Divergence functions are both highly useful and fundamental to many areas in information theory and machine learning, but require either parametric approaches or prior knowledge of labels on the full data set. This paper presents a method to estimate the divergence between two data sets in the absence of fully labeled data. This semi-labeled case is common in many domains where labeling data by hand is expensive or time-consuming, or wherever large data sets are present. The theory derived in this paper is demonstrated on a simulated example, and then applied to a feature selection and classification problem from pathological speech analysis.

The detection and characterization of transients in signals is important in many wide-ranging applications from computer vision to audio processing. Edge detection on images is typically realized using small, local, discrete convolution kernels, but this is not possible when samples are measured directly in the frequency domain. The concentration factor edge detection method was therefore developed to realize an edge detector directly from spectral data. This thesis explores the possibilities of detecting edges from the phase of the spectral data, that is, without the magnitude of the sampled spectral data. Prior work has demonstrated that the spectral phase contains particularly important information about underlying features in a signal. Furthermore, the concentration factor method yields some insight into the detection of edges in spectral phase data. An iterative design approach was taken to realize an edge detector using only the spectral phase data, also allowing for the design of an edge detector when phase data are intermittent or corrupted. Problem formulations showing the power of the design approach are given throughout. A post-processing scheme relying on the difference of multiple edge approximations yields a strong edge detector which is shown to be resilient under noisy, intermittent phase data. Lastly, a thresholding technique is applied to give an explicit enhanced edge detector ready to be used. Examples throughout are demonstrate both on signals and images.

In modern remote sensing, arrays of sensors, such as antennas in radio frequency (RF) systems and microphones in acoustic systems, provide a basis for estimating the direction of arrival of a narrow-band signal at the sensor array. A Uniform linear array (ULA) is the most well-studied array geometry in that its performance characteristics and limitations are well known, especially for signals originating in the far field. In some instances, the geometry of an array may be perturbed by an environmental disturbance that actually changes its nominal geometry; such as, towing an array behind a moving vehicle. Additionally, sparse arrays have become of interest again due to recent work in co-prime arrays. These sparse arrays contain fewer elements than a ULA but maintain the array length. The effects of these alterations to a ULA are of interest. Given this motivation, theoretical and experimental (i.e. via computer simulation) processes are used to determine quantitative and qualitative effects of perturbation and sparsification on standard metrics of array performance. These metrics include: main lobe gain, main lobe width and main lobe to side lobe ratio. Furthermore, in order to ascertain results/conclusions, these effects are juxtaposed with the performance of a ULA. Through the perturbation of each element following the first element drawn from a uniform distribution centered around the nominal position, it was found that both the theoretical mean and sample mean are relatively similar to the beam pattern of the full array. Meanwhile, by using a sparsification method of maintaining all the lags, it was found that this particular method was unnecessary. Simply taking out any three elements while maintaining the length of the array will produce similar results. Some configurations of elements give a better performance based on the metrics of interest in comparison to the ULA. These results demonstrate that a sparsified, perturbed or sparsified and perturbed array can be used in place of a Uniform Linear Array depending on the application.

The field of computed tomography involves reconstructing an image from lower dimensional projections. This is particularly useful for visualizing the inner structure of an object. Presented here is an imaging setup meant for use in computed tomography applications. This imaging setup relies on imaging electric fields through active interrogation. Models designed in Ansys Maxwell are used to simulate this setup and produce 2D images of an object from 1D projections to verify electric field imaging as a potential route for future computed tomography applications. The results of this thesis show reconstructed images that resemble the object being imaged using a filtered back projection method of reconstruction. This work concludes that electric field imaging is a promising option for computed tomography applications.

The recovery of edge information in the physical domain from non-uniform Fourier data is of importance in a variety of applications, particularly in the practice of magnetic resonance imaging (MRI). Edge detection can be important as a goal in and of itself in the identification of tissue boundaries such as those defining the locations of tumors. It can also be an invaluable tool in the amelioration of the negative effects of the Gibbs phenomenon on reconstructions of functions with discontinuities or images in multi-dimensions with internal edges. In this thesis we develop a novel method for recovering edges from non-uniform Fourier data by adapting the "convolutional gridding" method of function reconstruction. We analyze the behavior of the method in one dimension and then extend it to two dimensions on several examples.

The problem of detecting the presence of a known signal in multiple channels of additive white Gaussian noise, such as occurs in active radar with a single transmitter and multiple geographically distributed receivers, is addressed via coherent multiple-channel techniques. A replica of the transmitted signal replica is treated as a one channel in a M-channel detector with the remaining M-1 channels comprised of data from the receivers. It is shown that the distribution of the eigenvalues of a Gram matrix are invariant to the presence of the signal replica on one channel provided the other M-1 channels are independent and contain only white Gaussian noise. Thus, the thresholds representing false alarm probabilities for detectors based on functions of these eigenvalues remain valid when one channel is known to not contain only noise. The derivation is supported by results from Monte Carlo simulations. The performance of the largest eigenvalue as a detection statistic in the active case is examined, and compared to the normalized matched filter detector in a two and three channel case.

Multiple-channel detection is considered in the context of a sensor network where data can be exchanged directly between sensor nodes that share a common edge in the network graph. Optimal statistical tests used for signal source detection with multiple noisy sensors, such as the Generalized Coherence (GC) estimate, use pairwise measurements from every pair of sensors in the network and are thus only applicable when the network graph is completely connected, or when data are accumulated at a common fusion center. This thesis presents and exploits a new method that uses maximum-entropy techniques to estimate measurements between pairs of sensors that are not in direct communication, thereby enabling the use of the GC estimate in incompletely connected sensor networks. The research in this thesis culminates in a main conjecture supported by statistical tests regarding the topology of the incomplete network graphs.

Passive radar can be used to reduce the demand for radio frequency spectrum bandwidth. This paper will explain how a MATLAB simulation tool was developed to analyze the feasibility of using passive radar with digitally modulated communication signals. The first stage of the simulation creates a binary phase-shift keying (BPSK) signal, quadrature phase-shift keying (QPSK) signal, or digital terrestrial television (DTTV) signal. A scenario is then created using user defined parameters that simulates reception of the original signal on two different channels, a reference channel and a surveillance channel. The signal on the surveillance channel is delayed and Doppler shifted according to a point target scattering profile. An ambiguity function detector is implemented to identify the time delays and Doppler shifts associated with reflections off of the targets created. The results of an example are included in this report to demonstrate the simulation capabilities.

In many systems, it is difficult or impossible to measure the phase of a signal. Direct recovery from magnitude is an ill-posed problem. Nevertheless, with a sufficiently large set of magnitude measurements, it is often possible to reconstruct the original signal using algorithms that implicitly impose regularization conditions on this ill-posed problem. Two such algorithms were examined: alternating projections, utilizing iterative Fourier transforms with manipulations performed in each domain on every iteration, and phase lifting, converting the problem to that of trace minimization, allowing for the use of convex optimization algorithms to perform the signal recovery. These recovery algorithms were compared on a basis of robustness as a function of signal-to-noise ratio. A second problem examined was that of unimodular polyphase radar waveform design. Under a finite signal energy constraint, the maximal energy return of a scene operator is obtained by transmitting the eigenvector of the scene Gramian associated with the largest eigenvalue. It is shown that if instead the problem is considered under a power constraint, a unimodular signal can be constructed starting from such an eigenvector that will have a greater return.

The solution of the linear system of equations $Ax\approx b$ arising from the discretization of an ill-posed integral equation with a square integrable kernel is considered. The solution by means of Tikhonov regularization in which $x$ is found to as the minimizer of $J(x)=\{ \|Ax -b\|_2^2 + \lambda^2 \|L x\|_2^2\}$ introduces the unknown regularization parameter $\lambda$ which trades off the fidelity of the solution data fit and its smoothing norm, which is determined by the choice of $L$. The Generalized Discrepancy Principle (GDP) and Unbiased Predictive Risk Estimator (UPRE) are methods for finding $\lambda$ given prior conditions on the noise in the measurements $b$. Here we consider the case of $L=I$, and hence use the relationship between the singular value expansion and the singular value decomposition for square integrable kernels to prove that the GDP and UPRE estimates yield a convergent sequence for $\lambda$ with increasing problem size. Hence the estimate of $\lambda$ for a large problem may be found by down-sampling to a smaller problem, or to a set of smaller problems, and applying these estimators more efficiently on the smaller problems. In consequence the large scale problem can be solved in a single step immediately with the parameter found from the down sampled problem(s).