Mladen Victor Wickerhauser

Entropy-based algorithms for best basis selection

Adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals. It permits efficient compression of a variety of signals, such as sound and images. The predefined libraries of modulated waveforms include orthogonal wavelet-packets and localized trigonometric functions, and have reasonably well-controlled time-frequency localization properties. The idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost. The method relies heavily on the remarkable orthogonality properties of the new libraries: all expansions in a given library conserve energy and are thus comparable. Several cost functionals are useful; one of the most attractive is Shannon entropy, which has a geometric interpretation in this context. >

Wavelets and time-frequency analysis

We present a selective overview of time-frequency analysis and some of its key problems. In particular we motivate the introduction of wavelet and wavelet packet analysis. Different types of decompositions of an idealized time-frequency plane provide the basis for understanding the performance of the numerical algorithms and their corresponding interpretations within the continuous models. As examples we show how to control the frequency spreading of wavelet packets at high frequencies using nonstationary filtering and study some properties of periodic wavelet packets. Furthermore we derive a formula to compute the time localization of a wavelet packet from its indexes which is exact for linear phase filters, and show how this estimate deteriorates with deviation from linear phase.

Acoustic signal compression with wavelet packets

Abstract . The wavelet transform is generalized to produce a library of orthonormal bases of modulated wavelet packets, where each basis comes with a fast transform. These bases are similar to adaptive windowed Fourier transforms, and hence give rise to the notion of a “best basis” for a signal subject to a given cost function. This paper discusses some early results in acoustic signal compression using a simple counting cost function.

Improved predictability of two-dimensional turbulent flows using wavelet packet compression

Abstract We propose to use new orthonormal wavelet packet bases, more efficient than the Fourier basis, to compress two-dimensional turbulent flows. We define the “best basis” of wavelet packets as the one which, for a given enstrophy density, condenses the L2 norm into a minimum number of non-negligible wavelet packet coefficients. Coefficients below a threshold are discarded, reducing the number of degrees of freedom. We then compare the predictability of the original flow evolution with several such reductions, varying the number of retained coefficients, either from a Fourier basis, or from the best-basis of wavelet packets. We show that for a compression ratio of 1/2, we still have a deterministic predictability using the wavelet packet best-basis, while it is lost when using the Fourier basis. Likewise, for compression ratios of 1/20 and 1/200 we still have statistical predictability using the wavelet packet best-basis, while it is lost when using the Fourier basis. In fact, the significant wavelet packet coefficients in the best-basis appear to correspond to coherent structures. The weak coefficients correspond to vorticity filaments, which are only passively advected by the coherent structures. In conclusion, the wavelet packet best-basis seems to distinguish the low-dimensional dynamically active part of the flow from the high-dimensional passive components. It gives us some hope of drastically reducing the number of degrees of freedom necessary to the computation of two-dimensional turbulent flows.

Adapted local trigonometric transforms and speech processing

Uses an algorithm based on the adapted-window Malvar transform to decompose digitized speech signals into a local time-frequency representation. The authors present some applications and experimental results for a signal compression and automatic voiced-unvoiced segmentation. This decomposition provides a method of parameter simplification which appears to be useful for detecting fundamental frequencies, and characterizing formants. >

Sampling and Estimation

Multimedia signal processing begins with the taking of physical measurements to acquire sounds or images. To determine what measurements are needed, we use a mathematical model of the physical process. For example, speech recording begins with a model of the sound vibrations that a human being can produce. These result in slight fluctuations of the air pressure that can be measured by a microphone, producing a real-valued function of one real “time” variable. Similarly, taking a picture begins with a model of the light arriving at the camera. The amount that arrives during the acquisition period, or when the shutter is open, is a nonnegative real-valued function of two “space” variables. Finally, video can be modeled by a nonnegative function of one time plus two space variables.

Adapted waveform "de-noising" for medical signals and images

Our goal is to describe tools for adapting methods of analysis to various tasks occurring in harmonic and numerical analysis and signal processing. By choosing an orthonormal basis, in which space and frequency are suitably localized, one can achieve both understanding of structure and efficiency in computation. In fact, the search for computational efficiency is intimately related to efficiency in representation (i.e., compression) and to pattern extraction, or structural understanding. >

Inverse scattering for the heat operator and evolutions in

AbstractThe asymptotic behavior of functions in the kernel of the perturbed heat operator δ12−δ2−u(x) suffice to determineu(x). An explicit formula is derived using the

variables

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