Multiscale Hierarchical Decomposition of Images with Applications to Deblurring, Denoising and Segmentation

摘要

We extend the ideas introduced in TNV04 for hierarchical multiscale decompositions of images. Viewed as a function f 2 L2, a given image is hierarchically decomposed into the sum or product of simpler 'atoms' uk, where uk extracts a more refined information from the previous scale uk (number sign)8722;1. To this end, the uk's are obtained as dyadically scaled minimizers of standard functionals arising in image analysis. Thus, starting with v(number sign)8722; 1:= f and letting vk denote the residual at a given dyadic scale, k 2k, then the recursive step uk, vk = arginfQT 'vk (number sign)8722;1, k' leads to the desired hierarchical decomposition, f PTuk; here T is a blurring operator. We characterize such QT -minimizers 'by duality' and expand our previous energy estimates of the data f in terms of kukk. Numerical results illustrate applications of the new hierarchical multiscale decomposition for blurry images, images with additive and multiplicative noise and image segmentation.