Vision, Binocular

Strabismus surgery (group A)

Surgery is performed by specialized strabismus surgeons (Yan, Deng, Lin, Kang, Chen, Wu, Wang, Qiu, and Shen) according to agreed surgical formulae tailored to the clinical characteristics of the surgeon. Principles involved in the surgical procedure have been agreed as follows:

General anesthesia

Bilateral lateral rectus recession to be performed in divergence excessive IXT; unilateral recess/resect surgery to be performed in convergence insufficient IXT; either surgery in basic type of IXT

Standard sterile preparation of the operative sites

Conjunctival incisions

Standard isolation and cleaning of muscle to be operated

Muscle secured with 6/0 vicryl suture

Amount of recession and resection assessed on the basis of the maximum distance deviation angle (Table 1), modified according to standard practice of surgeon

Conjunctival incisions closed with 8/0 vicryl suture

Antibiotic ointments give at the end of procedure

Table 1

Surgical amounts for SOMIX trial

Deviation angle	Unilateral/bilateral recess surgery		Unilateral recess/resect surgery
(PD)	ULR recession (mm)	BLR recession (mm)	LR recession (mm)	MR resection (mm)
15	6
20	8		4	3
25		5.5	5	3.5
30		6	5.5	4
35		6.5	6	4.5
40		7	6.5	5
45		7.5	7	5
50		8	7	6

PD Prism diopter, ULR Unilateral lateral rectus, BLR Bilateral lateral rectus, LR Lateral rectus, MR Medial rectus

b) Follow-up visit    
The follow-up visits schedule of the study and corresponding clinical assessments for each group are showed in Fig. 2.Fig. 2

Standard protocol item. The follow-up visits schedule and corresponding clinical assessments. D, day; W, week; M, month; Y, year; BSV, binocular single vision; BCVA, best corrected visual acuity; PACT, prism and alternative cover test; NCS, Newcastle control score; IXTQ, intermittent exotropia questionnaire

Children in the observation group will be offered surgical treatment if a constant strabismus appears to be developing or parents request surgery and the responsible clinical team agrees that this is appropriate.

The binocular vision positioning theory is based on the geometric model of similar triangles in the binocular field of view. A mathematical model can be used to explain how the camera captures an image of the real object. The model comprises three coordinate systems: image, camera, and world. The two-dimensional (2D) image coordinate system has a plane identical to the physical imaging plane of the camera. The pixel layout determines the directions of the X and Y coordinate axes, with the unit of measurement in millimeters. The center of the physical imaging plane serves as the origin of the image coordinate system. The origin of the camera coordinate system is the camera optical center O. The XC and YC coordinate axes are parallel to the image coordinate system. The ZC coordinate axis coincides with the optical axis of the camera and is perpendicular to the physical imaging plane. The experimenter typically establishes the world coordinate system. The location of the origin and axis of the coordinates can be selected randomly; typically, it is placed on the measured object to facilitate description. Figure 1 shows the positional relationship of the three coordinate systems.
Assume a point (x_c,y_c,z_c) on the camera coordinate system. This point corresponds to a point (x′,y′) on the image coordinate system after the camera captures the image. The coordinate transformation relationship from the image coordinate system to the camera coordinate system can be expressed as Equation (1) [16 ]: $$Zc\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{cccc}f& 0& 0& 0\\ 0& f& 0& 0\\ 0& 0& 1& 0\end{array}\right]\left[\begin{array}{c}{x}_c\\ y_c\\ z_c\\ 1\end{array}\right],$$
where f is the focal length of the camera, determined during camera calibration, and Zc is a coefficient.
The conversion relationship between the coordinates (x_c,y_c,z_c) of the camera coordinate system and coordinates (x_n,y_n,z_n) of the world coordinate system can be expressed as Equation (2) [16 ]: $$\left[\begin{array}{c}\begin{array}{c}{x}_c\\ y_c\end{array}\\ \begin{array}{c}{z}_c\\ 1\end{array}\end{array}\right]=\left[\begin{array}{cc}R& t\\ 0^T& 1\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{x}_n\\ y_n\end{array}\\ \begin{array}{c}{z}_n\\ 1\end{array}\end{array}\right].$$
Equations (1) and (2) can be used to obtain the camera imaging model, where R and t are the rotation matrix and translation vector of the coordinate system transformation, respectively, and a_x, a_y, u₀, and v₀ are the internal parameters of the camera. The imaging model of the camera can be expressed as Equation (3) [16 ]: $$Zc\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{cccc}{a}_x& 0& u_0& 0\\ 0& a_y& v_0& 0\\ 0& 0& 1& 0\end{array}\right]\left[\begin{array}{cc}R& t\\ 0^T& 1\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{x}_n\\ y_n\end{array}\\ \begin{array}{c}{z}_n\\ 1\end{array}\end{array}\right].$$
When two cameras are used by the binocular vision system to capture photographs of the same target from different angles, the difference between the two images represents the information on the object’s three dimensions. Disparity can be calculated when the two image planes are completely coplanar, and the lines are aligned. This is known as the binocular stereoscopic correction process, dealing with the perspective transformation of the image. After the binocular stereoscopic correction, the three-dimensional coordinates of the space can be solved using the physical model shown in Figure 2: P is a point on the object, Z is the distance from point P to the focal plane, and f is the focal length. P_l and P_r are the projection points of point P on the focal planes of the left and right cameras with abscissas X_l and X_r, respectively, and T is the distance between the optical centers of the two cameras.
Based on the geometric relationship of similar triangles, the spatial coordinates (x/w, y/w, z/w) of point P in the camera coordinate system can be expressed as Equation (4) [17 ].
$$\left[\begin{array}{c}\begin{array}{c}x\\ y\end{array}\\ \begin{array}{c}z\\ w\end{array}\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& -s{u}_0\\ 0& 1& 0& -s{v}_0\\ 0& 0& 0& f\\ 0& 0& -\frac{1}{T}& 1\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{X}_r\\ Y_r\end{array}\\ \begin{array}{c}d\\ 1\end{array}\end{array}\right],$$
where (X_r,Y_r) are the projected point coordinates of P in the stereo-corrected image, s is the millimeter size of the pixel, d = X_l − X_r is the disparity between the coordinates of the two cameras, and w is the scale factor.
Any known coordinate point in the world coordinate system can be converted to the camera coordinate system using the aforementioned method. The unique conversion relationship between the world and camera coordinate systems can be solved if the four sets of matching coordinates in each of the two coordinate systems are known. The corresponding relationship between the deflection values of the galvo scanner and coordinates of the projected target is based on this conversion relationship.