## 2 Measurement Principles and Methods  
Per the principles of Fourier optical imaging, in a specific imaging system, when an ideal point light source serves as the input function, the resulting output is not a concentrated point but rather a distinct spot [27]. This phenomenon is known as the point spread function (PSF) [28], exemplified in Figure 1.  
The PSF characterizes the response of an imaging system to an ideal point source, illustrating the diffusion of this source on the imaging plane and thereby unveiling the system's spatial resolution. The optical transfer function (OTF) [29; 30], as delineated in Equation (1), emerges from the two-dimensional Fourier transform of the PSF [10]. It embodies the imaging system's response to various spatial frequencies and is composed of two components: the amplitude and the phase parts.  
\[OTF(u,v)=\mathcal{F}\{PSF(x,y)\} \tag{1}\]  
Here, \(\mathcal{F}\) is the Fourier transform, \(x,y\) are the spatial coordinates, and \(u,v\) are the frequency coordinates.  
The OTF provides a comprehensive description of an imaging system's capacity to transfer spatial details, encompassing both contrast and phase information [31]. The MTF, constituting the magnitude component of the OTF, specifically reflects the system's  
Figure 1: Formation of point spread function.  
profficiency in transferring contrast across various spatial frequencies. Accordingly, the MTF is formulized as shown in Equation (2):  
\[MTF(u,v)= |OTF(u,v)| \tag{2}\]  
Similarly, the LSF is the light intensity distribution captured on the image plane after imaging of a line source, either \(\delta(x)\) or \(\delta(y)\). The LSF is essentially the integral of the PSF along a specified direction, and its one-dimensional Fourier transform provides the cross-section profile of the MTF in that direction, as demonstrated in Equation (3):  
\[\begin{array}{l}LSF_{x}(x)=\int PSF(x,y)dy\\ LSF_{y}(y)=\int PSF(x,y)dx\end{array} \tag{3}\]  
When the input function is a step function, typically exemplified by the slanted-edge image region of a remote sensing image, its resultant output function is the ESF. This output represents the two-dimensional light intensity distribution post-optical system processing. The ESF can be viewed as the integral of the LSF, establishing a relationship between differentiation and integration, as illustrated in Equation (4). Furthermore, the one-dimensional Fourier transform of the LSF corresponds to the MTF for that specific direction [32], forming the theoretical foundation of the slanted-edge method used in MTF calculation.  
\[\begin{array}{l}ESF(x)=\int\limits_{-\infty}^{x}LSF(x^{\prime})dx^{\prime}\\ LSF(x)=\frac{d}{dx}ESF(x)\end{array} \tag{4}\]  
As delineated by the aforementioned theory, the interrelationship among PSF, LSF, ESF, and MTF is depicted in Figure 2. Consequently, by acquiring any one of LSF, PSF, or ESF, the MTF can be derived based on this established relationship [33; 34].  
The principal procedures for computing the MTF using the slanted-edge method [14], as applied in remote sensing imagery, are illustrated in Figure 3:  
1. Identify and capture an image of the slanted-edge calculation region in the remote sensing image, ensuring it conforms to specific criteria; the edge should be straight and form a slight angle with the detector's image element direction.
2. Conduct image preprocessing on the slanted-edge region to minimize random noise, followed by sub-pixel slanted-edge detection to ascertain the sub-pixel location of the slanted edge.
3. Gather data points adjacent to the slanted edge and perform multi-line alignment using the precise sub-pixel slanted-edge positions from step 2, resulting in the ESF of densely sampled points.  
Figure 2: Relationship between PSF, ESF, LSF, and MTF.