Signal processing for high-definition television

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995.Includes bibliographical references (p. 60-62).by Peter Monta.Ph.D

Subdivision schemes for curve design and image analysis

Subdivision schemes are able to produce functions, which are smooth up to pixel accuracy, in a few steps through an iterative process. They take as input a coarse control polygon and iteratively generate new points using some algebraic or geometric rules. Therefore, they are a powerful tool for creating and displaying functions, in particular in computer graphics, computer-aided design, and signal analysis. A lot of research on univariate subdivision schemes is concerned with the convergence and the smoothness of the limit curve, especially for schemes where the new points are a linear combination of points from the previous iteration. Much less is known for non-linear schemes: in many cases there are only ad hoc proofs or numerical evidence about the regularity of these schemes. For schemes that use a geometric construction, it could be interesting to study the continuity of geometric entities. Dyn and Hormann propose sufficient conditions such that the subdivision process converges and the limit curve is tangent continuous. These conditions can be satisfied by any interpolatory scheme and they depend only on edge lengths and angles. The goal of my work is to generalize these conditions and to find a sufficient constraint, which guarantees that a generic interpolatory subdivision scheme gives limit curves with continuous curvature. To require the continuity of the curvature it seems natural to come up with a condition that depends on the difference of curvatures of neighbouring circles. The proof of the proposed condition is not completed, but we give a numerical evidence of it. A key feature of subdivision schemes is that they can be used in different fields of approximation theory. Due to their well-known relation with multiresolution analysis they can be exploited also in image analysis. In fact, subdivision schemes allow for an efficient computation of the wavelet transform using the filterbank. One current issue in signal processing is the analysis of anisotropic signals. Shearlet transforms allow to do it using the concept of multiple subdivision schemes. One drawback, however, is the big number of filters needed for analysing the signal given. The number of filters is related to the determinant of the expanding matrix considered. Therefore, a part of my work is devoted to find expanding matrices that give a smaller number of filters compared to the shearlet case. We present a family of anisotropic matrices for any dimension d with smaller determinant than shearlets. At the same time, these matrices allow for the definition of a valid directional transform and associated multiple subdivision schemes

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 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

Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation

Wavelet-Based Audio Embedding & Audio/Video Compression

With the decline in military spending, the United States relies heavily on state side support. Communications has never been more important. High-quality audio and video capabilities are a must. Watermarking, traditionally used for copyright protection, is used in a new and exciting way. An efficient wavelet-based watermarking technique embeds audio information into a video signal. Several highly effective compression techniques are applied to compress the resulting audio/video signal in an embedded fashion. This wavelet-based compression algorithm incorporates bit plane coding, first difference coding, and Huffman coding. To demonstrate the potential of this audio embedding audio/video compression system, an audio signal is embedded into a video signal and the combined signal is compressed. Results show that overall compression rates of 15:1 can be achieved. The video signal is reconstructed with a median PSNR of nearly 33dB. Finally, the audio signal is extracted with out error

Wavelets and multirate filter banks : theory, structure, design, and applications

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2004.Includes bibliographical references (p. 219-230) and index.Wavelets and filter banks have revolutionized signal processing with their ability to process data at multiple temporal and spatial resolutions. Fundamentally, continuous-time wavelets are governed by discrete-time filter banks with properties such as perfect reconstruction, linear phase and regularity. In this thesis, we study multi-channel filter bank factorization and parameterization strategies, which facilitate designs with specified properties that are enforced by the actual factorization structure. For M-channel filter banks (M =/> 2), we develop a complete factorization, M-channel lifting factorization, using simple ladder-like structures as predictions between channels to provide robust and efficient implementation; perfect reconstruction is structurally enforced, even under finite precision arithmetic and quantization of lifting coefficients. With lifting, optimal low-complexity integer wavelet transforms can thus be designed using a simple and fast algorithm that incorporates prescribed limits on hardware operations for power-constrained environments. As filter bank regularity is important for a variety of reasons, an aspect of particular interest is the structural imposition of regularity onto factorizations based on the dyadic form uvt. We derive the corresponding structural conditions for regularity, for which M-channel lifting factorization provides an essential parameterization. As a result, we are able to design filter banks that are exactly regular and amenable to fast implementations with perfect reconstruction, regardless of the choice of free parameters and possible finite precision effects. Further constraining u = v ensures regular orthogonal filter banks,(cont.) whereas a special dyadic form is developed that guarantees linear phase. We achieve superior coding gains within 0.1% of the optimum, and benchmarks conducted on image compression applications show clear improvements in perceptual and objective performance. We also consider the problem of completing an M-channel filter bank, given only its scaling filter. M-channel lifting factorization can efficiently complete such biorthogonal filter banks. On the other hand, an improved scheme for completing paraunitary filter banks is made possible by a novel order-one factorization which allows greater design flexibility, resulting in improved frequency selectivity and energy compaction over existing state of the art methods. In a dual setting, the technique can be applied to transmultiplexer design to achieve higher-rate data transmissions.by Ying-Jui Chen.Ph.D