Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259-268]. Here, u 0 should capture "essential features" of f which are to be separated from the spurious components absorbed by v 0 , and λ 0 is a fixed threshold which dictates separation of scales. To proceed, we iterate the refinementWe focus our attention on the particular case of (X, Y ) = (BV, L 2 ) decomposition. The resulting hierarchical decomposition, f ∼ j u j , is essentially nonlinear. The questions of convergence, energy decomposition, localization, and adaptivity are discussed. The decomposition is constructed by numerical solution of successive Euler-Lagrange equations. Numerical results illustrate applications of the new decomposition to synthetic and real images. Both greyscale and color images are considered.

Abstract. 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.

Abstract. 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.