Applications of the XFT

  • Rafael G. Campos
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


The aim of this chapter is to show some uses and applications of the XFT (as an ordinary Fourier transform) ranging from digital steganography, to nonlinear partial differential equations and fractional differentiation. Most of the given examples are new applications, particularly the ones related to fractional differential problems. A new approach for computing the inversion of convolution operators is given in the last section.


XFT Translations Autostereograms Steganography Brain signals Edge detection Boundary/initial value problems Solitons KdV equations Burgers’ equation Fractional differentiation/integration Convolution 


  1. 1.
    R.P. Agarwal. Boundary Value Problems for Higher Order Differential Equations. World Scientific Publishing, Singapore, 1986.CrossRefzbMATHGoogle Scholar
  2. 10.
    E.R. Benton and G.W. Platzman. A table of solutions of the one-dimensional Burgers equation. Quart. Appl. Math., 30:195–212, 1972.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 11.
    A. Bergua and W. Skrandies. An early antecedent to modern random dot stereograms “the secret stereoscopic writing” of Ramón y Cajal. Int. J. Psychophysiology, 36:69–72, 2000.CrossRefGoogle Scholar
  4. 18.
    M. Bruschi, R.G. Campos, and E. Pace. On a method for computing eigenvalues and eigenfunctions of linear differential operators. Nuovo Cimento B, 105:131–163, 1990.MathSciNetCrossRefGoogle Scholar
  5. 20.
    J. Burgers. A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech., 1:171–199, 1948.MathSciNetCrossRefGoogle Scholar
  6. 21.
    J.C. Butcher. Numerical Methods for Ordinary Differential Equations. John Wiley and Sons, Inc., New Jersey, 2008.CrossRefzbMATHGoogle Scholar
  7. 22.
    D. Cafagna. Fractional calculus: A mathematical tool from the past for present engineers. IEEE Ind. Elect. Mag., 2:35–40, 2007.CrossRefGoogle Scholar
  8. 29.
    R.G. Campos. Time-reversal breaking in fractional differential problems. Eur. Phys. J. Plus, 130:121–125, 2015.CrossRefGoogle Scholar
  9. 35.
    R.G. Campos and C. Meneses. Differentiation matrices for meromorphic functions. Bol. Soc. Mat. Mexicana, 12:121–132, 2006.MathSciNetzbMATHGoogle Scholar
  10. 37.
    R.G. Campos and F.D. Mota. An implementation of the collocation method for initial value problems. Int. J. Model. Simul. Sci. Comput., 4:1350006, 2013.CrossRefGoogle Scholar
  11. 39.
    R.G. Campos and R.G. Ruiz. Fast integration of one-dimensional boundary value problems. Int. J. Mod. Phys. C, 24, 2013.MathSciNetCrossRefGoogle Scholar
  12. 45.
    L. Debnath. Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci., 54:3413–3442, 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 47.
    K. Diethelma, J.M. Ford, N.J. Ford, and M. Weilbeer. Pitfalls in fast numerical solvers for fractional differential equations. J. Comput. Appl. Math., 186:482–503, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 49.
    Ch. Epstein and J. Schotland. The bad truth about laplace’s transform. SIAM Review, 50:504–520, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 55.
    H.G. Feichtinger and T. Strohmer, editors. Advances is Gabor Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Basel, 2003.Google Scholar
  16. 56.
    C. Fox. The inversion of convolution transforms by differential operators. Proc. Amer. Math. Soc., 4:880–887, 1953.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 57.
    J. Fridrich. Steganography in Digital Media: Principles, Algorithms and Applications. Cambridge University Press, New York, 2010.zbMATHGoogle Scholar
  18. 64.
    G.H Golub and C.F. Van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore, 1996.Google Scholar
  19. 66.
    J.W. Goodman. Introduction to Fourier Optics. Roberts and Company Pub., Englewood, 2005.Google Scholar
  20. 68.
    I.S. Gradshteyn and I.M. Ryzhik. Table of Integrals, Series, and Products. Academic Press, San Diego, 1994.zbMATHGoogle Scholar
  21. 69.
    L. Greengard. Spectral integration and two-point boundary value problems. SIAM J. Numer. Anal., 28:1071–1080, 1991.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 70.
    K. Gröchening. Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, 2001.Google Scholar
  23. 74.
    H. Gzyl, A. Tagliani, and M. Milev. Laplace transform inversion on the real line is truly ill-conditioned. Appl. Math. Comput., 219:9805–9809, 2013.MathSciNetzbMATHGoogle Scholar
  24. 76.
    H. Han and Z. Xu. Numerical solitons of generalized Korteweg-de Vries equations. Appl. Math. Comput., 186:483–489, 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 79.
    I.I. Hirschman and D.V. Widder. Generalized inversion formulas for convolution transforms. Duke Math. J., 15:659–696, 1948.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 81.
    I.I. Hirschman and D.V. Widder. The Convolution Transform. Princeton University Press, New Jersey, 1955.zbMATHGoogle Scholar
  27. 83.
    K.T. Joseph and P.L. Sachdev. Initial boundary value problems for scalar and vector Burgers equations. Stud. Appl. Math., 106:481–505, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 84.
    F. Mainardi J.T. Machado, V. Kiryakova. A poster about the recent history of fractional calculus. Fract. Calc. Appl. Anal., 13:329–334, 2010.MathSciNetzbMATHGoogle Scholar
  29. 87.
    H.B. Keller. Numerical Solution of Two-Point Boundary Value Problems. SIAM Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 1990.Google Scholar
  30. 88.
    A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.zbMATHGoogle Scholar
  31. 89.
    R. Kimmel. 3d shape reconstruction from autostereograms and stereo. J. Vis. Comm. Image Rep., pages 324–333, 2002.Google Scholar
  32. 94.
    S.M. Kosslyn, W.L. Thompson, I.J. Kim, S.L. Rauch, and N.M. Alpert. Individual differences in cerebral blood flow in area 17 predict the time to evaluate visualized letters. J. Cogn. Neurosci., 8:78–82, 1996.CrossRefGoogle Scholar
  33. 97.
    M.S.K. Lau and C.P. Kwong. Analysis of echoes in single-image random-dot-stereograms. J. Math. Imaging Vis., 16:69–79, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 101.
    J. Lighthill. An Informal Introduction to Theoretical Fluid Mechanics. Oxford University Press, Oxford, 1988.CrossRefzbMATHGoogle Scholar
  35. 103.
    W.Y. Ma and B.S. Manjunath. Edge flow: a framework of boundary detection and image segmentation. In Computer Vision and Pattern Recognition, pages 744–749. IEEE, 1997.Google Scholar
  36. 104.
    J.T. Machado, V. Kiryakova, and F. Mainardi. Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simulat., 16:1140–115, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 105.
    R. Maini and H. Aggarwal. Study and comparison of various image edge detection techniques. Int. J. Image Process., 3:1–12, 2009.CrossRefGoogle Scholar
  38. 107.
    D. Marr and E. Hildreth. Theory of edge detection. Proc. R. Soc. London, 207:187–217, 1980.CrossRefGoogle Scholar
  39. 109.
    S. Momani and M. A. Noor. Numerical comparison of methods for solving a special fourth-order boundary value problem. Appl. Math. Comput., 191:218–224, 2007.MathSciNetzbMATHGoogle Scholar
  40. 113.
    S. Nerney, E.J. Schmahl, and Z.E. Musielak. Analytic solutions of the vector Burgers’ equation. Quart. Appl. Math., 54:63–71, 1996.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 115.
    P.L. Olson and M. Sivak. Perception-response time to unexpected roadway hazards. Hum. Factors, 28:91–96, 1986.CrossRefGoogle Scholar
  42. 116.
    M.D. Ortigueira. Fractional Calculus for Scientists and Engineers. Springer-Verlag, Dordrecht, 2011.CrossRefzbMATHGoogle Scholar
  43. 119.
    J.N. Pandey and A.H. Zemanian. Complex inversion for the generalized convolution transformation. Pac. J. Math., 25:147–157, 1968.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 123.
    F.A.P. Petitcolas, R.J. Anderson, and M.G. Kuhn. Information hiding-a survey. In Proceedings of the IEEE, volume 87, pages 1062–1078. IEEE, 1999.Google Scholar
  45. 124.
    I. Podlubny. Fractional Differential Equations. Academic Press, San Diego, 1999.zbMATHGoogle Scholar
  46. 126.
    P. Rosenau and J. M. Hyman. Compactons: solitons with finite wavelength. Phys. Rev. Lett., 70:564–567, 1993.CrossRefzbMATHGoogle Scholar
  47. 128.
    F. Rus and F.R. Villatoro. Self-similar radiation from numerical Rosenau-Hyman compactons. J. Comput. Phys., 227:440–454, 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  48. 129.
    S.G. Samko, A.A. Kilbas, and O.I. Marichev. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Switzerland, 1993.zbMATHGoogle Scholar
  49. 130.
    D.L. Schomer and F.H. Lopes da Silva. Niedermeyer’s Electroencephalography. Wolters Kluwer, Pennsylvania, 2011.Google Scholar
  50. 131.
    L.F. Shampine. Numerical Solution of Ordinary Differential Equations. Chapman and Hall, New York, 1994.zbMATHGoogle Scholar
  51. 132.
    L.F. Shampine, I. Gladwell, and S. Thompson. Solving ODEs with MATLAB. Cambridge University Press, Cambridge, 2003.CrossRefzbMATHGoogle Scholar
  52. 133.
    L.F. Shampine and H.A. Watts. Block implicit one-step methods. Math. Comp., 23:731–740, 1969.MathSciNetCrossRefzbMATHGoogle Scholar
  53. 134.
    S.S. Siddiqi and M. Iftikhar. Numerical solution of higher order boundary value problems. Abstr. Appl. Anal., 2013.Google Scholar
  54. 138.
    R. Szelinski. Computer Vision. Algorithms and Applications. Springer-Verlag, London, 2011.Google Scholar
  55. 139.
    Y. Tanno. On a class of convolution transforms. Tohoku Math. J., 18:156–173, 1966.MathSciNetCrossRefzbMATHGoogle Scholar
  56. 140.
    Y. Tanno. On a class of convolution transforms II. Tohoku Math. J., 19:168–186, 1967.MathSciNetCrossRefzbMATHGoogle Scholar
  57. 142.
    H.W. Thimbleby, S. Inglis, and I. H.Witten. Displaying 3d images: Algorithms for single-image random-dot stereograms. Computer, 27:38–48, 1994.CrossRefGoogle Scholar
  58. 145.
    V.V. Uchaikin. Fractional Derivatives for Physicists and Engineers. Springer-Verlag, Berlin, 2013.CrossRefzbMATHGoogle Scholar
  59. 146.
    A.B. Watson and A.J. Ahumada. Model of human visual-motion sensing. J. Opt. Soc. Am. A, 2:322–342, 1985.CrossRefGoogle Scholar
  60. 147.
    H.A. Watts and L.F. Shampine. A-stable block implicit one-step methods. BIT Numer. Math., 12:252–266, 1972.MathSciNetCrossRefzbMATHGoogle Scholar
  61. 148.
    J.A.C. Weideman and S.C. Reddy. A MATLAB differentiation matrix suite. ACM Trans. Math. Soft., 26:465–519, 2000.MathSciNetCrossRefGoogle Scholar
  62. 149.
    D.V. Widder. Inversion formulas for convolution transforms. Duke Math. J., 14:217–249, 1947.MathSciNetCrossRefzbMATHGoogle Scholar
  63. 150.
    D.V. Widder. The convolution transform. Bull. Amer. Math. Soc, 60:444–456, 1954.MathSciNetCrossRefzbMATHGoogle Scholar
  64. 155.
    A.H. Zemanian. Generalized Integral Transformations. Dover Publications, New York, 1987.zbMATHGoogle Scholar
  65. 156.
    D. Ziou and S. Tabbone. Edge detection techniques: an overview. Int. J. Pattern Recogn. Image Anal., 8:537–559, 1998.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Rafael G. Campos
    • 1
  1. 1.Science DepartmentUniversity of Quintana RooChetumalMexico

Personalised recommendations