Multiscale hierarchical decomposition of images with applications to deblurring, denoising, and segmentation
Suzanne Nezzar, Eitan Tadmor, Luminita A. Vese · Communications in Mathematical Sciences · 2008
We extend the ideas introduced in [33] for hierarchical multiscale decompositions of images.Viewed as a function f ∈ L 2 (Ω), a given image is hierarchically decomposed into the sum or product of simpler "atoms" u k , where u k extracts more refined information from the previous scale u k-1 .To this end, the u k 's are obtained as dyadically scaled minimizers of standard functionals arising in image analysis.Thus, starting with v -1 := f and letting v k denote the residual at a given dyadic scale, λ k ∼ 2 k , the recursive step [u k ,v k ] = arginf Q T (v k-1 ,λ k ) leads to the desired hierarchical decomposition, f ∼ P T u k ; here T is a blurring operator.We characterize such Q T -minimizers (by duality) and expand our previous energy estimates of the data f in terms of u k .Numerical results illustrate applications of the new hierarchical multiscale decomposition for blurry images, images with additive and multiplicative noise and image segmentation.