Daubechies, I.; Teschke, G. Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising. (English) Zbl 1079.68104 Appl. Comput. Harmon. Anal. 19, No. 1, 1-16 (2005). 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. Cited in 2 ReviewsCited in 45 Documents 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 Keywords:Contour and texture analysis; Near BV restoration; Nonlinear wavelet decomposition; Deblurring and denoising Citations:Zbl 1034.49039; Zbl 1051.49026 Software:DT-CWT × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Cohen, A.; Daubechies, I.; Feauveau, J.-C., Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 45, 485-560 (1992) · Zbl 0776.42020 [2] Cohen, A.; DeVore, R.; Petrushev, P.; Xu, H., Nonlinear approximation and the space \(BV(R^2)\), Am. J. Math., 121, 587-628 (1999) · Zbl 0931.41019 [3] Coifman, R. 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