Curvature Detection by Integral Transforms

In various fields of image analysis, determining the precise geometry of occurrent edges, e.g. the contour of an object, is a crucial task. Especially the curvature of an edge is of great practical relevance. In this thesis, we develop different methods to detect a variety of edge features, among them the curvature. We first examine the properties of the parabolic Radon transform and show that it can be used to detect the edge curvature, as the smoothness of the parabolic Radon transform changes when the parabola is tangential to an edge and also, when additionally the curvature of the parabola coincides with the edge curvature. By subsequently introducing a parabolic Fourier transform and establishing a precise relation between the smoothness of a certain class of functions and the decay of the Fourier transform, we show that the smoothness result for the parabolic Radon transform can be translated into a change of the decay rate of the parabolic Fourier transform. Furthermore, we introduce an extension of the continuous shearlet transform which additionally utilizes shears of higher order. This extension, called the Taylorlet transform, allows for a detection of the position and orientation, as well as the curvature and other higher order geometric information of edges. We introduce novel vanishing moment conditions which enable a more robust detection of the geometric edge features and examine two different constructions for Taylorlets. Lastly, we translate the results of the Taylorlet transform in R^2 into R^3 and thereby allow for the analysis of the geometry of object surfaces

Learning Theory and Approximation

Learning theory studies data structures from samples and aims at understanding unknown function relations behind them. This leads to interesting theoretical problems which can be often attacked with methods from Approximation Theory. This workshop - the second one of this type at the MFO - has concentrated on the following recent topics: Learning of manifolds and the geometry of data; sparsity and dimension reduction; error analysis and algorithmic aspects, including kernel based methods for regression and classification; application of multiscale aspects and of refinement algorithms to learning

Hybrid Projection Methods for Large-scale Inverse Problems with Mixed Gaussian Priors

When solving ill-posed inverse problems, a good choice of the prior is critical for the computation of a reasonable solution. A common approach is to include a Gaussian prior, which is defined by a mean vector and a symmetric and positive definite covariance matrix, and to use iterative projection methods to solve the corresponding regularized problem. However, a main challenge for many of these iterative methods is that the prior covariance matrix must be known and fixed (up to a constant) before starting the solution process. In this paper, we develop hybrid projection methods for inverse problems with mixed Gaussian priors where the prior covariance matrix is a convex combination of matrices and the mixing parameter and the regularization parameter do not need to be known in advance. Such scenarios may arise when data is used to generate a sample prior covariance matrix (e.g., in data assimilation) or when different priors are needed to capture different qualities of the solution. The proposed hybrid methods are based on a mixed Golub-Kahan process, which is an extension of the generalized Golub-Kahan bidiagonalization, and a distinctive feature of the proposed approach is that both the regularization parameter and the weighting parameter for the covariance matrix can be estimated automatically during the iterative process. Furthermore, for problems where training data are available, various data-driven covariance matrices (including those based on learned covariance kernels) can be easily incorporated. Numerical examples from tomographic reconstruction demonstrate the potential for these methods

Anisotropic Harmonic Analysis and Integration of Remotely Sensed Data

This thesis develops the theory of discrete directional Gabor frames and several algorithms for the analysis of remotely sensed image data, based on constructions of harmonic analysis. The problems of image registration, image superresolution, and image fusion are separate but interconnected; a general approach using transform methods is the focus of this thesis. The methods of geometric multiresolution analysis are explored, particularly those related to the shearlet transform. Using shearlets, a novel method of image registration is developed that aligns images based on their shearlet features. Additionally, the anisotropic nature of the shearlet transform is deployed to smoothly superrsolve remotely-sensed image with edge features. Wavelet packets, a generalization of wavelets, are utilized for a flexible image fusion algorithm. The interplay between theoretical guarantees for these mathematical constructions, and their effectiveness for image processing is explored throughout