| dc.description.abstract | The thesis deals with a framework for the analysis and synthesis of one- and two-dimensional signals, created by a new array of wavelets as basis functions, which are derived from generalized Hermite polynomials.
The signals under consideration could appear in various forms: they may be functions of time (like speech and sound) or space (like seismic data and images). Their information could be contained in the low-, medium-, and/or high-frequency parts of the spectrum. Signal analysis involves extraction of signal characteristics, like points of sudden change in signal values in the spatial/time domain, and boundaries between different components in the spectral/frequency domain. It is found that these characteristics reveal themselves at specific resolutions/scales. One of the important problems in this context is how to represent a signal so as to facilitate its analysis. More significantly, is there a framework which permits feature extraction directly from the representation of the signal? In addition to seeking a solution to these problems, an overriding motive in signal analysis is the localization of features in the spatial/time and spectral/frequency domains.
As far as the mathematical preliminaries are concerned, the signals under consideration are (i) assumed to be real, (ii) defined over
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(??,?) in both the spatial/time and spectral domains, and (iii) assumed to belong to the class of
L2
L
2
functions. The classical uncertainty principle says that a signal and its Fourier transform cannot both have compact support. Therefore, standard Fourier transform techniques, including their window (or short-time) versions (like the Gaussian and Gabor transforms), cannot meet localization requirements. Moreover, the size of the resolution cell in the phase-space is uniform.
In contrast, recent results, using what have come to be known as wavelets, point the way to overcoming the limitations of the classical techniques. However, there do exist certain difficulties in their exploitation. For instance, the wavelets currently used, like the Laplacian of the Gaussian and the Gabor function, are not orthogonal. On the other hand, orthogonal wavelets, like the cubic spline generated by recursion, are difficult to employ.
In an attempt to overcome the difficulties associated with the wavelets of the literature, a new array of wavelet-like functions is created from generalized Hermite polynomials. These functions are orthogonal and constitute a complete basis for the class of signals under consideration. They can be treated as a vector or a matrix array of wavelets for one- or two-dimensional signals, respectively. A significant outcome of this framework is that, in contrast with many of the results of the literature, signal representation is unique, and signal analysis can be handled explicitly in terms of the coefficient array. A further advantage of the proposed framework is that the one-dimensional Hermite polynomials can be used to generate (by tensor product) the two-dimensional versions, appropriately parametrized.
The new framework for signal analysis is endowed with the following properties:
The multilayered representation is stable.
The outputs of the various layers are independent of each other.
The zero-crossings of the outputs of the various layers can be obtained directly from the corresponding coefficient array.
The phase-space resolution cell has a variable shape, independent of its location (in the phase-space), unlike that of classical and wavelet transforms.
The thesis is organized as follows:
In Chapter 1, Fourier analysis of band-limited signals and window transforms are surveyed. The concepts of phase-space resolution cell and space/time and frequency localization are explained.
Chapter 2 contains a detailed analysis of the existing literature on wavelets, including a discussion of their advantages and limitations. The results of implementation of Mallat’s algorithm are also given.
The main contributions of the thesis are found in Chapters 3 and 4. In Chapter 3, a novel method for one-dimensional signal representation using a vector array of wavelets is presented. The properties of the representation are examined, and analytic expressions are derived for the space/time-bandwidth product and space/time-bandwidth ratio. These replace the uncertainty inequalities of the literature. The representation scheme is illustrated by examples.
Extensions to two-dimensional signal analysis are given in Chapter 4. Images, which constitute the most important class of such signals, are represented by using a matrix array of wavelets.
The thesis concludes with Chapter 5, which contains a summary of the work along with problems for further research.
The references contain only those items which have a direct bearing on or are related to the research work of the thesis and are easily accessible. Therefore, the references in French (hard to come by but still quoted by some researchers) are not included. It may also be mentioned here that the photographs were developed by a commercial photographer and hence are not entirely indicative of the actual results obtained. | |
