Journal of Scientific Computing

, Volume 50, Issue 3, pp 665–677 | Cite as

Wavelet-Based De-noising of Positron Emission Tomography Scans

  • Wolfgang Stefan
  • Kewei Chen
  • Hongbin Guo
  • Rosemary A. Renaut
  • Svetlana Roudenko


A method to improve the signal-to-noise-ratio (SNR)of positron emission tomography (PET) scans is presented. A wavelet-based image decomposition technique decomposes an image into two parts, one which primarily contains the desired restored image and the other primarily the remaining unwanted portion of the image. Because the method is based on a texture extraction model that identifies the desired image in the space of bounded variation, these restorations are approximations of piecewise constant images, and are referred to as the cartoon part of the image. Here an approximation using a wavelet decomposition is used which allows solutions to be computed very efficiently. To process 3-D volume data a slice by slice approach in all three directions is adopted. Using a redundant discrete wavelet transform, 3-D restorations can be efficiently computed on standard desktop computers. The method is illustrated for PET images which have been reconstructed from simulated data using the expectation maximization algorithm. When post-processed by the presented wavelet decomposition they show a significant increase in SNR. It is concluded that the new wavelet based method can be used as an alternative to the well established de-noising of PET scans by smoothing with a Gaussian point spread function. In particular, if the volume data are reconstructed using the EM algorithm with a larger number of iterations than the number of iterations that would be used without post-processing, the 3-D images are sharper and show more detail. A MATLAB® based graphical user interface is provided that allows easy exploration of the impact of parameter choices.


Wavelets Denoising Positron emission tomography ROF model Bounded variation 


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  1. 1.
    Statistical Parametric Mapping. Department of Imaging Neuroscience, University College London (2011).
  2. 2.
    Alexander, G.E., Chen, K., Pietrini, P., Rapoport, S.I., Reiman, E.M.: Longitudinal evaluation of cerebral metabolic decline in dementia: implications for using resting PET to measure outcome in long-term treatment studies of Alzheimer’s disease. Am. J. Psychiatr. 159, 738–745 (2002) CrossRefGoogle Scholar
  3. 3.
    Burrus, C.S., Gopinath, R.A., Guo, H.: Introduction to Wavelets and Wavelet Transforms: A Primer. Prentice Hall, Upper Saddle River (1998) Google Scholar
  4. 4.
    Chang, S.G., Yu, B., Vetterli, M.: Adaptive wavelet thresholding for image denoising and compression. IEEE Trans. Image Process. 9(9), 1532–1546 (2000) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chen, K., Lawson, M., Reiman, E.M., Feng, D., Huang, S.-C., Bandy, D., Ho, D., Yun, L.-S., Palant, A.: Generalized linear least square method for fast generation of myocardial blood flow parametric images with N-13 ammonia PET. IEEE Trans. Med. Imaging 17(2), 236–243 (1998) CrossRefGoogle Scholar
  6. 6.
    Cohen, A., Dahmen, W., Daubechies, I., DeVore, R.: Harmonic analysis of the space BV. Mat. Iberoam. 19, 235–263 (2003) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Cohen, A., Daubechies, I., Feauveau, J.-C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45(5), 485–560 (1992) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Cohen, A., DeVore, R., Petrushev, P., Xu, H.: Nonlinear Approximation and the Space \(\mathit{BV}(\mathcal{R}^{\in})\). Am. J. Math. 121, 587–628 (1999) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Combes, J.M., Grossmann, A., Tchamitchian, P. (eds.): Wavelets: Time-Frequency Methods and Phase Space: Proceedings of the International Conference, Marseille, France, 14–18 December 1987. Springer, Berlin, New York (1989) MATHGoogle Scholar
  10. 10.
    Daubechies, I.: Ten Lectures on Wavelets, p. 3600. SIAM, Philadelphia (1992). University City Science Center, Philadelphia, Pennsylvania MATHCrossRefGoogle Scholar
  11. 11.
    Daubechies, I., Teschke, G.: Wavelet based image decomposition by variational functionals. In: Frédéric, T. (ed.) Wavelet Applications in Industrial Processing. Proceedings of the SPIE, vol. 5266, pp. 94–105. SPIE Press, Bellingham (2004) Google Scholar
  12. 12.
    Daubechies, I., Teschke, G.: Variational image restoration by means of wavelets: simultaneous decomposition, deblurring, and denoising. Appl. Comput. Harmonic Anal. 19(1) (2005) Google Scholar
  13. 13.
    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. B 39, 1–38 (1977) MathSciNetMATHGoogle Scholar
  14. 14.
    Donoho, D.L., Johnstone, I.M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika (1994) Google Scholar
  15. 15.
    Fowler, J.E.: The redundant discrete wavelet transform and additive noise. IEEE Signal Process. Lett. 12(9), 629–632 (2005) CrossRefGoogle Scholar
  16. 16.
    Grant, M., Boyd, S., Ye, Y.: In: Disciplined Convex Programming. Nonconvex Optimization and Its Applications, pp. 155–210. Springer, Berlin (2006) Google Scholar
  17. 17.
    Grant, M., Boyd, S., Ye, Y.: CVX: Matlab Software for Disciplined Convex Programming, September 2007 Google Scholar
  18. 18.
    Lin, J.-W., Laine, A.F., Bergmann, S.R.: Improving PET-based physiological quantification through methods of wavelet denoising. IEEE Trans. Biomed. Eng. 48(2), 202–212 (2001) CrossRefGoogle Scholar
  19. 19.
    Kaufman, L.: Implementing and accelerating the EM algorithm for positron emission tomography. IEEE Trans. Med. Imaging M1-6(6), 37–51 (1987) CrossRefGoogle Scholar
  20. 20.
    Khlifa, N., Gribaa, N., Mbazaa, I., Hamruoni, K.: A based Bayesian wavelet thresholding method to enhance nuclear imaging. Int. J. Biomed. Imaging 2009, 506120 (2009) CrossRefGoogle Scholar
  21. 21.
    Kwan, K.S., Evans, A.C., Pike, G.B.: MRI simulation-based evaluation of image-processing and classification methods. IEEE Trans. Med. Imaging 18(11), 1085–1097 (1999) CrossRefGoogle Scholar
  22. 22.
    Lang, M., Guo, H., Odegard, J.E., Burrus, C.S., Wells, R.O. Jr.: Noise reduction using an undecimated discrete wavelet transform. IEEE Signal Process. Lett. 3(1), 10–12 (1996) CrossRefGoogle Scholar
  23. 23.
    Lefkimmiatis, S., Maragos, P., Papandreou, G.: Bayesian inference on multiscale models for Poisson intensity estimation: applications to photon-limited image denoising. IEEE Trans. Image Process. 18(8), 1724–1741 (2009) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lieu, L., Vese, L.: Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces. Appl. Math. Optim. 58(2), 167–193 (2008) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series, vol. 22. AMS (2002) Google Scholar
  26. 26.
    NRC: Mathematics and Physics of Emerging Biomedical Imaging. National Research Council, Institute of Medicine, National Academy Press, Washington (1996). Google Scholar
  27. 27.
    Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H 1 norm. Multiscale Model. Simul. 1(3), 349–370 (2003) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Phelps, M.E., Mazziotta, J., Schelbert, H.: Positron Emission Tomorgraphy and Autoradiography, Principles and Applications for the Brain and Heart. Raven Press, New York (1986) Google Scholar
  29. 29.
    Reiman, E.M., Caselli, R.J., Yun, L.-S., Chen, K., Bandy, D., Minoshima, S., Thibodeau, S., Osborne, D.: Preclinical evidence of a genetic risk factor for Alzhemer’s disease in apolipoprotein E type 4 homozygotes using positron emission tomography. N. Engl. J. Med. 334, 752–758 (1996) CrossRefGoogle Scholar
  30. 30.
    Roudenko, S.: Noise and texture detection in image processing. LANL report: W-7405-ENG-36 (2004) Google Scholar
  31. 31.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992) MATHCrossRefGoogle Scholar
  32. 32.
    Shepp, L.A., Vardi, Y.: Maximum likelihood reconstruction for emission tomography. IEEE Trans. Med. Imaging MI-1(2), 113–122 (1982) CrossRefGoogle Scholar
  33. 33.
    Shih, Y.Y., Chen, J.C., Liu, R.S.: Development of wavelet de-noising technique for PET images. Comput. Med. Imaging Graph. 29(4), 297–304 (2005) CrossRefGoogle Scholar
  34. 34.
    Silverman, D.H., Small, G.W., Chang, C.Y., Lu, C.S., Kung, D., Aburto, M.A., Chen, W., Czernin, J., Rapoport, S.I., Pietrini, P., Alexander, G.E., Schapiro, M.B., Jagust, W.J., Hoffman, J.M., Welsh-Bohmer, K.A., Alavi, A., Clark, C.M., Salmon, E., de Leon, M.J., Mielke, R., Cummings, J.L., Kowell, A.P., Gambhir, S.S., Hoh, C.K., Phelps, M.E., et al.: Positron emission tomography in evaluation of dementia: Regional brain metabolism and long-term outcome. JAMA J. Am. Med. Assoc. 286(17), 2120–2127 (2001) CrossRefGoogle Scholar
  35. 35.
    Stefan, W., Roudenko, S., Chen, K., Renaut, R.A., Guo, H.: Software for Denoising of 3D SPM Analyse Volumes. Arizona State University, Phoenix (2008). Google Scholar
  36. 36.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999). Special issue on Interior Point Methods (CD supplement with software), MathSciNetCrossRefGoogle Scholar
  37. 37.
    Toh, K., Tütüncü, R., Todd, M.: SDPT3 4.0 (beta) (software package). Technical report, Department of Mathematics National University of Singapore, September 2006.
  38. 38.
    Vardi, Y., Shepp, L.A., Kaufman, L.: A statistical model for positron emission tomography. J. Am. Stat. Assoc. 80, 8–20 (1985) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Wang, Y., Yin, W., Zhang, Y.: A fast algorithm for image deblurring with total variation regularization. SIAM J. Imaging Sci. 1(3), 248–272 (2008) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Zhao, C., Chen, Z., Ye, X., Zhang, Y., Aburano, T., Tian, M., Zhang, H.: Imaging a pancreatic carcinoma xenograft model with 11C-acetate: a comparison study with 18F-FDG. Nucl Med Commun (August) (2009). Epub ahead of print Google Scholar
  41. 41.
    Zubal, I.G., Harrell, C.R., Rattner, Z., Gindi, G., Hoffer, P.B.: Computerized three-dimensional segmented human anatomy. Med. Phys. 21(2), 299–302 (1994) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Wolfgang Stefan
    • 1
  • Kewei Chen
    • 2
  • Hongbin Guo
    • 3
  • Rosemary A. Renaut
    • 4
  • Svetlana Roudenko
    • 5
  1. 1.The University of Texas, MD AndersenHoustonUSA
  2. 2.Computational Image Analysis ProgramBanner Alzheimer’s Institute & and Banner Good Samaritan Medical CenterPhoenixUSA
  3. 3.Instarecon Inc.ChampaignUSA
  4. 4.School of Mathematical & Statistical SciencesArizona State UniversityPhoenixUSA
  5. 5.Department of MathematicsThe George Washington UniversityWashingtonUSA

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