## 2 Principles
### Light Plane Calibration Based on a Single Circle Target  
#### 2.2.1 Image Processing Algorithm  
Figure 2a displays the captured image of the circle target during calibration. The 2D data in structure is very important [21]. The light stripe must be extracted and the elliptic image profile of the circle target edge must be fitted. Hence, the first objective of this process is to detect (extract and fit) the ellipse because it contains a light stripe (note that the ellipse can be used as a mask to reduce the search area during light stripe extraction). The complete image processing algorithm is outlined in Figure 3, while the ellipse detection and light stripe extraction procedures are described in detail below. Figure 3: Flowchart of the complete image processing algorithm. Figure 2: Calibration images. (**a**) Captured image of the circle target. (**b**) Elliptic fitting profile. (**c**) Extracted light stripe. The existing ellipse detection methods can be grouped into Hough transform [22] and edge following [23] ones. The former techniques are relatively simple but not robust in complex scenarios. In contrast, the latter methods are very precise but require long computation times. Therefore, we have developed an efficient high-quality ellipse detection method [24], which is called the RALS method. Its algorithm consists of four steps: (1) arc-support line segment (LS) extraction, (2) arc-support group forming, (3) initial ellipse set generation, and (4) ellipse clustering and candidate verification. _Step 1. Arc-support LS extraction:_ Arc-support LS helps prune a straight LS while retaining the arc geometric cues. It has two critical parameters: direction and polarity. A detailed description of the corresponding procedure can be found in Ref. [25]. _Step 2. Arc-support groups forming:_ An elliptic curve may consist of several are-support LS, which require a link to form a group. Two consecutive linkable arc-support LS should satisfy the continuity and convexity conditions. The former condition states that the head of one arc-support LS and the tail of another one should be close enough. According to the convexity condition, the linked LS should move in the same direction (clockwise or anticlockwise). Iteratively, the linked arc-support LS that share the similar geometric properties are called an \"arc-support group\". _Step 3. Initial ellipse set generation:_ An arc-support group may contain all the arc-support LS of a curve or merely a separate arc-support LS. Thus, two complementary methods are utilized to generate the initial ellipse set. (1) From the local perspective, the arc-support LS group with a relatively high saliency score likely represents the dominant component of the polygonal approximation of an ellipse. (2) From the global perspective, the troublesome situation of the arc-support groups of a common ellipse is characterized by a polarity constraint, region restriction, and adaptive inlier criterion. _Step 4. Ellipse clustering and candidate verification:_ Owing to the presence of duplicates in the initial ellipse set, a hierarchical clustering method based on mean shift [26] has been developed. For convenient clustering, this method decomposes the five-dimensional ellipse parameter space into three low and cascaded dimensional spaces (centers, orientations, and semiaxes). Moreover, to ensure the high quality of the detected ellipses, ellipse candidate verification, which incorporates the stringent regulations for goodness measurements and elliptic geometric properties for refinement, is conducted. \(E\) is the elliptic fitting profile, as indicated in Figure 2b. After the ellipse detection, the light stripe within the ellipse is extracted as well. The automatic sub-pixel extraction process of the light stripe consists of three steps: (1) Gaussian filtering, (2) solving the Hesse matrix, and (3) extracting sub-pixel points. _Step 1. Gaussian filtering:_ A Gaussian filter is applied to the undistorted image. According to Ref.