Abstract
We propose a new model for image decomposition which separates an image into a cartoon, consisting only of geometric objects, and an oscillatory component, consisting of textures and noise. The model is given in a variational formulation with adaptive regularization norms for both the cartoon and texture part. The energy for the cartoon interpolates between total variation regularization and isotropic smoothing, while the energy for the textures interpolates between Meyer’s G norm and the H − − 1 norm. These energies are dual in the sense of the Legendre-Fenchel transform and their adaptive behavior preserves key features such as object boundaries and textures while avoiding staircasing in what should be smooth regions. Existence and uniqueness of a solution is established and experimental results demonstrate the effectiveness of the model for both grayscale and color image decomposition.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Problems 10(6), 1217–1229 (1994)
Aubert, G., Aujol, J.-F.: Modeling very oscillating signals. Application to image processing. Appl. Math. Optim. 51(2), 163–182 (2005)
Aujol, J.-F., Aubert, G., Blanc-Feraud, L., Chambolle, A.: Image decomposition application to sar images. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space 2003. LNCS, vol. 2695, pp. 297–312. Springer, Heidelberg (2003)
Aujol, J.-F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Computer Vision 63(1), 85–104 (2005)
Blomgren, P., Chan, T.F., Mulet, P., Vese, L., Wan, W.L.: Variational PDE models and methods for image processing. In: Numerical analysis 1999 (Dundee). CRC Res. Notes Math, vol. 420, pp. 43–67. Chapman & Hall, Boca Raton (2000)
Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vision 20(1-2), 89–97 (2004)
Chambolle, A., DeVore, R.A., Lee, N.-y., Lucier, B.J.: Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7(3), 319–335 (1998)
Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)
Chan, T.F., Esedoglu, S., Park, F.: A fourth order dual method for staircase reduction in texture extraction and image restoration problems. Technical Report, University of California, Los Angeles, CA, CAM05-28 (2005)
Chan, T.F., Esedoglu, S., Park, F.: Image decomposition combining staircase reduction and texture extraction. Technical Report, University of California, Los Angeles, CA, CAM05-18 (2005)
Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image processing. Submitted to SIAM J. Appl. Math. (2004)
Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90(432), 1200–1224 (1995)
Ekeland, I., Temam, R.: Convex analysis and variational problems. Translated from the French, Studies in Mathematics and it’s Applications, vol. 1. North-Holland Publishing Co., Amsterdam (1976)
Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
Gousseau, Y., Morel, J.-M.: Are natural images of bounded variation? SIAM J. Math. Anal. 33(3), 634–648 (2001) (electronic)
Le, T.M., Vese, L.: Image decomposition using total variation and div(bmo). CAM04-76, UCLA (2004)
Malgouyres, F.: Minimizing the total variation under a general convex constraint for image restoration. IEEE Trans. Image Process. 11(12), 1450–1456 (2002)
Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series, vol. 22. American Mathematical Society, Providence (2001); The fifteenth Dean Jacqueline B. Lewis memorial lectures
Misner, T., Ramsey, M., Arrowsmith, J.: Multi-frequency, multitemporal brush fire scar analysis in a semi-arid urban environment: Implications for future fire and flood hazards. In: 2002 meeting of the American Geological Society (2002)
Osher, S., Solé, A., Vese, L.: Image decomposition and restoration using total variation minimization and the \(H\sp {-1}\) norm. Multiscale Model. Simul. 1(3), 349–370 (2003) (electronic)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Vese, L.A., Osher, S.J.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(1-3), 553–572 (2003); Special issue in honor of the sixtieth birthday of Stanley Osher
Vese, L.A., Osher, S.J.: Image denoising and decomposition with total variation minimization and oscillatory functions. J. Math. Imaging Vision 20(1-2), 7–18 (2004); Special issue on mathematics and image analysis
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Levine, S.E. (2005). An Adaptive Variational Model for Image Decomposition. In: Rangarajan, A., Vemuri, B., Yuille, A.L. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2005. Lecture Notes in Computer Science, vol 3757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11585978_25
Download citation
DOI: https://doi.org/10.1007/11585978_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30287-2
Online ISBN: 978-3-540-32098-2
eBook Packages: Computer ScienceComputer Science (R0)