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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.

MSC:
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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