SUBDIVIDE AND CONQUER RESOLUTION

This contribution will be freewheeling in the domain of signal, image and surface processing and touch briefly upon some topics that have been close to the heart of people in our research group. A lot of the research of the last 20 years in this domain that has been carried out world wide is dealing with multiresolution. Multiresolution allows to represent a function (in the broadest sense) at different levels of detail. This was not only applied in signals and images but also when solving all kinds of complex numerical problems. Since wavelets came into play in the 1980's, this idea was applied and generalized by many researchers. Therefore we use this as the central idea throughout this text. Wavelets, subdivision and hierarchical bases are the appropriate tools to obtain these multiresolution effects. We shall introduce some of the concepts in a rather informal way and show that the same concepts will work in one, two and three dimensions. The applications in the three cases are however quite different, and thus one wants to achieve very different goals when dealing with signals, images or surfaces. Because completeness in our treatment is impossible, we have chosen to describe two case studies after introducing some concepts in signal processing. These case studies are still the subject of current research. The first one attempts to solve a problem in image processing: how to approximate an edge in an image efficiently by subdivision. The method is based on normal offsets. The second case is the use of Powell-Sabin splines to give a smooth multiresolution representation of a surface. In this context we also illustrate the general method of construction of a spline wavelet basis using a lifting scheme

Dynamic remeshing and applications

Triangle meshes are a flexible and generally accepted boundary representation for complex geometric shapes. In addition to their geometric qualities such as for instance smoothness, feature sensitivity ,or topological simplicity, intrinsic qualities such as the shape of the triangles, their distribution on the surface and the connectivity is essential for many algorithms working on them. In this thesis we present a flexible and efficient remeshing framework that improves these "intrinsic\u27; properties while keeping the mesh geometrically close to the original surface. We use a particle system approach and combine it with an iterative remeshing process in order to trim the mesh towards the requirements imposed by different applications. The particle system approach distributes the vertices on the mesh with respect to a user-defined scalar-field, whereas the iterative remeshing is done by means of "Dynamic Meshes\u27;, a combination of local topological operators that lead to a good natured connectivity. A dynamic skeleton ensures that our approach is able to preserve surface features, which are particularly important for the visual quality of the mesh. None of the algorithms requires a global parameterization or patch layouting in a preprocessing step, but works with simple local parameterizations instead. In the second part of this work we will show how to apply this remeshing framework in several applications scenarios. In particular we will elaborate on interactive remeshing, dynamic, interactive multiresolution modeling, semiregular remeshing and mesh simplification and we will show how the users can adapt the involved algorithms in a way that the resulting mesh meets their personal requirements

Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing