A frame is a possibly linearly dependent set of vectors in a Hilbert space that facilitates the decomposition and reconstruction of vectors. A Parseval frame is a frame that acts as its own dual frame. A Gabor frame comprises all translations and phase modulations of an appropriate window function. We show that the space of all frames on a Hilbert space indexed by a common measure space can be fibrated into orbits under the action of invertible linear deformations and that any maximal set of unitarily inequivalent Parseval frames is a complete set of representatives of the orbits. We show that all such frames are connected by transformations that are linear in the larger Hilbert space of square-integrable functions on the indexing space. We apply our results to frames on finite-dimensional Hilbert spaces and to the discretization of the Gabor frame with a band-limited window function.

Let $\Omega\subset\mathbb{R}^n$ be an open and connected subset of $\mathbb{R}^n$. We say that a function $F\colon \Omega-\Omega\to\mathbb{C}$, where $\Omega-\Omega=\{x-y\colon x,y\in\Omega\}$, is positive definite if for any $x_1,\ldots,x_m\in\Omega$ and any $c_1,\ldots,c_m\in \mathbb{C}$ we have that $\sum_{j,k=1}^m F(x_j-x_k)c_j\overline{c_k}\geq 0$. Let $F\colon\Omega-\Omega\to\mathbb{C}$ be a continuous positive definite function. We give necessary and sufficient conditions for $F$ to have an extension to a continuous and positive definite function defined on the entire Euclidean space $\mathbb{R}^n$. The conditions are formulated in terms of strong commutativity of some certain selfadjoint operators defined on a Hilbert space associated to our positive definite function.

This thesis compares and contrasts currents- and varifolds-based diffeomorphic image registration approaches for registering tree-like structures in the lung and surface of the lung. In these approaches, curve-like structures in the lung—for example, the skeletons of vessels and airways segmentation—and surface of the lung are represented by currents or varifolds in the dual space of a Reproducing Kernel Hilbert Space (RKHS). Currents and varifolds representations are discretized and are parameterized via of a collection of momenta. A momenta corresponds to a line segment via the coordinates of the center of the line segment and the tangent direction of the line segment at the center. A momentum corresponds to a mesh via the coordinates of the center of the mesh and the normal direction of the mesh at the center. The magnitude of the tangent vector for the line segment and the normal vector for the mesh are the length of the line segment and the area of the mesh respectively. A varifolds-based registration approach is similar to currents except that two varifolds representations are aligned independent of the tangent (normal) vector orientation. An advantage of varifolds over currents is that the orientation of the tangent vectors can be difficult to determine especially when the vessel and airway trees are not connected. In this thesis, we examine the image registration sensitivity and accuracy of currents- and varifolds-based registration as a function of the number and location of momenta used to represent tree like-structures in the lung and the surface of the lung. The registrations presented in this thesis were generated using the Deformetrica software package, which is publicly available at www.deformetrica.org.

An operator-theoretic formulation of the interpolation problem posed by Nevanlinna and Pick in the early twentieth century asks for conditions under which there exists a multiplier of a reproducing kernel Hilbert space that interpolates a specified set of data. Paul S. Muhly and Baruch Solel have shown that their theory for operator algebras built from W*-correspondences provides an appropriate context for generalizing this classic question. Their reproducing kernel W*-correspondences are spaces of functions that generalize the reproducing kernel Hilbert spaces. Their Nevanlinna-Pick interpolation theorem, which is proved using commutant lifting, implies that the algebra of multipliers of the reproducing kernel W*-correspondence associated with a certain W*-version of the classic Szegö kernel may be identified with their primary operator algebra of interest, the Hardy algebra. To provide a context for generalizing another familiar topic in operator theory, the study of the weighted Hardy spaces, Muhly and Solel have recently expanded their theory to include operator-valued weights. This creates a new family of reproducing kernel W*-correspondences that includes certain, though not all, classic weighted Hardy spaces. It is the purpose of this thesis to generalize several of Muhly and Solel's results to the weighted setting and investigate the function-theoretic properties of the resulting spaces. We give two principal results. The first is a weighted version of Muhly and Solel's commutant lifting theorem, which we obtain by making use of Parrott's lemma. The second main result, which in fact follows from the first, is a weighted Nevanlinna-Pick interpolation theorem. Other results, several of which follow from the two primary results, include the construction of an orthonormal basis for the nonzero tensor product of two W*-corrrespondences, a double commutant theorem, the identification of several function-theoretic properties of the elements in the reproducing kernel W*-correspondence associated with a weighted W*-Szegö kernel as well as the elements in its algebra of mutlipliers, and the presentation of a relationship between this algebra of multipliers and a weighted Hardy algebra. In addition, we consider a candidate for a W*-version of the complete Pick property and investigate the aforementioned weighted W*-Szegö kernel in its light.

This thesis is a preliminary study of the relative performance of the major speech compression techniques, Differential Pulse Code Modulation (DPCM) and Subband Coding (SBC) in the presence of transmission distortion. The combined effect of the channel distortions and the channel codec including error correction is represented by bursts of bit errors. While compression is critical since bandwidth is scarce in a wireless channel, channel distortions are greater and less predictable. Little to no work has addressed the impact of channel errors on perceptual quality of speech due to the complexity of the problem. At the transmitter, the input signal is compressed to 24 kbps using either DPCM or SBC, quantized, binary encoded and transmitted over the burst error channel. The reverse process is carried out at the receiver. DPCM achieves compression by removing redundant information in successive time domain samples, while SBC uses lower resolution quantizer to encode frequency bands of lower perceptual importance. The performance of these codecs is evaluated for BERs of 0.001 and 0.05, with the burst lengths varying between 4 and 64 bits. Two different speech segments - one voiced and one unvoiced are used in testing. Performance measures include two objective tests signal to noise ratio (SNR) & segmental SNR, and a subjective test of perceptual quality - the Mean Opinion Score (MOS). The results obtained show that with a fixed BER and increasing burst length in bits, the total errors reduce in the decoded speech thereby improving its perceptual quality for both DPCM and SBC. Informal subjective tests also demonstrate this trend as well as indicate distortion in DPCM seemed to be less perceptually degrading than SBC. / Graduation date: 1999

In the first part of this paper we show a similarity between the principle of Structural Risk Minimization Principle (SRM) (Vapnik, 1982) and the idea of Sparse Approximation, as defined in (Chen, Donoho and Saunders, 1995) and Olshausen and Field (1996). Then we focus on two specific (approximate) implementations of SRM and Sparse Approximation, which have been used to solve the problem of function approximation. For SRM we consider the Support Vector Machine technique proposed by V. Vapnik and his team at AT&T Bell Labs, and for Sparse Approximation we consider a modification of the Basis Pursuit De-Noising algorithm proposed by Chen, Donoho and Saunders (1995). We show that, under certain conditions, these two techniques are equivalent: they give the same solution and they require the solution of the same quadratic programming problem.

Light transport simulation aims at the numerical computation of the propagation of visible electromagnetic energy in macroscopic environments. In this thesis, we develop the foundations for a modern theory of light transport simulation, unveiling the geometric structure of the continuous theory and providing a formulation of computational techniques that furnishes remarkably efficacy with only local information. Utilizing recent results from various communities, we develop the physical and mathematical structure of light transport from Maxwell's equations by studying a lifted representation of electromagnetic theory on the cotangent bundle. At the short wavelength limit, this yields a Hamiltonian description on six-dimensional phase space, with the classical formulation over the space of "positions and directions" resulting from a reduction to the five-dimensional cosphere bundle. We establish the connection between light transport and geometrical optics by a non-canonical Legendre transform, and we derive classical concepts from radiometry, such as radiance and irradiance, by considering measurements of the light energy density. We also show that in idealized environments light transport is a Lie-Poisson system for the group of symplectic diffeomorphisms, unveiling a tantalizing similarity between light transport and fluid dynamics. Using Stone's theorem, we also derive a functional analytic description of light transport. This bridges the gap to existing formulations in the literature and naturally leads to computational questions. We then address one of the central challenges for light transport simulation in everyday environments with scattering surfaces: how are efficient computations possible when the light energy density can only be evaluated pointwise? Using biorthogonal and possibly overcomplete bases formed by reproducing kernel functions, we develop a comprehensive theory for computational techniques that are restricted to pointwise information, subsuming for example sampling theorems, interpolation formulas, quadrature rules, density estimation schemes, and Monte Carlo integration. The use of overcomplete representations makes us thereby robust to imperfect information, as is often unavoidable in practical applications, and numerical optimization of the sampling locations leads to close to optimal techniques, providing performance which considerably improves over the state of the art in the literature.

Light transport simulation aims at the numerical computation of the propagation of visible electromagnetic energy in macroscopic environments. In this thesis, we develop the foundations for a modern theory of light transport simulation, unveiling the geometric structure of the continuous theory and providing a formulation of computational techniques that furnishes remarkably efficacy with only local information. Utilizing recent results from various communities, we develop the physical and mathematical structure of light transport from Maxwell's equations by studying a lifted representation of electromagnetic theory on the cotangent bundle. At the short wavelength limit, this yields a Hamiltonian description on six-dimensional phase space, with the classical formulation over the space of "positions and directions" resulting from a reduction to the five-dimensional cosphere bundle. We establish the connection between light transport and geometrical optics by a non-canonical Legendre transform, and we derive classical concepts from radiometry, such as radiance and irradiance, by considering measurements of the light energy density. We also show that in idealized environments light transport is a Lie-Poisson system for the group of symplectic diffeomorphisms, unveiling a tantalizing similarity between light transport and fluid dynamics. Using Stone's theorem, we also derive a functional analytic description of light transport. This bridges the gap to existing formulations in the literature and naturally leads to computational questions. We then address one of the central challenges for light transport simulation in everyday environments with scattering surfaces: how are efficient computations possible when the light energy density can only be evaluated pointwise? Using biorthogonal and possibly overcomplete bases formed by reproducing kernel functions, we develop a comprehensive theory for computational techniques that are restricted to pointwise information, subsuming for example sampling theorems, interpolation formulas, quadrature rules, density estimation schemes, and Monte Carlo integration. The use of overcomplete representations makes us thereby robust to imperfect information, as is often unavoidable in practical applications, and numerical optimization of the sampling locations leads to close to optimal techniques, providing performance which considerably improves over the state of the art in the literature.

No description available.

Modern data collection methods are now frequently returning observations that should be viewed as the result of digitized recording or sampling from stochastic processes rather than vectors of finite length. In spite of great demands, only a few classification methodologies for such data have been suggested and supporting theory is quite limited. The focus of this dissertation is on discrimination and classification in this infinite dimensional setting. The methodology and theory we develop are based on the abstract canonical correlation concept of Eubank and Hsing (2005), and motivated by the fact that Fisher's discriminant analysis method is intimately tied to canonical correlation analysis. Specifically, we have developed a theoretical framework for discrimination and classification of sample paths from stochastic processes through use of the Loeve-Parzen isomorphism that connects a second order process to the reproducing kernel Hilbert space generated by its covariance kernel. This approach provides a seamless transition between the finite and infinite dimensional settings and lends itself well to computation via smoothing and regularization. In addition, we have developed a new computational procedure and illustrated it with simulated data and Canadian weather data.