## Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising.(English)Zbl 1079.68104

Summary: Inspired by papers of L. Vese and S. Osher [(*) “Modeling textures with total variation minimization and oscillating patterns in image processing”, J. Sci. Comput. 19, 553–572 (2003; Zbl 1034.49039)] and S. Osher, A. Solé and L. Vese [(**) “Image decomposition and restoration using total variation minimization and the $$H^{-1}$$ norm”, Multiscale Model. Simul. 1, 349–370 (2003; Zbl 1051.49026)], we present a wavelet-based treatment of variational problems arising in the field of image processing. In particular, we follow their approach and discuss a special class of variational functionals that induce a decomposition of images into oscillating and cartoon components and possibly an appropriate ‘noise’ component. In the setting of (*) and (**), the cartoon component of an image is modeled by a $$BV$$ function; the corresponding incorporation of $$BV$$ penalty terms in the variational functional leads to PDE schemes that are numerically intensive. By replacing the $$BV$$ penalty term by a $$B_1^1(L_1)$$ term (which amounts to a slightly stronger constraint on the minimizer), and writing the problem in a wavelet framework, we obtain elegant and numerically efficient schemes with results very similar to those obtained in (*) and (**). This approach allows us, moreover, to incorporate general bounded linear blur operators into the problem so that the minimization leads to a simultaneous decomposition, deblurring and denoising.

### MSC:

 68U10 Computing methodologies for image processing 49N90 Applications of optimal control and differential games 65T60 Numerical methods for wavelets 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory

### Citations:

Zbl 1034.49039; Zbl 1051.49026

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### References:

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