# Pavement crack characteristic detection based on sparse representation

- Xiaoming Sun
^{1}, - Jianping Huang
^{1}, - Wanyu Liu
^{1}Email author and - Mantao Xu
^{2}

**2012**:191

https://doi.org/10.1186/1687-6180-2012-191

© Sun et al.; licensee Springer. 2012

**Received: **8 March 2012

**Accepted: **31 July 2012

**Published: **30 August 2012

## Abstract

Pavement crack detection plays an important role in pavement maintaining and management. The three-dimensional (3D) pavement crack detection technique based on laser is a recent trend due to its ability of discriminating dark areas, which are not caused by pavement distress such as tire marks, oil spills and shadows. In the field of 3D pavement crack detection, the most important thing is the accurate extraction of cracks in individual pavement profile without destroying pavement profile. So after analyzing the pavement profile signal characteristics and the changeability of pavement crack characteristics, a new method based on the sparse representation is developed to decompose pavement profile signal into a summation of the mainly pavement profile and cracks. Based on the characteristics of the pavement profile signal and crack, the mixed dictionary is constructed with an over-complete exponential function and an over-complete trapezoidal membership function, and the signal is separated by learning in this mixed dictionary with a matching pursuit algorithm. Some experiments were conducted and promising results were obtained, showing that we can detect the pavement crack efficiently and achieve a good separation of crack from pavement profile without destroying pavement profile.

## Keywords

## Introduction

In the life cycle of pavement, there will be various pavement distresses due to the burden of vehicles and natural causes. The pavement distress will affect the lifespan of the pavement, vehicles energy assumption, transportation efficiency, and the transportation safety [1]. Among the various pavement distresses, pavement cracking data is the most important element for quantifying the condition of pavement surface [2, 3], so it is crucial to detect and recognize the pavement cracking automatically and accurately before repairing them.

There are some methods for detecting pavement cracking. The traditional method is to detect by human vision, but manual surface distress survey are subject to many limitation such as non-repeatability, subjectivity, and high personal costs [4]. Manual procedures are time-consuming, and present substantial differences between evaluations of different raters [5]. In recent years, several 2D image analysis-based pavement crack detection techniques were proposed [6–10]. One major issue with pure 2D video-based systems is their inability to discriminate dark areas not caused by pavement distress such as tire marks, oil spills, shadows, and recent fillings [11]. Moreover, the shadows and poor illumination are also major problems for daytime operation though they can be overcome using additional lighting systems or by acquiring data in the night after sunset [12].

Since each laser profile acquired by 3D laser detection system has its own characteristic in terms of profile shape, crack shape, the number of cracks and signal-to-noise ratio, the signal processing techniques are adapted to each laser profile to extract the features of the crack. As the technology protection, few literatures describe algorithms in detail in this area. Bursanescu used a special filter to avoid the noise and extract the crack. The filter uses an adaptive width mobile widow, and width is self-adjusting. We cannot find more details in his papers [5, 11, 14]. Laurent introduced the algorithm for the detection of cracks, which is the valley detection of candidate cracks in the individual pavement profiles [15–17]. This simple technique is fast and easy to implement, but it cannot achieve a good separation of crack from pavement profile.

For profile signal, the main profile signal is varying slowly, which spreads over the whole observing period; Crack signal has sharp edges and performs different shapes, which belongs to narrow-scale signal. Wavelets can detect the location of cracks accurately due to its good time-frequency characteristic, but the wavelet base is fixed which cannot match the crack shape well. So after analyzing the characteristics of main profile and cracking, we constructed a mixed over-complete dictionary according to main profile and crack characteristics and proposed a novel method based on sparse representation for crack detection and the mainly pavement profile extraction without noise. Some experiments were conducted and promising results were obtained, showing that we can detect the pavement crack accurately and achieve a good separation of crack from pavement profile.

## Pavement crack and profile detection based on sparse representation

### Sparse representation of signals

*D*∈

*R*

^{n× K}that contains

*K*atoms, {

*d*

_{ j }}

_{ j = 1 }

^{ K }, as its columns, it is assumed that a signal

*y*∈

*R*

^{ n }can be represented as a sparse linear combination of these atoms. The representation of

*y*may either be exact

*y*=

*Dx*, or approximate,

*y*≈

*Dx*, satisfying ‖

*y*−

*Dx*‖

_{2}≤ ϵ. The vector

*x*∈

*R*

_{ K }displays the representation coefficients of the signal

*y*. This, sparest representation, is the solution of either

where ‖.‖_{0} is the *l*^{0} norm , counting the non zero entries of a vector [20].

*f*contains crack signal

*s*

_{ C }and main profile signal

*s*

_{ Prof }two layers as a linear combination, we propose to seek the sparsest of all representation over the mixed dictionary. Thus we need to solve

where *Φ*_{
c
} is the crack dictionary, *Φ*_{
p
} is the main profile dictionary, *x*_{
c
} and *x*_{
p
} are the coefficients in the corresponding dictionaries.

### Characteristics of main profile signal and crack signal

*f*is profile signal,

*s*

_{ C }is crack signal,

*s*

_{ Prof }is main profile signal,

*s*

_{ n }is noise. where

*f*is profile signal,

*s*

_{Prof}is the main profile signal,

*s*

_{ R },

*s*

_{ C },

*s*

_{ bump },

*s*

_{Prot}, and

*s*

_{ n }represent the rut signal, crack signal, bump signal, pothole signal, and noise, respectively. This paper mainly studies the characteristics of cracks and main profiles and how the crack to be separated from the main profile.

Since longitudinal main profile signal *s*_{
Prof
} is used to calculate the international roughness index (IRI), it should not contain distress and noise. Therefore, it is necessary to analyze the characteristics of *s*_{
C
} and *s*_{
Prof
} and separate *s*_{
C
} from *f* without destroying *s*_{
Prof
}.

*s*

_{ C }has the following characteristics: clear and sharp edges, a direction below the horizontal surface, different narrow-scale shapes. In order to calculate the crack width and location accurately and achieve a good match with the shape of crack, we observed a large amount of crack data. Figure 5 clearly shows the basic shapes of crack, while most cracks perform asymmetrical form of these basic shapes due to the rain erosion, sand filling, etc.

*s*

_{ Prof }is a low-frequency curve as the red curve shown in Figure 6, spreads over all the observing periods and performs to be the profile of pavement without distress and noise.

### Mixed over-complete dictionary

The success of sparse representation application depends on how to pick the suitable dictionary which is employed to sparsely describe the signal. Based on the difference between *s*_{
C
} and *s* _{
Prof
}, it is possible to detect *s* _{
C
} and *s* _{
Prof
} by performing two different transformations. That is to say, we try to find two dictionaries *Φ*_{
c
} and *Φ*_{
p
} in line with the real signal to separate the *s*_{
C
} and *s*_{
Prof
} from *f* .

*x*, which depends on four scalar parameters

*a*,

*b*,

*c*, and

*d*. As shown in Figure 7, this function is flexible, which can construct different shapes of the cracks using four-point transform. The mathematical function model of the four-point curve is represented as follow:

Certainly, we construct the over-complete dictionary *Φ*_{
c
} with four-point curve function which can efficiently express the sharp edges and diversity of crack. The parameters *a* and *d* locate the “feet” of the trapezoid and the parameters *b* and *c* locate the “shoulders”. As for our work in this article, the scales ( *d* – *a*) of the four-point curve range from 1 to 5 and the shifts are densely sampled from 0 to *L* – 1 for each scale, where L is the length of signal.

*m*is the position information of exponential function,

*n*is the scale information,

*A*is a normalization factor. When

*m*and

*n*change in different areas separately, we can have different profiles. As for our work,

*m*ranges from 0 to

*L*– 1, where L is the length of signal, and the

*n*ranges from 1 to 800.

Finally, the mixed over-complete dictionary *Φ* is composed of exponential function *Φ*_{
p
} and trapezoidal membership function *Φ*_{
c
}, which can be denoted by *Φ*=*Φ*_{
p
} + *Φ*_{
c
} , and all the atoms in over-complete dictionary are normalized.

### Signal separation by the matching pursuit method

*Φ*. Figure.9 shows the framework of the algorithm based on MP used in this article. The more detailed algorithm is given below:

- 1.
Initialization: Set

*k*=1,*S*^{(0)}=0,*R*^{(0)}=*S*,*x*_{ ck }=0, where*k*is the number of iteration,*S*is the profile signal to be decomposed,*R*is the residual signal during the iterations; the superscript is the iteration number;*x*_{ ck }is the coefficients in*Φ*_{ c };*δ*_{c min}and*δ*_{c max}are the threshold of inner product between residual signal and each atom in crack dictionary*Φ*_{ c };*δ*_{ p }is the threshold of inner product between residual signal and each atom in crack dictionary*Φ*_{ p } - 2.Crack feature extraction with MP:
- 2.1
-find the atom in

*Φ*_{ c }with maximum inner product in each iteration, i.e. $\left|\u3008{R}^{\left(k-1\right)},{\Phi}_{\mathit{cj}}\u3009\right|=sup\left|\u3008{R}^{\left(k-1\right)},{\Phi}_{c}\u3009\right|$, ${x}_{\mathit{ck}}=ma{x}_{j}\left|\u3008{R}^{\left(k-1\right)},{\Phi}_{\mathit{cj}}\u3009\right|$ where*Φ*_{ cj }is the*j*th atom in*Φ*_{ c },*x*_{ ck }is the coefficients. - 2.2
-If ${\delta}_{min}\le \left|\u3008{R}^{\left(k-1\right)},{\Phi}_{\mathit{cj}}\u3009\right|\le {\delta}_{max}$, ${S}^{k}={S}^{\left(k-1\right)}+{x}_{\mathit{ck}}{\Phi}_{\mathit{ck}}$, ${R}^{\left(k\right)}=S-{S}^{\left(k\right)}$,k++, go to 2.1. Else, no crack, go to Step3.

- 2.3
-Through step 2.1 and step 2.2,

*S*can be expressed as $S={S}_{\mathit{residue}}+{\displaystyle \sum _{k=1}^{m}{x}_{\mathit{ck}}{\Phi}_{\mathit{ck}}}$*m*is the total number of iterations.

- 2.1
- 3.Main profile reconstruction with MP:
- 3.1
- Set

*k*=1, ${S}_{\mathit{residue}}=S-{\displaystyle \sum _{k=1}^{m}{x}_{\mathit{ck}}{\Phi}_{\mathit{ck}}}$, ${R}_{\mathit{residue}}^{\left(0\right)}={S}_{\mathit{residue}}$ - 3.2
-Project

*S*_{ residue }on a dictionary*Φ*_{ p }and find the atom in*Φ*_{ p }with maximum inner product in each iteration, i.e. $\left|\u3008{{R}_{\mathit{residue}}}^{\left(k-1\right)},{\Phi}_{\mathit{pj}}\u3009\right|=sup\left|\u3008{{R}_{\mathit{residue}}}^{\left(k-1\right)},{\Phi}_{p}\u3009\right|$, ${x}_{\mathit{pk}}={max}_{j}\left|\u3008{{R}_{\mathit{residue}}}^{\left(k-1\right)},{\Phi}_{\mathit{pj}}\u3009\right|$ where*Φ*_{ pj }is the*j*th atom in*Φ*_{ p },*x*_{ pk }is the coefficients. - 3.3
-If $\left|\u3008{{R}_{\mathit{residue}}}^{\left(k-1\right)},{\Phi}_{\mathit{pj}}\u3009\right|\le {\delta}_{p}$, ${{S}_{\mathit{residue}}}^{k}={{S}_{\mathit{residue}}}^{\left(k-1\right)}+{x}_{\mathit{pk}}{\Phi}_{\mathit{pk}}$, ${R}_{\mathit{residue}}^{\left(k\right)}={S}_{\mathit{residue}}-{S}_{\mathit{residue}}^{\left(k\right)}$,

*k*++, go to 3.2. Else, go to step 4.

- 3.1
- 4.
Finally,

*S*can be expressed as follow:$S={\displaystyle \sum _{k=0}^{m}{x}_{\mathit{ck}}{\Phi}_{\mathit{ck}}}+{\displaystyle \sum _{k=0}^{n}{x}_{\mathit{pk}}{\Phi}_{\mathit{pk}}}+\sigma $,

*m*is the iteration number of crack feature extraction,*n*is iteration number of main profile reconstruction,*σ*includes noise and approximation error.

## Experimental results

The following experiments are designed to examine the performance of the proposed approach for a good separation of the crack and main profile.

### Comparison experiment with wavelet and median filtering method

In order to verify the effectiveness of our method, we construct the simulation signal of pavement profile. The simulation signal is presented in Figure 8d, where the profile is composed of main profile simulated by exponential function (Figure 8a), pavement distress (Figure 8b) and white Gaussian noise (SNR = 8db).

**Disease characteristic parameter of the left crack in Figure**
8
**d (d)**

Parameter | Theoretical Value | Traditional method | Our method |
---|---|---|---|

Position(mm) | 161 | 161 | 161 |

Width(mm) | 4 | 6 | 4 |

Depth(mm) | 6 | 1.896 | 6 |

**Disease characteristic parameter of the right crack in Figure**
8
**d (d)**

Parameter | Theoretical Value | Traditional method | Our method |
---|---|---|---|

Position(mm) | 356 | 357 | 356 |

Width(mm) | 7 | 13 | 7 |

Depth(mm) | 4 | 2.442 | 4 |

From the above experiment results, it is clear that the sparse representation method outperforms the wavelet and median filtering method not only in crack detection and main profile reconstruction but also a good separation between crack and main profile.

### Comparison experiment with wavelet and Gabor dictionary

*γ*= (

*s*,

*u*,

*v*,

*w*) = (

*a*

^{ j },

*pa*

^{ j }

*Δu*,

*ka*

^{−j}

*Δv*,

*iΔw*) is time-frequency parameters of Gabor atom, in which

*a*= 2,

*Δu*= 1/2,

*Δv*=

*π*,

*Δw*=

*π*/6, 0 <

*j*≤ log

_{2}

*N*, 0 <

*p*≤

*N*2

^{−j+1}, 0 <

*k*≤ 2

^{j+1},0 ≤

*i*≤ 12; Figure 13 presents cracks obtained from Figure 8 d) by wavelet and Gabor dictionary and Figure 14 is an enlarged image of two cracks in Figure 13.

From Figures 13 and 14, it can be seen that the cracks detected by wavelet method and Gabor dictionary have distortion, which will affect the results of the subsequent main profile. It is observed that Gabor dictionary can match triangular shape of the crack very well, but when it detects trapezoidal shape of the crack, the match result is not good.

### Experiment result of actual pavement crack

*x*(

*t*), the crack waveform extracted by our algorithm is

*y*(

*t*), similarity

*R*can be expressed as:

*R*are all concentrated between 0.94 and 1, and the extracted crack waveform is good consistency with actual waveform.

### Accuracy experiment

**The first group of experiment accuracy with longitudinal direction**

The first group | Direction | The true position(mm) | The detected position(mm) | The true width(mm) | The detected width(mm) | Indication Error(mm) |
---|---|---|---|---|---|---|

1 | Longitudinal | 449 | 449 | 4 | 4.2856 | 0.2856 |

2 | Longitudinal | 430 | 430 | 4 | 4.2593 | 0.2593 |

3 | Longitudinal | 393 | 393 | 6 | 5.7726 | −0.2274 |

4 | Longitudinal | 298 | 298 | 6 | 5.6775 | −0.3225 |

5 | Longitudinal | 270 | 270 | 8 | 8.9032 | 0.9032 |

6 | Longitudinal | 228 | 228 | 8 | 8.2566 | 0.2566 |

7 | Longitudinal | 193 | 193 | 8 | 8.1504 | 0.1504 |

8 | Longitudinal | 180 | 180 | 6 | 6.2302 | 0.2302 |

9 | Longitudinal | 234 | 234 | 6 | 5.8847 | −0.1153 |

10 | Longitudinal | 231 | 231 | 6 | 5.7726 | −0.2274 |

**The second group of experiment accuracy with transverse direction**

The second group | Direction | The true position(mm) | The detected position(mm) | The true width (mm) | The detected width(mm) | Indication error(mm) |
---|---|---|---|---|---|---|

1 | Transverse | 141 | 141 | 6 | 6.5002 | 0.5002 |

2 | Transverse | 127 | 127 | 6 | 6.8589 | 0.8589 |

3 | Transverse | 141 | 141 | 6 | 5.9583 | −0.0417 |

4 | Transverse | 101 | 101 | 6 | 6.067 | 0.067 |

5 | Transverse | 143 | 143 | 6 | 6.3648 | 0.3648 |

6 | Transverse | 145 | 145 | 6 | 5.4348 | −0.5652 |

7 | Transverse | 147 | 147 | 6 | 6.2089 | 0.2089 |

8 | transverse | 157 | 157 | 8 | 8.3092 | 0.3092 |

9 | Transverse | 180 | 180 | 8 | 8.2698 | 0.2698 |

10 | Transverse | 144 | 144 | 8 | 7.5107 | −0.4893 |

## Conclusions

In this article, a novel method based on sparse representation is developed to detect pavement cracks and reconstruct the main pavement profile. The key for cracks separation from main profile is based on the features of the mixed over-complete dictionary, which consists of two kinds of atoms, one for crack representation and another for main profile representation. In this study, atoms of trapezoidal membership function are adopted to represent crack, and exponential function for main pavement profile. Compared to the wavelet and median filtering method, the cracks extracted by our method can match the shape of crack very well, which cannot damage the information of the main profile signal. Some outdoor and accuracy experiments were conducted and promising results were obtained, showing that this method cannot only detect the position of pavement crack efficiently and achieve a good separation of crack from pavement profile, but also reconstruct main profile very well. Because MP is very time consuming when the greedy exhaustive search in the whole huge over-complete dictionary adopted and it is still a challenging problem. In the future work, we will use computer grid technology to improve computational efficiency.

## Declarations

### Acknowledgments

This study was supported by International S&T Cooperation Project of China (2007DFB30320).

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.