結合三角基底正則化與分量式雅可比約束之局部全通方法於 X 光影像對位

Abstract

醫學影像配準是醫學影像分析中的核心課題，其目標在於建立不同時間點、視角或受試者之間影像的空間對應關係。傳統的強度式配準方法，如 B-spline 自由形變（free-form deformation, FFD）、大型變形微分同胚度量映射（large deformation diffeomorphic metric mapping, LDDMM）以及對稱正規化（Symmetric Normalization, SyN）演算法，雖能提供高準確度與良好的理論保證，但通常伴隨著高昂的計算成本，且對參數設定高度敏感。近年來，深度學習方法在計算效率與準確性方面展現顯著優勢，然而其在臨床環境中的泛化能力、對雜訊與偽影的穩健性，以及理論可解釋性方面仍存在限制。

局部全通（Local All-Pass, LAP）框架透過將影像配準問題重新表述為局部全通濾波關係，提供了一種高效率的替代方案。該方法避免了線性化近似與小位移假設，能夠快速且準確地估計位移場，並對單調的影像強度變化具有一定的穩健性。然而，既有的 LAP 方法主要依賴事後的平滑處理進行正則化，正則化能力有限，容易產生不穩定或不可逆的形變，進而限制其於實際醫學影像應用中的表現。

本論文提出三角基底正則化局部全通方法（Trigonometric Basis Regularized Local All-Pass, TBR-LAP），在 LAP 架構下以三角函數基底表示形變更新場，並引入明確的正則化項，包括平滑性約束與 Jacobian 行列式約束，以達到全域且可控的形變重建。此設計在保留 LAP 原有計算效率與估計準確性的同時，能有效提升對雜訊的穩健性，並降低形變場產生折疊的風險。

所提出之 TBR-LAP 方法強化了 LAP 框架於同模態醫學影像配準問題中的適用性，特別適合於 X 光影像之可變形配準，並在理論建模與實務應用層面上，為高效率且穩健的醫學影像配準方法提供新的研究方向。

Medical image registration is a fundamental task in medical image analysis, aiming to establish spatial correspondence between images acquired across different time points, viewpoints, or subjects. Classical intensity-based frameworks, including B-spline free-form deformation (FFD), large deformation diffeomorphic metric mapping (LDDMM), and the Symmetric Normalization (SyN) algorithm, provide high registration accuracy and strong theoretical guarantees, but often incur substantial computational cost and exhibit sensitivity to parameter selection. More recently, deep learning–based approaches have achieved remarkable computational efficiency; however, their generalization ability, robustness to noise and artifacts, and theoretical interpretability remain limited in clinical settings.

The Local All-Pass (LAP) framework offers an efficient alternative by reformulating image registration as a local all-pass filtering relation. By avoiding linearization assumptions, LAP enables accurate and fast displacement estimation and exhibits robustness to monotonic intensity variations. Nevertheless, existing LAP formulations rely on weak post hoc regularization, which can lead to unstable or non-invertible deformation fields and restrict their applicability in practice.

In this thesis, we propose the Trigonometric Basis Regularized Local All-Pass (TBR-LAP) method, which extends the LAP framework by representing deformation updates using trigonometric basis functions and incorporating explicit regularization terms, including smoothness penalties and Jacobian determinant constraints. This formulation enables global, regularized reconstruction of deformation fields while preserving the computational efficiency and estimation accuracy of LAP. The proposed regularization strategy improves robustness against noise and mitigates folding artifacts in challenging scenarios.

The proposed TBR-LAP method enhances the applicability of LAP-based registration to same-modality medical imaging problems, particularly deformable X-ray image alignment, and provides both theoretical and practical contributions to the development of robust and efficient medical image registration algorithms.