Skip to main content
SpringerLink
Log in

Fractional Fourier Transform and Fractional-Order Calculus-Based Image Edge Detection

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

Edge detection is an integral component of image processing to enhance the clarity of edges in an image. Detection of edges for an image may help for image segmentation, data compression, and image reconstruction. Edges of an image are considered a type of crucial information that can be extracted by applying detectors with different methodologies. Its main purpose is to simplify the image data in order to minimize the amount of data to be processed. There exist many rich classical edge detection techniques which make use of integer-order differentiation operators and can function in both spatial and frequency domains. In the case of integer-order differentiation operators, the gradient operator is identified by order ‘one’ and the Laplacian by order ‘two.’ This paper demonstrates a new kind of edge detector based on the ‘fractional’ (‘non-integer’)-order differentiation operation and through the usage of the ‘fractional Fourier transformation’ tool, so as to perform it in the fractional Fourier frequency domain, known as the edge detection based on fractional signal processing approach. It is shown through computer simulations that this approach can detect the edges precisely and efficiently. Finally, the performance of the proposed methodology is illustrated from the quantitative aspects of mean square error and peak signal-to-noise ratio through simulations. The experiments show that, for any grayscale image, this method can obtain better edge detection performance to satisfy human visual sense. Moreover, comparisons are also provided to prove that the proposed method outperforms the classical edge detection operators, interpreted in terms of robustness to noise.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. L.B. Almeida, The fractional Fourier transform and time–frequency representation. IEEE Trans. Signal Process. 42(11), 3084–3091 (1994)

    Article  Google Scholar 

  2. H. Brunner, L. Ling, M. Yamamoto, Numerical simulations of 2D fractional subdiffusion problems. J. Comput. Phys. 229(18), 6613–6622 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. X.H. Chen, X.D. Fei, Improved edge detection algorithm based on fractional differential approach. In: Proceedings of the International Conference on Image, Vision, and Computing, Singapore 50, (2012). doi:10.7763/IPCSIT.2012.V50.48

  4. Q. Chen, Z. Song, J. Dong, Z. Huang, Y. Hua, S. Yan, Contextualizing object detection and classification. IEEE Trans. Pattern Anal. Mach. Intell. 37(1), 13–27 (2015)

    Article  Google Scholar 

  5. K. Diethelm, The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics. (Springer, Berlin, 2010)

  6. M.E. Farmer, A.K. Jain, A wrapper-based approach to image segmentation and classification. IEEE Trans. Image Process. 14(2), 2060–2072 (2005)

    Article  Google Scholar 

  7. Z. Gan, H. Yang, Texture enhancement through multiscale mask based on RL fractional differential. Proc. Int. Conf. Inf. Netw. Autom. Kunming China 1, 333–337 (2010)

    Google Scholar 

  8. C. Gao, J. Zhou, W. Zhang, Edge detection based on the Newton interpolation’s fractional differentiation. Int. Arab J. Inf. Technol. 11(3), 223–228 (2014)

    Google Scholar 

  9. A. Ghaffari, E. Fatemizadeh, RISM: single-modal image registration via rank-induced similarity measure. IEEE Trans. Image Process. 24(12), 5567–5580 (2015)

    Article  MathSciNet  Google Scholar 

  10. R.C. Gonzalez, R.E. Woods, Digital Image Processing (Prentice-Hall, Englewood Cliffs, 2008)

    Google Scholar 

  11. F. He, S. Wang, Beyond \(\chi ^{2}\) difference: learning optimal metric for boundary detection. IEEE Signal Process. Lett. 22(1), 40–44 (2015)

    Article  Google Scholar 

  12. Z.J. Hou, G.W. Wei, A new approach to edge detection. Pattern Recogn. 35, 1559–15970 (2002)

    Article  MATH  Google Scholar 

  13. A.K. Jain, Fundamental of Digital Image Processing (Prentice-Hall, Upper Saddle River, 1989)

    MATH  Google Scholar 

  14. H.A. Jalab, R.W. Ibrahim, Denoising algorithm based on generalized fractional integral operator with two parameters. Discrete Dyn. Nat. Soc. 2012, 529849 (2012). doi:10.1155/2012/529849

    Article  MathSciNet  MATH  Google Scholar 

  15. H.A. Jalab, R.W. Ibrahim, Fractional masks based on generalized fractional differential operator for image denoising. Int. J. Comput. Inf. Syst. Control Eng. 7(2), 169–174 (2013)

    Google Scholar 

  16. Z.W. Ju, J.Z. Chen, J.L. Zhou, Image segmentation based on edge detection using K-means and an improved ant colony optimization. IEEE Int. Conf. Mach. Learn. Cybern. pp. 297–303 (2013)

  17. H.B. Kekre, S.D. Thepade, A. Maloo, Image retrieval using fractional coefficients of transformed image using DCT and Walsh transform. Int. J. Eng. Sci. Technol. 2(4), 362–371 (2010)

    Google Scholar 

  18. S. Kumar, K. Singh, R. Saxena, Closed-form analytical expression of fractional order differentiation in fractional Fourier transform domain. Circuits Syst. Signal Process. 32(4), 1875–1889 (2013)

    Article  MathSciNet  Google Scholar 

  19. S. Larnier, R. Mecca, Fractional–order diffusion for image reconstruction, in The 2012 IEEE International Conference on Acoustics, Speech and Signal Processing, Kyoto (2012), pp. 1057–1060

  20. K.N. Le, A mathematical approach to edge detection in hyperbolic-distributed and Gaussian-distributed pixel-intensity images using hyperbolic and Gaussian masks. Digit. Signal Process. 21(1), 162–181 (2011)

    Article  Google Scholar 

  21. C. Lee, W. Chao, S. Lee, J. Hone, A. Molnar, S.H. Hong, A low-power edge detection image sensor based on parallel digital pulse computation. IEEE Trans. Circuits Syst. II Exp. Br. 62(11), 1043–1047 (2015)

    Article  Google Scholar 

  22. M. Lehtomäki, A. Jaakkola, J. Hyyppä, J. Lampinen, H. Kaartinen, A. Kukko, E. Puttonen, H. Hyyppä, Object classification and recognition from mobile laser scanning point clouds in a road environment. IEEE Trans. Geosci. Remote Sens. 54(2), 1226–1239 (2016)

    Article  Google Scholar 

  23. C.Q. Li, H. Guo, Z.X. Qiong, A fractional differential approach to low contrast image enhancement. Int. J. Knowl. Lang. Process. 2(2), 20–29 (2012)

    Google Scholar 

  24. L. Ling, H. Peikang, W. Xiaohu, P. Xudong, Image edge detection based on beamlet transform. J. Syst. Eng. Electron. 20(1), 1–5 (2009)

    Google Scholar 

  25. D.Y. Liu, O. Gibaru, W. Perruquetti, T.M.L. Kirati, Fractional order differentiation by integration and error analysis in noisy environment. IEEE Transactions on Automatic Control 60(11), 2945–2960 (2015)

    Article  MathSciNet  Google Scholar 

  26. C. Lopez-Molina, H. Bustince, B. De Baets, Separability criteria for the evaluation of boundary detection benchmarks. IEEE Trans. Image Process. 25(3), 1047–1055 (2016)

    Article  MathSciNet  Google Scholar 

  27. C.L. MacDonald, N. Bhattacharya, B.P. Sprouse, G.A. Silva, Efficient computation of the Grünwald–Letnikov fractional diffusion derivative using adaptive time step memory. J. Comput. Phys. 297, 221–236 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. J.A.T. Machado, M.F. Silva, R.S. Barbosa, I.S. Jesus, C.M. Reis, M.G. Marcos, A.F. Galhano, Some applications of fractional calculus in engineering. Math. Probl. Eng. Article ID 639801 (2010). doi:10.1155/2010/639801

  29. S.K. Maji, H.M. Yahia, H. Badri, Reconstructing an image from its edge representation. Digit. Signal Proc. 23(6), 1867–1876 (2013)

    Article  Google Scholar 

  30. R. Marazzat, A.C Sparavigna, Astronomical image processing based on fractional calculus: the AstroFracTool. Instrum. Methods Astrophys. (2009) arXiv:0910.4637

  31. K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Academic, New York, 1974)

    MATH  Google Scholar 

  32. M.D. Ortigueira, Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering (Springer, 2011)

  33. H.M. Ozaktas, O. Arikan, M.A. Kutay, G. Bozdaği, Digital computation of the fractional Fourier transform. IEEE Trans. Signal Process. 44(9), 2141–2150 (1996)

    Article  Google Scholar 

  34. H.M. Ozaktas, M.A. Kutay, Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001)

    Google Scholar 

  35. S.C. Pei, J.J. Ding, Closed-form discrete fractional and affine Fourier transforms. IEEE Trans. Signal Process. 48(5), 1338–1353 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. S.C. Pei, M.H. Yeh, Two dimensional discrete fractional Fourier transform. Signal Process. 67(1), 99–108 (1998)

    Article  MATH  Google Scholar 

  37. Y.F. Pu, J.L. Zhou, X. Yuan, Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement. IEEE Trans. Image Process. 19(2), 491–511 (2010)

    Article  MathSciNet  Google Scholar 

  38. A. Sahin, H.M. Ozaktas, D. Mendlovic, Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters. Appl. Opt. 37(11), 2130–2141 (1998)

    Article  Google Scholar 

  39. A. Sahin, M.A. Kutay, H.M. Ozaktas, Nonseparable two-dimensional fractional Fourier transform. Appl. Opt. 37(23), 5444–5453 (1998)

    Article  Google Scholar 

  40. T. Sandev, R. Metzler, Ž. Tomovski, Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative. J. Phys. A Math. Theor. 44(25), 1–21 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. K. Singh, R. Saxena, S. Kumar, Caputo-based fractional derivative in fractional Fourier transform domain. IEEE J. Emerg. Sel. Topics Circuits Syst. 3(3), 330–337 (2013)

    Article  Google Scholar 

  42. A.C. Sparavigna, Using fractional differentiation in astronomy. Comput. Vis. Pattern Recogn. (2010) arXiv:0910.2381

  43. C.C. Tseng, Design of variable and adaptive fractional order FIR differentiators. Signal Process. 86(10), 2554–2566 (2006)

    Article  MATH  Google Scholar 

  44. C. Wang, L. Lan, S. Zhou, Grünwald–Letnikov based adaptive fractional differential algorithm on image texture enhancing. J. Comput. Inf. Syst. 9(2), 445–454 (2013)

    Google Scholar 

  45. J. Wang, Y. Ye, X. Pan, X. Gao, C. Zhuang, Fractional zero-phase filtering based on the Riemann–Liouville integral. Signal Process. 98(5), 150–157 (2014)

    Article  Google Scholar 

  46. J. Wang, Y. Ye, Y. Gao, S. Qian, X. Gao, Fractional compound integral with application to ECG signal denoising. Circuits Syst. Signal Process. 34(6), 1915–1930 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. J. Wang, Y. Ye, X. Pan, X. Gao, Parallel-type fractional zero-phase filtering for ECG signal denoising. Biomed. Signal Process. Control 18, 36–41 (2015)

    Article  Google Scholar 

  48. J. Wang, Y. Ye, X. Gao, Fractional 90-degree phase-shift filtering based on the double-sided Grunwald–Letnikov differintegrator. IET Signal Proc. 9(4), 328–334 (2015)

    Article  Google Scholar 

  49. D. Wei, Y.M. Li, Generalized sampling expansions with multiple sampling rates for lowpass and bandpass signals in the fractional Fourier transform domain. IEEE Trans. Signal Process. (2016). doi:10.1109/TSP.2016.2560148

  50. D. Wei, Novel convolution and correlation theorems for the fractional Fourier transform. Optik 127(7), 3669–3675 (2016)

    Article  Google Scholar 

  51. D. Wei, Y. Li, Sampling reconstruction of N-dimensional bandlimited images after multilinear filtering in fractional Fourier domain. Opt. Commun. 295, 26–35 (2013)

    Article  Google Scholar 

  52. D. Wei, Y.M. Li, Novel tridiagonal commuting matrices for Types I, IV, V, VIII DCT and DST matrices. IEEE Signal Process. Lett. 21(4), 483–487 (2014)

    Article  Google Scholar 

  53. J. Xu, L. Wang, Z. Shi, A switching weighted vector median filter based on edge detection. Signal Process. 98, 359–369 (2014)

    Article  Google Scholar 

  54. L. Yi, G. Zhang, Z. Wu, A scale-synthesis method for high spatial resolution remote sensing image segmentation. IEEE Trans. Geosci. Remote Sens. 50(10), 4062–4070 (2012)

    Article  Google Scholar 

  55. Z.J. Yu, C. Yan, H.X. Xiang, Edge detection of images based on improved Sobel operator and genetic algorithms. IEEE International Conference on Image Analysis and Signal Processing. pp. 31–35 (2009)

  56. A.I. Zayed, A convolution and product theorem for the fractional Fourier transform. IEEE Signal Process. Lett. 5(4), 101–103 (1998)

    Article  Google Scholar 

  57. D. Zosso, X. Bresson, J.P. Thiran, Geodesic active fields—a geometric framework for image registration. IEEE Trans. Image Process. 20(5), 1300–1312 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank the anonymous reviewers for their rigorous reviews, constructive comments, and valuable suggestions which greatly improved the quality and clarity of manuscript presentation.

Author information

Authors and Affiliations

  1. Department of Electronics and Communication Engineering, Thapar University, Patiala, Punjab, 147004, India

    Sanjay Kumar & Kulbir Singh

  2. Jaypee University Anoopshahr, Aligarh Road, Bulandshahr, Uttar Pradesh, 203390, India

    Rajiv Saxena

Authors
  1. Sanjay Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rajiv Saxena
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kulbir Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sanjay Kumar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, S., Saxena, R. & Singh, K. Fractional Fourier Transform and Fractional-Order Calculus-Based Image Edge Detection. Circuits Syst Signal Process 36, 1493–1513 (2017). https://doi.org/10.1007/s00034-016-0364-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-016-0364-x

Keywords

Search

Navigation