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.


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


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