Multiscale hierarchical decomposition methods for images corrupted by multiplicative noise

Recovering images corrupted by multiplicative noise is a well known challenging task. Motivated by the success of multiscale hierarchical decomposition methods (MHDM) in image processing, we adapt a variety of both classical and new multiplicative noise removing models to the MHDM form. On the basis of previous work, we further present a tight and a refined version of the corresponding multiplicative MHDM. We discuss existence and uniqueness of solutions for the proposed models, and additionally, provide convergence properties. Moreover, we present a discrepancy principle stopping criterion which prevents recovering excess noise in the multiscale reconstruction. Through comprehensive numerical experiments and comparisons, we qualitatively and quantitatively evaluate the validity of all proposed models for denoising and deblurring images degraded by multiplicative noise. By construction, these multiplicative multiscale hierarchical decomposition methods have the added benefit of recovering many scales of an image, which can provide features of interest beyond image denoising

Discretization of variational regularization in Banach spaces

Consider a nonlinear ill-posed operator equation $F(u)=y$ where $F$ is defined on a Banach space $X$. In general, for solving this equation numerically, a finite dimensional approximation of $X$ and an approximation of $F$ are required. Moreover, in general the given data \yd of $y$ are noisy. In this paper we analyze finite dimensional variational regularization, which takes into account operator approximations and noisy data: We show (semi-)convergence of the regularized solution of the finite dimensional problems and establish convergence rates in terms of Bregman distances under appropriate sourcewise representation of a solution of the equation. The more involved case of regularization in nonseparable Banach spaces is discussed in detail. In particular we consider the space of finite total variation functions, the space of functions of finite bounded deformation, and the $L^\infty$--space

Synthesis and photophysical characteristics of polyfluorene polyrotaxanes

Two alternating polyfluorene polyrotaxanes (3·TM-βCD and 3·TM-γCD) have been synthesized by the coupling of 2,7-dibromofluorene encapsulated into 2,3,6-tri-O-methyl-β- or γ-cyclodextrin (TM-βCD, TM-γCD) cavities with 9,9-dioctylfluorene-2,7-diboronic acid bis(1,3-propanediol) ester. Their optical, electrochemical and morphological properties have been evaluated and compared to those of the non-rotaxane counterpart 3. The influence of TM-βCD or TM-γCD encapsulation on the thermal stability, solubility in common organic solvents, film forming ability was also investigated. Polyrotaxane 3·TM-βCD exhibits a hypsochromic shift, while 3·TM-γCD displays a bathochromic with respect to the non-rotaxane 3 counterpart. For the diluted CHCl3 solutions the fluorescence lifetimes of all compounds follow a mono-exponential decay with a time constant of ≈0.6 ns. At higher concentration the fluorescence decay remains mono-exponential for 3·TM-βCD and polymers 3, with a lifetime τ = 0.7 ns and 0.8 ns, whereas the 3·TM-γCD polyrotaxane shows a bi-exponential decay consisting of a main component (with a weight of 98% of the total luminescence) with a relatively short decay constant of τ1 = 0.7 ns and a minor component with a longer lifetime of τ2 = 5.4 ns (2%). The electrochemical band gap (ΔEg) of 3·TM-βCD polyrotaxane is smaller than that of 3·TM-γCD and 3, respectively. The lower ΔEg value for 3·TM-βCD suggests that the encapsulation has a greater effect on the reduction process, which affects the LUMO energy level value. Based on AFM analysis, 3·TM-βCD and 3·TM-γCD polyrotaxane compounds exhibit a granular morphology with lower dispersity and smaller roughness exponent of the film surfaces in comparison with those of the neat copolymer 3

The Residual Method for Regularizing Ill-Posed Problems

Although the \emph{residual method}, or \emph{constrained regularization}, is frequently used in applications, a detailed study of its properties is still missing. This sharply contrasts the progress of the theory of Tikhonov regularization, where a series of new results for regularization in Banach spaces has been published in the recent years. The present paper intends to bridge the gap between the existing theories as far as possible. We develop a stability and convergence theory for the residual method in general topological spaces. In addition, we prove convergence rates in terms of (generalized) Bregman distances, which can also be applied to non-convex regularization functionals. We provide three examples that show the applicability of our theory. The first example is the regularized solution of linear operator equations on $L^p$-spaces, where we show that the results of Tikhonov regularization generalize unchanged to the residual method. As a second example, we consider the problem of density estimation from a finite number of sampling points, using the Wasserstein distance as a fidelity term and an entropy measure as regularization term. It is shown that the densities obtained in this way depend continuously on the location of the sampled points and that the underlying density can be recovered as the number of sampling points tends to infinity. Finally, we apply our theory to compressed sensing. Here, we show the well-posedness of the method and derive convergence rates both for convex and non-convex regularization under rather weak conditions.Comment: 29 pages, one figur