指數型態數據分析在許多科學和工程領域中佔有關鍵的地位,例如電子學,力學,流體力學,半導體物理學,化學,生物物理學,醫學成像等領域。我們研究在實際應用中，所面對有關指數型態數據分析的挑戰問題。我們的目標，是用最先進的數學成果，來處理科學與工程中的難題。

指數型態數據分析中的Prony方法，可以追溯在18世紀。這個經典的方法，將指數插值問題轉化為一個解根的問題，以及一個解一組線性方程式的問題。Prony 方法，可以從一組等距取樣的時間序列求值，解出一個背後是指數函數訊息中的頻率、震幅、相位差、阻尼因子。理論上，Prony 方法可以高分辨率的頻率資料。然而,在實際運用上,各種數值上的問題，限制了Prony方法的應用。即使到了20世紀運算效能隨著電腦科技大幅改進之際,基於Prony的方法並沒有像基於傅立葉的方法那般 被廣泛的應用。

近年來,Prony方法再度引起興趣。有一些最近的研究成果，是基於Prony方法與數值逼近理論領域中Pad ́e逼近法、以及符號運算中稀疏多項式插值的相關性。本論文旨在研究如何在一些實際的應用中，使用這些先進的指數分析技術。我們首先調查探討使用的指數訊號模型分析的不同應用領域。然後我們介紹一種短時距的Prony方法,用於檢測可能由電力系統產生的瞬態信號。我們還透過Prony方法和Pad ́e逼近法的相關性, 利用Pad ́e逼近理論中的一些收斂定理,來檢測信號中微弱與群聚的分量。這項技術的實用性，展現在電動機電流特徵分析(MCSA)、磁共振波譜 (MRS)、螢光生命週期成像(FLIM)以及磁共振成像(MRI) 的相關問題應用。最後，在數位浮水印的研究,我們提出了一種脆弱型數位浮水印方 案,可以用客觀的數學條件來驗證所有權,而且無需儲存原始浮水印資料。我們提出的浮水印內容，是基於跟指數分析中 Prony 方法習習相關、符號運算中的稀疏多項式差值。

Exponential analysis plays a crucial role in many scientific and engineering fields, such as electronics, mechanics, fluid mechanics, semi-conductor physics, chem- istry, 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 di cult 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. Theoret- ically, Prony’s method can deliver high-resolution frequency information. How- ever, 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 di↵erent application do- mains. 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 ap- proximation theory. The practicality is illustrated in motor current signature analysis (MCSA) and magnetic resonance spectroscopy (MRS), as well as relax- ometry 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 origi- nal 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.

Chapter 1
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