離散正交多項式於訊號及影像處理的應用

Abstract

在這篇碩士論文中，主要著重於離散正交多項式於訊號及影像處理的應用。這些應用概括了：時間序列的逼近與分段、時變自相關隨機程序模型的建立以及實作Savitzky-Golay濾波器並將其應用於訊號與影像處理。最後介紹了時域有限差分法(Finite Difference Time Domain)這個數值方法與其在路徑搜尋上的應用。 在架構上，首先，我會先介紹離散正交多項式的定義以及常見性質，並以其為基底函數應用於時間序列的逼近以及分段，且進一步地發展可快速求出基底函數權重的方法。接著同樣利用了正交多項式做為基底函數求取時變自相關隨機程序模型的參數以及實現Savitzky-Golay濾波器。其中對於Savitzky-Golay濾波器的性質與相關應用有較詳盡的整理，包含數位差分器、影像強化以及去除影像雜訊等等。並提出利用Savitzky-Golay濾波器的概念達到可兼顧影像品質以及雜訊強度的多重規模自適性除影像雜訊的方法。 希望這本論文能對您有幫助。

In this thesis, we mainly focus on the applications of discrete orthogonal polynomials, including time series approximation and segmentation, build of time-variant autoregressive (TVAR) process model and implementation of Savitzky-Golay filter (S-G filter) for image and signal processing. Besides, I also introduce the Finite difference time domain (FDTD) method, a numerical method used for solving electromagnetic wave equation originally, and its applications to path planning and image segmentation.

About the structure of this thesis, first, the definition and common properties of discrete orthogonal polynomials are introduced. Then using discrete orthogonal polynomials as bases, we apply them for time series approximation and segmentation with new developed fast update equation of weighting coefficients, estimation of parameters of time-variant autoregressive model and realization of Savitzky-Golay smoothing filter. Also, I summary the main applications and properties of Savitzky-Golay filter, such as digital differentiator, image enhancement and image denoising. I further proposed the multi-scale adaptive image denosing method based on the concept of Savitzky-Golay filter. Compared with traditional image denosing using Savitzky-Golay filter, the proposed method can maintain the detail of image better when denosing. May this thesis will be useful for you.