The image of the center of a 3D circular target is not the center of its imaging ellipse due to the imaging distortion of spatial circular targets, therefore, traditional 3D gaze estimation methods with multi-camera-multi-light-source systems generally replace the projection of the pupil center with the virtual image of the pupil center for 3D gaze estimation. However, this introduces a large gaze estimation error when the oblique angle between the optical axis of the eye and the camera optical axis is large. To eliminate the error caused by using the virtual image of the pupil center, a 3D gaze estimation method using a multi-camera-multilight-source system is developed in this study. We first estimate the cornea center, and then determine the matching points of the pupil imaging ellipse using the special polar plane. After the cornea radius and the kappa angle are calibrated, the optical axis of the eye is reconstructed by refraction planes constructed by the edge points of pupil imaging ellipses. Thus the 3D gaze is estimated by the real-time calculated transformation matrix represented the kappa angle. The feasibility and performance of the method has been analyzed via simulations and experiments. Since the estimation of spatial pupil center or the construction of refraction plane using the virtual image of the pupil center is not required, the proposed method mitigates the inherent errors caused by optical axis reconstruction in the traditional methods and simplifies the algorithm for gaze estimation, which has practical value.

The simplest configuration for estimating 3D line-of-sight based on corneal reflection and pupil refraction uses a single camera and two non-collinear light sources. The corneal radius and 3D corneal center can be determined by solving a system of six equations with six unknowns. However, when two light sources are collinear with the camera, the reflection planes of the two light sources are coplanar. In this case, the estimation of 3D corneal center is under-determined in the absence of any pre-known eyeball parameters (e.g., corneal radius). To solve this issue, this paper proposes to calibrate the corneal radius with corneal reflections from two light sources combined with an iris imaging model, so that the 3D corneal center is estimated with a known corneal radius. Then, the pupil features or iris features can be selected to estimate the 3D pupil center or 3D iris center, which uses an optimization solution for only the center parameters to avoid unnecessary errors. Thus, the optical axis of the eyeball is constructed with two spatial centers, and the kappa angle calculated in real time is used to convert the optical axis of the eyeball to the 3D line-of-sight. We simulated this gaze estimation process using pupil and iris features, and performed error analysis on the proposed method. In practical experiments, the proposed method was verified using iris features, and the gaze accuracy was 1.22° and 1.13° in the x- and y-directions, respectively, which is comparable to that of more sophisticated systems. A single-camera-two-collinear-light-source system is the most typical and common system configuration, and the realization of 3D gaze estimation under such a system configuration can effectively promote the wide application of eye-tracking interaction.

Computer-vision-based space circular target detection has a wide range of applications in visual measurement, object detection, and other fields. The space circular target is projected into an ellipse in the camera for localization. Traditional methods based on monocular vision use a precise calculation model to calculate the center coordinate and normal vector of the space circular target according to the image’s elliptic parameters. However, this accurate calculation method has the disadvantage of poor anti-interference ability in practical application. Aiming at the shortcomings of the above traditional calculation method, this paper proposes an optimization method for fitting the circular target in 3D space, where the image ellipse is projected back into 3D space and then detects the center coordinate and normal vector of the space circular target. Unlike the traditional method, this approach is not sensitive to the image’s elliptic parameters; it has stronger noise resistance performance and notable application value. The feasibility and effectiveness of the proposed method were verified by both simulation and practical experimental results.

The cross-ratio (CR)-based method exploits the invariance property of CRs in projective transformation to determine a screen point corresponding to the pupil center. However, this point is essentially the intersection of the eyeball optical axis (OA) and the screen, rather than the actual point-of-regard (POR). In addition, the premise of CR calculation is that the corneal reflection points of four on-screen light sources are coplanar with the 3D pupil center, but they are only assumed to be coplanar. To solve these issues, this paper proposes an improved CR-based gaze estimation method using weighted average and polynomial compensation. Under the configuration of a single camera and two light sources, the 3D corneal center and the normal vector of virtual pupil plane are first estimated using the eyeball imaging model, and then four reference planes parallel to the virtual pupil plane are determined based on the geometric model of pupil and screen corner points. The screen point corresponding to the intersection of each reference plane and the line connecting the camera optical center and the imaging pupil center is calculated using the conventional CR-based method. Thus, the point where the OA intersects the screen is determined by the weighted average of these four points. Finally, a polynomial is learned to compensate it to the POR. On the basis of simplifying the system configuration of CR-based methods, the proposed method avoids the non-coplanarity of 3D pupil center and corneal reflection plane, and considers the difference between the OA and the visual axis (VA) through polynomial compensation. The experimental results show that the gaze accuracy can reach 1.33°, which is competitive with the state-of-the-art methods using more complex systems.

Kappa-angle calibration shows its importance in gaze tracking due to the special structure of the eyeball. In a 3D gaze-tracking system, after the optical axis of the eyeball is reconstructed, the kappa angle is needed to convert the optical axis of the eyeball to the real gaze direction. At present, most of the kappa-angle-calibration methods use explicit user calibration. Before eye-gaze tracking, the user needs to look at some pre-defined calibration points on the screen, thereby providing some corresponding optical and visual axes of the eyeball with which to calculate the kappa angle. Especially when multi-point user calibration is required, the calibration process is relatively complicated. In this paper, a method that can automatically calibrate the kappa angle during screen browsing is proposed. Based on the 3D corneal centers and optical axes of both eyes, the optimal objective function of the kappa angle is established according to the coplanar constraint of the visual axes of the left and right eyes, and the differential evolution algorithm is used to iterate through kappa angles according to the theoretical angular constraint of the kappa angle. The experiments show that the proposed method can make the gaze accuracy reach 1.3${}^{°}$ in the horizontal plane and 1.34${}^{°}$ in the vertical plane, both of which are within the acceptable margins of gaze-estimation error. The demonstration of explicit kappa-angle calibration is of great significance to the realization of the instant use of gaze-tracking systems.