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

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

A 2D DWT architecture suitable for the Embedded Zerotree Wavelet Algorithm

Digital Imaging has had an enormous impact on industrial applications such as the Internet and video-phone systems. However, demand for industrial applications is growing enormously. In particular, internet application users are, growing at a near exponential rate. The sharp increase in applications using digital images has caused much emphasis on the fields of image coding, storage, processing and communications. New techniques are continuously developed with the main aim of increasing efficiency. Image coding is in particular a field of great commercial interest. A digital image requires a large amount of data to be created. This large amount of data causes many problems when storing, transmitting or processing the image. Reducing the amount of data that can be used to represent an image is the main objective of image coding. Since the main objective is to reduce the amount of data that represents an image, various techniques have been developed and are continuously developed to increase efficiency. The JPEG image coding standard has enjoyed widespread acceptance, and the industry continues to explore its various implementation issues. However, recent research indicates multiresolution based image coding is a far superior alternative. A recent development in the field of image coding is the use of Embedded Zerotree Wavelet (EZW) as the technique to achieve image compression. One of The aims of this theses is to explain how this technique is superior to other current coding standards. It will be seen that an essential part orthis method of image coding is the use of multi resolution analysis, a subband system whereby the subbands arc logarithmically spaced in frequency and represent an octave band decomposition. The block structure that implements this function is termed the two dimensional Discrete Wavelet Transform (2D-DWT). The 20 DWT is achieved by several architectures and these are analysed in order to choose the best suitable architecture for the EZW coder. Finally, this architecture is implemented and verified using the Synopsys Behavioural Compiler and recommendations are made based on experimental findings

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

Feasible Adaptation Criteria for Hybrid Wavelet - Large Margin Classifiers

In the context of signal classification, this paper assembles and compares criteria to easily judge the discrimination quality of a set of feature vectors. The quality measures are based on the assumption that a Support Vector Machine is used for the final classification. Thus, the ultimate criterion is a large margin separating the two classes. We apply the criteria to control the feature extraction process for signal classification. Adaptive features related to the shape of the signals are extracted by wavelet filtering followed by a nonlinear map. To be able to test many features, the criteria are easily computable while still reliably predicting the classification performance. We also present a novel approach for computing the radius of a set of points in feature space. The radius, in relation to the margin, forms the most commonly used error bound for Support Vector Machines. For isotropic kernels, the problem of radius computation can be reduced to a common Support Vector Machine classification problem

Digital Filters and Signal Processing

Digital filters, together with signal processing, are being employed in the new technologies and information systems, and are implemented in different areas and applications. Digital filters and signal processing are used with no costs and they can be adapted to different cases with great flexibility and reliability. This book presents advanced developments in digital filters and signal process methods covering different cases studies. They present the main essence of the subject, with the principal approaches to the most recent mathematical models that are being employed worldwide

Decay properties of spectral projectors with applications to electronic structure

Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.Comment: 63 pages, 13 figure

The Construction of Nonseparable Wavelet Bi-Frames and Associated Approximation Schemes

Wavelet analysis and its fast algorithms are widely used in many fields of applied mathematics such as in signal and image processing. In the present thesis, we circumvent the restrictions of orthogonal and biorthogonal wavelet bases by constructing wavelet frames. They still allow for a stable decomposition, and so-called wavelet bi-frames provide a series expansion very similar to those of pairs of biorthogonal wavelet bases. Contrary to biorthogonal bases, primal and dual wavelets are no longer supposed to satisfy any geometrical conditions, and the frame setting allows for redundancy. This provides more flexibility in their construction. Finally, we construct families of optimal wavelet bi-frames in arbitrary dimensions with arbitrarily high smoothness. Then we verify that the n-term approximation can be described by Besov spaces and we apply the theoretical findings to image denoising

Adaptive multiresolution visualization of large multidimensional multivariate scientific datasets

The sizes of today\u27s scientific datasets range from megabytes to terabytes, making it impossible to directly browse the raw datasets visually. This presents significant challenges for visualization scientists who are interested in supporting these datasets. In this thesis, we present an adaptive data representation model which can be utilized with many of the commonly employed visualization techniques when dealing with large amounts of data. Our hierarchical design also alleviates the long standing visualization problem due to limited display space. The idea is based on using compactly supported orthogonal wavelets and additional downsizing techniques to generate a hierarchy of fine to coarse approximations of a very large dataset for visualization. An adaptive data hierarchy, which contains authentic multiresolution approximations and the corresponding error, has many advantages over the original data. First, it allows scientists to visualize the overall structure of a dataset by browsing its coarse approximations. Second, the fine approximations of the hierarchy provide local details of the interesting data subsets. Third, the error of the data representation can provide the scientist with information about the authenticity of the data approximation. Finally, in a client-server network environment, a coarse representation can increase the efficiency of a visualization process by quickly giving users a rough idea of the dataset before they decide whether to continue the transmission or to abort it. For datasets which require long rendering time, an authentic approximation of a very large dataset can speed up the visualization process greatly. Variations on the main wavelet-based multiresolution hierarchy described in this thesis also lead to other multiresolution representation mechanisms. For example, we investigate the uses of norm projections and principal components to build multiresolution data hierarchies of large multivariate datasets. This leads to the development of a more flexible dual multiresolution visualization environment for large data exploration. We present the results of experimental studies of our adaptive multiresolution representation using wavelets. Utilizing a multiresolution data hierarchy, we illustrate that information access from a dataset with tens of millions of data values can be achieved in real time. Based on these results, we propose procedures to assist in generating a multiresolution hierarchy of a large dataset. For example, the findings indicate that an ordinary computed tomography volume dataset can be represented effectively for some tasks by an adaptive data hierarchy with less than 1.5% of its original size