Abstract:
We study covariant differential calculus on the quantum spheres S_q^{N-1} which are quantum homogeneous spaces with coactions of the quantum groups O_q(N). The first part of the paper is devoted to first order differential calculus. A classification result is proved which says that for N>=6 there exist exactly two covariant first order differential calculi on S_q^{N-1} which satisfy the classification constraint that the bimodule of one-forms is generated as a free left module by the differentials of the generators of S_q^{N-1}. Both calculi exist also for 3<=N<=5. The same calculi can be constructed using a method introduced by Hermisson. In case N=3, the result is in accordance with the known result by Apel and Schm\"udgen for the Podles sphere. In the second part, higher order differential calculus and symmetry are treated. The relations which hold for the two-forms in the universal higher order calculus extending one of the two first order calculi are given. A "braiding" homomorphism is found. The existence of an upper bound for the order of differential forms is discussed.

Abstract:
We investigate covariant first order differential calculi on the quantum complex projective spaces CP_q^{N-1} which are quantum homogeneous spaces for the quantum group SU_q(N). Hereby, one more well-studied example of covariant first order differential calculus on a quantum homogeneous space is given. Since the complex projective spaces are subalgebras of the quantum spheres S_q^{2N-1} introduced by Vaksman and Soibelman, we get also an example of the relations between covariant differential calculus on two closely related quantum spaces. Two approaches are combined in obtaining covariant first order differential calculi on CP_q^{N-1}: 1. restriction of covariant first order differential calculi from S_q^{2N-1}; 2. classification of calculi under appropriate constraints, using methods from representation theory. The main result is that under three reasonable settings of dimension constraints, covariant first order differential calculi on CP_q^{N-1} exist and are (for N >= 6) uniquely determined. This is a clear difference as compared to the case of the quantum spheres where several parametrical series of calculi exist. For two of the constraint settings, the covariant first order calculi on CP_q^{N-1} are also obtained by restriction from calculi on S_q^{2N-1} as well as from calculi on the quantum group SU_q(N).

Abstract:
We study covariant differential calculus on the quantum spheres S_q^2N-1. Two classification results for covariant first order differential calculi are proved. As an important step towards a description of the noncommutative geometry of the quantum spheres, a framework of covariant differential calculus is established, including a particular first order calculus obtained by factorization, higher order calculi and a symmetry concept.

Abstract:
We study the applicability of a set of texture descriptors introduced in recent work by the author to texture-based segmentation of images. The texture descriptors under investigation result from applying graph indices from quantitative graph theory to graphs encoding the local structure of images. The underlying graphs arise from the computation of morphological amoebas as structuring elements for adaptive morphology, either as weighted or unweighted Dijkstra search trees or as edge-weighted pixel graphs within structuring elements. In the present paper we focus on texture descriptors in which the graph indices are entropy-based, and use them in a geodesic active contour framework for image segmentation. Experiments on several synthetic and one real-world image are shown to demonstrate texture segmentation by this approach. Forthermore, we undertake an attempt to analyse selected entropy-based texture descriptors with regard to what information about texture they actually encode. Whereas this analysis uses some heuristic assumptions, it indicates that the graph-based texture descriptors are related to fractal dimension measures that have been proven useful in texture analysis.

Abstract:
Multivariate median filters have been proposed as generalisations of the well-established median filter for grey-value images to multi-channel images. As multivariate median, most of the recent approaches use the $L^1$ median, i.e.\ the minimiser of an objective function that is the sum of distances to all input points. Many properties of univariate median filters generalise to such a filter. However, the famous result by Guichard and Morel about approximation of the mean curvature motion PDE by median filtering does not have a comparably simple counterpart for $L^1$ multivariate median filtering. We discuss the affine equivariant Oja median and the affine equivariant transformation--retransformation $L^1$ median as alternatives to $L^1$ median filtering. We analyse multivariate median filters in a space-continuous setting, including the formulation of a space-continuous version of the transformation--retransformation $L^1$ median, and derive PDEs approximated by these filters in the cases of bivariate planar images, three-channel volume images and three-channel planar images. The PDEs for the affine equivariant filters can be interpreted geometrically as combinations of a diffusion and a principal-component-wise curvature motion contribution with a cross-effect term based on torsions of principal components. Numerical experiments are presented that demonstrate the validity of the approximation results.

Abstract:
Morphological amoebas are image-adaptive structuring elements for morphological and other local image filters introduced by Lerallut et al. Their construction is based on combining spatial distance with contrast information into an image-dependent metric. Amoeba filters show interesting parallels to image filtering methods based on partial differential equations (PDEs), which can be confirmed by asymptotic equivalence results. In computing amoebas, graph structures are generated that hold information about local image texture. This paper reviews and summarises the work of the author and his coauthors on morphological amoebas, particularly their relations to PDE filters and texture analysis. It presents some extensions and points out directions for future investigation on the subject.

Abstract:
In this paper, an iterative method for robust deconvolution with positivity constraints is discussed. It is based on the known variational interpretation of the Richardson-Lucy iterative deconvolution as fixed-point iteration for the minimisation of an information divergence functional under a multiplicative perturbation model. The asymmetric penaliser function involved in this functional is then modified into a robust penaliser, and complemented with a regulariser. The resulting functional gives rise to a fixed point iteration that we call robust and regularised Richardson-Lucy deconvolution. It achieves an image restoration quality comparable to state-of-the-art robust variational deconvolution with a computational efficiency similar to that of the original Richardson-Lucy method. Experiments on synthetic and real-world image data demonstrate the performance of the proposed method.

Abstract:
Subject of this paper is the theoretical analysis of structure-adaptive median filter algorithms that approximate curvature-based PDEs for image filtering and segmentation. These so-called morphological amoeba filters are based on a concept introduced by Lerallut et al. They achieve similar results as the well-known geodesic active contour and self-snakes PDEs. In the present work, the PDE approximated by amoeba active contours is derived for a general geometric situation and general amoeba metric. This PDE is structurally similar but not identical to the geodesic active contour equation. It reproduces the previous PDE approximation results for amoeba median filters as special cases. Furthermore, modifications of the basic amoeba active contour algorithm are analysed that are related to the morphological force terms frequently used with geodesic active contours. Experiments demonstrate the basic behaviour of amoeba active contours and its similarity to geodesic active contours.

Abstract:
We investigate possibilities to speed up iterative algorithms for non-blind image deconvolution. We focus on algorithms in which convolution with the point-spread function to be deconvolved is used in each iteration, and aim at accelerating these convolution operations as they are typically the most expensive part of the computation. We follow two approaches: First, for some practically important specific point-spread functions, algorithmically efficient sliding window or list processing techniques can be used. In some constellations this allows faster computation than via the Fourier domain. Second, as iterations progress, computation of convolutions can be restricted to subsets of pixels. For moderate thinning rates this can be done with almost no impact on the reconstruction quality. Both approaches are demonstrated in the context of Richardson-Lucy deconvolution but are not restricted to this method.

Abstract:
We investigate efficient algorithmic realisations for robust deconvolution of grey-value images with known space-invariant point-spread function, with emphasis on 1D motion blur scenarios. The goal is to make deconvolution suitable as preprocessing step in automated image processing environments with tight time constraints. Candidate deconvolution methods are selected for their restoration quality, robustness and efficiency. Evaluation of restoration quality and robustness on synthetic and real-world test images leads us to focus on a combination of Wiener filtering with few iterations of robust and regularised Richardson-Lucy deconvolution. We discuss algorithmic optimisations for specific scenarios. In the case of uniform linear motion blur in coordinate direction, it is possible to achieve real-time performance (less than 50 ms) in single-threaded CPU computation on images of $256\times256$ pixels. For more general space-invariant blur settings, still favourable computation times are obtained. Exemplary parallel implementations demonstrate that the proposed method also achieves real-time performance for general 1D motion blurs in a multi-threaded CPU setting, and for general 2D blurs on a GPU.