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Image decompositions using bounded variation and generalized homogeneous Besov spaces. (English) Zbl 1118.68176
The paper is devoted to the decomposition of an image into a piecewise-smooth component and an oscillatory component, in a variational approach. [Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations. The fifteenth Dean Jacqueline B. Lewis memorial lectures. University Lecture Series. 22. Providence, RI: American Mathematical Society (AMS). (2001; Zbl 0987.35003)] proposed refinements of the total variation model [L. I. Rudin, S. Osher, E. Fatemi, Physica D 60, No. 1–4, 259–268 (1992; Zbl 0780.49028)] that better represent the oscillatory part: the weaker spaces of generalized functions \(G=div(L^{\infty})\), \(F=\text{div}(\text{BMO})\) and \(E=\dot{B}^{-1}_{\infty, \infty}\) have been proposed to model the oscillatory part, instead of the standard \(L^2\) space, while keeping the piecewise-smooth part in BV, a function of bounded variation. Such new models separate better geometric structures from oscillatory structures, but it is difficult to realize them in practice. In the paper a generalization of Meyer’s \((\text{BV},E)\) model is proposed, using homogeneous Besov spaces \(\dot{B}^{\alpha}_{p, q}\), \(-2<\alpha<0\), \(1 \leq p, q \leq \infty\) to represent the oscillatory part. Theoretical, experimental results and comparisons to validate the proposed methods are presented.

68U10 Computing methodologies for image processing
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
49J10 Existence theories for free problems in two or more independent variables
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