Local scales and multiscale image decompositions

Article history: Received 17 April 2008 Revised 19 August 2008 Accepted 21 August 2008 Communicated by Charles K. Chui This paper is devoted to the study of local scales (oscillations) in images and use the knowledge of local scales for image decompositions. Denote by Kt(x)= ( e−2πt|ξ |2 )∨ (x), t > 0, the Gaussian (heat) kernel. Motivated from the Triebel–Lizorkin function space Ḟα p,∞ , we define a local scale of f at x to be t(x) 0 such that ∣∣S f (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 problem inf u∈BV { K(u)= |u|BV + λ ∥∥Kt̄(·) ∗ ( f − u)(·)∥∥L1}, where t̄(x) is some appropriate choice of a local scale to be captured at x in the oscillatory part v . © 2008 Elsevier Inc. All rights reserved.