## 1 Introduction
Then, from the geometric relations of the LED images that are created on the image sensor plane, the unknown position is calculated using a combination of vector estimation methods and least-square with the accuracy of decimeter level. Under the assumption that the image center is at the center of the pixel, this method does not require angular measurement. In practice, however, the center of the LED image rarely appears at the center of the pixel, so quantization errors are unavoidable. In addition, the separation between image sensors will largely determine the accuracy of positioning. Meanwhile, because of the existence of two image sensors, the complexity and quantization error of the two image sensors are at a high level. Therefore, in Luxapose [12], an AOA localization algorithm based on three or more LEDs is proposed, where a camera is regarded as an angle-of-arrival sensor. In [13], the authors put forward an indoor positioning algorithm using at least three LEDs, which realized an accuracy of 0.001 m in simulation. However, there are certain limitations: (i) The FOV of most smartphone cameras is very limited. As for 3-D positioning, with the increase of Rx's position, it would fail to position when there are fewer than three LEDs in the view. (ii) LED signal may be blocked by some objects such as walls and furniture in indoor environment, which causes the failure of positioning because the obstacles can lead to a situation where there are fewer than three LEDs in the visual of image sensor, as in the former case. Positioning may be interrupted or fail if fewer than three LEDs occur in the FOV of the camera, which greatly reduces the robustness and flexibility of the system. To reduce the number of required LEDs, in [14], the authors used two LED luminaires and an image sensor of camera as a receiver to realize the positioning accuracy of several cm in 3-D system. However, when only one LED is captured, the positioning fails. In [15], the authors adopted the localization algorithm from [14], and, if only one LED luminaire is captured, the position of the receiver would be simply located at the position of corresponding LED luminaires. In other words, it is just proximity. Hence, it causes great error of at least 82.5 cm in the 2-D positioning system of Reference [15] and the authors could not handle the situation where only one LED luminaire is captured. Meanwhile, the positioning algorithm of References [14; 15] seems to be collinear, but they are only adapted to the occasion such as corridor with straight line distribution of LEDs instead of indoor environment with four square lamps. Besides, the authors did not demonstrate their positioning method based on two LEDs, let alone how they obtained the coordinate of Rx. Their work also left some unresolved issues for follow-up research. Meanwhile, the authors of [14; 15] did not take the symmetry problem of angle into consideration. Employing angular sensors is another straightforward way to measure the receiver's orientation information, thereby \"compensating\" for the missing information caused by the reduced number of LED lamps. In [16; 17], the authors provided single LED based localization by employing angular sensors or gyro. However, the use of angle sensors poses another problem. Previous studies have shown that a major source of positioning and navigation errors is inaccuracy of azimuth angles or measurement by magnetometers and gyroscopes [18]. In [19; 20], the authors provided the techniques for position of the collinearity condition using single camera and LED street lights. In [19], the authors used the direct linear transformation and single value decomposition to solve the equation system for positioning. However, the robustness of the system is poor, and the algorithms are complicated where the scene is not collinear. In [20], the camera receives the visual information while the photodiode receives the visible light information, and the error of 3-D positioning is controlled at the decimeter level.