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

We extend the ideas introduced in [33] for hierarchical multiscale decompositions of images. Viewed as a function f ∈L(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 v−1 :=f and letting vk denote the residual at a given dyadic scale, λk ∼2 , the recursive step [uk,vk]=arginfQT (vk−1,λk) leads to the desired hierarchical decomposition, f ∼ P Tuk; 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 ‖uk‖. Numerical results illustrate applications of the new hierarchical multiscale decomposition for blurry images, images with additive and multiplicative noise and image segmentation.