Journal of Mathematical Imaging and Vision

, Volume 48, Issue 3, pp 566–582 | Cite as

A Framework for Moving Least Squares Method with Total Variation Minimizing Regularization

  • Yeon Ju Lee
  • Sukho Lee
  • Jungho Yoon


In this paper, we propose a computational framework to incorporate regularization terms used in regularity based variational methods into least squares based methods. In the regularity based variational approach, the image is a result of the competition between the fidelity term and a regularity term, while in the least squares based approach the image is computed as a minimizer to a constrained least squares problem. The total variation minimizing denoising scheme is an exemplary scheme of the former approach with the total variation term as the regularity term, while the moving least squares method is an exemplary scheme of the latter approach. Both approaches have appeared in the literature of image processing independently. By putting schemes from both approaches into a single framework, the resulting scheme benefits from the advantageous properties of both parties. As an example, in this paper, we propose a new denoising scheme, where the total variation minimizing term is adopted by the moving least squares method. The proposed scheme is based on splitting methods, since they make it possible to express the minimization problem as a linear system. In this paper, we employed the split Bregman scheme for its simplicity. The resulting denoising scheme overcomes the drawbacks of both schemes, i.e., the staircase artifact in the total variation minimizing based denoising and the noisy artifact in the moving least squares based denoising method. The proposed computational framework can be utilized to put various combinations of both approaches with different properties together.


Denoising Total variation Moving least squares Bregman iteration 



This work was supported by the Basic Science Research Programs 2012R1A1A2004518 (J. Yoon), 2010-0011689 (Y. Lee), 2010-0006567 (S. Lee), and the Priority Research Centers Program 2009-0093827 (Y. Lee and J. Yoon) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology.


  1. 1.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences, vol. 147. Springer, Berlin (2002) Google Scholar
  2. 2.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18(11), 2419–2434 (2009) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Blomgren, P., Chan, T., Mulet, P., Wong, C.K.: Total variation image restoration: numerical methods and extensions. In: IEEE ICIP, pp. 384–387 (1997) Google Scholar
  5. 5.
    Bose, N.K., Ahuja, N.A.: Superresolution and noise filtering using moving least squares. IEEE Trans. Image Process. 15(8), 2239–2248 (2006) CrossRefGoogle Scholar
  6. 6.
    Bougleux, S., Elmoataz, A., Melkemi, M.: Local and nonlocal discrete regularization on weighted graphs for image and mesh processing. Int. J. Comput. Vis. 84(2), 220–236 (2009) CrossRefGoogle Scholar
  7. 7.
    Breitkopf, P., Naceur, H., Passineux, A., Villon, P.: Moving least squares response surface approximation: formulation and metal forming applications. Comput. Struct. 83(17–18), 1411–1428 (2005) CrossRefGoogle Scholar
  8. 8.
    Brox, T., Kleinschmidt, O., Cremers, D.: Efficient nonlocal means for denoising of textural patterns. IEEE Trans. Image Process. 17(7), 1083–1092 (2008) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bruhn, A., Weickert, J.: Towards ultimate motion estimation: combining highest accuracy with real-time performance. In: IEEE Int. Conference on Computer Vis., vol. 1, pp. 749–755 (2005) Google Scholar
  10. 10.
    Bruhn, A., Weickert, J., Kohlberger, T., Schnörr, C.: A multigrid platform for real-time motion computation with discontinuity-preserving variational methods. Int. J. Comput. Vis. 70(3), 257–277 (2006) CrossRefGoogle Scholar
  11. 11.
    Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Caselles, V., Chambolle, A., Novaga, M.: The discontinuity set of solutions of the TV denoising problem and some extensions. Multiscale Model. Simul. 6(3), 879–894 (2007) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004) MathSciNetGoogle Scholar
  14. 14.
    Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Chan, T., Marquina, A., Mulet, P.: High-order total variation based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Chan, T., Esedoglu, S., Park, F., Yip, A.: Recent Developments in Total Variation Image Restoration. Mathematical Models in Computer Vision. Springer, Berlin (2005) Google Scholar
  18. 18.
    Chan, T., Esedoglu, S., Park, F.E.: Image decomposition combining staircase reduction and texture extraction. J. Vis. Commun. Image Represent. 18(6), 464–486 (2007) CrossRefGoogle Scholar
  19. 19.
    Chan, T., Esedoglu, S., Park, F.: A fourth order dual method for staircase reduction in texture extraction and image restoration problems. In: IEEE ICIP, pp. 4137–4140 (2010) Google Scholar
  20. 20.
    Chu, C.K., Glad, I.K., Godtliebsen, F., Marron, J.S.: Edge-preserving smoothers for image processing. J. Am. Stat. Assoc. 93(442), 526–556 (1998) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3D transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Deledalle, C.A., Duval, V., Salmon, J.: Non-local methods with shape-adaptive patches (NLM-SAP). IMA J. Numer. Anal. 43(2), 103–120 (2012) MATHMathSciNetGoogle Scholar
  23. 23.
    Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program., Ser. A 55(3), 293–318 (1992) CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Elad, M.: On the origin of the bilateral filter and ways to improve it. IEEE Trans. Image Process. 11(10), 1141–1151 (2002) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Esser, E.: Applications of Lagrangian-based alternating direction methods and connections to split-Bregman. UCLA CAM-Reports 09-31 (2009) Google Scholar
  26. 26.
    Esser, E., Zhang, X., Chan, T.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci. 3(4), 1015–1046 (2010) CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Fenn, M., Steidl, G.: Robust local approximation of scattered data. In: Geometric Properties from Incomplete Data, vol. 31, pp. 317–334 (2006) CrossRefGoogle Scholar
  28. 28.
    Gilboa, G., Osher, S.: Nonlocal linear image regularization and supervised segmentation. Multiscale Model. Simul. 6(2), 595–630 (2007) CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008) CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Goldstein, T., Osher, S.: The split Bregman method for L1 regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009) CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Gwosdek, P., Bruhn, A., Weickert, J.: Variational optic flow on the Sony PlayStation 3—accurate dense flow fields for real-time applications. J. Real-Time Image Process. 5(3), 163–177 (2010) CrossRefGoogle Scholar
  32. 32.
    Kamilov, U., Bostan, E., Unser, M.: Generalized total variation denoising via Augmented Lagrangian cycle spinning with Haar wavelets. In: IEEE ICASSP, pp. 909–912 (2012) Google Scholar
  33. 33.
    Kervrann, C., Boulanger, J.: Local adaptivity to variable smoothness for exemplar-based image regularization and representation. Int. J. Comput. Vis. 79(1), 45–69 (2008) CrossRefGoogle Scholar
  34. 34.
    Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comput. 37(155), 141–158 (1981) CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Levin, D.: The approximation power of moving least-squares. Math. Comput. 67(224), 1517–1531 (1998) CrossRefMATHGoogle Scholar
  36. 36.
    Lions, P.-L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. In: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, vol. 22. AMS, Providence (2001) Google Scholar
  38. 38.
    Mirzaei, D., Schaback, R., Dehghan, M.: On generalized moving least squares and diffuse derivatives. IMA J. Numer. Anal. 32(3), 983–1000 (2012) CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Mrazek, P., Meickert, J., Bruhn, A.: On robust estimation and smoothing with spatial and tonal kernels. In: Geometric Properties from Incomplete Data, vol. 31, pp. 335–352 (2006) CrossRefGoogle Scholar
  40. 40.
    Nayroles, B., Touzot, G., Villon, P.: Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10(5), 307–318 (1992) CrossRefMATHGoogle Scholar
  41. 41.
    Nikolova, M.: Local strong homogeneity of a regularized estimator. SIAM J. Appl. Math. 61(2), 633–658 (2000) CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Nikolova, M.: Weakly constrained minimization: application to the estimation of images and signals involving constant regions. J. Math. Imaging Vis. 21(2), 155–175 (2004) CrossRefMathSciNetGoogle Scholar
  43. 43.
    Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4(2), 460–489 (2005) CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Pizarro, L., Mrázek, P., Didas, S., Grewenig, S., Weickert, J.: Generalised nonlocal image smoothing. Int. J. Comput. Vis. 90(1), 62–87 (2010) CrossRefMathSciNetGoogle Scholar
  45. 45.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992) CrossRefMATHGoogle Scholar
  46. 46.
    Sapiro, G.: Geometric Partial Differential Equations and Image Processing. Cambridge University Press, Cambridge (2001) CrossRefGoogle Scholar
  47. 47.
    Schumaker, L.L.: Spline Functions: Basic Theory. Wiley, New York (1981) MATHGoogle Scholar
  48. 48.
    Setzer, S.: Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage. In: International Conf. Scale Space and Variational Methods in Computer Vis., vol. 5567, pp. 464–476. Springer, Berlin (2009) CrossRefGoogle Scholar
  49. 49.
    Setzer, S.: Operator splitting, Bregman methods and frame shrinkage in image processing. Int. J. Comput. Vis. 92(3), 265–280 (2011) CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Setzer, S.: Infimal convolution regularizations with discrete L1-type functionals. Commun. Math. Sci. 9(3), 797–827 (2011) CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    Stefan, W., Renaut, R.A., Gelb, A.: Improved total variation-type regularization using higher order edge detectors. SIAM J. Imaging Sci. 3(2), 232–251 (2010) CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Takeda, H., Farsiu, S., Milanfar, P.: Kernel regression for image processing and reconstruction. IEEE Trans. Image Process. 16(2), 349–366 (2007) CrossRefMathSciNetGoogle Scholar
  53. 53.
    Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Proc. IEEE Int. Conf. Computer Vision, pp. 839–846 (1998) Google Scholar
  54. 54.
    Unser, M., Blu, T.: Cardinal exponential splines: Part I: theory and filtering algorithms. IEEE Trans. Signal Process. 53(4), 1425–1438 (2005) CrossRefMathSciNetGoogle Scholar
  55. 55.
    Vogel, C.R., Oman, M.E.: Iterative methods for total variation denoising. SIAM J. Sci. Comput. 17(1), 227–238 (1996) CrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    Wang, C., Liu, Z.: Total variation for image restoration with smooth area protection. J. Signal Process. Syst. 61(3), 271–277 (2010) CrossRefGoogle Scholar
  57. 57.
    Wang, Y., Yin, W., Zhang, Y.: A fast algorithm for image deblurring with total variation regularization. CAAM Technical Report 07-10 (2007) Google Scholar
  58. 58.
    Winker, G., Aurich, V., Hahn, K., Martin, A., Rodenacker, K.: Noise reduction in images: some recent edge-preserving methods. Pattern Recognit. Image Anal. 9(4), 749–766 (1999) Google Scholar
  59. 59.
    Wu, C., Tai, X.-C.: Augmented Lagrangian method, dual methods, and split Bregman iterations for ROF, vectorial TV and higher order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010) CrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    Yoon, J., Lee, Y.: Nonlinear image upsampling method based on radial basis function interpolation. IEEE Trans. Image Process. 19(10), 2682–2692 (2010) CrossRefMathSciNetGoogle Scholar
  61. 61.
    Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM J. Imaging Sci. 3(3), 253–276 (2010) CrossRefMATHMathSciNetGoogle Scholar
  62. 62.
    Zhang, X., Burger, M., Osher, S.: A unified primal-dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46(1), 20–46 (2011) CrossRefMATHMathSciNetGoogle Scholar
  63. 63.
    Zhu, M., Chan, T.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration. UCLA CAM Reports 08-34 (2008) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute of Mathematical SciencesEwha W. Univ.SeoulSouth Korea
  2. 2.Dept. Software EngineeringDongseo Univ.BusanSouth Korea
  3. 3.Department of MathematicsEwha W. Univ.SeoulSouth Korea

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