The Time-Frequency and Time-Scale communities have recently developed an enormous number of overcomplete signal dictionaries -- wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decomposing a signal into an "optimal" superposition of dictionary elements. The optimization criterion is the l 1 norm of coefficients. The method has several advantages over Matching Pursuit and Best Ortho Basis, including super-resolution and stability. 1 Introduction Over the last five years or so, there has been an explosion of awareness of alternatives to traditional signal representations. Instead of just representing objects as superpositions of sinusoids (the traditional Fourier representation) we now have available alternate dictionaries -- signal representation schemes -- of which the Wavelets dictionary is only the most well-known. Wavelet dictionaries, Gabor dictionaries, Multi-scale...

During the last decade, CD-quality digital audio has essentially replaced analog audio. Emerging digital audio applications for network, wireless, and multimedia computing systems face a series of constraints such as reduced channel bandwidth, limited storage capacity, and low cost. These new applications have created a demand for high-quality digital audio delivery at low bit rates. In response to this need, considerable research has been devoted to the development of algorithms for perceptually transparent coding of high-fidelity (CD-quality) digital audio. As a result, many algorithms have been proposed, and several have now become international and/or commercial product standards. This paper reviews algorithms for perceptually transparent coding of CD-quality digital audio, including both research and standardization activities. The paper is organized as follows. First, psychoacoustic principles are described with the MPEG psychoacoustic signal analysis model 1 discussed in some detail. Next, filter bank design issues and algorithms are addressed, with a particular emphasis placed on the Modified Discrete Cosine Transform (MDCT), a perfect reconstruction (PR) cosine-modulated filter bank that has become of central importance in perceptual audio coding. Then, we review methodologies that achieve perceptually transparent coding of FM- and CD-quality audio signals, including algorithms that manipulate transform components, subband signal decompositions, sinusoidal signal components, and linear prediction (LP) parameters, as well as hybrid algorithms that make use of more than one signal model. These discussions concentrate on architectures and applications of

ABSTRACT This paper presents an overview of wavelet-based image coding. We develop the basics of image coding with a discussion of vector quantization. We motivate the use of transform coding in practical settings,and describe the properties of various decorrelating transforms. We motivate the use of the wavelet transform in coding using rate-distortion considerations as well as approximation-theoretic considerations. Finally,we give an overview of current coders in the literature. 1

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We examine the question of how to construct time-varying filter banks in the most general M -channel non-orthogonal case. We show that by associating with both analysis and synthesis operators a set of boundary filters, it is possible to make the analysis structure vary arbitrarily in time, and yet reconstruct the input with a similarly timevarying synthesis section. There is no redundancy or distortion introduced. This gives a solution to the problem of applying filter banks to finite length signals; it suffices to apply the boundary filters at the beginning and end of the signal segment. This also allows the construction of orthogonal and non-orthogonal bases with essentially any prescribed time and frequency localization, but which, nonetheless, are based on structures with efficient filter bank implementations. Work supported in part by the National Science Foundation under grant ECD-88-11111. The author was with the Dept. of Electrical Engineering, Columbia University, NY, he is...

In this work, a shifted wavelet packet (SWP) library, containing all the time shifted wavelet packet bases, is defined. A corresponding shift-invariant wavelet packet decomposition (SIWPD) search algorithm for a "best basis" is introduced. The search algorithm is representable by a binary tree, in which a node symbolizes an appropriate subspace of the original signal. We prove that the resultant "best basis" is orthonormal and the associated expansion, characterized by the lowest information cost, is shift-invariant. The shift-invariance stems from an additional degree of freedom, generated at the decomposition stage and incorporated into the search algorithm. The added dimension is a relative shift between a given parent-node and its respective children-nodes. We prove that for any subspace it suffices to consider one of two alternative decompositions, made feasible by the SWP library. These decompositions correspond to a zero shift and a 2 - relative shift where denotes the resolution level.

The matching pursuit algorithm can be used to derive signal decompositions in terms of the elements of a dictionary of time-frequency atoms. Using a structured overcomplete dictionary yields a signal model that is both parametric and signal-adaptive. In this paper, we apply matching pursuit to the derivation of signal expansions based on damped sinusoids. It is shown that expansions in terms of complex damped sinusoids can be efficiently derived using simple recursive filter banks. We discuss a subspace extension of the pursuit algorithm which provides a framework for deriving real-valued expansions of real signals based on such complex atoms. Furthermore, we consider symmetric and asymmetric two-sided atoms constructed from underlying one-sided damped sinusoids. The primary concern is the application of this approach to the modeling of signals with transient behavior such as music; it is shown that time-frequency atoms based on damped sinusoids are more suitable for representing trans...

In this paper, an extended library of smooth local trigonometric bases is defined, and an appropriate fast "best-basis" search algorithm is introduced. When compared with the standard local cosine decomposition (LCD), the proposed algorithm is advantageous in three respects. First, it leads to a best-basis expansion that is shift-invariant. Second, the resulting representation is characterized by a lower information cost. Third, the polarity of the folding operator is adapted to the parity properties of the segmented signal at the end-points. The shift-invariance stems from an adaptive relative shift of expansions in distinct resolution levels. We show that at any resolution level it suffices to examine and select one of two relative shift options a zero shift or a 2 -- shift. A variable folding operator, whose polarity is locally adapted to the parity properties of the signal, further enhances the representation. The computational complexity is manageable and comparable to that of the LCD.

In this paper we address the problem of finding the best time-varying filter bank treestructured representation for a signal. The tree is allowed to vary at regular intervals, and the spacing of these changes can be arbitrarily short. The question of how to choose tree-structured representations of signals based on filter banks has attracted considerable attention. Wavelets, and their adaptive versions, known as wavelet packets, represent one approach that has proved very popular. Wavelet packets are subband trees where the tree is chosen to match the characteristics of the signal. Variations where the tree varies over time have been proposed as the double tree, and the time-frequency tree algorithms. Time-variation adds a further level of adaptivity. In all of the approaches proposed so far the tree must be either fixed for the whole duration of the signal, or fixed for its dyadic subintervals (i.e. halves, quarters, etc). The solution that we propose, since it allows much more flexib...

In this thesis, we generalize the classical discrete wavelet transform, and construct wavelet transforms that are shift-invariant, time-varying, undecimated, and signal dependent. The result is a set of powerful and efficient algorithms suitable for a wide variety of signal processing tasks, e.g., data compression, signal analysis, noise reduction, statistical estimation, and detection. These algorithms are comparable and often superior to traditional methods. In this sense, we put wavelets in action.