## 3 Core Technologies and Algorithms
### Paraxial Region and Object Plane Perpendicularity Determination  
According to the theory of diffraction, there exists a paraxial region of each perpendicular object plane that allows the nanoparticle's imaging to satisfy an isoplanatic condition (Figure 4). In this paraxial region, the aberrations of the nanoparticle's imaging can be minimized, and the nanoparticle can be imaged ideally. Moreover, the images of the nanoparticles located in different positions on the paraxial region of some perpendicular object planes are considered to be identical. It is clear that there is one ideal imaging region corresponding to each of the paraxial regions on the imaging plane. Figure 4: Imaging of a nanoparticle on different object planes that are perpendicular to the optical axis. Figure 3: Schematic of the designed dual position-sensitive-detector (PSD)-based unit. Based on the diffraction imaging analysis, the spot intensity distribution and size of the nanoparticle imaging change with the object plane's variation. However, the intensity distribution of these spot images is very symmetrical when these images are within the paraxial region of each object plane, and those that are out of the paraxial region are asymmetric due to an aberration in the optical imaging system. In Figure 5, the curve surface was fit with the nanoparticle's intensity distribution image with a Gaussian function to obtain its central position. Then, we divided the nanoparticle's intensity distribution longitudinally into two parts by passing through the central position (as shown in Figure 5, where the splitting plane is parallel to the X-Z or Y-Z planes), where the curve surface fit each of the two parts with a Gaussian function. Next, we normalized the two fitting Gaussian surfaces of the two parts of the nanoparticle's intensity distribution. The full widths at half maximum (FWHM) of the two normalized Gaussian surfaces were obtained, and then the difference between the two FWHMs was also obtained. When the measured object plane is non-perpendicular to the optical axis, the images of a nanoparticle that move on this object plane can be considered to be a series of images on different object planes that are perpendicular to the optical axis (Figure 6a). The nanoparticle is moved linearly on the measured object plane to obtain a series of images of the linear position from one border position of the image plane to its opposite border position. For each image, the procedure shown in Figure 5 was run, and a series of differences between the two FWHMs was obtained for each image. Because the images' intensity distributions within the paraxial region had good symmetry, the differences between two FWHMs within the paraxial region were smaller than those that were not. These differences generally presented a \"U\" shape, as shown as Figure 6b, where the bottom of the line region is one line in the ideal imaging region of the measured object plane. Notably, the ideal imaging region of the measured object plane (Figure 6a) has some difference from the ideal imaging region of Figure 4 because this series of images was obtained from different perpendicular object planes. In the ideal imaging region of the measured object plane, although isoplanatic conditions could not be met (the sizes of these spot images were not the same), the symmetries of the intensity distribution of these spot images still performed well. In order to confirm that one object plane is perpendicular to the optical axis, the two boundary points of the linear region must be obtained within the ideal imaging region of the measured object plane. Here, a method of combining quartic polynomial curve fitting with logistic curve fitting was used to confirm the boundary point (Figure 6b). This process is called the polynomial combining logistic curve fitting method and is described as follows. In Figure 6b, the black star point is a series of the differences between the two FWHMs.