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 is an element of L-2 (Omega), 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, lambda(k) similar to 2(k), the recursive step [u(k), v(k)] - arginf Q(T) (v(k-1), lambda(k)) leads to the desired hierarchical decomposition, f similar to Sigma Tu(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 parallel to u(k)parallel to. Numerical results illustrate applications of the new hierarchical multiscale decomposition for blurry images, images with additive and multiplicative noise and image segmentation.