影像對位(image registration)在影像處理(image processing)的範疇中是一門極為基本且重要的技術，以探討如何整合並利用影像間所提供的資訊，透過一組適當的幾何轉換(geometric transformation)作為描述，達到其對位的目的。其涉及的應用範圍相當廣泛，包括不同角度、高度、儀器的遙測影像之整合(aerial imaging)，全景照之合成(panoramic imaging)，以及不同時間、儀器的醫學影像之追蹤或整合(medical imaging)等等，皆為此一技術的具體實現。
本研究的目的在於以幾何學的觀點提出一種連續分段仿射轉換(continuous piecewise affine transformation; CPA-transformation)，並以取得影像間彼此對應的特徵點為前提，將其應用於影像對位的議題中。本研究將針對所建立的方法同時進行理論層面與實務層面的探討：前者根據理論基礎做推衍與分析，文中提出兩個重要的性質－可逆性(invertibility)和傳遞性(transitivity)，並使其在輕度的條件下得以成立；後者則將我們的問題設計為一具有線性方程式限制的最小平方法之最佳化問題(least-squares optimization problem with linear constraints)，而這類問題在數學規劃法(mathematical programming)中已有許多發展完備的演算法，採用這些良好的演算法可確保計算過程的收斂性，並能有效節省電腦計算之負荷。其次，考量不同場合之運用，文中亦提出另一種問題設計方法，顯示本研究工作在應用上的適應性與彈性。此外，通篇論文以理論為主、實驗為輔，藉由電腦模擬實例的輔佐，更詳加驗證本研究工作的可行性與實用性。

Image registration is the problem of determining a geometric transformation to properly align the images of concern. This thesis presents a class of transformations – continuous piecewise affine transformations (abbreviated as CPA-transformations) – and its associated design methodologies for 2-D registration problem, by knowing the correspondence of point features as common information in the images to be registered. In our approach, both of theoretical and practical aspects are considered. With emphasis on an axiomatic theoretical development, two important properties – the invertibility and transitivity properties – are raised and proved under some mild conditions. In practical aspect, the CPA-transformation design problem is formulated as a least-squares optimization problem with linear constraints so that well-developed computational algorithms in mathematical programming can be applied directly. Furthermore, several experimental simulations are given in this thesis to demonstrate the contribution and applicability of our approach.

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