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

Abstract

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.

Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability

“…By the previous Remark 1, here and in what follows we will assume that |Ω| = 1 and Ω k(x)dx = 1. We prove in this section additional properties of minimizers of problem (8) related to uniqueness issues and a characterization using dual residual norm inspired from prior work [28], [4], [21] and [35]. Note that our functional in ( 8) is convex, but not strictly convex in the pair variable (u, g).…”

“…Jones, T.M. Le and the second author in [15] to model oscillatory components in natural images, in the case K = identity ( [15] is an alternative way to represent oscillatory details in images, in addition to other prior work by Aujol and collaborators [6], [7], [8], Le and collaborators [21], [16], [41], Starck et al [33], Levine, [22], or a hierarchical approach in Tadmor et al [34], [35], among others). We will make this choice to model the oscillatory component v of the recovered image, therefore the proposed deblurring model is a continuation of the work [15].…”

Multiscale hierarchical decomposition methods for ill-posed problems

“…Let us start with a decomposition result for the norm of the data f which, adapted to TVdeblurring, can be found in theorem 2.8 of [2]. Due to the flexibility of the penalty terms considered for this extension, the result can be transferred to related iterative schemes, as we will illustrate below.…”

“…In their influential works from 2004 and 2008, Tadmor et al [1,2] introduced a multiscale decomposition method for image denoising, deblurring and segmentation, based on the popular total variation (TV) model of Rudin, Osher and Fatemi (ROF) [3]. Recall that ROF decomposes an image f ∈ L 2 (Ω) in cartoon and texture as f = u λ + v λ such that Here Ω is a bounded and open set in R 2 .…”

“…Here T * : H → X * stands for the adjoint of the operator T. While for the related method (1.12) comprehensive convergence results exist in the corresponding references mentioned above, the situation is different for the MHDM defined with a non-quadratic penalty J in infinitedimensional spaces. Intriguingly, there is no convergence result for the sequence of iterates (x n ) apart from the denoising case (when T is the identity), which is a consequence of the weak-/strong convergence for the residual (Tx n − f) (see [1,2,5]) or of the residual error estimates in [15]. As regards the penalty J, this was assumed so far to be a (power of a) seminorm.…”