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研究生： 林漢倫
Han-Lun Lin
論文名稱： TILT 中基於參數變換的廣義影像轉換
Generalized Image Warping via Subtau-Based Transformations in TILT Processing
指導教授： 鄭經斅
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2025
畢業學年度： 113
語文別： 中文
論文頁數： 34
中文關鍵詞： TILT演算法 、幾何映射 、影像變換 、參數設計 、影像處理 、雅可比矩陣計算
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    • 本研究基於TILT演算法模型，提出一套可高度自訂的幾何變換流程，作為TILT
    操作中的影像轉換模組。傳統流程中常使用MATLAB內建的imtransform函數進行仿射
    與投影轉換，但其參數形式固定，難以處理較為複雜或非線性的幾何變形。為此，我
    們設計了新的影像轉換函數gsubtau，以一組參數向量τ 描述可調變換模型。該模型可
    支援單應性矩陣、相機校準的旋轉與平移（RT），以及額外的曲線扭曲項，提供更高的
    靈活度與應用擴展性。此外，我們重新設計了Jacobian計算方法，採用數值差分方式
    估算參數偏導數，以因應τ結構可變的特性。初步實驗結果顯示，此方法可維持TILT
    操作的收斂特性與變換方向，並支援多樣的變形組合。

    This research presents a highly customizable geometric transformation workflow based
    on the TILT algorithm, serving as the image transformation module within the TILT opera
    tion. Traditional workflows often utilize MATLAB’s built-in imtransform function for affine
    and projective transformations; however, its fixed parameter format hinders handling complex
    or nonlinear geometric deformations. To address this, we designed a new image transformation
    function, gsubtau, described by a parameter vector τ. This model supports homography ma
    trices, camera calibration rotation and translation (RT), and additional curve distortion terms,
    offering greater flexibility and application scalability. Furthermore, we redesigned the Jacobian
    calculation method, employing numerical differentiation to estimate parameter partial deriva
    tives to accommodate the variable structure of τ. Preliminary experimental results indicate that
    this method maintains the convergence characteristics and transformation direction of the TILT
    operation while supporting diverse deformation combinations.

    摘要iv Abstract v 目錄vi 一、緒論1 二、研究方法3 2.1低秩與稀疏分解模型.......................................................... 3 2.2具幾何變形之低秩紋理建模.................................................. 4 2.3幾何變換模型：針孔成像模型................................................ 5 2.4曲面展平模型................................................................. 7 2.5問題(2.2)的求解.............................................................. 8 2.5.1求解(2.2)的迭代式線性近似策略..................................... 8 2.5.2增廣拉格朗日法的介紹................................................ 9 2.6可自訂幾何變換設計.......................................................... 10 2.6.1原TILT幾何變換限制................................................. 10 2.6.2自訂參數向量τ設計.................................................. 11 2.6.3自訂影像變換函數gsubtau ............................................ 12 2.6.4幾何映射函數gmapping............................................... 13 2.6.5計算Jacobian........................................................... 16 三、實驗結果17 3.1幾何映射函數轉換效果....................................................... 17 3.1.1實驗設置............................................................... 17 3.1.2不同模式的轉換效果.................................................. 17 3.2改進TILT模型的演算法表現................................................. 18 3.2.1實驗設置與限制條件.................................................. 18 3.2.2角點模式............................................................... 19 3.2.3相機參數模式.......................................................... 20 3.2.4曲面模式............................................................... 21 3.2.5整體演算法表現評估.................................................. 22 四、總結24 參考文獻25

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