# Universal Behavior of the Image Resolution for Different Scanning Trajectories
## 1 Introduction  
Nowadays, scanning techniques are gaining popularity and becoming an essential component of a variety of devices, including microelectromechanical systems (MEMS), light detection and ranging (LiDAR) [1], atomic force microscopy (AFM) [2], magnetic resonance imaging (MRI) [3, 4, 5, 6], and magnetic particle imaging (MPI) [7, 8, 9, 10, 11], mapping and surveying mechanisms [12], and frequency modulated gyroscopes [13]. Due to its great performance and low cost, MEMS LiDAR is quickly becoming an essential sensor in vehicle environment sensing systems. The LiDAR method can be used in many different areas, such as agriculture [14], autonomous vehicles [15], and drones [16]. The three main technologies of MEMS LiDAR imaging are the beam-scanning trajectory, sampling scheme, and gridding. In order to obtain a denser scanning trajectory or pattern in a specific field of view (FOV) at the same scanning frequency, the sampling scheme should be enhanced. A universal sampling strategy that is unrelated to the beam scanning trajectory patterns is suggested by summarizing the laws of the Cartesian grid. Image resolution and the number of points per frame are both doubled compared to the already utilized sampling scheme with the same hardware setup and scanning frequency for the MEMS scanning mirror (MEMS-SM). This is advantageous for enhancing MEMS LiDAR's point cloud imaging capability [17]. Atomic force microscopy (AFM) has traditionally used triangular or saw-tooth signals for raster scanning. While this technology can offer workable answers during slow processes, it cannot provide adequate solutions during rapid scan regimes. High-speed atomic force microscopy is gaining popularity, especially for studying the dynamic behavior of biological molecules or their interactions with light. The raster-signal frequency is normally restricted to 1% of the first-resonance frequency of the positioning stage for an acceptable performance in traditional raster scanning. This restriction guarantees adequate monitoring of the raster signal's high-order harmonics while preventing the stage's vibration modes from being activated. As a result, the scan speed is slow, and it takes longer to completely scan the FOV of interest. Furthermore, in a highly resonant stage, monitoring sinusoidal reference signals with a narrow bandwidth is easier than tracking triangle or saw-tooth references at the same amplitude and fundamental frequency. Non-raster approaches, such as the Lissajous scanning trajectory, can be used instead of raster scanning to produce high-quality images at high scan rates [18]. Scanning trajectories are also used by medical imaging techniques like MRI [3; 4; 5; 6] and MPI [7; 8]. These imaging methods are essential because they can be used to look inside the human body without undergoing surgery and to diagnose, monitor, prevent, and treat a variety of diseases [19]. The purpose of these techniques is to scan the FOV to obtain an image that will be used later to analyze the patient's condition. The type of scanning trajectory chosen is crucial for obtaining accurate results. By choosing the appropriate or optimal trajectory, scanning time can be reduced and image quality can be enhanced in order to make an accurate diagnosis. This implies that a proper scanning trajectory can be very important for patient diagnosis and care [20]. It should be emphasized that all of the aforementioned methods are currently in the development stage, meaning that researchers are trying to improve them. Although the field is expanding, a fundamental problem with the present scanning techniques is the lack of a reliable estimation of the pixel size for a particular combination of scanning settings. For example, it is currently unknown what the pixel size of this scan will be given a chosen scanning trajectory to scan a FOV with a particular set of scanning parameters. Keep in mind that the quality of the reconstructed image from the scan is directly impacted by the choice of pixel size.