This paper addresses the problem of camera calibration and shape recovery using a single image of a reflectively symmetric object. Unlike existing methods requiring knowledge of 3D points or two images, this paper proposes to calibrate camera parameters using one image with known point distance ratios on 3D object. Specifically, we first recover the vanishing point of the symmetry plane normal. Then a set of candidate focal lengths are uniformly selected as the initial values, from which the pan and yaw angles of the camera can be obtained. To recover 3D points on the object, we recover the ratio of depth scale factors between symmetric 2D point pairs, then the ratio of depth scale factors between different symmetric points. Finally, constraints from 3D distance ratios on the object are used to refine the estimation of camera parameters. Both synthetic and real experiments demonstrate the feasibility of our method.

Camera Calibration plays an important role in 3D computer vision applications. Traditional methods restore camera parameters by capturing multiple images of a planar grid [25, 10, 9], which involves tedious manual works. Calibration with less preprocessing becomes a practical demand. To achieve this goal, researchers have proposed calibration methods under constrained motion, such as planar motion [11], pure rotation [19], turntable motion [23], and even general motion [13]. In addition, properties of object shapes are also explored, such as symmetry [8, 2], rotation invariance on surfaces of revolution (SOR) [20] or spheres [24].

Among them, the methods of using symmetry in [17, 15, 1, 12] describe how to calibrate camera extrinsics. Cao and Foroosh [5, 2] present methods for calibrating intrinsics. However, their works rely on an invariant cross ratio between four corresponding points of at least two images, which cannot be obtained when only one image is available. Wong et.al. [20] describes camera calibration using a imaged SOR.

In real experiment, ground-truth camera parameters are obtained using Zhang’s method [25]. A picture (see Fig. 3 (a)) of a cabinet is then taken with a Nikon D30 camera with an image resolution of 3120 × 2080. Following the way in [2], we manually selected 18 pairs of symmetric points with the Matlab control point selection tool. Using 2 or 28 known distance ratios, our method can calibrate 2 camera external, and 2 or 4 internal parameters, respectively. According to the obtained values of cost residuals, we ignore the results when |σ| ≥ 10−1. Table 2 shows the calibrated intrinsics, the calibrated angles and 3D reconstruction errors, respectively. Errors could be reduced by increasing the number of symmetric points. Figure 3 (b) shows the reconstructed 3D model. It can be seen that our method is practical and can be applied to 3D reconstruction with high accuracy.

This paper presents a method to calibrate camera parameters and reconstruct 3D geometry using only a single image of a symmetric object. Our method is able to calibrate one, two or even four parameters of the camera with knowledge of distance ratios on the object. This is achieved by using our newly discovered constraints for recovering ratios of depth scale factors during an iterative optimization process. By increasing the number of known distance ratios, four camera intrinsics can be obtained. Both synthetic and real experiments demonstrate the feasibility of our method.