Celebratio Mathematica — Daubechies

Complete Bibliography

Works connected to Gerd Teschke

Filter the Bibliography List

year	title	people

			clear

I. Daubechies and G. Teschke: “Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising,” Appl. Comput. Harmon. Anal. 19 : 1 (2005), pp. 1–16. MR 2147059 Zbl 1079.68104 article

Inspired by papers of Vese–Osher [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and Osher–Solé–Vese [Image decomposition and restoration using total variation minimization and the \( H^{-1} \) norm, Technical Report 02-57, 2002] we present a wavelet-based treatment of variational problems arising in the field of image processing. In particular, we follow their approach and discuss a special class of variational functionals that induce a decomposition of images into oscillating and cartoon components and possibly an appropriate ‘noise’ component. In the setting of [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and [Image decomposition and restoration using total variation minimization and the \( H^{-1} \) norm, Technical Report 02-57, 2002], the cartoon component of an image is modeled by a \( BV \) function; the corresponding incorporation of \( BV \) penalty terms in the variational functional leads to PDE schemes that are numerically intensive. By replacing the \( BV \) penalty term by a \( B_1^1(L_1) \) term (which amounts to a slightly stronger constraint on the minimizer), and writing the problem in a wavelet framework, we obtain elegant and numerically efficient schemes with results very similar to those obtained in [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and [Image decomposition and restoration using total variation minimization and the \( H^{-1} \) norm, Technical Report 02-57, 2002]. This approach allows us, moreover, to incorporate general bounded linear blur operators into the problem so that the minimization leads to a simultaneous decomposition, deblurring and denoising.

I. Daubechies, G. Teschke, and L. Vese: “Iteratively solving linear inverse problems under general convex constraints,” Inverse Probl. Imaging 1 : 1 (2007), pp. 29–46. MR 2262744 Zbl 1123.65044 article

We consider linear inverse problems where the solution is assumed to fulfill some general homogeneous convex constraint. We develop an algorithm that amounts to a projected Landweber iteration and that provides and iterative approach to the solution of this inverse problem. For relatively moderate assumptions on the constraint we can always prove weak convergence of the iterative scheme. In certain cases, i.e. for special families of convex constraints, weak convergence implies norm convergence. The presented approach covers a wide range of problems, e.g. Besov- or BV-restoration for which we present also numerical experiments in the context of image processing.

Luminita Aura Vese	Related
Gerd Teschke	Related