# Camera Calibration with Phase-Shifting Wedge Grating Array
## 1 Introduction  
Camera calibration has been extensively studied for few past decades, which attempts to determine the intrinsic and extrinsic parameters of cameras [1; 2; 3]. Through capturing calibration targets with known features, one can acquire an exact set of one-to-one correspondences between the world and the image coordinates. Camera parameters are then estimated from these correspondences. Thus, the feature detection will directly affect the calibration accuracy. Numerous studies have focused on developing patterns with distinctive features that can be accurately localized in the images [4]. Among various existing patterns, squares [5; 6; 7; 8], and circles [9; 10; 11; 12] have become popular due to their ease of use. In order to avoid the influence of the fabrication to the calibration results and further simplify the calibration procedures, the digital displays have been applied to the camera calibration, which provide an alternative way for planar targets [13; 14; 15; 16; 17; 18; 19]. When compared with the conventional calibration object (a planar target marked by printed patterns), digital displays have many advantages, including adjustable brightness, guaranteed flatness, and known pixel sizes in spite that they may be more expensive and inconvenient to move with power and data cables. Arbitrary features can be realized through simple programming on the computer. Most importantly, digital displays make active targets be possible. Therefore, more and more researchers used phase-shifting patterns, which that could solve the problem of the calibration imprecision that is caused by the coordinate extraction errors of the feature points [14; 17] as calibration targets with the advantages of digital displays. [PERSON] et al. [14] performed camera calibration with two phase-shifting sequences, one horizontal, and one vertical. [PERSON] et al. [15] also applied horizontal/vertical phase-shifting fringe patterns for camera calibration, and made improvement on feature detection and optimization. [PERSON] et al. [16] used horizontal/vertical fringe patterns as calibration targets, but extracted the two-dimensional (2D) phase-difference pulses as feature points. [PERSON] et al. [17] applied concentric circles and wedge gratings for solving the vanishing points, and then estimatedthe camera parameters that were based on the orthogonal vanishing point conjugate. [PERSON] et al. [18] proposed an accurate camera calibration method for defocused images. The calibration target included two pairs of horizontal/vertical binary patterns and one black pattern. [PERSON] et al. [19] encoded the feature points into the carrier phase of horizontal/vertical binary patterns, which can be accurately recovered even under severe defocus. These above methods could achieve sufficient accuracy with active targets; yet they require five-frame patterns at least. In other words, we need capture five or more images at each viewpoint, which is tedious and laborious. In order to obtain accurate calibration results and reduce the workload of capturing images marked by phase-shifting patterns. This paper presents an active calibration target that consists of phase-shifting wedge grating (PWG) arrays. The approach takes advantage of the high precision of the phase-shifting fringes extraction [14; 16; 17] and uses PWG arrays as calibration pattern to obtain the wrapped phase of the captured images. Meanwhile, the intersections of 2\(\pi\)-phase points that were regarded as feature points can be extracted accurately. The adoption of three-frame patterns could significantly reduce the workload and improve the efficiency of camera calibration. A liquid crystal display (LCD) is served as the planar target to display these patterns, which is flat, effortless and flexible. The three-step phase-shifting algorithm is used for the wrapped phase calculation. The 2\(\pi\)-phase points are roughly extracted with edge detection and are then optimized by windowed bicubic fitting with sub-pixel accuracy.