Image Restoration Using Partial Differential Equation

편미분 방정식을 이용한 이미지 복원

  • Published : 2006.12.30


This paper is concerned with simulation issues arising in the PDE-based image restoration such as the total variation minimization(TVM) and its generalizations. In particular, we study the issues of staircasing and excessive dissipation of TVM-like smoothing operators. A strategy of scaling the algebraic system and a non-convex minimization are considered respectively for anti-staircasing and anti-diffusion. Furthermore, we introduce a variable constraint parameter to better preserve image edges. The resulting algorithm has been numerically verified to be efficient and reliable in denoising. Various numerical results are shown to confirm the claim.

본 논문은 총 변화량 최소화와 같은 편 미분방정식을 기본으로 한 영상 복원에 제기된 이슈에 관련된다. 총 변화량 최소화방법과 같은 평활화 연산자의 과도한 분산과 계단화와 같은 문제점들에 대하여 특별히 연구한다. 계단화와 과도한 분산을 방지하기 위하여 대수시스템에서의 축척과 비 오목형 최소화 기법이 각각 고려된다. 더군다나 에지를 좀더 잘 보존하기 위한 다양한 제약 매개변수가 소개된다. 제안된 알고리즘이 소음제거에 있어서 효율적이고 합리적임이 수학적으로 증명되며 다양한 실험 결과가 보여진다.


  1. L. ALVAREZ, P LIONS, AND MOREL, image selective smoothing and edge detection by nonlinear diffusion. II. SIAM J. Numer. Anal., 29(1992), pp.845-866
  2. G. AUBERT AND P. KORNPROBST, Mathematical Problems in Image Processing no.147 in Applied Mathematics Sciences, Springer- Verlag, New York, 2002
  3. P. V. BLOMGREN, Total Variation Methods for Restoration of Vector Valued Images, (Ph.D. thesis), UCLA Dept. of Math. CAM 98-30,(1998)
  4. P.V.BLOMGREN AND T.F.CHAN, COLOR TV: Total Variation Methods for Restoration of Vector Valued Images, IEEE Trans. Image Processing, 7 (1998), pp 304-309
  5. F.CATTE, P.LIONS, M.MOREL, AND T.COLL, Image selective smoothing and edge detection by nonlinear diffusion., SIAM J.Numer. Anal., 29 (1992), pp.182-193
  6. A. CHAMBOLLE AND P.L. LIONS, Image recovery via Total Variational minimization and related problems, Numer. Math., 76(1997), pp.167-188
  7. T.CHAN, S. OSHER, AND J. SHEN The distal TV filter and nonlinear denoising, Technical Report #99-34, Department of Mathematics, University of California, Los Angleles, CA 900095-1555, October 1999
  8. T.F CHAN, S. H. KANG, AND J. SHEN, Euler's Elastica and Curvature Based Inpaintings, SIAM Journal on Applied Mathematic, 63 (2002), pp.564-592
  9. T. F. CHAN, SH. H. KANG, AND J.SHEN, Variational restoration of non-flat image features: Models and algorithms, SIAM Journal of Applied Mathematics, 61 (2000), pp.1388-1361
  10. T. F. CHAN, SH. H. KANG, AND J.SHEN, Mathematical Models for Local Deterministic Inpaintings, SIAM Journal of Applied Mathematics, 62 (2001), pp,1019-1043
  11. K.JOO AND S.KIM, PDE-based image restoration, II: Numerical schemes and color image denoising. (preprint)
  12. S.KlM, Edge-preserving noise removal, Part I: Second-order anisotropic diffusion. (Submitted to SIAM J.Sci. Comput.)
  13. S.KlM, I.IACOM, AND M. TYNAN, MinBAD: The minimum-biased anisotropic diffusion for noise removal, Technical Report #2002-06, Department of Mathematics, University of Kentcky, Lexington, KY 40506,2002
  14. S. KIM, AND S. H. KANG, Implicit procedures for PDE-based color image denoising via brightness-chromaticity decomposition, Technical Report #2002.Q7, Department of Mathematics, University of Kentucky, Lexington, KY 40506, 2002
  15. R. KIMMEL AND N.SOCHEN, Orientation Diffusion or How to Comb a Porcupine?, Special issue on PDEs in Image Processing, Computer Vision, and Computer Graphics, J. Visual Comm, Image Representation, 13 (2002), pp.238-248
  16. A. MARQUINA AND S. OSHER, Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal, SIAM J.Sci. Comput., 22(2000), pp.387-405
  17. S. MASNOU AND J. M. MOREL, Level Lines based Disocclusion, Proceedings of 5th IEEE Int'I Conf. on Image Process., Chicago, 3 (1998), pp.259-263
  18. M.NITZBERG AND T. SHIOTA, Nonlinear image filtering with edge and corner enhancement, IEEE Trans. on Pattern Anal. Mach. lntell., 14 (1992), pp. 826-833
  19. S.OSHER AND R. FEDKIW, Level Set Methods and Dynamic Implicit Surfaces Springer-Varlag New York, 2003
  20. P.PERONA AND J. MALIK, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. on Pattern Anal. Mach. lntell., 12 (1990), pp. 629-639
  21. L.RUDIN AND S. OSHER. Total variation based image restoration with free local constraints, Proc. 1st IEEE ICIP, 1(1994), PP. 31-35
  22. L.RUDlN, S. OSHER, AND E. FATEMI, Nonlinear total varivation based noise removal algorithms, Physica D, 60 (1992), pp.259-268
  23. G. SAPlRO AND D. RINGACH, Anisotropic diffusion of multi valued images with application to color filtering, IEEE Trans. Image Processing, 5 (1996), pp.1582-1586
  24. N. SOCHEN, R. KIMMEL, AND R. MALLADl, A geometrical framework for low level vision, IEEE Trans. Image Processing, 7 (1998), pp.310-318
  25. B. TANG, G. SAPIRO, AND V. CASELLES, Color image enhancement via Chromaticity Diffusion, Teclmical report, ECE University of Minnesota, (1999)
  26. R. WEINSTOCK, Calculus of Variations, Dover Phbilcations. Inc., New York, 1974
  27. Y. You, W. Xu, A, TANNENBAUM, AND M. KAVEH, Bechavioral analysis of anisotropic diffusion in image processing, IEEE Trans. Image Process., 5 (1996), pp.1539-1553