以傅立葉分析理論探討資訊重建的問題

Abstract

本論文的目的在於將傅立葉轉換的相角、或大小值，使用疊代的法則用於 腦波訊號和影像重建的問題上。眾所皆知的是，只有傅立葉轉換的相角﹝ 或大小﹞的資訊是不夠充份來用於重建一組﹝一個﹞訊號或影像的。因此 ，我們便試著想藉由低頻濾波器或切割方法的使用來改進這個問題。在本 論文裡，我們研究多頻道腦波訊號和二維影像重建的問題。探討只有相角 ﹝或大小﹞來重建的法則。在腦波分析上，我們使用了有十八個頻道、雙 極錄製的腦波放電訊號。由於我們使用隨機的大小函數，在以整段資料的 相角資訊來重建的實驗中，顯示出有高頻訊號的失真﹝小振幅和尖端的超 越量﹞。接下來，我們便設計了一個低頻濾波器來濾波。實驗結果告訴了 我們這個高頻失真的問題解決了。然後我們做進一步的研究，想利用切割 的方法來改善法則的應用結果。明顯可見的，利用此方法在腦波重建的問 題上，使用較小的切割長度可獲得較好及較省時快速的結果。我們更探討 了基於在多頻道腦波訊號的重建上其空間相角的彼此關聯。利用相角資訊 來重建影像會產生失去其真實灰值度的問題。利用切割的方法我們可以恢 復更多的灰值度資訊。同樣地，利用切割的方法而以大小資訊來重建影像 ；我們可以改善在法則上由於大小函數失去了大部份原有的資訊而難以重 建的問題。 This thesis aims to use iterative algorithms for reconstructing electroencephalogram (EEG) signals and images from the phase or magnitude of its Fourier transform. It is well known that only FT phase (or magnitude)information is insufficient to recover a signal or image. Therefore, we try to improve the reconstruction through the employment of low-pass filter or segmentation technique.In this thesis, we study the reconstruction problem for multi-channel EEG signals and 2-D images. Phase-only and magnitude-only algorithms are investigated. In EEG analysis, 18-channel, bipolar-derivation spike discharge EEG signals are utilized. Phase-only reconstruction from a whole segment ap-pears to have high- frequency distortion (small-amplitude ringing and peak overshoot). Next, a low-pass filter is designed and employed to the random magnitude function. Our experiment shows that the high frequency problem is improved. We further study the segmentation technique to see if the per-formance of the algorithm can be improved. It is observed that shorter seg-ment length leads to better quality of the reconstructed EEG signals and more efficiency computation. In addition,we explore the spatial phase correlation of multi- channel EEG signals based on the reconstruction approach. Image reconstruction from phase- only information has the problem of missing absolute gray-scale information. Applying of segmentation tech-nique,we are able to retrieve most of the gray-scale information. Similarly, application segmentation technique to magnitude-only reconstruction overcomes the limitation of the algorithm due to large information loss in magnitude function.