Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4605440 | Applied and Computational Harmonic Analysis | 2009 | 24 Pages |
Abstract
This paper is devoted to the study of local scales (oscillations) in images and use the knowledge of local scales for image decompositions. Denote byKt(x)=(eâ2Ït|ξ|2)â¨(x),t>0, the Gaussian (heat) kernel. Motivated from the Triebel-Lizorkin function space FËp,âα, we define a local scale of f at x to be t(x)⩾0 such that|Sf(x,t)|=|t1âα/2âKtâtâf(x)| is a local maximum with respect to t for some α<2. For each x, we obtain a set of scales that f exhibits at x. The choice of α and a local smoothing method of local scales via the nontangential control will be discussed. We then extend the work in [J.B. Garnett, T.M. Le, Y. Meyer, L.A. Vese, Image decomposition using bounded variation and homogeneous Besov spaces, Appl. Comput. Harmon. Anal. 23 (2007) 25-56] to decompose f into u+v, with u being piecewise-smooth and v being texture, via the minimization probleminfuâBV{K(u)=|u|BV+λâKt¯(â
)â(fâu)(â
)âL1}, where t¯(x) is some appropriate choice of a local scale to be captured at x in the oscillatory part v.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Peter W. Jones, Triet M. Le,