 Research Paper
 Open Access
Camera calibration with coplanar conics: a unified explanation and ambiguity analysis
 Shen Cai^{1}Email author and
 Zhanhao Wu^{1}
https://doi.org/10.1186/s410740180050y
© The Author(s) 2018
 Received: 13 June 2018
 Accepted: 9 October 2018
 Published: 8 November 2018
Abstract
In this paper, we propose a twostep method to give a unified explanation of camera calibration with two coplanar conics. Various kinds of conicsbased patterns in which often two parameters are unknown have been studied in previous literatures. The key in such algorithms is to adopt different strategies to compute the worldtoimage projective transformation (also called 2D homography). In the first step of our method, we show that two unknown parameters can always be computed in general cases by utilizing the underlying constraints on all parameters through the projective transformation (mathematically called projective invariants). The accompanied ambiguity problem is that the solutions of the unknown parameters are multiple. In the second step, the four intersection points (real or complex) of two totally known conics are utilized to compute the homography. The ambiguity in this step arises from the point correspondence problem. This results in multiple possibilities of correspondences followed by the ambiguous homographies. After analyzing the reasons of the two kinds of ambiguities, we apply the Centre Circle constraint to completely remove them. Finally, the experiments are shown to validate the proposed technique.
Keywords
 Homography
 Conics
 Projective invariants
 Ambiguity problems
 Centre Circle constraint
1 Introduction
Conic as an important image primitive has been studied very well in the early 1990s [1–3]. Many major problems in computer vision, such as reconstruction, motion estimation, and pose determination, can be solved with two coplanar conics. For instance, Forsyth et al. use projective invariants of coplanar conic pairs to recognize curved planar objects [4]. Rothwell et al. use four intersection points of two conics to obtain homography which in fact results in the solution of relative motion and pose [5]. However, the above works all assume that the conics’ parameters are known which is not always available in practice. If the a priori knowledge about the object conics is scarce or imprecise, there is no method to handle these cases. More importantly, the methods of computing homography in [1, 5] are not suitable for calibration because the proposed posterior rules of removing the correspondence ambiguity are only effective when the intrinsic parameters of the camera are given.
In the 2000s, people explored various 2D conics patterns, such as concentric circles, confocal conics, coplanar circles, PrincipalAxes Aligned (PAA) conics, conics with a common axis of symmetry, enclosing ellipses, and degenerate conics with double complex contact to calibrate a camera [6–13]. The core step in such algorithms is to determine the homography between the model plane and its image. But it is hard to extend their methods as they only find one special pattern. Moreover, due to lack of the natural link between different conditions, there is no explanation for the confusing question why this pattern is valid while others are not.
In recent years, Zhao [14] proposed a novel method of the 2D Euclidean structure recovery from the conic feature correspondences. The conic features are transformed from the homogeneous coordinates to the lifted coordinates to represent the geometric objects without considering the conic dual to the absolute points. Wang et al. [15] propose an algorithm which is efficient and easy to estimate the pose of camera based on the conic correspondences from world plane to image plane system. The above method needs more than two conics to work. However, it is not common to see many conics in the nature scene.
In fact, the key of camera calibration based on two coplanar conics is how to obtain homography using partial information of the conics in the model plane. In other words, what is the minimal condition used to compute the homography for two coplanar conics? Based on this idea, we propose a twostep method to give a unified solution of conicsbased camera calibration. The proposed method not only easily explains all existing conics patterns, but also finds other possibilities. The first step is to get the unknown parameters by using geometric invariance. In this step, we discuss which form of projective invariants should be chosen to construct the equations on the unknown parameters followed by the algebraic and geometric explanation of ambiguous solutions. This ambiguity problem results in the uncertain parameters of the conics. The second step is to compute homography by utilizing four intersection points of two conics and their correspondences in the image, no matter whether they are real, complex, or partially complex. This correspondence ambiguity will lead to multiple possibilities of correspondences and the ambiguous homographies. To obtain the only correct solution, these two kinds of ambiguities must be removed. Thereby, owing to the Centre Circle constraint [16, 17] which provides the geometrical explanation of the worldtoimage homography in perspective transformation, we set a cuboid bound for the camera center [18] to judge whether the computed homography is correct. Finally, the experiments with real and simulated data verify the correctness of the proposed method.

A unified explanation to the problem of calibration with coplanar conics is proposed.

We show how to use projective invariants to compute unknown parameters of conics.

We analyze the correspondence ambiguity of four intersection points of conics.

We apply the Centre Circle constraint to remove the above two ambiguities.
The paper is organized as follows. Section 2 includes some preliminaries. Section 3 gives the problem statements. Section 4 describes different invariant forms of conics and the algebraic and geometric reasons of ambiguous solutions. Section 5 first classifies the reasons of the pointscorrespondence ambiguity into three aspects. Then, several constraints are adopted to seriously reduce the number of the ambiguous correspondences. The degenerate situations are also discussed in this section. Section 6 provides our experimental results with simulated and real data.
2 Preliminaries
2.1 Basic equations
where (u_{0},v_{0}) are the pixel coordinates of the principal point and f_{u} and f_{v} are the scale factors in the image’s u and v axes. r_{1}, r_{2} are the first two vectors of rotation matrix R. t denotes the translation vector between the world coordinate system and the camera coordinate system. K is called the intrinsic parameter matrix and (R, t) are called the extrinsic parameters.
where h_{1}, h_{2} are the first two columns of H. It is well known in [19–21] that given the correct correspondences of four coplanar points, the 2D homography H can be solved.
2.2 Projective invariants related to two coplanar conics
Invariant theory as an important technique has been investigated in previous literatures [2, 4, 22, 23]. There are mainly three forms of invariants used by researchers as follows:
Note that Eq. 4 is closely related to Eq. 6. Detailed distinction in application is shown in Section 4.1 and more specific discussion is stated in [23].
2.3 The Centre Circle constraint and the cuboid bound for the optical center
Gurdjos et al. [16] proposed a Centre Circle theory to explain the constraints imposed from a 2D homography on the optical center. When a planar figure is the central projection of another planar figure, the center of projection then lies on a spatial circle (called Centre Circle) which is intersected by a sphere S (called Centre Sphere) and a plane perpendicular to this intersection (called Centre Plane). The two equations determining the position of the Centre Circle can be deduced from Eqs. (2) and (3) respectively.
3 Problem statements
where H denotes the homography. C_{1} and C_{2} are the matrices of the two conics in the model plane. Symbol \(\tilde {\mathbf {X}}\) denotes the image of the object X in the model plane. The symbol “ ≡” denotes the equivalence up to a scale factor.
Note that in the previous works [1, 5], C_{1} and C_{2} are totally known. Thus, there are 10 constraints on 8 unknowns in the process of computing homography and 10 constraints on 6 unknowns in pose estimation. By counting unknowns, if C_{1} and C_{2} are partially known, we can easily obtain the minimal condition of computing homography, i.e., C_{1} and C_{2} have two unknown parameters and the equations provide 10 constraints on 10 unknowns. Although the equations can be easily written, it is impossible to directly solve them due to the complicated nonlinearity.
Based on the above analysis, we can propose an important argument about the number of the unknowns and given constraints as follows.
Proposition 1
A necessary and sufficient condition of obtaining the worldtoimage homography up to a transformation for two coplanar conics is that four effective constraints on six parameters are given.
Proof
(⇒) Assume four effective constraints on the above six parameters are given (the simplest form is to give exact values of four parameters; the term “effective” means that the four constraints must be independent and different from the two constraints provided by geometric invariance), we can use two scalar invariants of two conics to establish two equations (be nonlinear in most cases) of two unknown parameters. By solving them, we can get all parameters. Thus, the four intersection points can be obtained followed by the computation of homography. (⇐) Assuming the homography up to a similarity transformation is given, we can compute backprojected points in the model plane. Note that the quadrangle formed by four intersection points of C_{1} and C_{2} actually has four DOF. Since the four independent DOF imposed on the quadrangle are invariant under the similarity transformation, we can obtain four equations in six unknowns, i.e., four effective constraints on six parameters. □
Now, we discuss confocal conics [9] which belong to PAA conics. As PAA conics already has three class constraints and one DOF, it needs to be given one effective and additional constraint to compute the homography. Therefore, the additional constraint for confocal conics is that two conics are confocal. Ying and Zha [7] also propose a condition of computing homography for PAA conics: giving the eccentricity of any one conic. This constraint could also be considered as a specially additional constraint.
For concentric circles, the analysis is a little complicated. Because of belonging to PAA conics, concentric circles belong to the degenerate conics system with double contact which cannot be explained by (9). On the conics system with double contact, interested readers could see [13] for the detailed properties. Here, we only give the conclusion: any two degenerate conics with double contact except concentric circles will result in a oneparameter family of homographies with an uncertain rotation parameter. For concentric circles, although the computed homography also has one unknown rotation parameter, the image of circular points is not influenced as the unknown variable is just in a similarity transformation under which the true homography is transformed into the computed one.
Although this question has been explained in principle, there still exist two problems that need to be solved. One is how to choose the suitable form of invariants to solve two unknown parameters and remove the ambiguous solutions and the other is after calculating four intersection points, how to automatically determine their correspondences in the image to compute the correct homography, especially for the case with complex intersections. These two significant problems will be demonstrated in the next two sections.
4 The ambiguous solutions of unknown parameters

Geometric invariants involving a pair of conics have various forms. Which form should be chosen to establish the equations of the unknown parameters?

Why ambiguous solutions occur?

How to remove the wrong solutions?
Gros and Quan [23] reveal that all the mentioned invariant forms in Section 2.2 are related to each other. They also imply that the complexities of the four invariant forms are different. Owing to that work, the first two questions can be answered. The removing method based on the reasonability of the intrinsic parameters is depicted in the last subsection.
4.1 Computing unknown parameters using the real invariant form
DoP and MNoS for all combinations of two unknowns
Comb  bd, ef  ab, cd, c(d)e(f)  ac(d), bf  ae(f), bc(e) 

DoP  2, 2  2, 3  3, 4  4, 4 
MNoS  4  6  12  16 
Here, we do not pay more attention to obtain the exact number of solutions of the polynomial equations for each combination of two unknowns, because we experimentally find that Matlab can directly solve these polynomial equations in two variables, which allow us not to care about the solving details. Notably, when some parameters equal to zero (corresponding to the five nongeneral classes mentioned in Section 3), the above polynomial equations will be further simplified and the number of solutions becomes smaller.
4.2 Reasons of the invariant ambiguity
It is easy to verify that the projective invariance is only a necessary condition, not a sufficient condition for a pair of plane conics. In other words, different pairs of coplanar conics may have the same projective invariants that will result in the ambiguity of solving unknown parameters. Now, we will explain the underlying reasons of the ambiguous solutions in geometry and algebra.
There is also a twofold ambiguity caused by the confusion of λ_{1} and λ_{2}. As pointed out in [23], two independent invariants, λ_{1}/λ_{3} and λ_{2}/λ_{3}, can be interpreted geometrically by the above two cross ratios. Therefore these two kinds of invariant forms arise from the uncertainty of correspondence in geometry.
Note that in the above expressions, λ_{1} and λ_{2} can be interchanged, which implies the real invariants conceal the geometrical ambiguity. Moreover, the real invariants also bring some new ambiguous solutions because of the increase of polynomial complexity. Therefore, the ambiguity produced in solving real invariant equations can be seen as a purely algebraic problem.
4.3 Ambiguity removal
Because of the invariant ambiguity, the conics whose configuration is wrong could be projected to the image of the correct conics. Thus, it is impossible to remove them in theory. In other words, for one image of two partially unknown conics, some ambiguous solutions are projectively correct. However, the principle of removing the invariant ambiguity is the same as the principle of removing the correspondence ambiguity described in Section 2.3. Given a homography, the Centre Circle which the camera center should lie in can be used to evaluate the reasonability of the homography.
5 Computation of homography between totally known conics and their images
After obtaining all parameters of the pair of conics, the four intersection points of two conics on the model plane and their projections can be computed. But consequently, the problem of how to find the correct correspondences arises. For real intersection points, people may manually select the correct correspondences in the image as the popular calibration tool does [25]. For complex intersection points without physical position, it is impossible to directly find their correspondences in the image.
Rothwell et al. [5] point out that there are 24 ways to match 4 image points to 4 object points. For real intersection points and complex intersection points, the possibilities can be reduced from 24 to 4 and 8 respectively. Then, they used several posterior rules to eliminate this ambiguity. The similar argument is given in [1]. However, these methods are only effective for estimating the pose as the intrinsic parameters must be given. If we use N images of object to calibrate the camera, there will be 8^{N} combinations to compute the intrinsic parameters. It is impossible to verify them in terms of such posterior rules. Although [6, 10] propose some methods to distinguish the circular points from the four complex intersections of two circles, it is still far away from the final solution to the general case.
In this section, we pay more attention to analyze the reasons of correspondence ambiguity and present the possibilities of correspondences (PoC) in theory for different cases of conics. In this way, the number of PoC we must deal with decreases obviously. Then, based on the Centre Circle constraint (as described in Section 2.3) which provides the geometrical explanation of the worldtoimage homography in perspective transformation, we set a cuboid bound for the optical center to remove wrong correspondences, no matter whether the intersections are complex or real.
5.1 Possibilities of correspondences for different cases

The correspondences of real lines. If the correspondences of two real lines in the quadrangle are clear, we can distinguish the PoC 1–4 from the PoC 5–8.

The adjacency of four complex intersections. Assuming that there indeed exists the adjacent relationship between the complex points no matter their order is clockwise or anticlockwise, the PoC 1,4,5,8 can be distinguished from the PoC 2,3,6,7 if this adjacency is known.

The order of points around the conic. The PoC 1,3,5,7 and 2,4,6,8 can be respectively treated as the clockwise order and the anticlockwise order.
Next, we will show how to utilize these three states to further reduce the number of PoC.
5.2 The adjacency of four complex points
Here, we will show that the adjacency of four complex intersection points can be found. Let us turn back to observe the geometric relationships of the separate conics and the enclosing conics both of which have four complex intersections (see Fig. 4). The quadrangle ABCD has two real lines AB and CD. Two complex lines AD and BC intersect at the real vertex Z of the selfpolar triangle XYZ. The complex lines BD and AC intersect at the real vertex Y. Assume the two conics are in visible halfspace segmented by the principal plane through the camera center parallel to the image plane (e.g., two ellipses). Thus, their four complex intersections also should be in visible halfspace and the possible points lying in invisible halfspace are X and Z.^{1} Because the real point Y always lies inside one conic, the line AC and BD intersect at Y both in the world plane and in the image. This implies that A should be adjacent to D, not to C. Therefore, as the position of the vertex Y intersected by two complex conjugate lines is projectively invariant, we obtain the adjacency of four complex intersections. Consequently, the number of PoC for these two cases is reduced to 4.
5.3 The correspondences of two real lines
The correspondence problem of two real lines is previously investigated in [6, 10]. Specifically, they propose to divide two coplanar circles with four complex intersections into two cases: the separate case and the enclosing case. Note that the terms “separate” and “enclosing” are visually suitable for a pair of circles or ellipses since they are closed and finite in Euclidean geometry. However, to the visually infinite conics, some adjustment should be given. Thus, we slightly extend the previous works to define the term “quasiseparate” and “quasienclosing” to cover all conics cases with four complex.
Definition 1
For two coplanar conics C_{1} and C_{2} with four distinguished complex intersections, let the two real lines consisted of the two pairs of complex conjugate points be l_{1} and l_{2}. We name
 (i)
C_{1} and C_{2} are quasiseparate if and only if they lie on the adjacent regions divided by l_{1} and l_{2}.
 (ii)
C_{1} and C_{2} are quasienclosing if and only if they lie on the same region or the opposite regions divided by l_{1} and l_{2}.
PoC for two cases with four complex intersections
Conics cases  Quasienclosing  Quasiseparate 

PoC  1, 4, 5, 8  1, 4 
PoC for two cases with real intersections
Conics cases  Tworealinter  Fourrealinter 

PoC  1, 4  1, 5 
5.4 The order of four points and geometric symmetry
The notion of adjacency or order is not existing in projective geometry as any complex point does not have the physical position. Thus, the order of four complex points cannot be distinguished and the PoC that arose from the order must be considered in general cases. Tables 2 and 3 together show the PoC of four intersection points in four general conics cases. Now, we will reveal the relationship between PoC and the geometric symmetry of the quadrangle ABCD.
The influence of centrosymmetric to PoC is obvious. When the quadrangle ABCD is centrosymmetric, the PoC ABCD and CDAB will generate the same homographies no matter the intersections are complex or real. As a result, we can pick up 2 PoC out of 4 to compute the homography.
When the quadrangle is axisymmetric, such as the trapezoid formed by two OPAS conics (see Fig. 2), the situation is a little complicated. Axisymmetry which lets the PoC ABCD and BADC generate the same homography can only influence the three cases without four real intersections. For the fourrealintersecting case, there is still 2 PoC needed to be considered.
The rectangle formed by PAA conics is both centrosymmetric and axisymmetric. Thus, for two enclosing PAA conics, all 4 PoC will generate the same homographies. It is also worth noting that, for a geometrically symmetric quadrangle, the number of the PoC is reduced at the cost of the ambiguous extrinsic parameters.
5.5 Summary

The adjacency of four intersections for all cases of two coplanar conics can always be distinguished.

The correspondences of real lines can be distinguished for the quasiseparate case and the tworealintersecting case.

For the geometrically symmetric conics, the number of PoC often can be further reduced.
Compared to the previous works, the number of the PoC we must deal with in theory declines obviously. For the residual PoC, we adopt the two steps described in Section 2.3 to judge whether the Centre Circle corresponding to one computed homography intersects with the cuboid bound around the optical center. Our experiments will verify this method can remove the invariant ambiguity and the correspondence ambiguity simultaneously.
The pseudo code of whole calibration method for two coplanar conics is illustrated in Algorithm 1.
5.6 Degenerate situations
After discussing the general case of two conics, we now analyze in which conditions the degenerate cases will occur. There are four types of degenerate conics systems which are the simplecontact system, the threepoint contact system, the system with double contact, and the system with fourpoint contact [24, pp.158160]. As the degenerate situations rarely happen in practice, we only give the conclusion about whether the degenerate conics can be used to compute a homography.
For the system with double contact (see Fig. 6c), only three points including two common points and the point intersected by the tangent l_{3} and l_{4} can be distinguished from the object plane and its image. The undetermined thing is the correspondence of another general point P of one conic which has one DOF. As a result, a oneparameter family of homographies will map this degenerate pattern to its image (interested readers could see the detailed derivation in [13]). Similarly, for the system with fourpoint contact, any line t passing through the common point has one DOF. The points M and N as the intersections of conics and t have one common unknown. Subsequently, the other two points intersected by the common tangent and the tangents passing through M and N still have one unknown. As a result, an incomplete homography with one parameter can be obtained.
6 Experiments
6.1 Computer simulations
Number of the residual PoC in theory. The symbol “ ⊗” denotes that this situation does not exist
Conics classes  PAA  OPAS  Concentric  CC and PAP and general 

Quasiseparate  ⊗  1  ⊗  2 
Quasienclosing  1  2  2  4 
Tworealinter  ⊗  1  ⊗  2 
Fourrealinter  1  2  1  2 
The two unknown parameters and the number of their solutions
Conics  C _{1}  C _{2}  C _{3}  C _{4} 

C _{5}  {a,b}, 4  {a,c}, 6  {a,f}, 12  {b,c}, 6 
C _{6}  {c,d}, 6  {b,f}, 6  {d,f}, 6  {e,f}, 4 
C _{7}  {b,d}, 4  {c,f}, 6  {b,e}, 6  {a,e}, 12 
Solutions of {c,d} for the conics pair {C_{1},C_{6}}
Unknown paras  c  d 

Correct  − 0.499  − 4.491 
Wrong1  − 14.933  − 0.698 
Wrong2  14.122  − 0.817 
Wrong3  1.635  − 1.529 
Wrong4  − 1.101+0.297i  3.060 −7.065i 
Wrong5  − 1.101 −0.297i  3.060+7.065i 
Range of the focal length computed from four PoC for each solution of {c,d}
PoC  1  2  3  4 

Correct paras  972.5–1387.3  0–4.7  0–12.3  0–43.2 
Wrong paras  0–10.2  0–3.0  0–2.2  0–4.3 
The number of the computed homographies satisfying the cuboid constraint
Conics  C _{1}  C _{2}  C _{3}  C _{4}  C _{7} 

C _{1}  ⊗  2  2  2  1 
C _{2}  2  ⊗  4  2  1 
C _{5}  1  1  2  2  1 
C _{6}  1  1  1  1  2 
6.2 Real images

Print a model plane with two general ellipses and a parabola as shown in Fig. 12. The left ellipse (denoted by E_{1}) can be obtained by rotating the ellipse x^{2}/16+y^{2}/9=1 30° anticlockwise. The equation of the right ellipse (denoted by E_{2}) is (x−8)^{2}/1+(y−6)^{2}/4=1. The equation of the parabola (denoted by P_{1}) is \(y5=\frac {3}{7}\left (x3\right)^{2}\).

Use a CCD camera (Point Grey FL208S2MC) with 4mm lens (uTron FV0420) to take 12 photos of the model plane. The image resolution is 1024×768.

Use Canny operator to detect the edge and Fitzgibbon’s conic fitting method [27] to obtain the conics.
Two solutions of the translation parameters
Conics pair  { E_{1}, E_{2}}  { E_{1}, P_{1}}  

Value  Mean  Theoretical  Mean  Theoretical 
d _{ x}  8.001  8  2.999  3 
d _{ y}  5.998  6  4.996  5 
dx′  8.500  8.499  − 3.490  − 3.496 
dy′  − 1.744  − 1.747  3.623  3.618 
Range of the focal length computed from two PoC for the correct and wrong parameters
PoC  1  2 

Correct paras  796.3.0–926.7  
Wrong paras  76.3–132.6  127.4–160.0 
Comparison of three results
Method  f _{ u}  f _{ v}  u _{0}  v _{0} 

Calibration toolbox  887.2  893.3  512.0  416.4 
Ours using { E_{1}, E_{2}}  883.3  875.3  519.7  388.6 
Ours using { E_{1}, P_{1}}  891.8  884.0  520.8  396.9 
Focal length calibration and AR The second experiment focuses on a specific AR application in practice.
Comparison of the focal length computed by our method and the Camera Calibration Toolbox
Method  Ours for (a)  Ours for (b)  Ours for (c)  Calibration toolbox 

f  3325.28  3870.98  3183.23  3255.14 
7 Conclusions
In this paper, a twostep method is developed to give a unified explanation of camera calibration based on coplanar conics. We also display a particular description of the invariant and correspondence ambiguity problems existing in the proposed method. Based on the analysis of the solutions of residual ambiguity in various cases, the wrong homographies can be removed by setting a cuboid bound for the camera center for each image and we can directly obtain the unambiguous calibration results. Furthermore, as the natural link between various conics patterns has been established, readers could get a full comprehension of the conicsbased calibration.
8 Appendix: Proof of Fig. 3
In this appendix, we use analytic geometry knowledge to verify the shape of the quadrangle formed by four intersection points of two conics for every case shown in Fig. 3.
The isogonal conjugate of the circumconic C_{1} with respect to the triangle ABC is the line l_{1}. Meanwhile, the isogonal conjugate of C_{2} with respect to the triangle ABC is the line l_{2}. For a given triangle, the direction of the isogonal conjugate line of a circumconic is only related to the direction of the axes of the circumconic. Therefore, l_{1} and l_{2} are parallel. The isogonal conjugate of the point D as a common point of C_{1} and C_{2} will be the point at infinity intersected by l_{1} and l_{2}. Moreover, as the isogonal conjugate of the circumcircle with respect to the triangle ABC is the line at infinity, D must be on the circumcircle.
For OPAS conics, as both of them are axisymmetric, the four intersections must be axisymmetric. Thus, the quadrangle formed by these four intersections can only be a trapezoid. In the similar way, two concentric conics are centrosymmetric, so are their intersections. Thus, the quadrangle can only be a parallelogram. For PAA conics, the quadrangle should be axisymmetric and centrosymmetric simultaneously. Thus, it is a rectangle.
Declarations
Funding
This work was partially supported by NSFC Program 61472075 and 61703092.
Authors’ contributions
SC proposed the ideas, deduced the formula, and was a major contributor in writing the manuscript. ZW did the experiment and complemented the manuscript. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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