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Title
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On the use of exponential analysis in science and industry
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Author
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Abstract
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| Exponential analysis plays a crucial role in many scientific and engineering fields, such as electronics, mechanics, fluid mechanics, semi-conductor physics, chemistry, biophysics, medical imaging, and so on. We investigate challenges related to exponential analysis from practical applications. Our goal is to use recent mathematical results to tackle difficult problems that may arise in science and engineering. In exponential analysis, Prony’s method can be traced back to the eighteenth century. This classic method turns an exponential interpolation problem into a root-finding problem and the solution of a set of linear equations. From a set of uniformly sampled time series data, it can extract frequencies, amplitudes, phases, and damping factors in a signal that is a sum of exponentials. Theoretically, Prony’s method can deliver high-resolution frequency information. However, in practice, various numerical issues have limited the application of Prony’s method. Ever since the rise of the digital computer in the twentieth century, Prony-based methods have not been implemented as widely as Fourier-based methods. In recent years, though, there has been a resurgence of interest in Prony’s method. Some of the latest developments are based on the connections between Prony’s method and Pad´e approximation from numerical approximation theory or sparse polynomial interpolation from computer algebra. This research aims to implement the latest developments in exponential analysis to some practical applications. We first survey exponential models in different application domains. Then, we introduce a short-time Prony’s method to detect transient signals that may arise from a power system. For detecting faint and clustered components in a signal, through the connection between Prony’s method and Pad´e approximation, we make use of some convergence theorems from Pad´e approximation theory. The practicality is illustrated in motor current signature analysis (MCSA) and magnetic resonance spectroscopy (MRS), as well as relaxometry in fluorescence lifetime imaging (FLIM) and magnetic resonance imaging (MRI). In digital watermarking, we present a fragile watermark scheme that can authenticate ownership by a mathematical criterion without storing the original watermark. The contents of our proposed fragile watermarks is based on sparse polynomial interpolation from computer algebra, which is closely related to Prony’s method in exponential analysis. |
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Language
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English
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Publication
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Antwerp
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University of Antwerp, Faculty of Sciences, Department of Mathematics and Computer Science
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2019
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Volume/pages
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154 p.
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Note
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Cuyt, Annie [Supervisor]
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Lee, Wen-Shin [Supervisor]
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Lee, Greg C. [Supervisor]
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Full text (publisher's version - intranet only)
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