Wavelets (Mathematics)

Reducing the amount of radiation in X-ray computed tomography has been an
active area of research in the recent years. The reduction of radiation has the downside of
degrading the quality of the CT scans by increasing the ratio of the noise. Therefore, some
techniques must be utilized to enhance the quality of images. In this research, we approach
the denoising problem using two class of algorithms and we reduce the noise in CT scans
that have been acquired with 75% less dose to the patient compared to the normal dose
scans.
Initially, we implemented wavelet denoising to successfully reduce the noise in
low-dose X-ray computed tomography (CT) images. The denoising was improved by
finding the optimal threshold value instead of a non-optimal selected value. The mean
structural similarity (MSSIM) index was used as the objective function for the
optimization. The denoising performance of combinations of wavelet families, wavelet
orders, decomposition levels, and thresholding methods were investigated. Results of this study have revealed the best combinations of wavelet orders and decomposition levels for
low dose CT denoising. In addition, a new shrinkage function is proposed that provides
better denoising results compared to the traditional ones without requiring a selected
parameter.
Alternatively, convolutional neural networks were employed using different
architectures to resolve the same denoising problem. This new approach improved
denoising even more in comparison to the wavelet denoising.

This study is performed in tandem with numerous experiments performed by the U.S. Navy to characterize the ocean environment in the South Florida region. The research performed in this study includes signal processing steps for isolating ocean phenomena, such as internal waves, in the magnetic field. Raw magnetometer signals, one on shore and one underwater, are processed and removed of common distortions. They are then run through a series of filtering techniques, including frequency domain cancellation (FDC). The results of the filtered magnetic residual are compared to similarly processed Acoustic Doppler Current Profiler (ADCP) data to correlate whether a magnetic signature is caused by ocean phenomena.

Ocean is home to a large population of marine mammals such as dolphins and whales and concerns over anthropogenic activities in the regions close to their habitants have been
increased. Therefore the ability to detect the presence of these species in the field, to
analyze and classify their vocalization patterns for signs of distress and distortion of their
communication calls will prove to be invaluable in protecting these species. The objective of this research is to investigate methods that automatically detect and classify vocalization patterns of marine mammals. The first work performed is the classification of bottlenose dolphin calls by type. The extraction of salient and distinguishing features from recordings is a major part of this endeavor. To this end, two strategies are evaluated with real datasets provided by Woods Hole Oceanographic Institution: The first strategy is to use contour-based features such as Time-Frequency Parameters and Fourier Descriptors and the second is to employ texture-based features such as Local Binary Patterns (LBP) and Gabor Wavelets. Once dolphin whistle features
are extracted for spectrograms, selection of classification procedures is crucial to the success of the process. For this purpose, the performances of classifiers such as K-Nearest Neighbor, Support Vector Machine, and Sparse Representation Classifier (SRC) are assessed thoroughly, together with those of the underlined feature extractors.

As ecosystems degrade globally, ecosystem services that support life are increasingly threatened.
Indications of degradation are occurring in the Northern Indian River Lagoon (IRL) estuary in east central
Florida. Factors associated with ecosystem degradation are complex, including climate and land use
change. Ecosystem research needs identified by the Millennium Ecosystem Assessment (MA) include the
need to: consider the social with the physical; account for dynamism and change; account for complexity;
address issues of scale; and focus on ecosystem structure and process. Ecosystems are complex, self-organizing, multi-equilibrial, non-linear, middle-number systems that exist in multiple stable states. Results found are relative to the observation and the frame of analysis, requiring multi-scaled analytical techniques. This study addresses the identified ecosystem research needs and the complexity of the associated factors given these additional constraints. Relativity is addressed through univariate analysis of dissolved oxygen as a measure of the general health of the Northern IRL. Multiple spatial levels are employed to associate social process scales with physical process scales as basin, sub-basins, and watersheds. Scan statistics return extreme value clusters in space-time. Wavelet transforms decompose time-scales of cyclical data using varying window sizes to locate change in process scales in space over time. Wavelet transform comparative methods cluster temporal process scales across space. Combined these methods describe the space-time structure of process scales in a complex ecosystem relative to the variable examined, where the highly localized results allow for connection to unexamined variables.

Society's increased demand for communications requires searching for techniques that preserve bandwidth. It has been observed that much of the time spent during telephone communications is actually idle time with no voice activity present. Detecting these idle periods and preventing transmission during these idle periods can aid in reducing bandwidth requirements during high traffic periods. While techniques exist to perform this detection, certain types of noise can prove difficult at best for signal detection. The use of wavelets with multi-resolution subspaces can aid detection by providing noise whitening and signal matching. This thesis explores its use and proposes a technique for detection.

This study deals with applying the wavelet transform to mainly two different areas of signal processing: adaptive signal processing, and signal detection. It starts with background information on the theory of wavelets with an emphasis on the multiresolution representation of signals by the wavelet transform in Chapter 1. Chapter 2 begins with an overview of adaptive filtering in general and extends it to transform domain adaptive filtering. Later in the chapter, a novel adaptive filtering architecture using the wavelet transform is introduced. The performance of this new structure is evaluated by using the LMS algorithm with variations in step size. As a result of this study, the wavelet transform based adaptive filter is shown to reduce the eigenvalue ratio, or condition number, of the input signal. As a result, the new structure is shown to have faster convergence, implying an improvement in the ability to track rapidly changing signals. Chapter 3 deals with signal detection with the help of the wavelet transform. One scheme studies signal detection by projecting the input signal onto different scales. The relationship between this approach and that of matched filtering is established. Then the effect of different factors on signal detection with the wavelet transform is examined. It is found that the method is robust in the presence of white noise. Also, the wavelets are analyzed as eigenfunctions of a certain random process, and how this gives way to optimal receiver design is shown. It is further demonstrated that the design of an optimum receiver leads to the wavelet transform based adaptive filter structure described in Chapter 2.

The problems encountered in development and implementation of orthonormal two dimensional wavelet bases and their filter banks in polar coordinates are addressed. These wavelets and filter banks have possible applications in processing signals that are collected by sensors working in the polar coordinate system, such as biomedical and radar generated signals. The relationship between the space of measurable, square-integrable functions on the punctured polar coordinate system L^2(P) and space of measurable, square-integrable functions on the rectangular plane L^2(R^2) is developed. This allows us to develop complete wavelet bases in a more convenient and familiar surrounding of L^2(R^2) and to transport this theory to L^2(P). Corresponding filter banks are also developed. The implementation of wavelet analysis of punctured polar plane is discussed. An example of wavelet bases, filter banks, and implementation is provided.

This thesis presents a comprehensive analysis of a relatively new transform for discrete time signals, called the Discrete Wavelet Transform (DWT). We find how this transform is connected with the already existing theory of perfect reconstruction filter banks and the recently introduced theory of multiresolution analysis. We use the conditions obtained from these two theories in order to understand the construction of wavelet filters, which also generate continuous functions that prove to constitute an orthonormal basis for the L$\sp2$ space. We also investigate the connection of this transform to the sampled wavelet series of nonorthogonal functions with good time-frequency localization properties. Finally, we see the way that the DWT maps a discrete signal in the phase plane and the applications that such representations incorporate.

In this thesis we applied wavelet transforms to image and video coding. First, a survey of various wavelets and their features is presented, including continuous, discrete, and orthogonal wavelets. Theories and concepts underlying one and two-dimensional wavelet transforms are introduced and compared to Fourier transform and sub-band coding. The core of the thesis is the implementation of two-dimensional and three-dimensional codec architectures and their application to coding images and videos, respectively. We studied performance of the wavelet codec by comparing it to DCT and JPEG coding techniques. We applied these techniques for compression of a variety of test images and videos. We also analyzed the adaptability and scalability of 2D and 3D codec. Experimental results, presented in the thesis, illustrate the superior performance of wavelets compared to other coding techniques.

The need for reliable underwater communication at Florida Atlantic University is critical in transmitting data to and from Autonomous Underwater Vehicles (AUV) and remote sensors. Since a received signal is corrupted with ambient ocean noise, the nature of such noise is investigated. Furthermore, we establish connection between ambient ocean noise and fractal noise. Since the matched filter is designed under the assumption that noise is white, performance degradation of the matched filter due non-white noise is investigated. We show empirical results that the wavelet transform provides an approximate Karhunen-Loeve expansion for 1/f-type noise. Since whitening can improve only broadband signals, a new method for synchronization signal design in wavelet subspaces with increased energy-to-peak amplitude ratio is presented. The wavelet detector with whitening of fractal noise and detection in wavelet subspace is shown. Results show that the wavelet detector improves detectability, however this is below expectation due to differences between fractal noise and ambient ocean noise.