# Quantitative Inspection of Remanence of Broken Wire Rope Based on Compressed Sensing

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## Abstract

**:**

## 1. Introduction

## 2. Acquiring MFL Signal of Remanence

## 3. Data Processing

#### 3.1. Signal Pre-Processing

_{o}(n) and h

_{l}(n) are the decomposing filters ${\overline{h}}_{o}\left(k\right)={h}_{l}\left(-k\right)$; and ${\widehat{x}}_{{A}_{j}}$ is the approximate signal whose baseline and highest noise are eliminated.

#### 3.2. Denoising Based on CSWF

- (1)
- For the pre-processed signal ${\widehat{x}}_{{A}_{j}}$, the Mallat decomposition algorithm is used and the wavelet coefficients W
_{j}under each scale j are obtained. - (2)
- The appropriate random measurement matrix $\mathsf{\Phi}$ (here is a 350 × 1024 Gaussian matrix), is selected, and the wavelet coefficients of linear measurements y under the measurement matrix $\mathsf{\Phi}:y=\mathsf{\Phi}{W}_{j}$ are calculated.
- (3)
- Through the OMP algorithm, the most-sparse wavelet coefficient ${\widehat{W}}_{j}$ is reconstructed; the algorithm steps are as follows:Step One: residue, ${r}_{t}{|}_{t=0}=y$, and index set, ${A}_{t}=\varphi $ (empty set), are initializedFor iteration, t is 1 to K (K is the sparse degree; here it is 8.)BeginStep Two: the inner product is calculated $\langle {r}_{t}\u25cf\mathsf{\Phi}\rangle $Then, the column of whose inner product is the maximum in $\mathsf{\Phi}$ is obtained: ${\lambda}_{t}=arg\underset{t=1~N}{\mathrm{max}}|\langle {r}_{t-1}\u25cf{\mathsf{\Phi}}_{t}\rangle |$;The subscript ${A}_{t}=\left[{A}_{t-1},{A}_{{\lambda}_{t}}\right]$ is stored, and the most orthogonal column of Φ: ${\mathsf{\Phi}}_{t}={\mathsf{\Phi}}_{t-1}\cup \left\{{\mathsf{\Phi}}_{{\lambda}_{t}}\right\}$, the selected column of $\mathsf{\Phi}$, is set to
**0**;Step Three: The least-squares method is used ${\omega}_{t}=argmin\parallel y-{\mathsf{\Phi}}_{t}{\omega}_{t}{\parallel}_{2}={\left({\mathsf{\Phi}}_{t}^{H}{\mathsf{\Phi}}_{t}\right)}^{-1}{\mathsf{\Phi}}_{t}^{H}y$;Step Four: Approximation ${y}_{t}={\mathsf{\Phi}}_{t}{\omega}_{t}={\mathsf{\Phi}}_{t}{\left({\mathsf{\Phi}}_{t}^{H}{\mathsf{\Phi}}_{t}\right)}^{-1}{\mathsf{\Phi}}_{t}^{H}y$ is updated;The residue, ${r}_{t}=y-{y}_{t}$, is updated;End - (4)
- Using approximate wavelet coefficients ${\widehat{W}}_{j}\left({A}_{j}\right)={\omega}_{t}$ the MFL signals are reestablished.

#### 3.3. MFL Image Processing

#### 3.3.1. Defect Image Extraction

**–**255 and, the size of local MFL is 200 × 200 pixels.

#### 3.3.2. Defect Characteristic Exactions

_{1}–Φ

_{7}).

- 1.
- Basic Description of Image Shape

_{1}), and minor axis (L

_{2}). All these descriptions are as follows.

_{i}is followed by c

_{1}, c

_{2}, c

_{3}, …, c

_{n}and the perimeter can be presented as follows:

_{1}is defined as the maximum distance between any two points in the outer boundary, and L

_{2}is defined as the longest straight line which is vertical to L

_{1}, as shown in Figure 9.

_{1}(x

_{1},y

_{1}) and α

_{2}(x

_{2},y

_{2}). The endpoints of the vertical line are v

_{1}(m

_{1},n

_{1}) and v

_{2}(m

_{2},n

_{2}) to L1. The L

_{1}and L

_{2}are calculated as follows:

- 2.
- Characteristic Descriptions of Shape

_{1}and the minor axis L

_{2}:

- 3.
- Characteristics of Invariant Moment

_{pq}, will change when (x,y) is translated. To reduce and eliminate unfavorable effects, the central moments are defined as:

## 4. Quantitative Recognition

## 5. Comment and Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Framework of the detection device and detection method diagram; (

**b**) Excitation source; and (

**c**) Signal acquisition system board.

**Figure 10.**Different hidden nodes recognition graphs: (

**a**) Identification ratio graph of 21 hidden layer nodes; (

**b**) Identification ratio graph of 24 hidden layer nodes; (

**c**) Identification ratio graph of 27 hidden layer nodes; and (

**d**) Identification ratio graph of 30 hidden layer nodes.

**Figure 11.**Training performance graphs for different hidden layer numbers: (

**a**) 21 hidden nodes; (

**b**) 24 hidden nodes; (

**c**) 27 hidden nodes; and (

**d**) 30 hidden nodes.

Broken Wires | G | F | e | Φ_{1} | Φ_{2} | Φ_{3} | Φ_{4} | Φ_{5} | Φ_{6} | Φ_{7} |
---|---|---|---|---|---|---|---|---|---|---|

1 | 4.22 | 0.864 | 0.647 | 6.69 × 10^{10} | 4.48 × 10^{21} | 2.88 × 10^{20} | 2.35 × 10^{20} | 4.02 × 10^{37} | −2.26 × 10^{28} | 6.27 × 10^{39} |

2 | 6.06 | 0.392 | 0.537 | 6.71 × 10^{10} | 4.51 × 10^{21} | 2.66 × 10^{20} | 2.36 × 10^{20} | −2.76 × 10^{42} | −4.03 × 10^{29} | −1.24 × 10^{42} |

3 | 11.9 | 0.935 | 0.623 | 6.70 × 10^{10} | 4.49 × 10^{21} | 7.26 × 10^{21} | 2.57 × 10^{21} | 3.83 × 10^{42} | 2.51 × 10^{28} | −4.9 × 10^{42} |

4 | 19.6 | 1.150 | 0.642 | 6.58 × 10^{10} | 4.32 × 10^{21} | 2.11 × 10^{21} | 1.51 × 10^{21} | −2.01 × 10^{43} | −6.24 × 10^{30} | 3.15 × 10^{43} |

5 | 7.15 | 0.364 | 0.499 | 6.58 × 10^{10} | 4.33 × 10^{21} | 5.52 × 10^{21} | 5.62 × 10^{21} | −1.19 × 10^{44} | −7.64 × 10^{30} | −7.68 × 10^{42} |

7 | 1.64 | 0.811 | 0.592 | 6.65 × 10^{10} | 4.42 × 10^{21} | 6.67 × 10^{21} | 4.95 × 10^{21} | −3.16 × 10^{43} | 7.42 × 10^{29} | −9.84 × 10^{43} |

Hidden Layer Number | Iteration Time (s) | Maximum Error for Training Set (%) | Maximum Error for Test Set (%) | Train Sample Number |
---|---|---|---|---|

21 | 138 | 1.353 | 2.521 | 55 |

24 | 128 | 1.606 | 2.734 | 55 |

27 | 173 | 1.479 | 4.211 | 55 |

30 | 121 | 1.075 | 2.732 | 55 |

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**MDPI and ACS Style**

Zhang, J.; Tan, X.
Quantitative Inspection of Remanence of Broken Wire Rope Based on Compressed Sensing. *Sensors* **2016**, *16*, 1366.
https://doi.org/10.3390/s16091366

**AMA Style**

Zhang J, Tan X.
Quantitative Inspection of Remanence of Broken Wire Rope Based on Compressed Sensing. *Sensors*. 2016; 16(9):1366.
https://doi.org/10.3390/s16091366

**Chicago/Turabian Style**

Zhang, Juwei, and Xiaojiang Tan.
2016. "Quantitative Inspection of Remanence of Broken Wire Rope Based on Compressed Sensing" *Sensors* 16, no. 9: 1366.
https://doi.org/10.3390/s16091366