以精準計算想法輔助一維訊號與圖片的擴展與壓縮

Abstract

在訊號分析的領域中，高頻區的分析時常會遇到比較大的誤差，例如吉布斯現象。為了嘗試解決這些問題，本論文先提出了精準計算的概念，在有限項傅立葉級數中加入額外的修正項輔助分析訊號高頻區，並成功降低吉布斯現象，讓分析結果能夠更加正確。基於精準計算中區分訊號高低頻的想法，本論文透過自定義的方法將包含心電圖與腦電圖在內的一維訊號分為高頻區與低頻區，再分別透過離散餘弦轉換與勒壤德多項式展開的組合進行訊號分解，與其他方法進行壓縮效果的比較，發現當一維訊號的高頻區與低頻區之間的區分明顯且交雜不多時，本論文所提出的運算方法能夠在一維訊號壓縮中有卓越的表現，但在高低頻混雜的訊號就比較難以展現理想的結果。 接著，本論文亦將相關結果擴展到二維影像處理中，比較自定義方法與其他方法的壓縮效果，並嘗試將壓縮後的影像重建為新影像，同樣發現當圖片的高頻區與低頻區之間的區分明顯且交雜不多時，圖片的壓縮與重建效果非常理想。若圖片的高低頻混雜比較明顯，重建的圖片就會出現比較明顯的失真。

In the field of signal analyzing, the error often become significant in high frequency parts, such as Gibbs phenomenon. In order to solve the problem, the thesis advances the conception of precision calculation, adding extra correction items to support the analyzation of the high frequency parts of the signals, which reduces Gibbs phenomenon successfully. Based on the conception of separating the signal’s high frequency parts and low frequency parts in precision calculation, the thesis divides 1D signals, including ECG and EEG, into high frequency parts and low frequency parts by self-defined methods. Afterwards, we decompose the signal through the combination of discrete Fourier transform(DCT) and Legendre polynomial expansion respectively, then compare the compression efficiency of the proposed methods with other methods. We find out that the proposed methods work perfectly when the signal’s high frequency parts and low frequency parts are separated clearly without severe hybridization. However, the methods fail to show ideal outcome in severe hybridization ones. Moreover, we expand our proposed methods to 2D image compression, compare the compression efficiency of the proposed methods with other methods, then attempt to recover the new images from the compressed ones. Similar to 1D signals, when the signal’s high frequency parts and low frequency parts are separated clearly without severe hybridization, the recovery of the image shows ideal result, and distortions will be observed in severe hybridization images.