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The polynomial discrete Radon transform

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Abstract

This paper presents a new approach called polynomial discrete Radon transform (PDRT), regarded as a generalization of the classical finite discrete Radon transform. Specifically, the PDRT transforms an image into Radon space by summing the pixels according to polynomial curves. The PDRT can be applied on square \(p \times p\) images where \(p\) is assumed to be a prime number. It is based on a simple arithmetic operations and requires no data interpolation. An interesting property of the PDRT is its exact inversion. This means that an image can be transformed and then perfectly reconstructed. Through this study, we show that the new approach can be applied for some pattern recognition applications.

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References

  1. Beylkin, G.: Discrete Radon transform. IEEE Trans. Acoust. Speech Signal Process. 35, 162–172 (1987)

    Article  MathSciNet  Google Scholar 

  2. Deans, S.: Hough transform from the Radon transform. IEEE Trans. Pattern Anal. Mach. Intell. PAMI–3(2), 185–188 (1981)

    Article  Google Scholar 

  3. Deans, S.R.: The Radon Transform and Some of Its Applications, revised edition. Krieger Publishing Co., Malabar, FL (1993)

  4. Duda, R., Hart, P.: Pattern Classification and Scene Analysis. Wiley Inter-Science, New York (1973)

    MATH  Google Scholar 

  5. Guil, N., Zapata, E.: Lower order circle and ellipse Hough transform. Pattern Recognit. 30(10), 1729–1744 (1997)

    Article  Google Scholar 

  6. Hendriks, C.L.L., van Ginkel, M., Verbeek, P.W., van Vliet, L.J.: The generalized Radon transform: Sampling, accuracy and memory considerations. Pattern Recognit. 38(12), 2494–2505 (2005)

    Article  Google Scholar 

  7. Hough, P.: Methods and means for recognizing complex patterns. U.S. Patent (1962)

  8. Kesidis, A.: A gray-scale inverse Hough transform algorithm. In: The 2000 IEEE International Symposium on Circuits and Systems, Geneva, vol. 5, pp. 297–300 (2000)

  9. Kesidis, A.L., Papamarkos, N.: A window-based inverse Hough transform. Pattern Recognit. 33(6), 1105–1117 (2000)

  10. Kesidis, A., Papamarkos, N.: On the inverse Hough transform. IEEE Trans. Pattern Anal. Mach. Intell. 21, 1329–1342 (1999)

    Article  Google Scholar 

  11. Kingston, A.: Orthogonal discrete radon transform over \(p^n\times p^n\) images. Signal Process. 86, 2040–2050 (2006)

    Article  MATH  Google Scholar 

  12. Kingston, A., Svalbe, I.: Generalized finite radon transform for \(n \times n\) images. Image Vis. Comput. 25, 1620–1630 (2007)

    Article  Google Scholar 

  13. Li, H., Lavin, M., LeMaster, R.: Fast Hough transform: a hierarchical approach. Comput. Vis. Graph. Image Process. 36(12), 139–161 (1986)

    Article  Google Scholar 

  14. Lun, D., Chen, T., Hsung, T., Feng, D., Chan, Y.: Efficient blind image restoration using discrete periodic Radon transform. IEEE Trans. Image Process. 13, 188–200 (2004)

    Article  Google Scholar 

  15. Lun, D., Hsung, T., Chen, T.: Orthogonal discrete periodic Radon transform. Part ii: applications. Signal Process. 83, 957–971 (2003)

    Article  MATH  Google Scholar 

  16. Lun, D., Hsung, T., Chen, T.: Orthogonal discrete periodic Radon transform. Part i: theory and realization. Signal Process. 83, 941–955 (2003)

    Article  MATH  Google Scholar 

  17. Lun, D., Hsung, T., Shen, T.: Orthogonal discrete periodic Radon transform. Part i: theory and realization. Signal Process. 83, 941–955 (2003)

  18. Matus, F., Flusser, J.: Image representation via a finite Radon transform. IEEE Trans. Pattern Anal. Mach. Intell. 15(10), 996–1006 (1993)

  19. Minh, N., Vetterli, M.: Image denoising using orthonormal finite ridgelet transform. In: Proceedings of the SPIE Wavelet Application in Signal and Image Processing, vol. 4119, pp. 831–842 (2000)

  20. Minh, N., Vetterli, M.: Orthonormal finite ridgelet transform for image compression. In: Proceedings of the International Conference on Image Processing, vol. 2, pp. 367–370 (2000)

  21. Murphy, L.M.: Linear feature detection and enhancement in noisy images via the Radon transform. Pattern Recognit. Lett. 4, 279–284 (1986)

    Article  Google Scholar 

  22. Princen, J., Illingworth, J., Kittler, J.: Hypothesis testing: a framework for analyzing and optimizing Hough transform performance. IEEE Trans. Pattern Anal. Mach. Intell. 4(10), 329–341 (1994)

    Article  Google Scholar 

  23. Radon, J.: Ufiber die bestimmung von funktionen durch ihre integral-werte langs gewisser mannigfaltigkeiten. Berichte Sachsische Akademie der Wissenschaften, Leipzig, Math-Phys. Kl 62, 262–267 (1917)

  24. Shepp, L., Krustal, J.: Computerized tomography: the new medical x-ray technology. Am. Math. Mon. 85, 420–439 (1978)

    Article  MATH  Google Scholar 

  25. Svalbe, I., van der Spek, D.: Reconstruction of tomographic images using analog projections and the digital Radon transform. Linear Algebra Appl. 339, 125–145 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Tofts, P.: The Radon Ttransform: Theory and Implementation. Ph.D. Thesis (1996)

  27. Yuen, S., Ma, C.: An investigation of the nature of parameterization for the Hough transform. Pattern Recognit. 6, 1009–1040 (1997)

    Article  Google Scholar 

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Authors and Affiliations

  1. Laboratoire d’Informatique, Programmation Algorithmique et Heuristiques (LIPAH), Computer Technology Department, Faculté des Sciences de Tunis, Tunis, Tunisia

    Ines ELouedi & Atef Hamouda

  2. Laboratoire d’Images, Signaux et Systèmes Intelligents (LISSI), Université Paris Est-Creteil, Paris, France

    Régis Fournier & Amine Naït-Ali

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  1. Ines ELouedi
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  2. Régis Fournier
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  3. Amine Naït-Ali
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  4. Atef Hamouda
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Correspondence to Ines ELouedi.

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ELouedi, I., Fournier, R., Naït-Ali, A. et al. The polynomial discrete Radon transform. SIViP 9 (Suppl 1), 145–154 (2015). https://doi.org/10.1007/s11760-014-0727-3

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  • DOI: https://doi.org/10.1007/s11760-014-0727-3

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