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A Variational Model to Remove the Multiplicative Noise in Ultrasound Images

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Abstract

In this paper we study a variational model to deal with the speckle noise in ultrasound images. We prove the existence and uniqueness of the minimizer for the variational problem, and derive the existence and uniqueness of weak solutions for the associated evolution equation. Furthermore, we show that the solution of the evolution equation converges weakly in BV and strongly in L 2 to the minimizer as t→∞. Finally, some numerical results illustrate the effectiveness of the proposed model for multiplicative noise removal.

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References

  1. Aubert, G., Aujol, J.F.: A variational approach to removing multiplicative noise. SIAM J. Appl. Math. 68, 925–946 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brezis, H.: Operateures Maximaux Monotone. North-Holland, Amsterdam (1993)

    Google Scholar 

  3. Chen, Y.M., Rao, M.: Minimization problems and associated flows related to weighted p energy and total variation. SIAM J. Math. Anal. 34(5), 1084–1104 (2002)

    Article  MathSciNet  Google Scholar 

  4. Dutt, V., Greenleaf, J.: Adaptative speckle reduction filter for log-compressed b-scan images. IEEE Trans. Med. Imag. 15(6), 802–813 (1996)

    Article  Google Scholar 

  5. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  6. Giusti, F.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Math., vol. 80. Birkhäuser, Basel (1984)

    MATH  Google Scholar 

  7. Huang, Y., Ng, M., Wen, Y.: A new total variation method for multiplicative noise removal. SIAM J. Imaging Sci. 2(1), 20–40 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Jin, Z.M., Yang, X.P.: Analysis of a new variation model for multiplicative noise removal. J. Math. Anal. Appl. 362, 415–426 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kaplan, D., Ma, Q.: On the statistical characteristics of the log-compressed Rayleigh signals: theoretical formulation and experimental results. J. Acoust. Soc. Am. 95, 1396–1400 (1994)

    Article  Google Scholar 

  10. Kornprobst, P., Deriche, R., Aubert, G.: Image sequence analysis via partial differential equations. J. Math. Imaging Vis. 11(1), 5–26 (1999)

    Article  MathSciNet  Google Scholar 

  11. Krissian, K., Kikinis, R., Westin, C.F., Vosburgh, K.: Speckle-constrained filtering of ultrasound images. IEEE Comput. Vis. Pattern Recogn. 547–552 (2005)

  12. Lin, F.H., Yang, X.P.: Geometric Measure Theory—An Introduction. Science Press/International Press, Beijing/Boston (2002)

    MATH  Google Scholar 

  13. Loupas, A.: Digital image processing for noise reduction in medical ultrasonics. PhD thesis, University of Edinburgh, UK (1988)

  14. Lysaker, M., Lundervold, A., Tai, X.C.: Noise removal using fourth-order partial differential equation with application to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003)

    Article  Google Scholar 

  15. Rudin, L., Lions, P.L., Osher, S.: Multiplicative denoising and deblurring: theory and algorithms. In: Osher, S., Paragios, N. (eds.) Geometric Level Sets in Imaging, Vision, and Graphics, pp. 103–119. Springer, Berlin (2003)

    Chapter  Google Scholar 

  16. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Article  MATH  Google Scholar 

  17. Tur, M., Chin, C., Goodman, J.W.: When is speckle noise multiplicative? Appl. Opt. 21(7), 1157–1159 (1982)

    Article  Google Scholar 

  18. Tuthill, T.A., Sperry, R.H., Parker, K.J.: Deviation from Rayleigh statistics in ultrasonic speckle. Ultrason. Imag. 10, 81–90 (1988)

    Article  Google Scholar 

  19. Zhou, X.: An evolution problem for plastic antiplanar shear. Appl. Math. Optim. 25, 263–285 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210046, Jiangsu, P.R. China

    Zhengmeng Jin

  2. School of Science, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, P.R. China

    Zhengmeng Jin & Xiaoping Yang

Authors
  1. Zhengmeng Jin
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  2. Xiaoping Yang
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Correspondence to Zhengmeng Jin.

Additional information

This work is supported by the National Natural Science Foundation of China (No. 10926193) and the Doctoral Programme Foundation of Education Ministry of China (N0.2003028802) and the Scientific Research Foundation of NUPT (NY209025).

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Jin, Z., Yang, X. A Variational Model to Remove the Multiplicative Noise in Ultrasound Images. J Math Imaging Vis 39, 62–74 (2011). https://doi.org/10.1007/s10851-010-0225-3

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  • DOI: https://doi.org/10.1007/s10851-010-0225-3

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