Journal
APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS
Volume 19, Issue 1, Pages 1-16Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.acha.2004.12.004
Keywords
contour and texture analysis; near BV restoration; nonlinear wavelet decomposition; deblurring and denoising
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Inspired by papers of Vese-Osher [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and Osher-Sole-Vese [Image decomposition and restoration using total variation minimization and the H-1 norm, Technical Report 02-57, 2002] 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 [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and [Image decomposition and restoration using total variation minimization and the H-1 norm, Technical Report 02-57, 20021, 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 [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and [Image decomposition and restoration using total variation minimization and the H-1 nonn, Technical Report 02-57, 2002]. 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. (c) 2004 Elsevier Inc. All rights reserved.
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