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

Abstract

We extend the ideas introduced in Viewed as a function f L 2 (), 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, k 2 k , the recursive step [u k ,v k ] = arginf Q T (v k-1 , 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.

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