Tracking multiple surgical instruments in a near-infrared optical system

Abstract

Surgical navigation systems can assist doctors in performing more precise and more efficient surgical procedures to avoid various accidents. The near-infrared optical system (NOS) is an important component of surgical navigation systems. However, several surgical instruments are used during surgery, and effectively tracking all of them is challenging. A stereo matching algorithm using two intersecting lines and surgical instrument codes is proposed in this paper. In our NOS, the markers on the surgical instruments can be captured by two near-infrared cameras. After automatically searching and extracting their subpixel coordinates in the left and right images, the coordinates of the real and pseudo markers are determined by the two intersecting lines. Finally, the pseudo markers are removed to achieve accurate stereo matching by summing the codes for the distances between a specific marker with the other two markers on the surgical instrument. Experimental results show that the markers on the different surgical instruments can be automatically and accurately recognized. The NOS can accurately track multiple surgical instruments.

Keywords:

• Multiple surgical instruments
• near-infrared
• stereo matching
• surgical navigation

Introduction

In traditional surgery, both surgical planning and implementation rely on the knowledge of the lesion region and the clinical experience of the doctors. Moreover, given some lesion regions, like the brain and spine, are indistinct, doctors perform surgery only according to the pathological features of the lesion regions. However, the uncertainty of this method may result in complicated surgical procedures, huge operation wounds and long recovery times. With the developments in science and technology, surgery has gradually become more precise and less invasive. Over the past few years, computer technology, precision instruments and digital image technology have been rapidly developed. These technologies have been introduced to the medical field, breaking new ground in surgery. Consequently, computer-aided surgery (CAS) is launched.[1–5] CAS follows a certain procedure. First, with the use of advanced imaging devices in medicine, such as MRI and CT, multi-mode image data of a patient is obtained and subjected to processing, such as image segmentation, image fusion, or three-dimensional reconstruction, with the assistance of computers. Second, a rational and quantitative operation plan to achieve minimal invasion is developed based on the clinical experience of the doctor and the analysis of the pathological features of the lesion region. Finally, after simulating the surgical procedure, the doctor can actually perform the operation aided by a video monitoring system and stereotaxic device. With the rapid development in CAS, it has been used in neurosurgery,[6,7] plastic surgery,[8,9] osteopathy [10–12] and otolaryngology,[13] among other medical fields. As an important tool in CAS, a surgical navigation system (SNS) can show the location and direction of surgical instruments relative to the lesion region to guide surgical operations with real-time images.[14] Since the introduction of the first SNS more than 20 years ago, SNS technologies have likewise undergone rapid development, significantly improving the accuracy of lesion localization, reducing surgical injuries, optimizing surgical pathways and increasing operative success rates.[15–17] Tracking technology, particularly near-infrared tracking technology, is one of the key technologies used in SNSs.[18,19] It is extensively applied to SNSs because of its convenience, reliability and accuracy. Near-infrared cameras are also used in SNSs to capture surgical instruments.[20] These cameras can filter ambient light and reduce noises caused by lights to detect markers on surgical instruments precisely. During tracking, the three-dimensional coordinates of each marker can be calculated by finding the matching relationship between the markers on a surgical instrument and the points in the right and left images. The space coordinates of the tip of the surgical instrument can be obtained by computing the coordinates of each marker. Therefore, matching markers is a crucial function of a near-infrared optical system (NOS). However, multiple surgical instruments are always used in CAS; as a result, numerous points appear in the right and left images when the instruments are captured. The multitude of points usually leads to identification difficulties, making correct matching difficult to achieve. Thus, an effective, precise and quick stereo matching algorithm is imperative.

Stereo matching refers to the search for corresponding relations between pixels of the same object in the same space but from different viewpoints. In stereoscopic vision, stereo matching means finding the corresponding relations between targets of interest in the right and left images captured by two or more cameras. Unfortunately, geometric information, grayscale distortion and noise jamming occur in the two images when stereo matching is implemented, complicating stereo matching. Stereo matching is currently implemented under specific conditions. Over the past several years, various algorithms and strategies have been proposed for stereo matching.[21–25] Stereo matching has two main modes: matching based on characteristics [26,27] and matching based on regions.[28,29] Both modes can be used for stereo matching of images or markers, but they differ in terms of accuracy, effectiveness and speed. Furthermore, only multiple points exist in the images of surgical instruments, which are captured in an NOS, and these points have few edge differences and are in grayscale. As a result, matching points by the two traditional modes of stereo matching is difficult. Methods of ordering constraints and epipolar constraints have been adopted in stereo matching for points in binocular vision.[30,31] However, surgical instruments move in different directions and occupy different positions during surgeries. As a result, the locations of the corresponding points in the right and left images are inconsistent. Therefore, correctly matching all points by methods of ordering constraints and epipolar constraints is difficult. At present, the clinical measuring equipment for NOSs tends to avoid the process of stereo matching. For example, Northern Digital Inc. (NDI) offered a solution[32]: a control emitter is installed in the tracking system to activate points one by one within a certain time interval, that is, only one point is activated each time, and near-infrared light is emitted. With only one point presented in the pair of images, the space coordinates of the point can be directly computed. However, this procedure is considerably complex. The timing sequence must be synchronous with the camera. Furthermore, this solution will increase the development cost of the navigation system and prolong the development period. Moreover, the physical connection between markers and the emitter is inconvenient when performing surgeries. Therefore, the stereo matching of points and the automatic recognition of surgical instruments are crucial to realize the function of NOSs fully.

Methods

The NOS construction

The NOS in an SNS is required to track an instrument and provide real-time positioning information. Near-infrared filter technology is used in ordinary optical camera to construct the NOS more easily at a reduced cost. The NOS is divided mainly into two: the software component and hardware part (Figure 1). The NOS works as follows. First, when the system is initiated, the markers on the surgical instruments emit light waves. Second, the cameras capture all the markers and send the image data to the computer via USB interface. Third, the computer automatically finds all points and determines the point centers at the subpixel level. Fourth, the space coordinates of the markers are computed based on the internal and external parameters of the camera.[18] Fifth, the locations and directions of the tips of the surgical instruments are determined. Finally, the surgical instruments can be tracked in real time as they register in the coordinate systems.

Figure 1. NOS: (A) system hardware; (B) navigation software.

Stereo matching and recognition of surgical instruments

The core functions of an SNS are to obtain the real-time locations of the surgical instruments and show the relative positions of instruments and surgical sites to guide doctors during surgeries. In the tracking of surgical instruments, that is, computing their coordinates in real time, the markers should first be stereo matched. Therefore, the stereo matching of markers is a crucial step. In this study, an NOS aided by two near-infrared cameras is developed. The system model is shown in Figure 2.

Figure 2. System model.

As shown in Figure 2, Ω^l(X_l, Y_l, Z_l), Ω^r(X_r, Y_r, Z_r) and Ω^g(X_g, Y_g, Z_g) stand for the coordinates in the right and left cameras and the world coordinates, respectively. X_r and Y_r are on the image plane of the right camera, and Z_r overlaps with the optical axis of the camera lens. O_r is the center point of the optical axis of the right camera. The same method is applied to identify the coordinate system of the left camera. O_g is the original point of the left camera O_l, and the linking line between O_l and O_r is in the direction of X_g of the world coordinates. X_g, Z_l, Z_g and X_g are located on the same plane. The relations of these three coordinates can be expressed as follows: (1) (2) (3)

In Equations (1)–(3), p_g, p_l and p_r represent the coordinates of a point in the three-coordinate systems; R_lg and R_rg and T_lg and T_rg represent the rotation matrices and translation matrices, respectively, which are the transformation from Ω^l and Ω^r to Ω^g. R_lr and T_lr represent the rotation matrix and translation matrix, respectively, which are the transformation from Ω^l to Ω^r.

Optical relationship

Let p be a marker in Ω^g. The subpixel coordinates of p in the right and left images are a_l and a_r, respectively. Stereo matching is finding the corresponding relation between a_l and a_r. For the left camera, matrix G_l is obtained after calibration. Assuming that a_l= [u_al, v_al]^T, the normalized coordinate vector c_l= [u_cl, v_cl]^T and a_l satisfy the following equation: (4)

If the lens distortion of the near-infrared camera is considered,[18] the corresponding relation between c_l and the ideal normalized coordinate vector f_l= [x_l, y_l]^T can be expressed as (5) (6) where e_x is the tangential distortion vector, r_l² = x_l² + y_l², and β_li is the lens distortion coefficient. First, f_l is initialized to c_l, and then the approximate value of f_l is obtained by iterative calculation. Given f_l is the normalized image projection, coordinate p_l of p in Ω^l satisfies the following relation: (7) where κ_l is an arbitrary value. Assuming W_l= [x_l, y_l, 1]^T, Equation (7)(7) can be simplified as follows: (8)

In Figure 2, straight lines O_lp_l in Ω^l can be represented by Equation (8)(8) . Similarly, straight lines O_rp_r in Ω^r can be represented by (9)

Substituting Equation (8)(8) into Equation (2)(2) derives the following equation: (10)

Similarly, substituting Equation (9)(9) into Equation (3)(3) derives the following equation: (11)

The direction vectors of straight lines O_lp_l and O_rp_r in Ω^g are denoted by j_l=R_lgW_l and j_r=R_rgW_r. The direction vector of O_lO_r in the X-axis of Ω^g is denoted by j_o. Straight lines O_lp_l and O_lO_r constitute plane Π_lpr, whose normal vector n can be derived using the following equation: (12)

Given both p_r and p_l originate from p, that is, straight lines O_lp_l and O_rp_r cross at p in Ω^g, O_rp_r is also on plane Π_lpr; thus, (13)

According to Equations (4)–(12), a subpixel coordinate point represents a straight line in Ω^r. If two lines corresponding to the two subpixel coordinates in the right and left images originate from the same marker, then they intersect and satisfy Equation (13)(13) . However, the satisfaction of Equation (13)(13) does not mean that both lines absolutely originate from the same marker. Moreover, both lines certainly do not originate from the same marker if they do not satisfy Equation (13)(13) .

Stereo matching based on codes

Assuming four markers represented by p_i (i = 1, 2, 3, 4) exist, as shown in Figure 3(A), and p₁, p₂ and p₃ are located on the same surgical instrument, the instrument can be tracked by tracking p₁, p₂ and p_3. The subpixel coordinates of marker p_i in the right and left images are a_li and a_ri, respectively. According to Equations (4)–(11), the straight lines of a_li and a_ri in Ω^g are O_lp_li and O_rp_ri, whose direction vectors are j_li and j_ri, respectively. To illustrate the principle of stereo matching based on codes, matching p₃ is taken as an example. The subpixel coordinates of p₃ are a_l3 and a_r3. Space matching for p₃ means finding the correct corresponding relation between a_l3 and a_r3 in all the subpixel coordinates. The direction vectors of straight lines O_lp_l3 and O_rp_r3 in Ω^g are denoted by j_l3 and j_r3. Line O_lp_l3 and line O_lO_r constitute plane Π_l3pr.

Figure 3. Stereo matching: (A) p₁, p₂ and p₄ are beyond plane Π_l3pr; (B) p₁ is on plane Π_l3pr.

(1) When p₁, p₂ and p₄ are beyond plane Π_l3pr (Figure 3A)

According to Equations (4)–(11), the direction vectors of markers p₁, p₂ and p₄ in Ω^g are j_r1, j_r2 and j_r4, and the normal vector of Π_l3pr is n₃ according to Equation (12)(12) . Into the following equation, j_r1, j_r2, j_r3 and j_r4 are incorporated to obtain the value of μ: (14)

Owing to the influence of calculation errors, even if p_i is on plane Π_l3pr, μ cannot be zero but can only be approximated to zero. Therefore, if μ is less than the stipulated threshold value μ_c, then μ satisfies Equation (14)(14) , that is, the subpixel coordinate a_r of j_r in the right image and the subpixel coordinate a_l in the left image originate from the same p. Therefore, only j_r3 satisfies Equation (14)(14) , whereas j_r1, j_r2 and j_r4 do not. As a result, the correct matching relation for p₃ between the right and the left images can be obtained.

(2) When at least one among p₁, p₂ and p₄ is on plane Π_l3pr (Figure 3B)

Assuming that p₁ exists on plane Π_l3pr, that is, points p₁, p₃, O_l and O_r are all on the same plane, the normal vector of the plane is perpendicular to all the straight lines inside the plane when matching p₁ and p₃. Moreover j_r1 and j_r3 satisfy Equation (14)(14) . Two real markers, i.e. p₁ and p₃, and two pseudo markers, i.e. p₁′ and p₃′, can be obtained. The surgical instruments should be assigned codes, i.e. the markers on the instruments and the distance between them should be assigned codes to eliminate pseudo markers.

The distances between two markers on the surgical instruments are already known. At least three markers should be installed on a surgical instrument, and the distance between these markers should be unequal. If the distance between two markers is d, then D is used to represent the set of all distances between markers (d). The unicity of the distances between the markers on the surgical instruments has to be represented with codes to distinguish the distances between markers. Represented by k, the method of coding is as follows: (15)

The sum of the codes for two distances is the code for the marker they share. As shown in Equation (15)(15) , the code for each distance is unique; thus, the codes for the markers determined by the distance codes are likewise unique. The steps in matching subpixel coordinates in the right and left images by distance restriction are as follows:

Step 1: On the basis of Equation (10)(10) and (11)(11) , the coordinates of all the markers (including real markers and pseudo markers) that satisfy Equation (14)(14) are determined. The distances between markers are then computed using the Euclidean distance. Finally, the distance matrix W is obtained, with each distance denoted by w.

Step 2: The distance matrix W is scanned. The w and d values of the surgical instruments are substituted into the following equation: (16)

In consideration of the measuring errors, φ is set to be the permissible error range. If w and d satisfy Equation (16)(16) , then the two markers related to distance w are temporarily considered being on a certain instrument, and then distance code j_n is recorded. Finally, after scanning, the upper triangular matrix U, which involves all distance codes, is obtained, and S=U+U^T.

Step 3: The sum of every two distance codes in each line or column of the diagonal matrix S is determined and then compared with the codes for the markers. If they are equal, the marker is real. Through this step, the system can distinguish all real markers; achieve correct matching between points in the right and left images; determine the matching relation between the markers on the surgical instruments and their points in the images; and record the corresponding distance codes during the process of scanning, that is, confirm which codes belong to which instrument for instrument recognition and tracking.

Results and discussion

Single surgical instrument

To verify the tracking accuracy of our NOS, we have designed a surgical instrument equipped with three near-infrared markers. As shown in Figure 4, the distances between two markers are 81.626, 91.496 and 36.482 mm. The distances are obtained by measuring with a vernier caliper with an accuracy of 0.02 mm and computing the center of the physical markers for a dozen times.

Figure 4. Distances between two markers on a surgical instrument.

First, the instrument is placed within the view of the NOS. Next, 20 pairs of images of the instrument are taken from different positions. Finally, the subpixel coordinates of the three markers are determined. The measurement results are shown in Table 1, in which a_li(x, y) and a_ri(x, y) (i = 1, 2, 3) stand for the subpixel coordinates of all the points in the right and left images, respectively. Afterward, the correct matching relation of the three markers can be obtained.

Table 1. Subpixel coordinates and matching relation of all the points.

i	al (x,y)		ar (x,y)		Correct matching relation
1	(920.1028	313.4239)	(986.4667	283.9251)	1 < −−−− > 1
2	(994.8413	372.2210)	(1076.0962	341.3278)	2 < −−−− > 2
3	(786.7019	458.6429)	(701.5292	446.6803)	3 < −−−− > 3

Through Equations (4)–(11), plane Π_lpr1, plane Π_lpr2 and plane Π_lpr3, which are constructed by O_lO_r with O_lp₁, O_lp₂ and O_lp₃, respectively, can be obtained. Through the same equations, the direction vectors of O_rp₁, O_rp₂ and O_rp₃ can be described as j_r1, j_r2 and j_r3, respectively. With a_l1 as an example, the normal vector of plane Π_lpr1 is n₁ according to Equation (12)(12) . Incorporating j_r1, j_r2, j_r3 and n₁ into Equation (14)(14) derives μ₁₁, μ₁₂ and μ_13, whose values are shown in Table 2.

Table 2. Values of μ of points in pairs of images.

μij	1	2	3
1	0.0002	0.0284	0.0700
2	0.0279	0.0002	0.0424
3	0.0707	0.0427	0.0004

Bold values meet the requirements.

Owing to calculation error, μ<μ_c=1.0×e⁻³. Only μ₁₁, μ₂₂ and μ₃₃ satisfy Equation (13)(13) ; thus, a_l1 and a_r1 originate from the same marker, p₁, which can be correctly matched. Similarly, the corresponding relations between a_l2 and a_r2, and between a_l3 and a_r3 can be confirmed. Thus, markers p₁, p₂ and p₃ can be correctly matched. Then, the coordinates of these three markers in the system can be obtained based on Equations (10)(10) and (11)(11) and are shown in Table 3.

Table 3. Coordinates of markers.

i	pi (x, y, z)
1	[125.1197, −72.6015, 622.8587]
2	[154.1098, −51.4452, 614.7087]
3	[110.4676, 18.2491, 617.2883]

According to the results in Table 3, the distances between two markers among these three markers are 36.588, 81.845 and 91.694 mm. These results are consistent with the real values within the permissive error range.

Multiple surgical instruments

However, multiple instruments are tracked in an SNS. Thus, different instruments should be recognized. To verify the algorithm proposed in this paper, we have designed three surgical instruments I_i (i = 1, 2, 3) with nine markers, as shown in Figure 5. First, the instruments are placed within the view of the NOS. Then, the locations and directions of the instruments are adjusted. Finally, the system captures the pairs of images of all the markers on the surgical instruments, as shown in Figure 5(B and C). The subpixel coordinates of all points, a_li/a_ri (i = 1, 2, … , 9), are determined.

Figure 5. Three surgical instruments: (A) surgical instruments with near-infrared markers; (B) markers captured by the left camera; (C) markers captured by the right camera.

After manual analysis, the correct matching relations of the nine markers on these three surgical instruments can be obtained, as shown in Figure 6. The corresponding relations of the points in Figure 6 are regarded as the standard in this experiment.

Figure 6. Correct matching relations of all points.

The values of μ_ij (i = 1, 2, … , 9; j = 1, 2, … , 9) can be obtained using Equations (4)–(14) and are reported in Table 4. Given μ < μ_c=1.0×e⁻³, a_l1 in the left image and a_r1 in the right originate from the same marker in Table 4, i.e. the points are a right match. After the comparative analysis of all the points, the matching relations of the other markers are found to be correct.

Table 4. Values of μ of points in pairs of images.

μij	1	2	3	4	5	6	7	8	9
1	0.0003	0.0792	0.0568	0.1166	0.1026	0.1791	0.2232	0.2237	0.2538
2	0.0568	0.0179	0.0003	0.0552	0.0450	0.1202	0.1632	0.1644	0.1971
3	0.0740	0.0005	0.0167	0.0367	0.0276	0.1024	0.1450	0.1465	0.1799
4	0.1009	0.0297	0.0435	0.0075	0.0002	0.0743	0.1163	0.1181	0.1526
5	0.1083	0.0376	0.0509	0.0005	0.0073	0.0666	0.1084	0.1102	0.1451
6	0.1716	0.1062	0.1140	0.0695	0.0720	0.0002	0.0400	0.0427	0.0802
7	0.2090	0.1470	0.1514	0.1106	0.1105	0.0400	0.0009	0.0022	0.0412
8	0.2111	0.1492	0.1535	0.1128	0.1126	0.0422	0.0032	0.0000	0.0390
9	0.2481	0.1897	0.1907	0.1537	0.1509	0.0820	0.0441	0.0406	0.0001

Bold values meet the requirements.

However, pseudo markers still exist in practical applications. Thus, a board with 15 markers is designed in the experiment, as shown in Figure 7.

Figure 7. Board with 15 markers: (A) markers captured by the near-infrared camera; (B) the geometry of markers.

As shown in Figure 7(A), five surgical instruments I_i (i = 1, 2, …, 5) are simulated, resulting in 15 markers, which are represented by p_i (i = 1, 2, …, 15). Each instrument consists of three markers, as shown in Figure 7(B)_. The distances between two markers of these five instruments are each unique. To obtain these distances, we capture six pairs of images of the luminous panel. After manual analysis, we have obtained the matching relations of all the markers. Subsequently, all the distances between two markers are measured using the intrinsic and extrinsic parameters of the NOS. The results are shown in Table 5.

Table 5. Distances between two markers of five instruments (unit: mm).

Ii	La	Lb	Lc
1	20.3104	28.6190	27.0618
2	28.4081	32.0648	47.3987
3	33.4089	48.0840	37.4404
4	50.9139	47.8183	29.4804
5	35.5860	38.3802	67.3612

As shown in Table 5, the distances between two random markers on each instrument are unequal; thus, each distance can easily be distinguished with a unique code. According to Equation (15)(15) , the distances (edges) of the five instruments are assigned codes, as shown in Figure 8(A). The codes for the distances of the instruments are independent of the coding order of the surgical instruments. As shown in Figure 7(B), the coding order of the instruments is from I₁ to I_5. Each marker is then given a code by summing the codes for the distances between the marker and the other two markers on the surgical instrument. The coding results for the markers are shown in Figure 8(B).

Figure 8. Codes for surgical instruments: (A) codes for distances; (B) codes for markers.

The right and left images of the board were captured by the NOS. The subpixel coordinates of the 15 points in the pairs of images are extracted and shown in Table 6. After manual analysis, the matching relations of these 15 pairs of points are determined, as shown in Table 6, and regarded as the standard to verify the algorithm. After the subpixel coordinates of all the points in the pairs of images are obtained, we can derive the linear equations for the markers by using Equations (4)–(12), and all the matching results satisfy Equation (13)(13) , as shown in Figure 9(A). However, given more than one marker exists on the same plane, 15 correct matches and 23 pseudo markers are derived among all the corresponding relations.

Figure 9. Matching results: (A) all matching results satisfying Equation (13)(13) ; (B) experimental results.

Table 6. Subpixel coordinates and matching relations of the 15 pairs of points.

No.	al (x,y)	ar (x,y)	Corresponding relation
1	[497.4878, 468.1911]	[332.5146,464.7727]	(1,1)
2	[562.3381, 469.1564]	[386.9572, 464.6878]	(2,2)
3	[643.5982, 471.8993]	[458.5801, 466.2507]	(3,3)
4	[733.7264, 475.9549]	[542.3738, 468.6908]	(4,4)
5	[836.1024, 480.6134]	[644.0919, 471.7770]	(5,5)
6	[535.0917, 551.4058]	[363.1766, 544.6413]	(6,6)
7	[778.1924, 567.6032]	[584.9238, 562.4122]	(8,8)
8	[616.5449, 570.8992]	[433.5192, 564.1320]	(7,7)
9	[671.9316, 620.5589]	[321.9727, 613.0733]	(12,9)
10	[647.6483, 620.4255]	[461.5875, 613.6088]	(10,10)
11	[784.0688,621.7798]	[483.6134, 614.1908]	(9,11)
12	[485.1006,623.3484]	[591.0010, 618.2507]	(11,12)
13	[878.0323, 625.4812]	[687.7490, 624.3831]	(13,13)
14	[612.2428, 707.3566]	[430.9462, 698.7278]	(14,14)
15	[859.2526, 712.7435]	[669.0534, 716.1281]	(15,15)

An additional step of analyzing these 38 matching relations should be taken to identify the 15 correct matches by using the coding information of the surgical instruments. According to the distance codes and marker codes, we match the subpixel coordinates between the pairs of images. In Equation (14)(14) , μ<μ_c=0.0001 and φ = 0.05 mm. Finally, we obtain the results, which are shown in Figure 9(B). The comparative analysis results of the matching relations in Figure 9(B) and the results in Table 6 show that the matching relations in the experiment are in accordance with the results in Table 6. Thus, the proposed algorithm is verified to be accurate in matching markers on multiple surgical instruments.

Conclusion

The near-infrared optical systems have been widely applied to surgical navigation. They perform a major function in clinical medicine. However, when multiple instruments are considered in an SNS, matching all the markers on the surgical instruments becomes a challenge. Thus, an effective, precise and rapid algorithm is necessary for stereo matching. This paper proposed a stereo matching and recognition algorithm using two intersecting lines and marker codes. The two intersecting lines could determine a point, and each marker code was the sum of the codes for the distances between the marker and the other two markers on a surgical instrument. To verify the algorithm, we conducted several stereo matching experiments on a single surgical instrument, multiple surgical instruments and multiple surgical instruments under a special condition. The experiment on a single instrument showed that the topological structure of its points in the pair of images were very distinct and could be correctly matched. Similarly, the experiment on three surgical instruments obtained good matching results. In both experiments, a one-to-one relationship existed. We also designed a board involving five surgical instruments for the third experiment. The experiment showed that one-to-one relationships did not exist between the subpixel points in the left and right images. To eliminate pseudo markers, we encoded the instruments by using the codes for the distances and markers. Finally, correct matching was achieved. Therefore, our algorithm can realize accurate matching of the markers on multiple surgical instruments and provide SNSs with precise positional information. Our future efforts will focus on improving the performance of the algorithm by clinical application.

Funding information

This research was funded by the State Scholarship Fund under Grant CSC No. 201408440326, the Pearl River S&T Nova Program of Guangzhou under Grant No. 2014J2200049 and No. 201506010035, the Project of Outstanding Young Teachers' Training in Colleges and Universities of Guangdong under Grant No. YQ2015091, the Features Innovative Program in Colleges and Universities of Guangdong under Grant No. 2015KTSCX069, the Guangdong Provincial Science and Technology Program under Grant No. 2013B090600057, No. 2014A020215006, the Fundamental Research Funds for the Central Universities under Grant No. 2014ZG003D.

Disclosure statement

The authors declare no conflict of interest in this study.