@article {IOPORT.05995322,
    author       = {Garnett, John B. and Jones, Peter W. and Le, Triet M. and Vese, Luminita A.},
    title        = {Modeling oscillatory components with the homogeneous spaces ${B\dot MO}^{-\alpha}$ and $\dot W^{-\alpha,p}$.},
    year         = {2011},
    journal      = {Pure and Applied Mathematics Quarterly},
    volume       = {7},
    number       = {2},
    issn         = {1558-8599},
    pages        = {275-318},
    publisher    = {International Press, Somerville, MA},
    abstract     = {Summary: This paper is devoted to the decomposition of an image $f$ into $u+v$, with $u$ a piecewise-smooth or ``cartoon" component, and $v$ an oscillatory component (texture or noise), in a variational approach. The cartoon component $u$ is modeled by a function of bounded variation, while $v$, usually represented by a square integrable function, is now being modeled by a more refined and weaker texture norm, as a distribution. Generalizing the idea of {\it Y. Meyer} [Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series. 22. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0987.35003)], where $v\in F=\text{div}(BMO)=B\dot MO^{-1}$, we model here the texture component by the action of the Riesz potentials on $v$ that belongs to $BMO$ or to $L^p$. In an earlier work [{\it T. M. Le} and {\it L. A. Vese}, Multiscale Model. Simul. 4, No. 2, 390--423 (2005; Zbl 1093.94006)], the authors proposed energy minimization models to approximate $(BV,F)$ decompositions explicitly expressing the texture as divergence of vector fields in $BMO$. In this paper, we consider an equivalent more isotropic norm of the space $F$ in terms of the Riesz potentials, and study models where the Riesz potentials of oscillatory components belong to $BMO$ or to $L^p$, $1\le p<\infty$ (thus we consider oscillatory components in $B\dot MO^\alpha$ or in $\dot W^{\alpha,p}$, with $\alpha<0)$. Theoretical, experimental results and comparisons to validate the proposed methods are presented.},
    identifier   = {05995322},
}