Detection of Edge with the Aid of Mollification Based on Wavelets

Abstract

In preceding papers, the present authors proposed the application of the mollification based on wavelets to the calculation of the fractional derivative (fD) or the derivative of a function involving noise. We study here the application of that method to the detection of edge of a function. Mathieu et al. proposed the CRONE detector for a detection of an edge of an image. For a function without noise, we note that the CRONE detector is expressed as the Riesz fractional derivative (fD) of the derivative. We study here the application of the mollification to the calculation of the Riesz fD of the derivative for a data involving noise, and compare the results with the results obtained by our method of applying simple derivative to mollified data.

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Morita, T. and Sato, K. (2014) Detection of Edge with the Aid of Mollification Based on Wavelets. Applied Mathematics, 5, 2849-2861. doi: 10.4236/am.2014.518271.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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http://dx.doi.org/10.1002/9781118033210
[2] Morita, T. and Sato, K. (2011) Mollification of Fractional Derivatives Using Rapidly Decaying Harmonic Wavelet. Fractional Calculus and Applied Analysis, 14, 284-300.
http://dx.doi.org/10.2478/s13540-011-0017-5
[3] Morita, T. and Sato, K. (2011) Mollification of the Gibbs Phenomenon Using Orthogonal Wavelets. Proceedings of the Multimedia Technology (ICMT), 2011 International Conference, Hangzhou, 26-28 July 2011, 6441-6444.
http://dx.doi.org/10.1109/ICMT.2011.6002341
[4] Morita, T. and Sato, K. (2013) Mollification Based on Wavelets. Axioms, 2, 67-84.
http://dx.doi.org/10.3390/axioms2020067
[5] Mathieu, B., Melchior, P., Oustaloup, A. and Ceyrat, Ch. (2003) Fractional Differentiation for Edge Detection. Signal Processing, 83, 2421-2432.
http://dx.doi.org/10.1016/S0165-1684(03)00194-4
[6] Oustaloup, A. (1995) La dérivation non entière théorie synthèse et applications. Hermes, Paris.
[7] Chui, C.K. (1992) An Introduction to Wavelets. Academic Press, Inc., New York.
[8] Hamming, R.W. (1998) Digital Filters. Dover Publications Inc., Mineola, New York.
[9] Morita, T. and Sato, K. Asymptotics of Fractional Derivatives with Application to Confluent Hyper-Geometric Function. (in preparation)
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[12] Ortigueira, M.D. (2011) Fractional Calculus for Scientists and Engineers. Springer, Dordrecht.
http://dx.doi.org/10.1007/978-94-007-0747-4
[13] Murio, D.A. (1993) The Mollification Method and the Numerical Solution of Ill-Posed Problems. John Wiley, New York.
http://dx.doi.org/10.1002/9781118033210
[14] Morita, T. and Sato, K. (2011) Mollification of Fractional Derivatives Using Rapidly Decaying Harmonic Wavelet. Fractional Calculus and Applied Analysis, 14, 284-300.
http://dx.doi.org/10.2478/s13540-011-0017-5
[15] Morita, T. and Sato, K. (2011) Mollification of the Gibbs Phenomenon Using Orthogonal Wavelets. Proceedings of the Multimedia Technology (ICMT), 2011 International Conference, Hangzhou, 26-28 July 2011, 6441-6444.
http://dx.doi.org/10.1109/ICMT.2011.6002341
[16] Morita, T. and Sato, K. (2013) Mollification Based on Wavelets. Axioms, 2, 67-84.
http://dx.doi.org/10.3390/axioms2020067
[17] Mathieu, B., Melchior, P., Oustaloup, A. and Ceyrat, Ch. (2003) Fractional Differentiation for Edge Detection. Signal Processing, 83, 2421-2432.
http://dx.doi.org/10.1016/S0165-1684(03)00194-4
[18] Oustaloup, A. (1995) La dérivation non entière théorie synthèse et applications. Hermes, Paris.
[19] Chui, C.K. (1992) An Introduction to Wavelets. Academic Press, Inc., New York.
[20] Hamming, R.W. (1998) Digital Filters. Dover Publications Inc., Mineola, New York.
[21] Morita, T. and Sato, K. Asymptotics of Fractional Derivatives with Application to Confluent Hyper-Geometric Function. (in preparation)
[22] Butzer, M.L. (1971) Fourier Analysis and Approximation, Vol. I, One-Dimensional Theory. Birkhäuser Verlag, Basel.
[23] Rubin, B. (1996) Fractional Integrals and Potentials. Addison, Wesley and Longman, Edinburgh Gate, Harlow.
[24] Ortigueira, M.D. (2011) Fractional Calculus for Scientists and Engineers. Springer, Dordrecht.
http://dx.doi.org/10.1007/978-94-007-0747-4
[25] Murio, D.A. (1993) The Mollification Method and the Numerical Solution of Ill-Posed Problems. John Wiley, New York.
http://dx.doi.org/10.1002/9781118033210
[26] Morita, T. and Sato, K. (2011) Mollification of Fractional Derivatives Using Rapidly Decaying Harmonic Wavelet. Fractional Calculus and Applied Analysis, 14, 284-300.
http://dx.doi.org/10.2478/s13540-011-0017-5
[27] Morita, T. and Sato, K. (2011) Mollification of the Gibbs Phenomenon Using Orthogonal Wavelets. Proceedings of the Multimedia Technology (ICMT), 2011 International Conference, Hangzhou, 26-28 July 2011, 6441-6444.
http://dx.doi.org/10.1109/ICMT.2011.6002341
[28] Morita, T. and Sato, K. (2013) Mollification Based on Wavelets. Axioms, 2, 67-84.
http://dx.doi.org/10.3390/axioms2020067
[29] Mathieu, B., Melchior, P., Oustaloup, A. and Ceyrat, Ch. (2003) Fractional Differentiation for Edge Detection. Signal Processing, 83, 2421-2432.
http://dx.doi.org/10.1016/S0165-1684(03)00194-4
[30] Oustaloup, A. (1995) La dérivation non entière théorie synthèse et applications. Hermes, Paris.
[31] Chui, C.K. (1992) An Introduction to Wavelets. Academic Press, Inc., New York.
[32] Hamming, R.W. (1998) Digital Filters. Dover Publications Inc., Mineola, New York.
[33] Morita, T. and Sato, K. Asymptotics of Fractional Derivatives with Application to Confluent Hyper-Geometric Function. (in preparation)
[34] Butzer, M.L. (1971) Fourier Analysis and Approximation, Vol. I, One-Dimensional Theory. Birkhäuser Verlag, Basel.
[35] Rubin, B. (1996) Fractional Integrals and Potentials. Addison, Wesley and Longman, Edinburgh Gate, Harlow.
[36] Ortigueira, M.D. (2011) Fractional Calculus for Scientists and Engineers. Springer, Dordrecht.
http://dx.doi.org/10.1007/978-94-007-0747-4
[37] Murio, D.A. (1993) The Mollification Method and the Numerical Solution of Ill-Posed Problems. John Wiley, New York.
http://dx.doi.org/10.1002/9781118033210
[38] Morita, T. and Sato, K. (2011) Mollification of Fractional Derivatives Using Rapidly Decaying Harmonic Wavelet. Fractional Calculus and Applied Analysis, 14, 284-300.
http://dx.doi.org/10.2478/s13540-011-0017-5
[39] Morita, T. and Sato, K. (2011) Mollification of the Gibbs Phenomenon Using Orthogonal Wavelets. Proceedings of the Multimedia Technology (ICMT), 2011 International Conference, Hangzhou, 26-28 July 2011, 6441-6444.
http://dx.doi.org/10.1109/ICMT.2011.6002341
[40] Morita, T. and Sato, K. (2013) Mollification Based on Wavelets. Axioms, 2, 67-84.
http://dx.doi.org/10.3390/axioms2020067
[41] Mathieu, B., Melchior, P., Oustaloup, A. and Ceyrat, Ch. (2003) Fractional Differentiation for Edge Detection. Signal Processing, 83, 2421-2432.
http://dx.doi.org/10.1016/S0165-1684(03)00194-4
[42] Oustaloup, A. (1995) La dérivation non entière théorie synthèse et applications. Hermes, Paris.
[43] Chui, C.K. (1992) An Introduction to Wavelets. Academic Press, Inc., New York.
[44] Hamming, R.W. (1998) Digital Filters. Dover Publications Inc., Mineola, New York.
[45] Morita, T. and Sato, K. Asymptotics of Fractional Derivatives with Application to Confluent Hyper-Geometric Function. (in preparation)
[46] Butzer, M.L. (1971) Fourier Analysis and Approximation, Vol. I, One-Dimensional Theory. Birkhäuser Verlag, Basel.
[47] Rubin, B. (1996) Fractional Integrals and Potentials. Addison, Wesley and Longman, Edinburgh Gate, Harlow.
[48] Ortigueira, M.D. (2011) Fractional Calculus for Scientists and Engineers. Springer, Dordrecht.
http://dx.doi.org/10.1007/978-94-007-0747-4
[49] Murio, D.A. (1993) The Mollification Method and the Numerical Solution of Ill-Posed Problems. John Wiley, New York.
http://dx.doi.org/10.1002/9781118033210
[50] Morita, T. and Sato, K. (2011) Mollification of Fractional Derivatives Using Rapidly Decaying Harmonic Wavelet. Fractional Calculus and Applied Analysis, 14, 284-300.
http://dx.doi.org/10.2478/s13540-011-0017-5
[51] Morita, T. and Sato, K. (2011) Mollification of the Gibbs Phenomenon Using Orthogonal Wavelets. Proceedings of the Multimedia Technology (ICMT), 2011 International Conference, Hangzhou, 26-28 July 2011, 6441-6444.
http://dx.doi.org/10.1109/ICMT.2011.6002341
[52] Morita, T. and Sato, K. (2013) Mollification Based on Wavelets. Axioms, 2, 67-84.
http://dx.doi.org/10.3390/axioms2020067
[53] Mathieu, B., Melchior, P., Oustaloup, A. and Ceyrat, Ch. (2003) Fractional Differentiation for Edge Detection. Signal Processing, 83, 2421-2432.
http://dx.doi.org/10.1016/S0165-1684(03)00194-4
[54] Oustaloup, A. (1995) La dérivation non entière théorie synthèse et applications. Hermes, Paris.
[55] Chui, C.K. (1992) An Introduction to Wavelets. Academic Press, Inc., New York.
[56] Hamming, R.W. (1998) Digital Filters. Dover Publications Inc., Mineola, New York.
[57] Morita, T. and Sato, K. Asymptotics of Fractional Derivatives with Application to Confluent Hyper-Geometric Function. (in preparation)
[58] Butzer, M.L. (1971) Fourier Analysis and Approximation, Vol. I, One-Dimensional Theory. Birkhäuser Verlag, Basel.
[59] Rubin, B. (1996) Fractional Integrals and Potentials. Addison, Wesley and Longman, Edinburgh Gate, Harlow.
[60] Ortigueira, M.D. (2011) Fractional Calculus for Scientists and Engineers. Springer, Dordrecht.
http://dx.doi.org/10.1007/978-94-007-0747-4

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