Local scales and multiscale image decompositions

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)=(e−2πt|ξ|2)∨(x),t>0, the Gaussian (heat) kernel. Motivated from the Triebel-Lizorkin function space F˙p,∞α, we define a local scale of f at x to be t(x)⩾0 such that|Sf(x,t)|=|t1−α/2∂Kt∂t∗f(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 probleminfu∈BV{K(u)=|u|BV+λ‖Kt¯(⋅)∗(f−u)(⋅)‖L1}, where t¯(x) is some appropriate choice of a local scale to be captured at x in the oscillatory part v.