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X-Ray CT Image Reconstruction via Wavelet Frame Based Regularization and Radon Domain Inpainting

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Abstract

X-ray computed tomography (CT) has been playing an important role in diagnostic of cancer and radiotherapy. However, high imaging dose added to healthy organs during CT scans is a serious clinical concern. Imaging dose in CT scans can be reduced by reducing the number of X-ray projections. In this paper, we consider 2D CT reconstructions using very small number of projections. Some regularization based reconstruction methods have already been proposed in the literature for such task, like the total variation (TV) based reconstruction (Sidky and Pan in Phys. Med. Biol. 53:4777, 2008; Sidky et al. in J. X-Ray Sci. Technol. 14(2):119–139, 2006; Jia et al. in Med. Phys. 37:1757, 2010; Choi et al. in Med. Phys. 37:5113, 2010) and balanced approach with wavelet frame based regularization (Jia et al. in Phys. Med. Biol. 56:3787–3807, 2011). For most of the existing methods, at least 40 projections is usually needed to get a satisfactory reconstruction. In order to keep radiation dose as minimal as possible, while increase the quality of the reconstructed images, one needs to enhance the resolution of the projected image in the Radon domain without increasing the total number of projections. The goal of this paper is to propose a CT reconstruction model with wavelet frame based regularization and Radon domain inpainting. The proposed model simultaneously reconstructs a high quality image and its corresponding high resolution measurements in Radon domain. In addition, we discovered that using the isotropic wavelet frame regularization proposed in Cai et al. (Image restorations: total variation, wavelet frames and beyond, 2011, preprint) is superior than using its anisotropic counterpart. Our proposed model, as well as other models presented in this paper, is solved rather efficiently by split Bregman algorithm (Goldstein and Osher in SIAM J. Imaging Sci. 2(2):323–343, 2009; Cai et al. in Multiscale Model. Simul. 8(2):337–369, 2009). Numerical simulations and comparisons will be presented at the end.

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References

  1. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, pp. 417–424. ACM/Addison-Wesley, New York/Reading (2000)

    Google Scholar 

  3. Bezdek, J.C., Hathaway, R.J.: Convergence of alternating optimization. Neural Parallel Sci. Comput. 11(4), 351–368 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Bushberg, J.T., Seibert, J.A., Leidholdt, E.M., Jr, Boone, J.M.: The Essential Physics of Medical Imaging. Williams & Wilkins, Baltimore (2002)

    Google Scholar 

  5. Cai, J.F., Chan, R.H., Shen, Z.: A framelet-based image inpainting algorithm. Appl. Comput. Harmon. Anal. 24(2), 131–149 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cai, J.F., Osher, S., Shen, Z.: Linearized Bregman iterations for frame-based image deblurring. SIAM J. Imaging Sci. 2(1), 226–252 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cai, J.F., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8(2), 337–369 (2009)

    Article  MathSciNet  Google Scholar 

  8. Cai, J.F., Chan, R., Shen, L., Shen, Z.: Tight frame based method for high-resolution image reconstruction. In: Damlamian, A., Jaffard, S. (eds.) Contemporary Applied Mathematics, pp. 1–36. Higher Education Press, Beijing (2010)

    Google Scholar 

  9. Cai, J.F., Dong, B., Osher, S., Shen, Z.: Image restorations: total variation, wavelet frames and beyond. Preprint (2011)

  10. Chan, T.F., Shen, J.: Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. Society for Industrial Mathematics, Philadelphia (2005)

    Book  MATH  Google Scholar 

  11. Chan, R.H., Chan, T.F., Shen, L., Shen, Z.: Wavelet algorithms for high-resolution image reconstruction. SIAM J. Sci. Comput. 24(4), 1408–1432 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chan, R.H., Riemenschneider, S.D., Shen, L., Shen, Z.: High-resolution image reconstruction with displacement errors: a framelet approach. Int. J. Imaging Syst. Technol. 14(3), 91–104 (2004)

    Article  MathSciNet  Google Scholar 

  13. Chan, R.H., Riemenschneider, S.D., Shen, L., Shen, Z.: Tight frame: an efficient way for high-resolution image reconstruction. Appl. Comput. Harmon. Anal. 17(1), 91–115 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chan, R.H., Shen, L., Shen, Z.: A framelet-based approach for image inpainting. Res. Rep. 4, 325 (2005)

    Google Scholar 

  15. Choi, K., Wang, J., Zhu, L., Suh, T.S., Boyd, S., Xing, L.: Compressed sensing based cone-beam computed tomography reconstruction with a first-order method. Med. Phys. 37, 5113 (2010)

    Article  Google Scholar 

  16. Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2006)

    Article  MathSciNet  Google Scholar 

  17. Daubechies, I.: Ten Lectures on Wavelets. CBMS-NSF Lecture Notes, SIAM, vol. 61. Society for Industrial Mathematics, Philadelphia (1992)

    Book  MATH  Google Scholar 

  18. Daubechies, I., Han, B., Ron, A., Shen, Z.: Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14(1), 1–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dong, B., Shen, Z.: MRA Based Wavelet Frames and Applications. IAS Lecture Notes Series, Summer Program on “The Mathematics of Image Processing”. Park City Mathematics Institute, Salt Lake City (2010)

    Google Scholar 

  20. Dong, B., Shen, Z.: Wavelet frame based surface reconstruction from unorganized points. J. Comput. Phys. 230(22), 8247–8255 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dong, B., Chien, A., Shen, Z.: Frame based segmentation for medical images. Commun. Math. Sci. 9(2), 551–559 (2011)

    MathSciNet  MATH  Google Scholar 

  22. Donoho, D.L.: De-noising by soft-thresholding. IEEE Trans. Inf. Theory 41(3), 613–627 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(1), 293–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Elad, M., Starck, J.L., Querre, P., Donoho, D.L.: Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA). Appl. Comput. Harmon. Anal. 19(3), 340–358 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Esser, E.: Applications of Lagrangian-based alternating direction methods and connections to split Bregman. CAM Rep. 9, 31 (2009)

    Google Scholar 

  26. Feldkamp, L.A., Davis, L.C., Kress, J.W.: Practical cone-beam algorithm. J. Opt. Soc. Am. A 1(6), 612–619 (1984)

    Article  Google Scholar 

  27. Friedman, J., Hastie, T., Höfling, H., Tibshirani, R.: Pathwise coordinate optimization. Ann. Appl. Stat. 1(2), 302–332 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl. 2(1), 17–40 (1976)

    Article  MATH  Google Scholar 

  29. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. Society for Industrial Mathematics, Philadelphia (1989)

    Book  MATH  Google Scholar 

  31. Goldstein, T., Osher, S.: The split Bregman algorithm for L1 regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gordon, R., Bender, R., Herman, G.T.: Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and X-ray photography* 1. J. Theor. Biol. 29(3), 471–481 (1970)

    Article  Google Scholar 

  33. Islam, M.K., Purdie, T.G., Norrlinger, B.D., Alasti, H., Moseley, D.J., Sharpe, M.B., Siewerdsen, J.H., Jaffray, D.A.: Patient dose from kilovoltage cone beam computed tomography imaging in radiation therapy. Med. Phys. 33, 1573 (2006)

    Article  Google Scholar 

  34. Jia, X., Lou, Y., Li, R., Song, W.Y., Jiang, S.B.: GPU-based fast cone beam CT reconstruction from undersampled and noisy projection data via total variation. Med. Phys. 37, 1757 (2010)

    Article  Google Scholar 

  35. Jia, X., Dong, B., Lou, Y., Jiang, S.B.: GPU-based iterative cone-beam CT reconstruction using tight frame regularization. Phys. Med. Biol. 56, 3787–3807 (2011)

    Article  Google Scholar 

  36. Luo, Z.Q., Tseng, P.: On the convergence of the coordinate descent method for convex differentiable minimization. J. Optim. Theory Appl. 72(1), 7–35 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mao, Y., Fahimian, B.P., Osher, S.J., Miao, J.: Development and optimization of regularized tomographic reconstruction algorithms utilizing equally-sloped tomography. IEEE Trans. Image Process. 19(5), 1259–1268 (2010)

    Article  MathSciNet  Google Scholar 

  38. Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures. Am. Math. Soc., Providence (2001)

    MATH  Google Scholar 

  39. Osher, S., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Berlin (2003)

    MATH  Google Scholar 

  40. Patch, S.K.: Computation of unmeasured third-generation VCT views from measured views. IEEE Trans. Med. Imaging 21(7), 801–813 (2002)

    Article  Google Scholar 

  41. Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Sächs. Akad. Wiss. 69, 262–267 (1917)

    Google Scholar 

  42. Rockafellar, R.T., Roger, J.B.W.: Variational Analysis. Springer, Berlin (1997)

    Google Scholar 

  43. Ron, A., Shen, Z.: Affine systems in L 2(ℝd): the analysis of the analysis operator. J. Funct. Anal. 148(2), 408–447 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  45. Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge Univ Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  46. Segars, W.P., Lalush, D.S., Tsui, B.M.W.: A realistic spline-based dynamic heart phantom. IEEE Trans. Nucl. Sci. 46(3), 503–506 (1999)

    Article  Google Scholar 

  47. Segars, W.P., Lalush, D.S., Tsui, B.M.W.: Modeling respiratory mechanics in the MCAT and spline-based MCAT phantoms. IEEE Trans. Nucl. Sci. 48(1), 89–97 (2001)

    Article  Google Scholar 

  48. Segars, W.P., Tsui, B.M., Lalush, D.S., Frey, E.C., King, M.A., Manocha, D.: Development and Application of the New Dynamic Nurbs-Based Cardiac-Torso (NCAT) Phantom. Biomedical 5 Engineering. University of North Carolina, Chapel Hill (2001)

    Google Scholar 

  49. Setzer, S.: Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage. In: Scale Space and Variational Methods in Computer Vision, pp. 464–476 (2009)

    Chapter  Google Scholar 

  50. Shen, Z.: Wavelet frames and image restorations. In: Proceedings of the International Congress of Mathematicians, Hyderabad, India (2010)

    Google Scholar 

  51. Siddon, R.L.: Fast calculation of the exact radiological path for a 3-dimensional ct array. Med. Phys. 12, 252–255 (1985)

    Article  Google Scholar 

  52. Sidky, E.Y., Pan, X.: Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization. Phys. Med. Biol. 53, 4777 (2008)

    Article  Google Scholar 

  53. Sidky, E.Y., Kao, C.M., Pan, X.: Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT. J. X-Ray Sci. Technol. 14(2), 119–139 (2006)

    Google Scholar 

  54. Starck, J.L., Elad, M., Donoho, D.L.: Image decomposition via the combination of sparse representations and a variational approach. IEEE Trans. Image Process. 14(10), 1570–1582 (2005)

    Article  MathSciNet  Google Scholar 

  55. Tai, X.C., Wu, C.: Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. In: Scale Space and Variational Methods in Computer Vision, pp. 502–513 (2009)

    Chapter  Google Scholar 

  56. Tseng, P.: Convergence of a block coordinate descent method for nondifferentiable minimization. J. Optim. Theory Appl. 109(3), 475–494 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  57. Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Program. 117(1), 387–423 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  58. Wang, Y., Yin, W., Zhang, Y.: A fast algorithm for image deblurring with total variation regularization. Technical Report TR07-10, Rice University CAAM (2007)

  59. Yu, H., Wei, Y., Hsieh, J., Wang, G.: Data consistency based translational motion artifact reduction in fan-beam CT. IEEE Trans. Med. Imaging 25(6), 792–803 (2006)

    Article  Google Scholar 

  60. Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM J. Imaging Sci. 3, 253–276 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank Dr. Xun Jia (Department of Radiation Oncology, University of California, San Diego) for providing us with the MATLAB program of Siddon’s algorithm, the real CT data, and many valuable discussions on the subject.

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  1. Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., Tucson, AZ, 85721-0089, USA

    Bin Dong

  2. Department of Mathematics, National University of Singapore, Singapore, 117542, Singapore

    Jia Li & Zuowei Shen

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  1. Bin Dong
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  2. Jia Li
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Dong, B., Li, J. & Shen, Z. X-Ray CT Image Reconstruction via Wavelet Frame Based Regularization and Radon Domain Inpainting. J Sci Comput 54, 333–349 (2013). https://doi.org/10.1007/s10915-012-9579-6

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