International Journal of Computer Vision

, Volume 67, Issue 1, pp 111–136 | Cite as

Structure-Texture Image Decomposition—Modeling, Algorithms, and Parameter Selection

  • Jean-François AujolEmail author
  • Guy Gilboa
  • Tony Chan
  • Stanley Osher


This paper explores various aspects of the image decomposition problem using modern variational techniques. We aim at splitting an original image f into two components u and ρ, where u holds the geometrical information and ρ holds the textural information. The focus of this paper is to study different energy terms and functional spaces that suit various types of textures. Our modeling uses the total-variation energy for extracting the structural part and one of four of the following norms for the textural part: L2, G, L1 and a new tunable norm, suggested here for the first time, based on Gabor functions. Apart from the broad perspective and our suggestions when each model should be used, the paper contains three specific novelties: first we show that the correlation graph between u and ρ may serve as an efficient tool to select the splitting parameter, second we propose a new fast algorithm to solve the TVL1 minimization problem, and third we introduce the theory and design tools for the TV-Gabor model.


image decomposition restoration parameter selection BV G L1 Hilbert space projection total-variation Gabor functions 


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  1. Adams, R. 1975. Sobolev Spaces. Pure and applied Mathematics. Academic Press, Inc.Google Scholar
  2. Aliney, S. 1997. A property of the minimum vectors of a regularizing functional defined by means of the absolute norm. IEEE Transactions on Signal Processing, 45(4):913–917.Google Scholar
  3. Aubert, G. and Aujol, J.F. 2005. Modeling very oscillating signals. Application to image processing. Applied Mathematics and Optimization, 51(2):163–182.MathSciNetGoogle Scholar
  4. Aubert, G. and Kornprobst, P. 2002. Mathematical Problems in Image Processing, vol. 147 of Applied Mathematical Sciences. Springer-Verlag, 2002.Google Scholar
  5. Aujol, J.F., Aubert, G., Blanc-Féraud, L., and Chambolle, A. 2005. Image decomposition into a bounded variation component and an oscillating component. Journal of Mathematical Imaging and Vision, 22(1):71–88.CrossRefGoogle Scholar
  6. Aujol, J.F., Aubert, G., Blanc-Féraud, L., and Antonin Chambolle. 2003. Decomposing an image: Application to SAR images. In Scale-Space '03, volume 2695 of Lecture Notes in Computer Science, 2003.Google Scholar
  7. Aujol, J.F. and Chambolle, A. 2005. Dual norms and image decomposition models. International Journal on Computer Vision, 63(1):85–104.Google Scholar
  8. Aujol, J.F. and Gilboa, G. 2004. Implementation and parameter selection for BV-Hilbert space regularizations, 2004. UCLA CAM Report 04-66.Google Scholar
  9. Aujol, J.F., Gilboa, G., Chan, T., and Osher, S. 2005. Structure-texture image decomposition—modeling, algorithms, and parameter selection, 2005. UCLA CAM Report 05-10,
  10. Aujol, J.F. and Kang, S.H. 2004. Color image decomposition and restoration, 2004. UCLA CAM Report 04-73, to appear in Journal of Visual Communication and Image Representation.Google Scholar
  11. Bect, J., Aubert, G., Blanc-Féraud, L., and Chambolle, A. 2004. A l1 unified variational framwork for image restoration. In ECCV 2004.Google Scholar
  12. Chambolle, A. 2004. An algorithm for total variation minimization and applications. JMIV, 20:89–97.MathSciNetGoogle Scholar
  13. Chambolle, A. and Lions, P.L. 1997. Image recovery via total variation minimization and related problems. Numerische Mathematik, 76(3):167–188.MathSciNetGoogle Scholar
  14. Chambolle, A., De Vore, R.A., Lee, N., and Lucier, B.J. 1998. Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage. IEEE Transcations on Image Processing, 7(3):319–335.Google Scholar
  15. Chan, T. and Esedoglu, S.2004. Aspects of total variation regularized L1 function approximation, 2004. CAM report 04-07, to appear in SIAM Journal on Applied Mathematics.Google Scholar
  16. Combettes, P.L. and Wajs, V.R. 2004. Theoretical analysis of some regularized image denoising methods. In ICIP 04, vol. 1, pp. 969–972.Google Scholar
  17. Combettes, P.L. and Wajs, V.R. 2005. Signal recovery by proximal forward-backward splitting. SIAM Journal on Multiscale Modeling and Simulation, in press.Google Scholar
  18. Combettes, P.L. and Luo, J. 2002. An adaptative level set method for nondifferentiable constrained image recovery. IEEE Transactions on Image Processing, 11(11):1295–1304.CrossRefMathSciNetGoogle Scholar
  19. Daubechies, I. and Teschke, G. 2004. Variational image restoration by means of wavelets: simultaneous decomposition, deblurring and denoising, submitted.Google Scholar
  20. Donoho, D.L. and Johnstone, M. 1995. Adapting to unknown smoothness via wavelet shrinkage. Journal of the American Statistical Association, 90(432):1200–1224.MathSciNetGoogle Scholar
  21. Dunn, D. and Higgins, W.E. 1995. Optimal Gabor filters for texture segmentation. IEEE Transactions on Image Processing, 4(7):947–964.Google Scholar
  22. Ekeland, I. and Temam, R. 1974. Analyse convexe et problmes variationnels. Etudes Mathématiques. Dunod, 1974.Google Scholar
  23. Gabor, D. 1946. Theory of communication. J. Inst. of Electrical Engineering, 93(3):429–457.Google Scholar
  24. Gilboa, G., Sochen, N., and Zeevi, Y.Y. submitted. Estimation of optimal PDE-based denoising in the SNR sense.Google Scholar
  25. Gilboa, G., Sochen, N., and Zeevi, Y.Y. submitted. Variational denoising of partly-textured images by spatially varying constraints.Google Scholar
  26. Gilboa, G., Sochen, N., and Zeevi, Y.Y. 2003. Texture preserving variational denoising using an adaptive fidelity term. In Proc. VLSM 2003, Nice, France, pp. 137–144.Google Scholar
  27. Gilboa, G., Sochen, N., and Zeevi, Y.Y. 2004. Estimation of optimal PDE-based denoising in the SNR sense, 2004. CCIT report No. 499, Technion, August, see
  28. Gilboa, G., Sochen, N., and Zeevi, Y.Y. 2005. Estimation of the optimal variational parameter via SNR analysis. In Scale-Space '05, volume 3459 of Lecture Notes in Computer Science, pp. 230–241.Google Scholar
  29. Groetsch, C. and Scherzer, O. 2001. Inverse scale space theory for inverse problems. In Scale-Space '01, volume 2106 of Lecture Notes in Computer Science, pp. 317–325.Google Scholar
  30. Hintermuller, M. and Kunisch, K. 2004. Total bounded variation regularization as bilaterally constrained optimization problem. SIAM Journal on Applied Mathematics, 64(4):1311–1333.MathSciNetGoogle Scholar
  31. Jain, A.K. and Farrokhnia, F. 1991. Unsupervised texture segmentation using Gabor filters. Pattern Recognition, 24(12):1167–1186.CrossRefGoogle Scholar
  32. Le, T. and Vese, L. 2004. Image decomposition using total variation and div(BMO), 2004. UCLA CAM Report 04-36.Google Scholar
  33. Malgouyres, F. 2002. Mathematical analysis of a model which combines total variation and wavelet for image restoration. Journal of information processes, 2(1):1–10.Google Scholar
  34. Malgouyres, F. 2002. Minimizing the total variation under a general convex constraint for image restoration. IEEE transactions on image processing, 11(12):1450–1456.CrossRefMathSciNetGoogle Scholar
  35. Mallat, S.G. 1989. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7):674–693.Google Scholar
  36. Mallat, S.G. 1998. A Wavelet Tour of Signal Processing. Academic Press, 1998.Google Scholar
  37. Meyer, Y. 2001. Oscillating patterns in image processing and in some nonlinear evolution equations, March 2001. The Fifteenth Dean Jacquelines B. Lewis Memorial Lectures.Google Scholar
  38. Morosov, V.A. 1966. On the solution of functional equations by the method of regularization. Soviet Math. Dokl., 7:414–417.MathSciNetGoogle Scholar
  39. Mrázek, P. and Navara, M. 2003. Selection of optimal stopping time for nonlinear diffusion filtering. IJCV, 52(2/3):189–203.CrossRefGoogle Scholar
  40. Nikolova, M. 2004. A variational approach to remove outliers and impulse noise. JMIV, 20(1–2):99–120.MathSciNetGoogle Scholar
  41. Osher, S., Burger, M., Goldfarb, D., Xu, J. and Yin, W. 2004. An iterative regularization method for total variation based image restoration, 2004. CAM report 04-13, to appear in SIAM Journal on Multiscale Modeling and Simulation.Google Scholar
  42. Osher, S.J., Sole, A. and Vese, L.A. 2003. Image decomposition and restoration using total variation minimization and the H-1 norm. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 1(3):349–370.MathSciNetGoogle Scholar
  43. Porat, M. and Zeevi, Y.Y. 1988. The generalized Gabor scheme of image representation in biological and machine vision. IEEE Transactions on Pattern Analysis and Machine Intelligence, 10(4):452–468.Google Scholar
  44. Rudin, L., Osher, S., and Fatemi, E. 1992. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268.CrossRefGoogle Scholar
  45. Starck, J.L., Elad, M., and Donoho, D.L. 2003. Image decomposition: separation of texture from piecewise smooth content, 2003. To appear in IEEE Transactions on Image Processing.Google Scholar
  46. Steidl, G., Didas, S., and Neumann, J. 2005. Relations between higher order TV regularization and support vector regression. In Scale-Space and PDE methods in computer vision, R. Kimmel, N. Sochen, J. Weickert Eds, volume LNCS 3459, pp. 515–527.Google Scholar
  47. Steidl, G., Weickert, J., Brox, T., Mrazek, P., and Welk, M. 2004. On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and sides. SIAM J. Numer. Anal., 42:686–658.CrossRefMathSciNetGoogle Scholar
  48. Strong, D., Aujol, J.F., and Chan, T.F. 2005. Scale recognition, regularization parameter selection, and Meyer's G norm in total variation regularization, January 2005. UCLA CAM Report 05-02.Google Scholar
  49. Tadmor, E., Nezzar, S., and Vese, L. 2004. A multiscale image representation using hierarchical (BV,L2) decompositions. SIAM Journal on Multiscale Modeling and Simulation, 2(4):554–579.CrossRefMathSciNetGoogle Scholar
  50. Vese, L.A. and Osher, S.J. 2003. Modeling textures with total variation minimization and oscillating patterns in image processing. Journal of Scientific Computing, 19:553–572.CrossRefMathSciNetGoogle Scholar
  51. Yin, W., Goldfarb, D., and Osher, S. 2005. Total variation based image cartoon-texture decomposition, 2005. Columbia University CORC Report TR-2005-01, UCLA CAM Report 05-27.Google Scholar
  52. Zibulski, M. and Zeevi, Y.Y. 1997. Analysis of multi-window Gabor-type schemes by frame methods. J. Appl. Comp. Harmon. Anal. 4(2):188–221.MathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Jean-François Aujol
    • 1
    • 2
    Email author
  • Guy Gilboa
    • 1
  • Tony Chan
    • 1
  • Stanley Osher
    • 1
  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.CMLA (CNRS UMR 8536)ENS CachanFrance

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