衛星影像正射化校正流程的目的在於使影像表現出正確的地表幾何形狀，並且使影像能夠與其他地圖套疊。影像正射化可分為光線追蹤(top-down)及逆轉換(bottom-up)兩種方法，本研究採用逆轉換，利用有理函數模型(rational function model, RFM)表示物像關係。衛星會透過載體的星曆資料解算出RFM所需的參數，描述三維大地坐標和二維衛星像元坐標之間的關係。RFM以兩個三階多項式函數相除的形式呈現大地坐標和像元坐標之間的轉換，並透過此關係式進行影像正射化流程。然而該轉換在處理部分案例時，僅用三階多項式組成的有理函數模型依然不足以描述細微幾何形變。這類變形可能因衛星定位不精確、衛星軌道振盪以及大氣狀況等因素產生，導致最終正射影像套疊不準確，產生如地物位移、形狀扭曲等現象。本研究提出了一套影像正射化的改善流程，透過在RFM後插入非線性的映射模型，使流程可以分別藉由RFM捕捉全域誤差，以及非線性映射模型捕捉局部誤差。對於非線性映射模型，研究中評估了三種不同函數：兩種最小二乘配置法（least squares collocation, LSC）和一種徑向基函數（radial basis function, RBF），前者藉由協變方矩陣、後者藉由核函數的線性組合來捕捉局部變形。由於這三種函數複雜性較高而使得計算所需資源及時間增加，故本研究引入了兩種加速技術：GPU加速和網格化策略，前者藉由平行運算提高硬體的使用效率；後者減少冗餘的計算量且同時維持幾何穩定性，大幅提高模型效率。在有雲影像、衛星軌道定位不精確的案例中，傳統使用最小二乘來重新解算RFM的方法其檢核點精度會達8個像素，而使用本研究提出的改善流程其檢核點精度可少於2個像素。此外，在整合前述加速策略後，計算一張衛星影像坐標轉換的所需時間由3天進步到少於30秒，時間效率提高了95%以上。

The purpose of the satellite image orthorectification process is to render the images to correctly represent the geometric features of the Earth's surface, thereby enabling them to overlay with other maps. Two principal methodologies can be used for this purpose: ray-tracing (top-down) and inverse transformation (bottom-up). This study adopted the inverse transformation approach, using the rational function model (RFM) to represent object-image relationships. Satellite carrier ephemeris data were utilized to calculate the parameters required by RFM, thus defining the relationship between 3D geodetic coordinates and 2D satellite pixel coordinates. RFM employs a quotient of two cubic polynomial functions to represent the transformation between geodetic and pixel coordinates, which then forms the basis for the orthorectification process. However, in handling certain cases, the RFM, composed only of cubic polynomials, still falls short in describing subtle geometric deformation. This deformation can be induced by factors such as imprecise satellite positioning, satellite orbital fluctuation, and atmospheric condition, which may cause inaccuracies in the final orthoimage overlay, resulting in phenomena such as object displacement and shape distortion.
In response to this, this study proposes an improved orthorectification process, which inserting a non-linear mapping model after RFM, enabling the capture of global errors via RFM and local errors via the non-linear mapping model. For the non-linear mapping model, three different functions were evaluated: two least squares collocation (LSC) and one radial basis function (RBF), the former using covariance matrices, and the latter using a linear combination of kernel functions to capture local deformation. Due to the high complexity of these three functions, they require increased computational resources and time. Therefore, two acceleration techniques were introduced in this research: GPU acceleration and gridding strategy. The former leverages parallel computation to enhance hardware utilization, while the latter reduces redundant computations while maintaining geometric stability, thus significantly improving model efficiency. In cases involving cloud images and imprecise satellite orbit positioning, the traditional method of using least squares to re-generate RFM yields a checkpoint accuracy of up to 8 pixels. However, using the improved process proposed in this research, the checkpoint accuracy can be reduced to less than 2 pixels. Moreover, after integrating the acceleration strategies, the time required for the coordinate transformation of a single satellite image has been reduced from 3 days to less than 30 seconds, resulting in a time efficiency improvement of over 95%.

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