# Image Reconstruction of Internal Defects in Wood Based on Segmented Propagation Rays of Stress Waves

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Ellipse-Based Spatial Interpolation and Image Reconstruction

#### 2.2. Modified Image Reconstruction Algorithm

_{1}is the long axis of the ellipse, and b

_{1}is the short axis of the ellipse. Therefore, c

_{1}is the control coefficient of the elliptical shape. For convenience of calculation, each original ray is discretized into a set of points. If any point of a certain ray is in the range of the ellipse, it indicates that the ray passes through the ellipse. Whether a certain point is in the elliptical zone can be identified as follows,

_{xb}

_{1}is the distance between a certain discretized point and the short axis of the ellipse, and D

_{ya}

_{1}is the distance between a certain discretized point and the long axis of the ellipse. If f(x,y) = 1, it means that the point is within the ellipse. Then, the original ray that the point belongs to is recorded. When all the rays have been identified, the estimated value of the segmented ray can be calculated as

_{Rk}is the velocity value of a certain ray in the elliptical neighborhood, and M is the total number of the recorded rays. An original ray will be segmented many times to get a more precise rays graph for spatial interpolation, and the termination condition for iterative segmentation can be given as,

_{Ri}is the length of a certain ray after segmentation, and l

_{min}is the shortest length of all original propagation rays. Therefore, each ray has the opportunity to be segmented and optimized.

_{2}is the long axis of the ellipse, and b

_{2}is the short axis of the ellipse. Therefore, c

_{2}is the control coefficient of the second ellipse for interpolation.

_{SE}(point start to point end) and L

_{PE}(a certain point in ellipse to point end). If g(x,y) = 1, it means that the grid cell is within a segmented area of the ellipse. Then, the grid cells in one ellipse can be can be divided into different affected zones. When all ellipses are segmented, if a grid cell is in several affected zones at the same time, the value of the grid cell is determined as follows,

_{xy}is the estimated velocity value of a certain grid cell, v’

_{k}is the value of a certain zone that affects the specific grid cell, and N is the total number of zones that affect the specific grid cell, simultaneously.

- Step 1:
- Normalizing all collected values of velocity rays with respect to the range of velocity values, visualizing them, and generating the grid graph.
- Step 2:
- Segmenting a certain ray repeatedly, constructing elliptical neighborhood of segmented rays with Equation (2), and estimating the values of segmented rays with Equation (3), until the termination condition (Equation (4)) is reached.
- Step 3:
- Repeating Step 2 until every original ray has been processed, and generating the modified rays graph.
- Step 4:
- Constructing all of the elliptical affected zones with Equations (6), and estimating the values of a certain grid cell with Equations (7) and (8).
- Step 5:
- Repeating Step 4 until every grid cell has been processed.
- Step 6:
- Reconstructing the image of internal defects in wood using the estimated values of grid cells with a certain color scale.

#### 2.3. Materials and Data Acquisition

_{ij}represents the measured distance between sensor number i and sensor number j. t

_{ij}represents the acquired propagation time from sensor number i to sensor number j. Therefore, S

_{ij}represents the velocity from sensor number i to sensor number j. J is the total number of sensors. The velocities are the entries of the matrix S, and matrix S is the input for the proposed method.

## 3. Results and Discussion

#### 3.1. Results Based on Simulation Data

_{1}is the control coefficient of the elliptical shape in RSEN algorithm, and c

_{2}is the control coefficient of the ellipse for interpolation in the SISE algorithm.

_{1}have been studied (that is, from 0 to 1). To clearly display the results, we selected three representative values of c

_{1}for each sample (min, mid, and max). When c

_{1}is less than min or more than max for each sample, the corresponding rays graph begins to become worse. Value mid is the optimal choice of the value of the parameter c

_{1}. For samples 1–3, the difference between the results of the RSEN method with different control coefficients is not obvious. It indicates that the algorithm is not sensitive to the control coefficient for defect patterns with single circle distribution or edge distribution. In addition, for samples 4 and 5, when the control coefficient is greater than 0.75, the results begin to become worse. Overall, the coefficient c

_{1}with a value in the range of 0.65–0.75 is suggested for ray segmentation.

_{2}is not obvious. For sample 4, when the control coefficient is less than 0.9, the result begins to become worse. In general, the coefficient c

_{2}with a value in the range of 0.85–0.95 is suggested for image reconstruction.

#### 3.2. Results Based on Experiment Data

_{1}with a value in the range of 0.65–0.8 is suggested. The experimental results, as shown in Figure 9 (the scales of the false colors range from 0 to 1), prove that the improvement of the rays graph is beneficial to the subsequent interpolation algorithm. Hole defects and knot defects are clearly shown in reconstructed images, and even the complex crack defects can be approximately reflected. The coefficient c

_{2}with a value in the range of 0.85–0.95 is suggested for the SISE algorithm.

#### 3.3. Defect Area Analysis

#### 3.4. Defect Shape Analysis

## 4. Conclusions

- (1)
- The original rays graphs are all improved significantly by the RSEN (ray segmentation by elliptical neighborhood) algorithm. Compared with the original graph of rays, the segmented graph of rays can better reflect the potential spatial distribution of defects and benefits the subsequent spatial interpolation. In addition, the RSEN algorithm is not sensitive to control coefficient c
_{1}for defect patterns with single circle distribution or edge distribution. For other defects, when c_{1}is less than 0.65 or more than 0.85, the corresponding results of rays graphs begin to become worse. The coefficient c_{1}with a value in the range of 0.65–0.8 is suggested. - (2)
- The images reconstructed by the SISE (spatial interpolation by segmented ellipse) algorithm can reflect the size and shape of defects inside wood. In addition, the SISE algorithm is also not sensitive to control coefficient c
_{2}for defect patterns with single circle distribution or edge distribution. For other defects, when c_{2}is less than 0.85, the corresponding results of reconstructed images begin to become worse. The coefficient c_{2}with a value in the range of 0.85–0.95 is suggested. - (3)
- The defective area proportion from the reconstructed image using the proposed method is closer to the actual defective area, and the contour extracted from the reconstructed image using the proposed method is much more similar to the actual contour.
- (4)
- The average accuracy of the proposed algorithm is 8.9% higher than that of the EBSI (ellipse based spatial interpolation) algorithm, and the average precision of the proposed algorithm is 12.8% higher than that of the EBSI algorithm.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) An example of the rays graph; (

**b**) Gridding of the imaging area; (

**c**) Illustration of the EBSI (ellipse-based spatial interpolation) method.

**Figure 2.**Illustration of the proposed method. RSEN (ray segmentation by elliptical neighborhood), SISE (spatial interpolation by segmented ellipse).

**Figure 3.**Illustration of the key steps in RSEN and SISE: (

**a**) Strategy for estimating the value of the segmented ray; (

**b**) strategy for selection of affected regions according to the segmented rays.

**Figure 4.**Simulation data (graph of defect distribution and graph of rays) and experimental data (real image of sample and graph of rays): (

**a**) Single circle distribution; (

**b**) single edge distribution; (

**c**) two-edge distribution; (

**d**) two-rectangle distribution; (

**e**) three-circle distribution; (

**f**) firmiana timber with hole defects; (

**g**) chinaberry timber with knot defects; (

**h**) pecan timber with crack defects.

**Figure 6.**Original rays graph and segmented rays graph using the RSEN method from simulation data with different control coefficients.

**Figure 7.**Images reconstructed using the SISE method from simulation data with different control coefficients.

**Figure 8.**Original rays graph and segmented rays graph using the RSEN method from the real data with different control coefficients.

**Figure 9.**Images reconstructed using the SISE method from the real data with different control coefficients.

**Figure 11.**Proportion of defective area from the reconstructed images using different methods for all samples.

Sensor | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | null | 0.80 | 0.80 | 0.70 | 0.56 | 0.40 | 0.22 | 0.35 | 0.52 | 0.67 | 0.78 | 0.80 |

2 | 0.80 | null | 0.80 | 0.80 | 0.71 | 0.57 | 0.41 | 0.23 | 0.35 | 0.53 | 0.67 | 0.79 |

3 | 0.80 | 0.80 | null | 0.80 | 0.80 | 0.72 | 0.57 | 0.41 | 0.22 | 0.36 | 0.54 | 0.69 |

4 | 0.70 | 0.80 | 0.80 | null | 0.80 | 0.80 | 0.72 | 0.57 | 0.40 | 0.21 | 0.38 | 0.55 |

5 | 0.56 | 0.71 | 0.80 | 0.80 | null | 0.80 | 0.80 | 0.71 | 0.56 | 0.38 | 0.20 | 0.39 |

6 | 0.40 | 0.57 | 0.72 | 0.80 | 0.80 | null | 0.80 | 0.80 | 0.70 | 0.54 | 0.37 | 0.22 |

7 | 0.22 | 0.41 | 0.57 | 0.72 | 0.80 | 0.80 | null | 0.80 | 0.80 | 0.68 | 0.53 | 0.36 |

8 | 0.35 | 0.23 | 0.41 | 0.57 | 0.71 | 0.80 | 0.80 | null | 0.80 | 0.79 | 0.67 | 0.52 |

9 | 0.52 | 0.35 | 0.22 | 0.40 | 0.56 | 0.70 | 0.80 | 0.80 | null | 0.80 | 0.78 | 0.66 |

10 | 0.67 | 0.53 | 0.36 | 0.21 | 0.38 | 0.54 | 0.68 | 0.79 | 0.80 | null | 0.80 | 0.77 |

11 | 0.78 | 0.67 | 0.54 | 0.38 | 0.20 | 0.37 | 0.53 | 0.67 | 0.78 | 0.80 | null | 0.80 |

12 | 0.80 | 0.79 | 0.69 | 0.55 | 0.39 | 0.22 | 0.36 | 0.52 | 0.66 | 0.77 | 0.80 | null |

S1 | S2 | S3 | S4 | S5 | S6 | S7 | S8 | |
---|---|---|---|---|---|---|---|---|

c_{1} | 0.65 | 0.65 | 0.75 | 0.65 | 0.65 | 0.75 | 0.65 | 0.8 |

c_{2} | 0.85 | 0.85 | 0.85 | 0.95 | 0.95 | 0.95 | 0.95 | 0.85 |

**Table 3.**Four kinds of classification in confusion matrix. TP indicates the area is correctly predicted as defects. FN indicates the area is incorrectly predicted as normal wood. FP indicates the area is incorrectly predicted defects. TN indicates the area is correctly predicted as normal wood.

Predict Defects | Predict Wood | |
---|---|---|

Real defects | TP | FN |

Real wood | FP | TN |

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## Share and Cite

**MDPI and ACS Style**

Du, X.; Li, J.; Feng, H.; Chen, S.
Image Reconstruction of Internal Defects in Wood Based on Segmented Propagation Rays of Stress Waves. *Appl. Sci.* **2018**, *8*, 1778.
https://doi.org/10.3390/app8101778

**AMA Style**

Du X, Li J, Feng H, Chen S.
Image Reconstruction of Internal Defects in Wood Based on Segmented Propagation Rays of Stress Waves. *Applied Sciences*. 2018; 8(10):1778.
https://doi.org/10.3390/app8101778

**Chicago/Turabian Style**

Du, Xiaochen, Jiajie Li, Hailin Feng, and Shengyong Chen.
2018. "Image Reconstruction of Internal Defects in Wood Based on Segmented Propagation Rays of Stress Waves" *Applied Sciences* 8, no. 10: 1778.
https://doi.org/10.3390/app8101778