Efficient Design of One and Two Dimensional Wavelet Filters Using All-pass Filters

近年來，小波、濾波器組與多解析訊號分析等理論已經被整合一個獨特的理論。已經有許多研究成果顯示小波基底可以由完美重建的濾波器組來得到。換句話說，完美重建濾波器組可以用來運算離散小波轉換，且可以得到連續的小波基底。 本論文使用數位全通濾波器為基礎來建構雙通道的小波濾波器組。藉由在相位響應上形成適當的近似，可以針對L2 、L∞ 或最大平坦設計準則來得到符合我們預期的濾波器組表現。並且利用簡單的頻譜轉換的方法得到二維小波濾波器。 本論文中有許多設計實例，呈現令人滿意的結果，設計方法簡單且方便，可以驗證此論文的方法可行性。Recently, wavelet, filter banks, and multiresolution signal analysis theorems have been converged to an unique one. Many researches have shown that wavelet bases can be generated by Perfect Reconstruction Filter Banks（PRFB）. In other words, discrete wavelet transform can be computed by using PRFB, and continuous wavelet bases can also be obtained based on PRFB. In this thesis, we construct two-channel PRFB using digital all-pass filters. By appropriately approximating the desired phase responses derived from the proposed structure, we can solve the design problem optimally in the L2,L∞, and maximally flat sense respectively to obtain the expected performance of filter banks. By using frequency mapping technique, we can get 2-D wavelet filters easily. Many simulation examples are presented in this thesis to show the effectiveness of the proposed method. From the comparison of design results, we observe that the proposed method provides the satisfactory design results