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Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising. (English) Zbl 1079.68104
Summary: Inspired by papers of {\it L. Vese} and {\it 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 {\it S. Osher, A. Solé} and {\it 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.

68U10Image processing (computing aspects)
49N90Applications of optimal control and differential games
65T60Wavelets (numerical methods)
94A08Image processing (compression, reconstruction, etc.)
Full Text: DOI
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