## 1 Introduction  
The 9/11 terrorist attack in New York in 2001 has led to tremendous growth in security-related markets. Most of the key security products that employ optical design technology (e.g., fingerprint identification systems and cameras) are used for building security [1]. These devices capture optical images of people approaching a building to help identify possible intruders among numerous visitors [2]. These cameras often use a zoom lens because normal monitoring is performed in wide mode and zooming allows a person or an object to be observed in tele mode by changing the field of view (FOV) [1]. However, a zoom lens not only has a limited FOV, but also requires mechanical movement if the need-to-be recognized target is beyond its FOV [3, 4]. In such cases, security may be compromised if the target approaches swiftly while the camera is moving. To overcome this weakness, an optical system with an FOV of 180\({}^{\circ}\) must be used to monitor a wide FOV range with no mechanical movement. Unfortunately, a system with an FOV of 180\({}^{\circ}\) or greater does not usually provide satisfactory resolution because the increase in FOV leads to a sharp rise in aberrations such as coma, astigmatism, and field curvature, all of which affect the image quality of the off-axis field [5]. Due to this resolution degradation, a clear image cannot be obtained, nor can the target be accurately recognized even with a high-pixel image sensor [6]. An optical system with a higher off-axis resolution can be designed to overcome this issue by obstructing the center while giving more weight to the off-axis area than the center [7].  
If the FOV of a system is increased to greater than 180\({}^{\circ}\) and positioned vertically, a 360\({}^{\circ}\) wide view perpendicular to the optical axis is possible. This optical system is called an omni-directional optical system [8]. Despite its center being obstructed, there is no functional issue because the screen center is facing the sky. Meanwhile, the system's effective focal length (EFL), the key physical parameter of an optical system, must be measured to determine the modulation transfer function, distortion, etc. There are four official EFL measurement methods [9] and of these, the nodal slide and magnification methods are the most frequently used. Another method is to use an object at a very far distance to extrapolate data. However, the principles of this method are similar to those of the abovementioned nodal slide and magnification methods. Moreover, these two methods are rarely used in practice [9]. The nodal slide method is based on the principle that chart images never move when a system rotates on its nodal point. Once the nodal point is located, the EFL can be determined. The magnification method, in contrast, uses a chart of known size and a collimator of known focal length to calculate the EFL, using the imaging magnification of the chart that is obtained from the image when the chart passes through the collimator and the test optical system [9]. These methods are characterized by the use of the paraxial property near the screen center, which cannot be applied to a center-obstructed omni-directional optical system. This study investigates a method for measuring the EFL of a center-obstructed omni-directional optical system.