# Non-Destructive Detection of Wire Rope Discontinuities from Residual Magnetic Field Images Using the Hilbert-Huang Transform and Compressed Sensing

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## Abstract

**:**

## 1. Introduction

## 2. RMF Detection

#### 2.1. Platform Design

#### 2.2. Data Acquisition

#### 2.3. RMF Image

## 3. Signal Processing

#### 3.1. Reprocessing Theory

- The average number of maxima and minima of an IMF component must be equivalent to the number of 0 crossings, or they differ by 1 at most.
- The average of the maxima and minima, as defined by the envelope, should be 0 at any given moment.

- (1)
- First, extend the raw signal to obtain $\tilde{x}(t)$, and initialize the residual signal ${r}_{n}$, IMFs set ${c}_{i}$ as ${c}_{i}=\varnothing ,{r}_{n}=\tilde{x}(t)$.
- (2)
- Add Gaussian white noise $w(t)$ to ${r}_{n}$:$$y(t)=\tilde{x}(t)+w(t).$$
- (3)
- Implement EMD for $y(t)$ to obtain IMF ${c}_{ij}$:$$x(t)={\displaystyle \sum _{i=1}^{n}{c}_{ij}+{r}_{n}}.$$
- (4)
- Repeat steps (2) and (3) k times, obtaining an IMF set ${c}_{ij}(i\le n,j\le k)$, and calculate the average of the IMFs. Update ${r}_{n}$ as follows:$$\{\begin{array}{c}{c}_{i}(t)=\frac{1}{k}{\displaystyle \sum _{j=1}^{k}{c}_{ij}}\\ {r}_{n}={r}_{n-1}-{c}_{ij}\end{array}$$
- (5)
- If i > n or ${r}_{n}$ cannot be further decomposed, the decomposition is complete. Otherwise, i = i + 1, and return to step (2).

#### 3.2. Compressed Sensing Theory

^{M×N}(M ≪ N) and a noisy signal $x\in {R}^{N}$ (which is not a sparse signal), the signal cannot be reconstructed using CS theory. However, if the signal is sparse in the transform domain, it can still be reconstructed into the raw signal $\widehat{x}$ [29]. Thus, we assume that a transformation basis $\psi \in {R}^{N\times N}$ exists, where, under this basis, the transformation of $x$ is sparse, and the transformation coefficient of the noise is much smaller than that of the actual signal. The transformation is:

#### 3.3. Description of the De-Noising Algorithm

- i
- The EEMD described in Section 3.1 is applied to the raw data, and the reprocessing signal $\widehat{X}$ is obtained.
- ii
- Apply CSWF to the re-processed signal of the i-th channel:
- (1)
- The Mallat decomposition algorithm is applied, and the sparse expression ${W}_{j}$ of signal ${\widehat{x}}_{i}$ is obtained for each scale $j$.
- (2)
- Randomly generate a Gaussian matrix $\mathrm{\Phi}$ and calculate the linear measure under the matrix $\mathrm{\Phi}$: ${y}_{j}=\mathrm{\Phi}{W}_{j}$.
- (3)
- Implement the OMP algorithm and reconstruct the most-sparse wavelet coefficient ${\widehat{W}}_{j}$. These procedures are as follows:
- Step One: initialize residue, ${r}_{t}{|}_{t=0}=y$, and index set, ${A}_{t}=\varphi $ (empty set);
- For each iteration t from 1 to K (here, K = 8);
- Begin;
- Step Two: the inner product is calculated $\langle {r}_{t}\mathrm{\Phi}\rangle $;
- Then, the column whose inner product is the maximum in $\mathrm{\Phi}$ is obtained: ${\lambda}_{t}=arg\underset{t=1~N}{\mathrm{max}}\left|\langle {r}_{t-1}\xb7{\mathrm{\Phi}}_{t}\rangle \right|$; The subscript ${A}_{t}=\left[{A}_{t-1},{A}_{{\lambda}_{t}}\right]$ is stored, and the most orthogonal column of Φ: ${\mathrm{\Phi}}_{t}={\mathrm{\Phi}}_{t-1}\cup \left\{{\mathrm{\Phi}}_{{\lambda}_{t}}\right\}$, the selected column of $\mathrm{\Phi}$, is set to
**0**; - Step Three: The least-squares method ${\omega}_{t}=argmin\Vert y-{\mathrm{\Phi}}_{t}{\omega}_{t}{\Vert}_{2}={\left({\mathrm{\Phi}}_{t}^{H}{\mathrm{\Phi}}_{t}\right)}^{-1}{\mathrm{\Phi}}_{t}^{H}y$ is implemented;
- Step Four: Approximation ${y}_{t}={\mathrm{\Phi}}_{t}{\omega}_{t}={\mathrm{\Phi}}_{t}{\left({\mathrm{\Phi}}_{t}^{H}{\mathrm{\Phi}}_{t}\right)}^{-1}{\mathrm{\Phi}}_{t}^{H}y$ is updated;
- The residue, ${r}_{t}=y-{y}_{t}$, is updated;
- End.

- (4)
- Utilize the inverse wavelet transform for the approximate coefficients ${\widehat{\mathrm{W}}}_{\mathrm{j}}({\mathrm{A}}_{\mathrm{j}})={\mathsf{\omega}}_{\mathrm{t}}$, and the RMF signal is then re-established.

- iii
- If the channel number i < 18, return to step iv or end the process.

## 4. RMF Image Processing

#### 4.1. Morphological Processing and Defect Location Detection

#### 4.2. Normalization and Resolution Enhancement

- The position $({i}_{c},{j}_{c})$ of the minimum of defect is obtained by searching the modulus maximum of a target region image in multiplied image. Then, the axial center is $({i}_{c},{j}_{c})$.
- In the target region, the defect image $g$ can be expressed as:$$g(i,j)=\{g(i,j)|i\in [{i}_{c}-99,{i}_{c}+100],{j}_{c}\in [0,N-1]\},$$
- If ${j}_{c}<N/2$, $g(i,j)$ is given as follows:$$\{\begin{array}{c}g(i,j+N/2-{j}_{c})=g(i,j),\text{\hspace{1em}}0\le j\le N-{j}_{c}-1\hfill \\ g(i,j-N+{j}_{c})=g(i,j),\text{\hspace{1em}}N-{j}_{c}\le j\le N-1\hfill \end{array}\text{}$$If ${j}_{c}>N/2$, $g(i,j)$ is given as follows:$$\{\begin{array}{c}g(i,j+{j}_{c}-N/2)=g(i,j),\text{\hspace{1em}}0\le j\le {j}_{c}-N/2\hfill \\ g(i,j-{j}_{c}+N/2)=g(i,j),\text{\hspace{1em}}{j}_{c}-N/2<j\le N-1\hfill \end{array}$$

## 5. Detection of Broken Wires

#### 5.1. Extracting Artificial Image Characteristics

_{3}is zero. If the skewness tends to the right, the value is positive, but, if it tends to the left, the value is negative. The conformance is defined as:

#### 5.2. Quantitative Defect Detection

## 6. Results and Discussion

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) Schematic diagram of the excitation device; (

**b**) The printed circuit boards employed in the detection system, including the control board and a sensor array board.

**Figure 5.**(

**a**) The 18 circumferential data points obtained from GMR sensors for a single pulse plotted in polar coordinates (the radius represents the voltage); (

**b**) Partial raw data expanded image.

**Figure 6.**(

**a**) One-dimensional waveforms of three channels of raw data (the data is derived from Figure 4b; (

**b**) Data processing flow diagram.

**Figure 8.**Image of a defect signal with system noise suppressed by means of the proposed HHT and CSWF algorithm (raw data obtained from Figure 5b).

**Figure 9.**Mapped binary data (

**a**) and original filtered wire rope RMF image (

**b**), indicating that the mapped data provides the locations of defects.

**Figure 10.**Images of broken wires (top) and high-resolution gray-level RMF images of the corresponding broken wires (bottom): (

**a**) one broken wire; (

**b**) two broken wires; (

**c**) three broken wires; (

**d**) five broken wires; (

**e**) seven broken wires; (

**f**) warping two wires.

**Figure 11.**Performance of the designed RBF classification network: (

**a**) training performance; (

**b**) plot of the detection error.

Number of Broken Wires | 1 | 2 | 3 | 4 | 5 | 7 |
---|---|---|---|---|---|---|

$m$ | 102 | 254 | 231 | 164 | 251 | 174 |

$\sigma $ | 17.3 | 4.12 | 20 | 19.4 | 15.4 | 28.57 |

$R$ | 4.60 × 10^{−3} | 2.61 × 10^{−4} | 6.11 × 10^{−3} | 5.73 × 10^{−3} | 3.61 × 10^{−3} | 1.24 × 10^{−2} |

${\mu}_{3}$ | −0.076 | −0.009 | −0.35 | −0.112 | −0.267 | −0.549 |

$U$ | 0.023 | 0.947 | 0.071 | 0.02 | 0.758 | 0.068 |

$e$ | 5.91 | 0.33 | 4.99 | 6.1 | 1.35 | 5.65 |

M1 | 1.71 × 10^{−3} | 5.66 × 10^{−4} | 7.33 × 10^{−4} | 1.04 × 10^{−3} | 6.48 × 10^{−4} | 1.01 × 10^{−3} |

M2 | 7.42 × 10^{−9} | 1.89 × 10^{−11} | 8.08 × 10^{−11} | 1.72 × 10^{−10} | 5.39 × 10^{−12} | 3.18 × 10^{−10} |

M3 | 1.82 × 10^{−12} | 6.70 × 10^{−15} | 1.04 × 10^{−13} | 6.27 × 10^{−13} | 5.68 × 10^{−16} | 4.22 × 10^{−15} |

M4 | 2.62 × 10^{−12} | 8.36 × 10^{−15} | 6.85 × 10^{−14} | 1.08 × 10^{−12} | 5.37 × 10^{−15} | 1.50 × 10^{−13} |

M5 | 4.79 × 10^{−24} | 2.60 × 10^{−29} | −5.03 × 10^{−27} | 8.05 × 10^{−25} | 9.31 × 10^{−3}° | 3.09 × 10^{−27} |

M6 | 1.95 × 10^{−15} | −1.14 × 10^{−18} | −2.39 × 10^{−17} | 4.37 × 10^{−17} | 1.49 × 10^{−18} | 4.55 × 10^{−17} |

M7 | −3.11 × 10^{−24} | 5.68 × 10^{−29} | 2.84 × 10^{−27} | 3.69 × 10^{−25} | −9.77 × 10^{−31} | −2.19 × 10^{−27} |

**Table 2.**Performance of the various RBF classification networks when the absolute limiting error was 2 wires.

Spread | Maximum Error | Average Broken Wires Error | Training Accuracy | Recognition |
---|---|---|---|---|

0.05 | 5 | 1.25 | 1 | 78.13% |

0.10 | 5 | 1.0313 | 96.70% | 84.38% |

0.12 | 5 | 0.7813 | 95.60% | 93.75% |

0.15 | 5 | 1 | 86.81% | 87.50% |

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**MDPI and ACS Style**

Zhang, J.; Tan, X.; Zheng, P.
Non-Destructive Detection of Wire Rope Discontinuities from Residual Magnetic Field Images Using the Hilbert-Huang Transform and Compressed Sensing. *Sensors* **2017**, *17*, 608.
https://doi.org/10.3390/s17030608

**AMA Style**

Zhang J, Tan X, Zheng P.
Non-Destructive Detection of Wire Rope Discontinuities from Residual Magnetic Field Images Using the Hilbert-Huang Transform and Compressed Sensing. *Sensors*. 2017; 17(3):608.
https://doi.org/10.3390/s17030608

**Chicago/Turabian Style**

Zhang, Juwei, Xiaojiang Tan, and Pengbo Zheng.
2017. "Non-Destructive Detection of Wire Rope Discontinuities from Residual Magnetic Field Images Using the Hilbert-Huang Transform and Compressed Sensing" *Sensors* 17, no. 3: 608.
https://doi.org/10.3390/s17030608