

**a**) One iteration. (

**b**) Two iterations. (

**c**) Three iterations. (

**d**) Four iterations.

**Figure 21.**Hydraulic parts. (

**a**) Application of hydraulic parts in aircraft landing gear. (

**b**) The tested hydraulic part.

**Figure 24.**The 150th reconstructed image slice. (

**a**) The original reconstructed image. (

**b**) The 150th image after processing. (

**c**) The 150th image after edge extraction.

Parameters | Values |
---|---|

detector inner diameter | 190 mm |

axial length | 108 mm |

spatial resolution | 0.99 mm |

Energy resolution | 12.83% at 511 KeV |

Time resolution | 1.53 ns |

sensitivity | 7.12% at 350–650 KeV |

Platform (Type, Frequency) | Consumption Time for Reconstructing 52 Slices | |||
---|---|---|---|---|

One Iteration | Two Iterations | Three Iterations | Four Iterations | |

FPGA (XC7A100T, 125 MHz) | 5.37 | 10.79 | 16.12 | 22.95 |

CPU (Core i7-4790, 3.6 GHz) | 259.38 | 362.73 | 458.75 | 561.54 |

Acceleration ratio | 48.3× | 33.6× | 28.5× | 24.4× |

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**MDPI and ACS Style**

Yao, M.; Luo, G.; Zhao, M.; Guo, R.; Liu, J.
Fast γ Photon Imaging for Inner Surface Defects Detecting. *Sensors* **2021**, *21*, 8134.
https://doi.org/10.3390/s21238134

**AMA Style**

Yao M, Luo G, Zhao M, Guo R, Liu J.
Fast γ Photon Imaging for Inner Surface Defects Detecting. *Sensors*. 2021; 21(23):8134.
https://doi.org/10.3390/s21238134

**Chicago/Turabian Style**

Yao, Min, Guangdong Luo, Min Zhao, Ruipeng Guo, and Jian Liu.
2021. "Fast γ Photon Imaging for Inner Surface Defects Detecting" *Sensors* 21, no. 23: 8134.
https://doi.org/10.3390/s21238134