# The Self-Calibration Method for the Vertex Distance of the Elliptical Paraboloid Array
## 1 Introduction  
Sensor arrays are widely used in modern scientific research and industrial production [1; 2; 3]. Depending on the application requirements, sensor arrays can be designed with different geometries, including those that are linear [4], circular [5], planar [6], L-shaped [7], and so on. [PERSON] et al. [8] proposed a capacitance-sensor-array-based imaging system to detect water leakage inside insulating slabs with porous cells. [PERSON] et al. [9] developed a giant magneto resistance sensor array which included various types of small gaps, curling wires, wide fractures, and abrasion to detect defects in various types of wire rope. [PERSON] et al. [10] utilized a large area sinusoidal grid surface array as the measurement reference of a surface encoder for multi-axis position measurement. [PERSON] et al. [11] presented a novel instrumentation system, including an infrared laser source and a photodiode sensor array, to provide an accurate measurement of the gas void fraction of two-phase CO\({}_{2}\) flow. [PERSON] [12] demonstrated a novel sensor array based on \(SnO_{2}\), Cu\(O\), and \(WO_{2}\) nanowires, which is able to discriminate four typical compounds added to food products. [PERSON] et al. [13] demonstrated that the bioimpedance sensor array has the ability to achieve long-term monitoring of intact skin and acute wound healing from beneath primary dressings. [PERSON] et al. [14] developed a fluorometric sensor array for the detection of TNT(trinitrotoluene), DNT (dinitrotoluene), and TNP (trinitrophenyl), which demonstrated a promising capacity to detect structurally similar nitroaromatics in mixtures and the complex media of soil and groundwater samples. The advantage of using a sensor array instead of a single sensor is that the array adds new dimensions to the observations, helping to estimate more parameters and improve the estimation performance. Of course, the calibration of relevant parameters is a requirement that cannot be ignored before using the sensor array. [PERSON] et al. [15] described a method for calibrating the distances between the balls of a 1-D ball array using a specially designed device with a laser interferometer, which gives the distances between the balls in two directions. [PERSON] et al. [16] proposed a new alternative method for calibrating the ball array based on a CMM (coordinate measuring machine) and a gage block. To obtain a high accuracy in ball calibration, repeated measurements of the balls must be taken at least 10 times. [PERSON] et al. [17] introduced a self-calibration method for a ball plate, through which not only the pitch position of the balls, but also their radial and height position on the circular ball plate, were calibrated. Through computing the intersections of 2\(\pi\)-phase lines to detect the phase-shifting wedge grating arrays centers, [PERSON] et al. [18] completed camera calibration. [PERSON] et al. [19] presented a calibration method for a camera array and a rectification method for generating a light field image from the captured images. [PERSON] et al. [20] used a high-precision machine developed by the research of [PERSON] to measure the aperture diameter and aperture distance of a microstructure array. [PERSON] et al. [21] presented a calibration model that can be extended to uncalibrated replicas of sensor arrays without acquiring new samples, favoring mass-production applications for gas sensor arrays. [PERSON] et al. [22] purposed a new array geometry calibration method for underwater compact arrays to improve the robustness and accuracy of the calibration results. [PERSON] et al. [23] researched a calibration method for an array complementary metal-oxide-semiconductor photodetector using a black-box calibration device and an electrical analog delay method to address the disadvantages of traditional PMD (photonic mixer device) solid-state array lidar calibration methods. Through the above research, it can be concluded that the calibration of sensor arrays is helpful to improve their accuracy and practicability.