# A Novel Edge Detection Method Based on the Regularized Laplacian Operation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Regularized Laplacian Operation

## 3. The Edge Detection Algorithm

_{i}includes pixels which have been identified as edges by at least i different edge maps.

_{i}is compared with each edge map D

_{j}, and it generates four different probabilities:${\mathrm{TP}}_{{\mathrm{PGT}}_{i},\text{}{\mathrm{D}}_{j}},\text{}{\mathrm{FP}}_{{\mathrm{PGT}}_{i},\text{}{\mathrm{D}}_{j}},\text{}{\mathrm{TN}}_{{\mathrm{PGT}}_{i},\text{}{\mathrm{D}}_{j}},\text{}{\mathrm{FN}}_{{\mathrm{PGT}}_{i},\text{}{\mathrm{D}}_{j}}.$ Among them, TP

_{A,B}(true positive) means the probability of pixels which have been determined as edges in both edge maps A and B; FP

_{A,B}(false positive) means the probability of pixels determined as edges in A, but non-edges in B; TN

_{A,B}(true negative) means the probability of pixels determined as non-edges in both A and B; and FN

_{A,B}(false negative) means the probability of pixels determined as edges in B, but non-edges in A.

_{i}, we average the four probabilities over all edge maps D

_{j}, and get ${\overline{\mathrm{TP}}}_{{\mathrm{PGT}}_{i}},\text{}{\overline{\mathrm{FP}}}_{{\mathrm{PGT}}_{i}},\text{}{\overline{\mathrm{TN}}}_{{\mathrm{PGT}}_{i}},\text{}{\overline{\mathrm{FN}}}_{{\mathrm{PGT}}_{i}}$, where ${\overline{\mathrm{TP}}}_{{\mathrm{PGT}}_{i}}=\frac{1}{N}{\displaystyle \sum _{j=1}^{N}{\mathrm{TP}}_{{\mathrm{PGT}}_{i},{\mathrm{D}}_{j}}}$, and the expressions of other probabilities are similar. Then, a statistical measurement of each PGT

_{i}is given by the Chi-square test:

_{i}with the highest ${\chi}_{{\mathrm{PGT}}_{i}}^{2}$ is considered as the estimated ground truth (EGT).

_{j}is then matched to the EGT by four new probabilities: ${\mathrm{TP}}_{{\mathrm{D}}_{j},\mathrm{EGT}},\text{}{\mathrm{FP}}_{{\mathrm{D}}_{j},\mathrm{EGT}},\text{}{\mathrm{TN}}_{{\mathrm{D}}_{j},\mathrm{EGT}},\text{}{\mathrm{FN}}_{{\mathrm{D}}_{j},\mathrm{EGT}}.$ The Chi-square measurements ${\chi}_{{\mathrm{D}}_{j}}^{2}$ are obtained by the same way as in Step 3. Then, the best edge map is the one which gives the highest ${\chi}_{{\mathrm{D}}_{j}}^{2}$, and the corresponding regularization parameter ${\alpha}_{j}$ is the one we want.

## 4. Experiments and Results

_{6}. Compared with the EGT, the BDM of each edge map D

_{j}is shown in Figure 3b, from which we can see the best edge map is D

_{6}. Hence, the regularization parameter is chosen as $\alpha =0.05$. The choice of the scale parameter in the LoG detector is carried out similarly. It does not need any parameters in the LED.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Basu, M. Gaussian-based edge-detection methods—A survey. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev.
**2002**, 32, 252–260. [Google Scholar] [CrossRef] - Marr, D.; Hildreth, E. Theory of edge detection. Proc. R. Soc. Lond. B
**1980**, 207, 187–217. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Canny, J. A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell.
**1986**, 8, 679–698. [Google Scholar] [CrossRef] [PubMed] - Sarkar, S.; Boyer, K. Optimal infinite impulse response zero crossing based edge detectors. CVGIP Image Underst.
**1991**, 54, 224–243. [Google Scholar] [CrossRef] - Demigny, D. On optimal linear filtering for edge detection. IEEE Trans. Image Process.
**2002**, 11, 728–737. [Google Scholar] [CrossRef] [PubMed] - Kang, C.C.; Wang, W.J. A novel edge detection method based on the maximizing objective function. Pattern Recognit.
**2007**, 40, 609–618. [Google Scholar] [CrossRef] - Wang, X. Laplacian operator-based edge detectors. IEEE Trans. Pattern Anal. Mach. Intell.
**2007**, 29, 886–890. [Google Scholar] [CrossRef] [PubMed] - Lopez-Molina, C.; Bustince, H.; Fernandez, J.; Couto, P.; De Baets, B. A gravitational approach to edge detection based on triangular norms. Pattern Recognit.
**2010**, 43, 3730–3741. [Google Scholar] [CrossRef] - Murio, D.A. The Mollification Method and the Numerical Solution of Ill-Posed Problems; Wiley-Interscience: New York, NY, USA, 1993; pp. 1–5. ISBN 0-471-59408-3. [Google Scholar]
- Wan, X.Q.; Wang, Y.B.; Yamamoto, M. Detection of irregular points by regularization in numerical differentiation and application to edge detection. Inverse Probl.
**2006**, 22, 1089–1103. [Google Scholar] [CrossRef] [Green Version] - Xu, H.L.; Liu, J.J. Stable numerical differentiation for the second order derivatives. Adv. Comput. Math.
**2010**, 33, 431–447. [Google Scholar] [CrossRef] - Huang, X.; Wu, C.; Zhou, J. Numerical differentiation by integration. Math. Comput.
**2013**, 83, 789–807. [Google Scholar] [CrossRef] - Wang, Y.C.; Liu, J.J. On the edge detection of an image by numerical differentiations for gray function. Math. Methods Appl. Sci.
**2018**, 41, 2466–2479. [Google Scholar] [CrossRef] - Gonzalez, R.C.; Woods, R.E. Digital Image Processing, 3rd ed.; Pearson: London, UK, 2007; pp. 158–162. ISBN 978-0-13-168728-8. [Google Scholar]
- Gunn, S.R. On the discrete representation of the Laplacian of Gaussian. Pattern Recognit.
**1999**, 32, 1463–1472. [Google Scholar] [CrossRef] - Xu, H.L.; Liu, J.J. On the Laplacian operation with applications in magnetic resonance electrical impedance imaging. Inverse Probl. Sci. Eng.
**2013**, 21, 251–268. [Google Scholar] [CrossRef] - Yitzhaky, Y.; Peli, E. A method for objective edge detection evaluation and detector parameter selection. IEEE Trans. Pattern Anal. Mach. Intell.
**2003**, 25, 1027–1033. [Google Scholar] [CrossRef] [Green Version] - Gu, C.H.; Li, D.Q.; Chen, S.X.; Zheng, S.M.; Tan, Y.J. Equations of Mathematical Physics, 2nd ed.; Higher Education Press: Beijing, China, 2002; pp. 80–86. ISBN 7-04-010701-5. (In Chinese) [Google Scholar]
- Lopez-Molina, C.; Baets De, B.; Bustince, H. Quantitative error measures for edge detection. Pattern Recognit.
**2013**, 46, 1125–1139. [Google Scholar] [CrossRef] - Baddeley, A.J. An error metric for binary images. In Proceedings of the IEEE Workshop on Robust Computer Vision, Bonn, Germany, 9–11 March 1992; Wichmann Verlag: Karlsruhe, Germany, 1992; pp. 59–78. [Google Scholar]
- Fernández-García, N.L.; Medina-Carnicer, R.; Carmona-Poyato, A.; Madrid-Cuevas, F.J.; Prieto-Villegas, M. Characterization of empirical discrepancy evaluation measures. Pattern Recognit. Lett.
**2004**, 25, 35–47. [Google Scholar] [CrossRef] - Heath, M.D.; Sarkar, S.; Sanocki, T.A.; Bowyer, K.W. A robust visual method for assessing the relative performance of edge detection algorithms. IEEE Trans. Pattern Anal. Mach. Intell.
**1997**, 19, 1338–1359. [Google Scholar] [CrossRef]

**Figure 1.**The reference edge image and three polluted edge maps: (

**a**) reference edge ${E}_{R}$; (

**b**) polluted edge map ${E}_{1}$; (

**c**) polluted edge map ${E}_{2}$; (

**d**) polluted edge map ${E}_{3}$.

**Figure 3.**The figure of BDMs: (

**a**) the BDM of ${\Delta}_{{\mathrm{PGT}}_{i}}^{1},\text{}i\in \{1,2,\dots ,11\}$; (

**b**) the BDM of ${\Delta}_{{\mathrm{D}}_{j}}^{1},\text{}i\in \{1,2,\dots ,11\}$.

**Figure 4.**Edge detection results of the airplane image: (

**a**) the ground truth; (

**b**) the edge detected by the regularized edge detector (RED); (

**c**) the Laplacian of Gaussian (LoG); (

**d**) the Laplacian-based edge detector (LED).

**Figure 5.**Edge detection results of the elephant image: (

**a**) the ground truth; (

**b**) the edge detected by the RED; (

**c**) the LoG; (

**d**) the LED.

**Table 1.**The Baddeley’s delta metrics (BDMs) between the reference edge image ${E}_{R}$ and the polluted edge maps ${E}_{i}\text{}(i=1,2,3)$ with the different choices of parameters c and k.

Parameter Sets | ${\mathbf{\Delta}}^{\mathit{k}}({\mathit{E}}_{\mathit{R}},{\mathit{E}}_{1})$ | ${\mathbf{\Delta}}^{\mathit{k}}({\mathit{E}}_{\mathit{R}},{\mathit{E}}_{2})$ | ${\mathbf{\Delta}}^{\mathit{k}}({\mathit{E}}_{\mathit{R}},{\mathit{E}}_{3})$ |
---|---|---|---|

k = 1, c = 2 | 0.0566 | 0.0937 | 0.1256 |

k = 1, c = 3 | 0.0950 | 0.1879 | 0.2461 |

k = 1, c = 4 | 0.1397 | 0.2614 | 0.3305 |

k = 2, c = 2 | 0.2182 | 0.3307 | 0.3637 |

k = 2, c = 3 | 0.2753 | 0.4925 | 0.6313 |

k = 2, c = 4 | 0.3317 | 0.6159 | 0.8021 |

Images | LED | LoG | RED |
---|---|---|---|

Airplane | 0.7515 | 0.1270 | 0.1232 |

Elephant | 0.6619 | 0.3041 | 0.2593 |

Turtle | 0.4430 | 0.1226 | 0.1323 |

Brush | 0.5790 | 0.1883 | 0.1673 |

Tiger | 0.9239 | 0.2854 | 0.2748 |

Grater | 0.5537 | 0.2353 | 0.2143 |

Pitcher | 0.5032 | 0.2584 | 0.2296 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xu, H.; Xiao, Y.
A Novel Edge Detection Method Based on the Regularized Laplacian Operation. *Symmetry* **2018**, *10*, 697.
https://doi.org/10.3390/sym10120697

**AMA Style**

Xu H, Xiao Y.
A Novel Edge Detection Method Based on the Regularized Laplacian Operation. *Symmetry*. 2018; 10(12):697.
https://doi.org/10.3390/sym10120697

**Chicago/Turabian Style**

Xu, Huilin, and Yuhui Xiao.
2018. "A Novel Edge Detection Method Based on the Regularized Laplacian Operation" *Symmetry* 10, no. 12: 697.
https://doi.org/10.3390/sym10120697