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

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Non-separable 2D wavelets with two-row filters

In the literature 2D (or bivariate) wavelets are usually constructed as a tensor product of 1D wavelets. Such wavelets are called separable. However, there are various applications, e.g. in image processing, for which non-separable 2D wavelets are preferable. In this paper, we investigate the class of compactly supported orthonormal 2D wavelets that was introduced by Belogay and Wang [2]. A characteristic feature of this class of wavelets is that the support of the corresponding filter comprises only two rows. We are concerned with the biorthogonal extension of this kind of wavelets. It turns out that the 2D wavelets in this class are intimately related to some underlying 1D wavelet. We explore this relation in detail, and we explain how the 2D wavelet transforms can be realized by means of a lifting scheme, thus allowing an efficient implementation. We also describe an easy way to construct wavelets with more rows and shorter columns

From spline wavelet to sampling theory on circulant graphs and beyond– conceiving sparsity in graph signal processing

Graph Signal Processing (GSP), as the field concerned with the extension of classical signal processing concepts to the graph domain, is still at the beginning on the path toward providing a generalized theory of signal processing. As such, this thesis aspires to conceive the theory of sparse representations on graphs by traversing the cornerstones of wavelet and sampling theory on graphs. Beginning with the novel topic of graph spline wavelet theory, we introduce families of spline and e-spline wavelets, and associated filterbanks on circulant graphs, which lever- age an inherent vanishing moment property of circulant graph Laplacian matrices (and their parameterized generalizations), for the reproduction and annihilation of (exponen- tial) polynomial signals. Further, these families are shown to provide a stepping stone to generalized graph wavelet designs with adaptive (annihilation) properties. Circulant graphs, which serve as building blocks, facilitate intuitively equivalent signal processing concepts and operations, such that insights can be leveraged for and extended to more complex scenarios, including arbitrary undirected graphs, time-varying graphs, as well as associated signals with space- and time-variant properties, all the while retaining the focus on inducing sparse representations. Further, we shift from sparsity-inducing to sparsity-leveraging theory and present a novel sampling and graph coarsening framework for (wavelet-)sparse graph signals, inspired by Finite Rate of Innovation (FRI) theory and directly building upon (graph) spline wavelet theory. At its core, the introduced Graph-FRI-framework states that any K-sparse signal residing on the vertices of a circulant graph can be sampled and perfectly reconstructed from its dimensionality-reduced graph spectral representation of minimum size 2K, while the structure of an associated coarsened graph is simultaneously inferred. Extensions to arbitrary graphs can be enforced via suitable approximation schemes. Eventually, gained insights are unified in a graph-based image approximation framework which further leverages graph partitioning and re-labelling techniques for a maximally sparse graph wavelet representation.Open Acces

Wavelet methods in speech recognition

In this thesis, novel wavelet techniques are developed to improve parametrization of speech signals prior to classification. It is shown that non-linear operations carried out in the wavelet domain improve the performance of a speech classifier and consistently outperform classical Fourier methods. This is because of the localised nature of the wavelet, which captures correspondingly well-localised time-frequency features within the speech signal. Furthermore, by taking advantage of the approximation ability of wavelets, efficient representation of the non-stationarity inherent in speech can be achieved in a relatively small number of expansion coefficients. This is an attractive option when faced with the so-called 'Curse of Dimensionality' problem of multivariate classifiers such as Linear Discriminant Analysis (LDA) or Artificial Neural Networks (ANNs). Conventional time-frequency analysis methods such as the Discrete Fourier Transform either miss irregular signal structures and transients due to spectral smearing or require a large number of coefficients to represent such characteristics efficiently. Wavelet theory offers an alternative insight in the representation of these types of signals. As an extension to the standard wavelet transform, adaptive libraries of wavelet and cosine packets are introduced which increase the flexibility of the transform. This approach is observed to be yet more suitable for the highly variable nature of speech signals in that it results in a time-frequency sampled grid that is well adapted to irregularities and transients. They result in a corresponding reduction in the misclassification rate of the recognition system. However, this is necessarily at the expense of added computing time. Finally, a framework based on adaptive time-frequency libraries is developed which invokes the final classifier to choose the nature of the resolution for a given classification problem. The classifier then performs dimensionaIity reduction on the transformed signal by choosing the top few features based on their discriminant power. This approach is compared and contrasted to an existing discriminant wavelet feature extractor. The overall conclusions of the thesis are that wavelets and their relatives are capable of extracting useful features for speech classification problems. The use of adaptive wavelet transforms provides the flexibility within which powerful feature extractors can be designed for these types of application