Abstract
The curvelet is more suitable for image processing than the wavelet and able to represent smooth and edge parts of image with sparsity. Based on this, we present a new model for image restoration and decomposition via curvelet shrinkage. The new model can be seen as a modification of Daubechies-Teschke’s model. By replacing the B β p,q term by a G β p,q term, and writing the problem in a curvelet framework, we obtain elegant curvelet shrinkage schemes. Furthermore, the model allows us to incorporate general bounded linear blur operators into the problem. Various numerical results on denoising, deblurring and decomposition of images are presented and they show that the model is valid.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Rudin, L., Osher, S., Fatemi, E.: Nolinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series, vol. 22. American Mathematical Society, Providence (2001)
Vese, L., Osher, S.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(1–3) (2003)
Vese, L., Osher, S.: Image denoising and decomposition with total variation minimization and oscillatory functions. J. Math. Imaging Vis. (1–2), 7–18 (2004)
Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H −1 norm. SIAM J. Multiscale Model. Simul. 1–3, 349–370 (2003)
Lieu, L., Vese, L.: Image restoration and decompostion via bounded total variation and negative Hilbert-Sobolev spaces. Applied Mathematics and Optimization (2007, in press)
Daubechies, I., Teschke, G.: Wavelet based image decomposition by variational functionals. In: Proc. SPIE, Wavelet Applications in Industrial Processing, vol. 5266, pp. 94–105 (2004)
Candès, E.J., Donoho, D.L.: Curvelets—a surprisingly effective nonadaptive representation for objects with edges. In: Rabut, C., Cohen, A., Schumaker, L.L. (eds.) Curves and Surfaces, pp. 105–120. Vanderbilt University Press, Nashville (2000)
Candès, E.J., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with piecewise-C 2 singularities. Commun. Pure Appl. Math. 57(2), 219–266 (2004)
Borup, L., Nielsen, M.: Frame decomposition of decomposition spaces. J. Fourier Anal. Appl. 13(1), 39–70 (2007)
Mumford, D., Gidas, B.: Stochastic models for generic images. Q. Appl. Math. 59, 85–111 (2001)
Daubechies, I., Defrise, M., DeMol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57, 1413–1541 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jiang, L., Feng, X. & Yin, H. Variational Image Restoration and Decomposition with Curvelet Shrinkage. J Math Imaging Vis 30, 125–132 (2008). https://doi.org/10.1007/s10851-007-0051-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-007-0051-4