在數位影像與多維數據迅速擴張的時代中，先進影像處理技術的需求日益增加。本論文針對三維(3D)曲面重建相關挑戰進行探討，專注於數據獲取、處理及提升影像品質。傳統方法主要為二維(2D)場景所設計，在應用於超解析(SR)3D重建時往往無法滿足需求。本研究著重於常見的問題，例如低解析度獲取及數據缺失，這些因素對後續3D重建的造成影響。同時，系統地研究X光、太赫茲(THz)及磁振造影(MRI)等成像模式所固有的限制，從輻射風險到慢速數據獲取及區域解析度限制。

本論文的核心貢獻在於推進數學框架以重建3D曲面輪廓。首創性的工作是開發一種新型以歐拉-彈性為主的公式，這在多個成像模式中首次被引入。此創新補充了使用變分框架的隱式表示方法探索。這些新方法由針對變分公式量身定製的數值演算法所支撐，特別是投影梯度下降法(PGDM)和交替方向乘子法(ADMM)。為了定量評估這些創新模型，提出並應用了一套全面的指標，包括高斯和平均曲率的標準差以及多尺度結構相似度指數(MS-SSIM)。通過在X光、磁振造影和太赫茲成像應用中的嚴格驗證，這些方法的有效性和多樣性得到展示，為醫療影像和電腦視覺領域帶來了有希望的進展。這包括提升影像解析度、縮短獲取時間，以及改善醫療診斷和治療計劃的清晰度和可用性。

作為計算創新的補充，本論文還展示了硬體能力的重大進展。本論文最終介紹了以擴散器為輔助的次太赫茲成像系統(DaISy)，一種新型將太赫茲攝像感測器陣列與高功率次太赫茲源的組合，從而顯著提高了影像品質。為了對抗相干成像中普遍存在的斑塊瑕疵，提供並應用了一套適應性光學相干理論。這一理論進化不僅是提供了硬體突破的補充，而且是推動相干成像技術在一系列科學和工業領域部署的關鍵驅動力，提供了可能影響未來研究和應用的重大貢獻。

In an era where digital imagery and multidimensional data are rapidly expanding, the demand for advanced image processing techniques becomes more pressing. This thesis specifically addresses the challenges associated with three-dimensional (3D) surface reconstruction, focusing on data acquisition, processing, and enhancement of image quality. Traditional methods, primarily designed for two-dimensional (2D) scenarios, often fall short when applied to super-resolution (SR) 3D reconstructions. This work critically examines common obstacles, such as low-resolution capture and missing data, which significantly affect subsequent 3D reconstructions. The limitations intrinsic to X-ray, TeraHertz (THz), and Magnetic Resonance Imaging (MRI) modalities, from radiation risks to slow acquisition speeds and regional resolution constraints, are methodically examined.

The core contribution of this thesis lies in the advancement of mathematical frameworks for 3D surface profile reconstruction. A pioneering effort is the development of a novel Euler-Elastica-based formulation, introduced here for the first time across multiple imaging modalities. Complementing this innovation is an exploration of the implicit representation approach using variational frameworks. These novel methodologies are underpinned by the development of robust numerical algorithms tailored to variational formulations, notably the Projected Gradient Descent Method (PGDM) and the Alternating Direction Method of Multipliers (ADMM). To quantitatively evaluate these innovations, a comprehensive suite of metrics, including the standard deviation of Gaussian and mean curvatures, as well as the Multi-Scale Structural Similarity Index Measure (MS-SSIM), is proposed and employed. Through rigorous validation in X-ray, MRI, and THz imaging applications, the effectiveness and versatility of these approaches are demonstrated, heralding promising advancements in medical imaging and computer vision. These include enhanced image resolution, reduced acquisition times, and improved clarity and usability for medical diagnostics and treatment planning.

Complementing computational innovations, substantial strides in hardware capabilities are presented. This thesis culminates in the exposition of the Diffuser-aided sub-THz Imaging System (DaISy), a novel assembly that amalgamates a THz camera sensor array with a high-power sub-THz source to markedly elevate image quality. To counter the prevalent speckle artefacts in coherent imaging, an adapted optical coherence theory is formulated and applied. This theoretical evolution is not just an adjunct to the hardware breakthroughs but a pivotal driver for the deployment of coherent imaging techniques across a spectrum of scientific and industrial arenas, offering substantial contributions that may influence future research and applications.

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