A Linear Image Reconstruction Framework Based on Sobolev Type Inner Products

Abstract

Exploration of information content of features that are present in images has led to the development of several reconstruction algorithms. These algorithms aim for a reconstruction from the features that is visually close to the image from which the features are extracted. Degrees of freedom that are not fixed by the constraints are disambiguated with the help of a so-called prior (i.e. a user defined model). We propose a linear reconstruction framework that generalizes a previously proposed scheme. The algorithm greatly reduces the complexity of the reconstruction process compared to non-linear methods. As an example we propose a specific prior and apply it to the reconstruction from singular points. The reconstruction is visually more attractive and has a smaller L 2 -error than the reconstructions obtained by previously proposed linear methods.

Linear Image Reconstruction by Sobolev Norms on the Bounded Domain

“…The upper row of Fig. 5 shows from left to right: the image from which the features were extracted, a reconstruction by the unbounded domain method (Janssen et al 2006) (parameters: γ = 50, k = 1), and a reconstruction by the newly introduced bounded domain method using Dirichlet boundary conditions (parameters: γ = 50, k = 1). Features that were used are up to second order derivatives measured at the singular points (see Sect.…”

“…However, scale space representations of natural images do contain singular points attached to generic topological transitions (Damon 1995). If one endows these points with suitable attributes one can obtain a reconstruction that is visually close to the initial image (Janssen et al 2006).…”

“…So the mirroring has nothing to do with Fig. 1 The left image shows a reconstruction from differential structure obtained from the right image using the unbounded domain reconstruction method as presented in our previous work (Janssen et al 2006). The upper circle in the right image shows a detail of an area near the center of the image and the circle on the right shows a corresponding area near the boundary of the image.…”

“…Several linear and non-linear methods (Nielsen and Lillholm 2001;Janssen et al 2006;Lillholm et al 2003) appeared in literature which all search for an image that (1) is indistinguishable from its original when observed through the filters the features were extracted with, and (2) simultaneously minimizes a certain prior. If such a prior is a norm of Sobolev type on the unbounded domain one can obtain visually attractive reconstructions while retaining linearity, as we have shown in earlier work (Janssen et al 2006;Duits 2005). However, boundaries often cause problems to signal processing algorithms (see e.g.…”

“…The input image is depicted on the right of Fig. 1 and is still considered as a compactly supported element in L 2 (R 2 ), as required in our previous work (Janssen et al 2006). So the mirroring has nothing to do with Fig.…”