Image modelling, inpainting, decomposition and restoration by redundant representations and variational calculus

• Applicant: Professor Dr. Gerd Teschke
• Subject Area: Mathematics
• Term: from 2006 to 2010
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 27869400

The project will address investigations on the usage of frames (stable redundant nonorthogonal expansions in Banach spaces) and their use in numerical applications in the field of inverse problems (inpainting, restoration and decomposition tasks in image processing). In inverse problems, and in particular in image processing, one of the very first tasks is to identify the underlying model function spaces that embed the problem into the right setting. When dealing with spaces of (special) bounded variation functions - which is a very favoured space in image processing - the problem is often that the associated PDE (Euler Lagrange) schemes to approximate the solution are numerically very intensive and time consuming. It would be desirable to bypass this drawback and to derive the solution in some numerically thrifty way. Moreover, in certain applications the solution is often assumed to have a (super) sparse expansion, or it is required to express the solution in a very space saving way. To this end, we aim to answer here the question on how we can characterize the function spaces and variational problems under consideration and/or how we can replace them by other easier to handle frameworks (e.g. embeddings of SBV functions into Besov and oscillation spaces). In particular, we ask for proper characterizations of the underlying function spaces (or its replacements) by means of frames that allow an adequate discretization/decomposition. Once this is achieved, we may rewrite the variational problems (where we hope being not too far off the original problem) and aim to construct schemes for the frame coefficients that easily provide tools to derive the solution numerically. Here we essentially focus on two applications: firstly, solving nonlinear problems in its variational form where the solution is assumed to have (super) sparse frame expansion and, secondly, investigating the use of frames for modelling free discontinuity problems such as Mumford Shah like functionals as they appear in the field of image inpainting/restoration.

• DFG Programme: Research Grants