Multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation

Abstract

We extend the ideas introduced in [33] for hierarchical multiscale decompositions of images. Viewed as a function f ∈ L2 (Ω), a given image is hierarchically decomposed into the sum or product of simpler "atoms" uk, where uk extracts more refined information from the previous scale uk-1. To this end, the uk's are obtained as dyadically scaled minimizers of standard functionals arising in image analysis. Thus, starting with ν-1:= f and letting νk denote the residual at a given dyadic scale, λk∼2k, the recursive step [uk,νk] = arginfT (νk-1, λk) leads to the desired hierarchical decomposition, f∼∑Tuk; here T is a blurring operator. We characterize such T-minimizers (by duality) and expand our previous energy estimates of the data f in terms of ∥uk∥. Numerical results illustrate applications of the new hierarchical multiscale decomposition for blurry images, images with additive and multiplicative noise and image segmentation. © 2008 International Press.