# Crack Detection in Concrete Tunnels Using a Gabor Filter Invariant to Rotation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. System Description

## 4. Image Sensors for Crack Detection

_{WA}.

_{D}, namely the distance between the camera’s optical center and the tunnel surface. In general, this working distance is recommended to be lower than the radius of the tunnel, R

_{T}, plus the radius of the working area, R

_{WA}, and greater than the radius of the tunnel, R

_{T}, minus the radius of the working area, R

_{WA}:

_{C}is the number of cameras, P

_{T}is the tunnel perimeter, W

_{C}

^{min}is the minimum crack width, and N

_{P}is the number of pixels of the camera sensor. Moreover, a (small) overlap between the field of view of two adjacent cameras is recommended to avoid gaps.

## 5. Methodology for Spatial-Frequency Image Analysis by Gabor Filters Invariant to Rotation

#### 5.1. Gabor Filter

_{x}and σ

_{y}are the smoothing parameters (standard deviations of the Gaussian envelope in the direction of the wave and orthogonal to it). The frequency domain formulation of the Gabor filter is,

_{x}and σ

_{y}. We actually aim to enhance the cracks in images, so the filtered image should offer high values in the presence of cracks and low values otherwise. This can be achieved by selecting the frequency value so the corresponding wavelength (2π/F) approaches the width of the crack, and an orientation value so it matches that of the crack. In turn, smoothing parameters serve to increase or decrease the range of frequencies and orientations to which the filter reacts.

#### 5.2. Gabor Filters Invariant to Rotation

_{θ}, and a set of values for F, σ

_{x}, and σ

_{y}parameters is selected. Then, a filtered image is obtained for each orientation in n

_{θ}

_{¡}and the remaining parameter values. The resulting filtered-images are combined to obtain a final, single image by taking the maximum value at each pixel,

^{F}is the final filtered image, I

^{n}is the image filtered by the Gabor filter with the n

^{th}orientation, and i and j are the columns and rows of the image, respectively. Finally, this resulting image, I

^{F}, is segmented by computing a suitable threshold value.

^{−1}, σ

_{x}= 2.08 pixels and σ

_{y}= 17.16 pixels (see Section 6 for further details on these values).

#### 5.3. Selection of Optimal Parameters by Differential Evolution Algorithm

_{x}, and σ

_{y}parameter values of the Gabor filters is essential to obtain accurate results. In the present work, the use of a genetic algorithm based on the so-called Differential Evolution optimization method is proposed. This algorithm was first introduced by Storn and Price [22], and allows good convergence results to be achieved, to a large extent avoiding falls in local minimums. The algorithm operates in four main steps:

**Initialization**: Generate N_{p}vectors randomly within a given parameter space. Each vector is composed of the variables of the function to be minimized, i.e., the parameters of the image segmentation method whose optimization is looked for. The segmentation method is applied to a set of training images using each of these vectors, and the corresponding result is evaluated.**Mutation**: N_{p}new vectors are created by applying,$${v}_{i,g}={x}_{best,g}+F({x}_{r1,g}-{x}_{r2,g})$$_{p}−1 range, and F is a mutation scale vector. The original algorithm proposed by Storn and Price used another random vector, r3, instead of the one leading to the best result, best, and r1, r2, r3, and i should be different.**Recombination**: In this step, the mutated vector is recombined with the unmutated one by applying,$${u}_{ji,g}=\{\begin{array}{cc}{v}_{ji,g}& {\mathrm{if}\mathrm{rand}}_{\mathrm{j}}\left[0,1\right]\le CR\hfill \\ {x}_{ji,g}& \mathrm{otherwise}\hfill \end{array}$$**Selection**: Finally, the recombined vectors are evaluated and the best performing one, from the original and recombined vectors, are assigned to the next generation as follows:$${x}_{i,g+1}=\{\begin{array}{cc}{u}_{i,g}& \mathrm{if}f({u}_{i,g})f({x}_{i,g})\hfill \\ {x}_{i,g}& \mathrm{otherwise}\hfill \end{array}$$

## 6. Experimental Results

#### 6.1. Inspection Platform Prototype

_{MAX}is the maximum admissible speed of the vehicle, R

_{A}is the resolution in the direction of movement of the vehicle, T

_{Exp}is the exposure time, and W

_{C}

^{min}is the minimum crack width. A correction factor of 0.9 is then applied to the result in order to ensure that the maximum speed is never exceeded. For example, if exposure time is set to 100 µs and W

_{C}

^{min}is 0.2 mm, it results in S

_{MAX}being 1 m/s. Appling the correction factor means that the vehicle speed should never be greater than 0.9 m/s.

#### 6.2. Crack Detection Results

- The first generation has been generated by defining a bank of filters covering the entire frequency space (instead of just taking a random selection). The parameters of the filter bank are: filter size (N × M) = 128 × 128 pixels, number of frequencies (n
_{F}) = 12, number of orientations (n_{θ}) = 16, frequency band (B_{F}) = 0.5 octaves, overlapping constant along x (K_{x}) = 1, overlapping constant along y (K_{y}) = 1, Max. Center Frequency (F_{M}) = 0.427 pixels^{−1}. - There are 12 individuals in each generation, N
_{p}= 12. This value matches the number of frequencies of the filter bank described in the previous section. - Mutation probability has been established at 50%, CR = 0.5.
- The mutation scale factor F is not a fixed value, but a function of the current generation and the number of generations without improvement, according to,$$F=1-\frac{n}{{n}_{max}}+\frac{{n}^{wi}}{{n}_{max}^{wi}}$$
_{max}is the maximum number of generations, n^{wi}is the number of generations without improvement, and n^{wi}_{max}is the maximum number of generations without improvement. Thus, the scale factor decreases as the number of current generation increases, and increases through non-improving generations. The value of F is always greater than 0 and smaller than 2. - The classifier threshold has been computed by minimizing the classifier weighted error (1 minus balanced accuracy). To do this, the response of all the training samples is ordered, and the weighted error is computed for each value between two samples. The threshold leading to the lowest weighted error is finally selected.
- The stop criterion has been set to stop searching when the maximum number of generations, n
_{max}, (set to 500) is reached, or the maximum number of generations without improvement, n^{wi}_{max}, (set to 50) is achieved. There is an improvement when the classifier weighted error of one individual in the present generation is smaller than the smallest previous classifier weighted error.

^{−1}), dispersion along X = 2.0813 (pixels), dispersion along Y = 17.1607 (pixels), threshold = 3.7301.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Mechanical defects in tunnels: (

**a**) fissures; (

**b**) detachments on precasts; (

**c**) seen steel frames.

**Figure 3.**Two examples of camera position distribution, and the portion of the tunnel section covered by each camera. (The field-of-view of the cameras is shadowed in blue). (

**a**) Working distance (W

_{D}) is smaller than the radius of the tunnel (R

_{T}) (

**b**) Working distance (W

_{D}) is greater than the radius of the tunnel (R

_{T}), so only half of the cameras can be placed in the same plane.

**Figure 4.**Example of the application of the rotation-invariant Gabor filter. The original image and the result of applying the rotation-invariant Gabor filter to it are shown on the left. The Gabor filters applied along 16 orientations (from 0° to 180°) are shown on the right, along with their maximum values (

**a**–

**p**).

**Figure 6.**Evolution of the weighted error through the increase in the number of generations of the Gabor filter invariant to rotation.

**Figure 7.**Example of application of the rotation-invariant Gabor filter. (

**a**,

**d**,

**g**) are the normalized original images. (

**b**,

**e**,

**h**) are the filtered images; (

**c**,

**f,i**) the segmented images.

**Figure 8.**Result of applying the proposed algorithm to a large area of the tunnel. (

**a**) Original image; (

**b**) filtered image; (

**c**) segmented image.

References | Vision Techniques |
---|---|

[1,2,3,4] | Thresholding original images |

[2,5,6,7,8,9,10,11] | Thresholding processed images |

[6,12,13] | More elaborated threshold techniques |

[14,18] | Seeds |

[15] | CNN |

[9,10,11,16] | Edge detectors |

[17] | Genetic algorithms |

[3,18] | Models |

[19] | SVM |

[20] | Gabor filters |

Crack Type | Subtype | Opening (mm) | Description |
---|---|---|---|

0 | <0.1 | Usually not taken into account | |

A | 0.1–0.4 | Do not reach the intrados steel They are of an aesthetic nature | |

B | 0.4–2 | Pass through the first reinforcement layer They are stabilized | |

C | C1 | <0.4 | Multicracked precasts Cracks without risk of fall or detachment |

C2 | >0.4 | Multicracked precasts Cracks with risk of detachment |

Crack Size | Sensor Resolution (Pixels) | Electricity Tunnel | Underground Tunnel | High-Speed Train Tunnel | ||||||
---|---|---|---|---|---|---|---|---|---|---|

No. of Cameras | Focal Length (mm) | Nb of Cameras | Focal Length (mm) | No. of Cameras | Focal Length (mm) | |||||

1 Pass | 2 Pass | 1 Pass | 2 Pass | 1 Pass | 2 Pass | |||||

0.1 mm | 12,288 | 14 | 7 | 80 | 22 | 11 | 105 | – | – | – |

0.4 mm | 8194 | 6 | 3 | 50 | 8 | 4 | 50 | – | 7 | 135 |

6144 | 7 | 4 | 11 | 6 | – | 9 | ||||

1 mm | 6144 | – | 2 | 28 | – | 3 | 28 | – | 4 | 50 |

2 mm | 4096 | – | – | – | 2 | 28 | – | 3 | 28 | |

1024 | 9 | 5 | 12 | – | – | – | – | – | – |

**Table 4.**Confusion matrix obtained in the detection of cracks in tunnels using the Gabor filters invariant to rotation.

Confusion Matrix | Expert Classification | ||
---|---|---|---|

Positive | Negative | ||

Classifier classification | Positive | TP = 19,050 | FP = 3772 |

Negative | FN = 950 | TN = 76,228 |

**Table 5.**Sensitivity, specificity, accuracy, precision, and balanced accuracy in the detection of cracks in tunnels using the Gabor filters invariant to rotation.

Sensitivity | Specificity | Accuracy | Precision | Balanced Accuracy |
---|---|---|---|---|

0.9525 | 0.9529 | 0.9373 | 0.9876 | 0.9527 |

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**MDPI and ACS Style**

Medina, R.; Llamas, J.; Gómez-García-Bermejo, J.; Zalama, E.; Segarra, M.J.
Crack Detection in Concrete Tunnels Using a Gabor Filter Invariant to Rotation. *Sensors* **2017**, *17*, 1670.
https://doi.org/10.3390/s17071670

**AMA Style**

Medina R, Llamas J, Gómez-García-Bermejo J, Zalama E, Segarra MJ.
Crack Detection in Concrete Tunnels Using a Gabor Filter Invariant to Rotation. *Sensors*. 2017; 17(7):1670.
https://doi.org/10.3390/s17071670

**Chicago/Turabian Style**

Medina, Roberto, José Llamas, Jaime Gómez-García-Bermejo, Eduardo Zalama, and Miguel José Segarra.
2017. "Crack Detection in Concrete Tunnels Using a Gabor Filter Invariant to Rotation" *Sensors* 17, no. 7: 1670.
https://doi.org/10.3390/s17071670