Linear Image Reconstruction from a Sparse Set of α-Scale Space Features by Means of Inner Products of Sobolev Type

Abstract

AbstractInner products of Sobolev type are extremely useful for image reconstruction of images from a sparse set of α-scale space features. The common (non)-linear reconstruction frameworks, follow an Euler Lagrange minimization. If the Lagrangian (prior) is a norm induced by an inner product of a Hilbert space, this Euler Lagrange minimization boils down to a simple orthogonal projection within the corresponding Hilbert space. This basic observation has been overlooked in image analysis for the cases where the Lagrangian equals a norm of Sobolev type, resulting in iterative (non-linear) numerical methods, where already an exact solution with non-iterative linear algorithm is at hand. Therefore we provide a general theory on linear image reconstructions and metameric classes of images. By applying this theory we obtain visually more attractive reconstructions than the previously proposed linear methods and we find connected curves in the metameric class of images, determined by a fixed set of linear features, with a monotonic increase of smoothness. Although the theory can be applied to any linear feature reconstruction or principle component analysis, we mainly focus on reconstructions from so-called topological features (such as top-points and grey-value flux) in scale space, obtained from geometrical observations in the deep structure of a scale space.KeywordsScale SpaceSobolev SpacesGelfand TriplesTikhonov RegularizationTop Point ReconstructionDeep StructureFlux Features