The classical Fourier transform is, in essence, a way to take data and extract components (in the form of complex exponentials) which are invariant under cyclic shifts. We consider a case in which the components must instead be invariant under automorphisms of a binary tree. We present a technique by which a slightly relaxed form of the generalized Fourier transform in this case can eventually be computed using only simple tools from linear algebra, which has possible advantages in computational efficiency.

Hilbert-Huang Transform (HHT) is a data analysis tool, first developed in 1998, which can be used to extract the periodic components embedded within oscillatory data. This thesis is dedicated to the understanding, application, and development of this tool. First, the background theory of HHT will be described and compared with other spectral analysis tools. Then, a number of applications will be presented, which demonstrate the capability for HHT to dissect and analyze the periodic components of different oscillatory data. Finally, a new algorithm is presented which expands HHT ability to analyze discontinuous data. The sum result is the creation of a number of useful tools developed from the application of HHT, as well as an improvement of the HHT tool itself.

No description available.

The construction of the generalized continuous wavelet transform (GCWT) on Hilbert spaces is a special case of the coherent state transform construction, where the coherent state system arises as an orbit of an admissible vector under a strongly continuous unitary representation of a locally compact group. In this thesis we extend this construction to the setting of Hilbert C*-modules. In particular, we define a coherent state transform and a GCWT on Hilbert modules. This construction gives a reconstruction formula and a resolution of the identity formula analogous to those found in the Hilbert space setting. Moreover, the existing theory of standard normalized tight frames in finite countably generated Hilbert modules can be viewed as a discrete case of this construction We also show that the image space of the coherent state transform on Hilbert module is a reproducing kernel Hilbert module. We discuss the kernel and the intertwining property of the group coherent state transform.

ABSTRACT The quest to model wave propagation in various physical systems has produced a large set of diverse nonlinear equations. Nonlinear singular integro-differential equations rank amongst the intricate nonlinear wave equations available to study the classical problem of wave propagation in physical systems. Integro-differential equations are characterized by the simultaneous presence of integration and differentiation in a single equation. Substantial interest exists in nonlinear wave equations that are amenable to the Inverse Scattering Transform (IST). The IST is an adroit mathematical technique that delivers analytical solutions of a certain type of nonlinear equation: soliton equation. Initial value problems of numerous physically significant nonlinear equations have now been solved through elegant and novel implementations of the IST. The prototype nonlinear singular integro-differential equation receptive to the IST is the Intermediate Long Wave (ILW) equation, which models one-dimensional weakly nonlinear internal wave propagation in a density stratified fluid of finite total depth. In the deep water limit the ILW equation bifurcates into a physically significant nonlinear singular integro-differential equation known as the 'Benjamin-Ono' (BO) equation; the shallow water limit of the ILW equation is the famous Korteweg-de Vries (KdV) equation. Both the KdV and BO equations have been solved by dissimilar implementations of the IST. The Modified Korteweg-de Vries (MKdV) equation is a nonlinear partial differential equation, which was significant in the historical development of the IST. Solutions of the MKdV equation are mapped by an explicit nonlinear transformation known as the 'Miura transformation' into solutions of the KdV equation. Historically, the Miura transformation manifested the intimate connection between solutions of the KdV equation and the inverse problem for the one-dimensional time independent Schroedinger equation. In light of the MKdV equation's significance, it is natural to seek 'modified' versions of the ILW and BO equations. Solutions of each modified nonlinear singular integro-differential equation should be mapped by an analogue of the original Miura transformation into solutions of the 'unmodified' equation. In parallel with the limiting cases of the ILW equation, the modified version of the ILW equation should reduce to the MKdV equation in the shallow water limit and to the modified version of the BO equation in the deep water limit. The Modified Intermediate Long Wave (MILW) and Modified Benjamin-Ono (MBO) equations are the two nonlinear singular integro-differential equations that display all the required attributes. Several researchers have shown that the MILW and MBO equations exhibit the signature characteristic of soliton equations. Despite the significance of the MILW and MBO equations to soliton theory, and the possible physical applications of the MILW and MBO equations, the initial value problems for these equations have not been solved. In this thesis we use the IST to solve the initial value problems for the MILW and MBO equations on the real-line. The only restrictions that we place on the initial values for the MILW and MBO equations are that they be real-valued, sufficiently smooth and decay to zero as the absolute value of the spatial variable approaches large values.

Thesis (Sc. D.)--Massachusetts Institute of Technology and Woods Hole Oceanographic Institution, 1987. / "January 1987." Bibliography: p. 193-200.

No description available.

Spectral decomposition is a modern tool that utilizes seismic data to generate additional useful information in seismic exploration for hydrocarbon detection, lithology identification, stratigraphic interpretation, filtering and others. Different spectral decomposition methods with applications to seismic data were reported and investigated in past years. Many methods usually do not consider the non-stationary features of seismic data and, therefore, are not likely to give satisfactory results. S-transform developed in recent years is able to provide time-dependent frequency analysis while maintaining a direct relationship with the Fourier spectrum, a unique property that other methods of spectral decomposition may not have. In this thesis, I investigated the feasibility and efficiency of using S-transform for hydrocarbon detection and time-varying surface wave filtering. S-transform was first applied to two seismic data sets from a clastic reservoir in the North Sea and a deep carbonate reservoir in the Sichuan Basin, China. Results from both cases demonstrated that S-transform decomposition technique can detect hydrocarbon zones effectively and helps to build the relationships between lithology changes and high frequency variation and between hydrocarbon occurrence and low-frequency anomaly. However, its time resolution needs to be improved. In the second part of my thesis, I used S-transform to develop a novel Time-frequency-wave-number-domain (T-F-K) filtering method to separate surface wave from reflected waves in seismic records. The S-T-F-K filtering proposed here can be used to analyze surface waves on separate f-k panels at different times. The method was tested using hydrophone records of four-component seismic data acquired in the shallow-water Persian Gulf where the average water depth is about 10m and Scholte waves and other surfaces wave persistently strong. Results showed that this new S-T-F-K method is able to separate and sttenuate surface waves and to improve greatly the quality of seismic reflection signals that are otherwise completely concealed by the aliased surface waves.

The evolution of technology in the textile industry reached the design of embroidery patterns for machine embroidery. In order to create quality designs the shapes to be embroidered need to be segmented into regions that define different parts. One of the objectives of our research is to develop a method to automatically segment the shapes and by doing so making the process faster and easier. Shape analysis is necessary to find a suitable method for this purpose. It includes the study of different ways to represent shapes. In this thesis we focus on shape representation through its skeleton. We make use of a shape's skeleton and the shape's boundary through the so-called feature transform to decide how to segment a shape and where to place the segment boundaries. The direction of stitches is another important specification in an embroidery design. We develop a technique to select the stitch orientation by defining direction lines using the skeleton curves and information from the boundary. We compute the intersections of segment boundaries and direction lines with the shape boundary for the final definition of the direction line segments. We demonstrate that our shape segmentation technique and the automatic placement of direction lines produce sufficient constrains for automated embroidery designs. We show examples for lettering, basic shapes, as well as simple and complex logos.

In this note, we study the inverse of the Abel transform for the symmetric space ${\bf SU}^{\star}(2n)/{\bf Sp}(n)$. We start by giving a recursive formula for the dual of the Abel transform on the root system $A_{n-1}$. This formula allows us to consider a transmutation property on the generalized Abel transform associated to $A_{n-1}$. / This paper has not been published. However, it has been cited in peer reviewed venues.