# A Low Redundancy Wavelet Entropy Edge Detection Algorithm

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. First Order

#### 2.2. Second Order

#### 2.3. Entropy Based

#### 2.4. Wavelet Based

#### 2.5. Deep Learning/Machine Learning Based

## 3. Methodology

#### 3.1. Wavelet Decomposition

#### 3.2. Wavelet Decomposition Level Selection

#### 3.3. Entropy Thresholding

## 4. Results

#### 4.1. Computational Efficiency

Algorithm 1: LRWEEDA edge detection |

#### 4.2. Noise Resilience

#### 4.3. Performance against Standard Edge Detection Metrics

- The Boundary F1 score is defined as the harmonic mean (F1-measure) of the precision and recall values which measure the matching weight for the predicted boundary and the ground truth boundary, as$$BFS=2\xb7precision\xb7recall/(recall+precision).$$
- The Jaccard coefficient for two sets is defined as the size of the intersection of the two sets divided by the size of their union as$$JC=\frac{TP}{(TP+FP+FN)}.$$
- Pratt’s FOM uses Euclidean distance to compare two edge images [60]. It multiplies a scale factor ∝ to the Euclidean distance calculated between the two images to penalize displaced edges, as$$Prat{t}^{\prime}sFOM=\frac{1}{max({I}_{A},{I}_{B})}\sum _{i=1}^{{I}_{A}}\frac{1}{1+\propto {d}_{i}^{2}}$$

- Qualitative results of the proposed algorithm were obtained and compared with similar edge detection algorithms (Figure 9)
- Ten images were used to calculate the average processing times of the algorithms (Figure 10).
- Noise resilience of the proposed algorithm was analyzed by using four images and compared with Canny (Figure 9).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Kim, C.M.; Hong, E.J.; Park, R.C. Chest X-ray Outlier Detection Model using Dimension Reduction and Edge Detection. IEEE Access
**2021**, 9, 86096–86106. [Google Scholar] [CrossRef] - Dai, W.; Na, J.; Huang, N.; Hu, G.; Yang, X.; Tang, G.; Xiong, L.; Li, F. Integrated edge detection and terrain analysis for agricultural terrace delineation from remote sensing images. Int. J. Geogr. Inf. Sci.
**2020**, 34, 484–503. [Google Scholar] [CrossRef] - Gafton, P.; Maraz, E. 2D image relighting with image-to-image translation. arXiv
**2020**, arXiv:2006.07816. [Google Scholar] - Li, M.; Lin, Z.; Mech, R.; Yumer, E.; Ramanan, D. Photo-sketching: Inferring contour drawings from images. In Proceedings of the 2019 IEEE Winter Conference on Applications of Computer Vision (WACV), Waikoloa, HI, USA, 7–11 January 2019; pp. 1403–1412. [Google Scholar]
- Khan, N.H.; Adnan, A. Urdu optical character recognition systems: Present contributions and future directions. IEEE Access
**2018**, 6, 46019–46046. [Google Scholar] [CrossRef] - Hirz, M.; Walzel, B. Sensor and object recognition technologies for self-driving cars. Comput.-Aided Des. Appl.
**2018**, 15, 501–508. [Google Scholar] [CrossRef] [Green Version] - Ziou, D.; Tabbone, S. Edge detection techniques-an overview. Pattern Recognit. Image Anal. C/C Raspoznavaniye Obraz. Anal. Izobr.
**1998**, 8, 537–559. [Google Scholar] - Nadernejad, E.; Sharifzadeh, S.; Hassanpour, H. Edge detection techniques: Evaluations and comparison. Appl. Math. Sci.
**2008**, 2, 1507–1520. [Google Scholar] - Arbelaez, P.; Maire, M.; Fowlkes, C.; Malik, J. Contour Detection and Hierarchical Image Segmentation. IEEE Trans. Pattern Anal. Mach. Intell.
**2011**, 33, 898–916. [Google Scholar] [CrossRef] [Green Version] - Xie, S.; Tu, Z. Holistically-nested edge detection. In Proceedings of the IEEE International Conference on Computer Vision, Santiago, Chile, 13–16 December 2015; pp. 1395–1403. [Google Scholar]
- Orujov, F.; Maskeliūnas, R.; Damaševičius, R.; Wei, W. Fuzzy based image edge detection algorithm for blood vessel detection in retinal images. Appl. Soft Comput.
**2020**, 94, 106452. [Google Scholar] [CrossRef] - Canny, J. A Computational Approach to Edge Detection. IEEE Trans. Pattern Anal. Mach. Intell.
**1986**, PAMI-8, 679–698. [Google Scholar] [CrossRef] - Prewitt, J.M. Object enhancement and extraction. Pict. Process. Psychopictorics
**1970**, 10, 15–19. [Google Scholar] - Roberts, L.G. Machine Perception of Three-Dimensional Soups. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA, 1963. [Google Scholar]
- Kanopoulos, N.; Vasanthavada, N.; Baker, R.L. Design of an image edge detection filter using the Sobel operator. IEEE J. Solid-State Circuits
**1988**, 23, 358–367. [Google Scholar] [CrossRef] - Basu, M. Gaussian-based edge-detection methods-a survey. IEEE Trans. Syst. Man Cybern. Part C
**2002**, 32, 252–260. [Google Scholar] [CrossRef] [Green Version] - Liu, Y.; Cheng, M.M.; Hu, X.; Wang, K.; Bai, X. Richer convolutional features for edge detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, HI, USA, 22–25 July 2017; pp. 3000–3009. [Google Scholar]
- Poma, X.S.; Riba, E.; Sappa, A. Dense extreme inception network: Towards a robust cnn model for edge detection. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, Snowmass Village, CO, USA, 1–5 March 2020; pp. 1923–1932. [Google Scholar]
- Gong, X.Y.; Su, H.; Xu, D.; Zhang, Z.T.; Shen, F.; Yang, H.B. An overview of contour detection approaches. Int. J. Autom. Comput.
**2018**, 15, 656–672. [Google Scholar] [CrossRef] - Siuzdak, J. A single filter for edge detection. Pattern Recognit.
**1998**, 31, 1681–1686. [Google Scholar] [CrossRef] - Peli, T.; Malah, D. A study of edge detection algorithms. Comput. Graph. Image Process.
**1982**, 20, 1–21. [Google Scholar] [CrossRef] - Maini, R.; Aggarwal, H. Study and comparison of various image edge detection techniques. Int. J. Image Process. (IJIP)
**2009**, 3, 1–11. [Google Scholar] - Waghule, D.R.; Ochawar, R.S. Overview on edge detection methods. In Proceedings of the 2014 International Conference on Electronic Systems, Signal Processing and Computing Technologies (ICESC), Nagpur, India, 9–11 January 2014; pp. 151–155. [Google Scholar]
- Joshi, S.R.; Koju, R. Study and comparison of edge detection algorithms. In Proceedings of the 2012 Third Asian Himalayas International Conference on Internet (AH-ICI), Kathmundu, Nepal, 23–25 November 2012; pp. 1–5. [Google Scholar]
- Torre, V.; Poggio, T.A. On Edge Detection. IEEE Trans. Pattern Anal. Mach. Intell.
**1986**, PAMI-8, 147–163. [Google Scholar] [CrossRef] - Deriche, R. Using Canny’s criteria to derive a recursively implemented optimal edge detector. Int. J. Comput. Vis.
**1987**, 1, 167–187. [Google Scholar] [CrossRef] - Van Vliet, L.J.; Young, I.T.; Verbeek, P.W. Recursive Gaussian derivative filters. In Proceedings of the Fourteenth International Conference on Pattern Recognition (Cat. No. 98EX170), Brisbane, QLD, Australia, 20 August 1998; Volume 1, pp. 509–514. [Google Scholar]
- Pal, N.R.; Pal, S.K. Entropic thresholding. Signal Process.
**1989**, 16, 97–108. [Google Scholar] [CrossRef] - Pun, T. Entropic thresholding, a new approach. Comput. Graph. Image Process.
**1981**, 16, 210–239. [Google Scholar] [CrossRef] [Green Version] - Shiozaki, A. Edge extraction using entropy operator. Comput. Vision, Graph. Image Process.
**1986**, 36, 1–9. [Google Scholar] [CrossRef] - Gull, S.; Skilling, J. Maximum entropy method in image processing. IEE Proc. Commun. Radar Signal Process.
**1984**, 131, 646–659. [Google Scholar] [CrossRef] - El-Sayed, M.A. A new algorithm based entropic threshold for edge detection in images. arXiv
**2012**, arXiv:1211.2500. [Google Scholar] - Kapur, J.N.; Sahoo, P.K.; Wong, A.K. A new method for gray-level picture thresholding using the entropy of the histogram. Comput. Vision Graph. Image Process.
**1985**, 29, 273–285. [Google Scholar] [CrossRef] - Yang, C. A new operator for detecting edges in images based on modified Tsallis entropy. In Proceedings of the 2011 International Conference on Consumer Electronics, Communications and Networks (CECNet), Xianning, China, 16–18 April 2011; pp. 4671–4674. [Google Scholar] [CrossRef]
- Chang, C.I.; Chen, K.; Wang, J.; Althouse, M.L. A relative entropy-based approach to image thresholding. Pattern Recognit.
**1994**, 27, 1275–1289. [Google Scholar] [CrossRef] - Medina-Carnicer, R.; Madrid-Cuevas, F.J.; Fernández-García, N.; Carmona-Poyato, A. Evaluation of global thresholding techniques in non-contextual edge detection. Pattern Recognit. Lett.
**2005**, 26, 1423–1434. [Google Scholar] [CrossRef] - Shannon, C.E. A mathematical theory of communication. ACM SIGMOBILE Mob. Comput. Commun. Rev.
**2001**, 5, 3–55. [Google Scholar] [CrossRef] - Singh, B.; Singh, A.P. Edge Detection in Gray Level Images based on the Shannon Entropy. J. Comput. Sci.
**2008**, 3, 186–191. [Google Scholar] [CrossRef] [Green Version] - Li, J. A Wavelet Approach to Edge Detection. Master’s Thesis, Sam Houston State University, Huntsville, TX, USA, 2003. [Google Scholar]
- Liu, W.; Ma, Z. Wavelet image threshold denoising based on edge detection. In Proceedings of the IMACS Multiconference on Computational Engineering in Systems Applications, Beijing, China, 4–6 October 2006; Volume 1, pp. 72–78. [Google Scholar]
- Schmeelk, J. Wavelet transforms and edge detectors on digital images. Math. Comput. Model.
**2005**, 41, 1469–1478. [Google Scholar] [CrossRef] - Shih, M.Y.; Tseng, D.C. A wavelet-based multiresolution edge detection and tracking. Image Vis. Comput.
**2005**, 23, 441–451. [Google Scholar] [CrossRef] - Siddique, J.; Barner, K.E. Wavelet-based multiresolution edge detection utilizing gray level edge maps. In Proceedings of the ICIP 98, 1998 International Conference on Image Processing, Chicago, IL, USA, 7 October 1998; Volume 2, pp. 550–554. [Google Scholar]
- Zhang, L.; Bao, P. Edge detection by scale multiplication in wavelet domain. Pattern Recognit. Lett.
**2002**, 23, 1771–1784. [Google Scholar] [CrossRef] - Jiang, W.; Lam, K.M.; Shen, T.Z. Efficient edge detection using simplified Gabor wavelets. IEEE Trans. Syst. Man Cybern. Part B Cybern.
**2009**, 39, 1036–1047. [Google Scholar] [CrossRef] - Hao, Y.; Changshun, L.; Lei, P. An improved method of image edge detection based on wavelet transform. In Proceedings of the 2011 IEEE International Conference on Computer Science and Automation Engineering (CSAE), Shanghai, China, 10–12 June 2011; Volume 3, pp. 678–681. [Google Scholar]
- Elaraby, A.E.A.; Owny, E.; Ahmed, H.B.; Heshmat, M.; Hassaballah, M.; Rardy, A.S.A. A Novel Algorithm for Edge Detection of Noisy Medical Images. Int. J. Signal Process. Image Process. Pattern Recognit.
**2013**, 6, 365–374. [Google Scholar] [CrossRef] [Green Version] - Xishan, T. A Novel Image Edge Detection Algorithm based on Prewitt Operator and Wavelet Transform. Int. J. Adv. Comput. Technol.
**2012**, 4, 73–82. [Google Scholar] - Srivastava, G.; Verma, R.; Mahrishi, R.; Rajesh, S. A novel wavelet edge detection algorithm for noisy images. In Proceedings of the ICUMT ’09, International Conference on Ultra Modern Telecommunications Workshops, St. Petersburg, Russia, 12–14 October 2009; pp. 1–8. [Google Scholar] [CrossRef]
- Wu, Y.; He, Y.; Cai, H. Optimal threshold selection algorithm in edge detection based on wavelet transform. Image Vis. Comput.
**2005**, 23, 1159–1169. [Google Scholar] [CrossRef] - Vetterli, M.; Kovacevic, J. Wavelets and Subband Coding; Prentice-Hall: Englewood Cliffs, NJ, USA, 1995. [Google Scholar]
- Bertasius, G.; Shi, J.; Torresani, L. Deepedge: A multi-scale bifurcated deep network for top-down contour detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Boston, MA, USA, 8–10 June 2015; pp. 4380–4389. [Google Scholar]
- Shen, W.; Wang, X.; Wang, Y.; Bai, X.; Zhang, Z. Deepcontour: A deep convolutional feature learned by positive-sharing loss for contour detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Boston, MA, USA, 8–10 June 2015; pp. 3982–3991. [Google Scholar]
- He, J.; Zhang, S.; Yang, M.; Shan, Y.; Huang, T. Bi-directional cascade network for perceptual edge detection. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, Long Beach, CA, USA, 16–20 June 2019; pp. 3828–3837. [Google Scholar]
- Gonzalez, R.C.; Woods, R.E. Digital Image Processing, 3rd ed.; Prentice-Hall, Inc.: Upper Saddle River, NJ, USA, 2006. [Google Scholar]
- Graps, A. An introduction to wavelets. IEEE Comput. Sci. Eng.
**1995**, 2, 50–61. [Google Scholar] [CrossRef] - Chakrabarti, C.; Vishwanath, M. Efficient realizations of the discrete and continuous wavelet transforms: From single chip implementations to mappings on SIMD array computers. IEEE Trans. Signal Process.
**1995**, 43, 759–771. [Google Scholar] [CrossRef] - Nelson, M.; Gailly, J.L. The Data Compression Book; M&T Books: New York, NY, USA, 1996; Volume 2. [Google Scholar]
- Wei, D.; Rajashekar, U.; Bovik, A.C. 3.4—Wavelet Denoising for Image Enhancement. In Handbook of Image and Video Processing, 2nd ed.; Bovik, A., Ed.; Communications, Networking and Multimedia; Academic Press: Burlington, MA, USA, 2005; pp. 157–165. [Google Scholar] [CrossRef]
- Csurka, G.; Larlus, D.; Perronnin, F.; Meylan, F. What is a good evaluation measure for semantic segmentation? Bmvc
**2013**, 27, 10–5244. [Google Scholar] - Abdou, I.E.; Pratt, W.K. Quantitative design and evaluation of enhancement/thresholding edge detectors. Proc. IEEE
**1979**, 67, 753–763. [Google Scholar] [CrossRef]

**Figure 4.**Global structure measurement using Shannon entropy. Both images are 512 × 512 pixels in size.

**Figure 5.**Entropy per decomposition level where vertical and horizontal components are added and normalised using the test image from Figure 3.

**Figure 7.**Shannon entropy curve for threshold selection and Coiflet wavelet. Output images are 258 × 258 pixels in size.

**Figure 9.**The original image is shown in the first column of the image grid. Other columns correspond to different edge detection algorithms, and the algorithm name is mentioned in the bottom of the column. For the LRWEEDA algorithm, the following parameters were used (from the first row to the last row): (1) “Lena” image: LRWEEDA (output image size of 256 × 256 pixels) using Haar, j = 1, $\mathsf{\Lambda}$ = 0.480; (2) “House” image: LRWEEDA (258 × 258 pixels) using Coiflet, j = 1, $\mathsf{\Lambda}$ = 0.635; (3) “Mandril” image: LRWEEDA (258 × 258 pixels) using Coiflet, j = 1, $\mathsf{\Lambda}$ = 0.520; (4) “Peppers” image: LRWEEDA (131 × 131 pixels) using Coiflet, j = 2, $\mathsf{\Lambda}$ = 0.480; (5) “Cameraman” image: LRWEEDA (131 × 131 pixels) using Coiflet, j = 2, $\mathsf{\Lambda}$ = 0.530; and (6) “Jetplane” image: LRWEEDA (256 × 256 pixels) using Haar, j = 1, $\mathsf{\Lambda}$ = 0.555.

**Figure 10.**Average computation time for different edge detection algorithms. A dataset of ten images was used for the analysis.

**Figure 11.**The original image is shown in the first column of the image grid. Other columns correspond to different noise levels. The first, third, fifth, and seventh rows correspond to Canny processed images (grouped in green) and the second, fourth, sixth and eighth rows correspond to LRWEEDA processed images (grouped in red). The corresponding LRWEEDA parameters are shown below the images of red groups. The LRWEEDA processed output images sizes are 483 × 483 pixels when $j=2$; 244 × 244 pixels when $j=3$; 124 × 124 pixels when $j=4$; 64 × 64 pixels when $j=5$ and 34 × 34 pixels when $j=6$.

**Figure 12.**A synthetic image was used to calculate the Dice coefficient of the proposed algorithm. The original synthetic image (270 × 238 pixels) and the ground truth edges (270 × 238 pixels) are shown in (

**a**). The edges obtained by LRWEEDA, Canny, Prewitt, Sobel, Roberts and Zero cross algorithms are shown from left to right in (

**b**). In (

**b**), all the images are 270 × 238 pixels. The edges were compared with the ground truth in (

**c**). The edges calculated by each algorithm and the ground truth edges are shown in green and purple, respectively. The overlap between the calculated edges and the ground truth is shown in white. The Dice coefficient for each algorithm is shown under each image. The red color box in (

**a**) is enlarged for the analysis purpose in (

**d**).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tao, Y.; Scully, T.; Perera, A.G.; Lambert, A.; Chahl, J.
A Low Redundancy Wavelet Entropy Edge Detection Algorithm. *J. Imaging* **2021**, *7*, 188.
https://doi.org/10.3390/jimaging7090188

**AMA Style**

Tao Y, Scully T, Perera AG, Lambert A, Chahl J.
A Low Redundancy Wavelet Entropy Edge Detection Algorithm. *Journal of Imaging*. 2021; 7(9):188.
https://doi.org/10.3390/jimaging7090188

**Chicago/Turabian Style**

Tao, Yiting, Thomas Scully, Asanka G. Perera, Andrew Lambert, and Javaan Chahl.
2021. "A Low Redundancy Wavelet Entropy Edge Detection Algorithm" *Journal of Imaging* 7, no. 9: 188.
https://doi.org/10.3390/jimaging7090188