Local scales and multiscale image decompositions

• Published in: Applied and computational harmonic analysis Vol. 26; no. 3; pp. 371 - 394
• Main Authors: Jones, Peter W., Le, Triet M.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.05.2009
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2008.08.001

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 K t ( x ) = ( e − 2 π t | 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 | S f ( x , t ) | = | t 1 − α / 2 ∂ K t ∂ 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 ‖ K t ¯ ( ⋅ ) ∗ ( f − u ) ( ⋅ ) ‖ L 1 } , where t ¯ ( x ) is some appropriate choice of a local scale to be captured at x in the oscillatory part v.