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Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods.

Water diffusion is anisotropic in organized tissues such as white matter and muscle. Diffusion tensor imaging (DTI), a non-invasive MR technique, measures water self-diffusion rates and thus gives an indication of the underlying tissue microstructure. The diffusion rate is often expressed by a second-order tensor. Insightful DTI visualization is challenging because of the multivariate nature and the complex spatial relationships in a diffusion tensor field. This chapter surveys the different visualization techniques that have been developed for DTI and compares their main characteristics and drawbacks. We also discuss some of the many biomedical applications in which DTI helps extend our understanding or improve clinical procedures. We conclude with an overview of open problems and research directions.

In the context of intelligent vehicles, we perform a comparative study on recursive Bayesian filters for pedestrian path prediction at short time horizons (

Structure tensors are a common tool for orientation estimation in image processing and computer vision. We present a generalization of the traditional second-order model to a higher-order structure tensor (HOST), which is able to model more than one significant orientation, as found in corners, junctions, and multi-channel images. We provide a theoretical analysis and a number of mathematical tools that facilitate practical use of the HOST, visualize it using a novel glyph for higher-order tensors, and demonstrate how it can be applied in an improved integrated edge, corner, and junction

Many different measures have been proposed to compute similarities and distances between diffusion tensors. These measures are commonly used for algorithms such as segmentation, registration, and quantitative analysis of diffusion Tensor Imaging data sets. The results obtained from these algorithms are extremely dependent on the chosen measure. The measures presented in literature can be of complete different nature, and it is often difficult to predict the behavior of a given measure for a specific application. In this chapter, we classify and summarize the different measures that have been presented in literature. We also present a framework to analyze and compare the behavior of the measures according to several selected properties. We expect that this framework will help in the initial selection of a measure for a given application and to identify when the generation of a new measure is needed. This framework will also allow the comparison of new measures with existing ones.

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 generalises a previously proposed scheme. As an example we propose a specific prior and apply it to the reconstruction from singular point s. The reconstruction is visually more attractive and has a smaller $\mathbb{L}_{\rm 2}$-error than the previously proposed linear methods.

In previous work, singular points (or top points) in the scale space representation of generic images have proven valuable for image matching. In this paper, we propose a construction that encodes the scale space description of top points in the form of a directed acyclic graph. This representation allows us to utilize graph matching algorithms for comparing images represented in terms of top point configurations instead of using solely the top points and their features in a point matching algorithm, as was done previously. The nodes of the graph represent the critical paths together with their top points. The edge set will capture the neighborhood distribution of vertices in scale space, and is constructed through a Delaunay triangulation scheme. We also will present a many-to-many matching algorithm for comparing such graph-based representations. This algorithm is based on a metric-tree representation of labelled graphs and their low-distortion embeddings into normed vector spaces via spherical encoding. This is a two-step transformation that reduces the matching problem to that of computing a distribution-based distance measure between two such embeddings. To evaluate the quality of our representation, two sets of experiments are considered. First, the stability of this representation under Gaussian noise of increasing magnitude is examined. In the second set of experiments, a series of recognition experiments is run on a small face database.

A second order partial differential operator is applied to an image function. By using a multigrid operator known from the so-called approximation property, we derive a new type of multiresolution decomposition of the image. As an example, the Poisson case is treated in-depth. Using the new transform we devise an algorithm for image fusion. The actual recombination is performed on the image functions on which the partial differential operator has been applied first. A fusion example is elaborated upon. Other applications can be envisaged as well.

The probability distribution on the set of naturally occurring images is sparse with most of the probability mass on a small subset of all possible images, hence not all images are equally likely to be seen in nature. This can indirectly be observed by studying the marginal statistics of filter responses on natural images. Intensity differences, or equivalently responses of linear filters, of natural images have a spiky distribution with heavy tails, which puts a large proportion of the probability mass on small intensity differences, but at the same time giving a reasonable probability on large differences. This is due to the fact that images consist mostly of smooth regions separated by discontinuous boundaries. We propose to model natural images as stochastic Lévy processes with a kernel distributed intensity differences. We will argue that the scale invariant a kernels of the recently proposed a scale space theory provides a promising model of the intensity difference distribution (or in general linear filter responses) in conjunction with the Lévy process model of natural images.

This paper presents an algorithm for computing stability of top-points in scale-space. The potential usefulness of top-points in scale-space has already been shown for a number of applications, such as image reconstruction and image retrieval. In order to improve results only reliable top-points should be used. The algorithm is based on perturbation theory and noise propagation.

Segmentation based on color, instead of intensity only, pro- vides an easier distinction between materials, on the condition that ro- bustness against irrelevant parameters is achieved, such as illumination source, shadows, geometry and camera sensitivities. Modeling the phys- ical process of the image formation provides insight into the effect of different parameters on object color.In this paper, a color differential geometry approach is used to detect material edges, invariant with respect to illumination color and imaging conditions. The performance of the color invariants is demonstrated by some real-world examples, showing the invariants to be successful in discounting shadow edges and illumination color.

The behaviour of critical points of Gaussian scale-space im- ages is mainly described by their creation and annihilation. In existing literature these events are determined in so-called canonical coordinates. A description in a user-defined Cartesian coordinate system is stated, as well as the results of a straightforward implementation. The location of a catastrophe can be predicted with subpixel accuracy. An example of an annihilation is given. Also an upper bound is derived for the area where critical points can be created. Experimental data of an MR, a CT, and an artificial noise image satisfy this result.

This paper deals with parallelization methods for time-dependent problems where the time steps are shared out among the processors. A Full Multigrid technique serves as solution algorithm, hence information of the preceding time step and of the coarser grid is necessary to compute the solution at each new grid level. Applying the usual extrapolation formula to process this information, the parallelization will not be very efficient. We developed another extrapolation technique which causes a much higher parallelization effect. Test examples show that no essential loss of exactness appears, such that the method presented here shall be well-applicable.

Nonsteady Navier-Stokes equations represent a differential-algebraic system of strangeness index one after any spatial discretization. Since such systems are hard to treat in their original form, most approaches use some kind of index reduction. Processing this index reduction it is important to take care of the manifolds contained in the differential-algebraic equation (DAE). We investigate for several discretization schemes for the Navier-Stokes equations how the consideration of the manifolds is taken into account and propose a variant of solving these equations along the lines of the theoretically best index reduction. Applying this technique, the error of the time discretisation depends only on the method applied for solving the DAE.