On the Multiple-view Triangulation Problem with Perspective and

Non-perspective Cameras

A Virtual Reprojection-based Approach

Graziano Chesi

Department of Electrical and Electronic Engineering, University of Hong Kong, Pokfulam Road, Hong Kong

Keywords:

Vision System, Multiple-view, Perspective Camera, Non-perspective Camera, Triangulation.

Abstract:

This paper considers the multiple-view triangulation problem in a vision system with perspective and non-

perspective cameras. In particular, cameras that can be modeled through a spherical projection followed by

a perspective one, such as perspective cameras and ﬁsheye cameras, are considered. For this problem, an

approach based on reprojecting the available image points onto virtual image planes is proposed, which has

the advantage of transforming the original problem into a new one for which the existing methods for multiple-

view triangulation with perspective cameras can be used. In particular, algebraic and geometric errors of such

methods are now evaluated on the virtual image planes, and the solution of the new problem exactly approaches

the sought scene point as image noise and calibration errors tend to zero. The proposed approach is illustrated

by several numerical investigations with synthetic and real data.

1 INTRODUCTION

It is well-known that the multiple-view triangulation

problem is of fundamental importance in computer vi-

sion and robotics. Speciﬁcally, this problem consists

of recovering a scene point from its available image

projections on two or more cameras located in the

scene. Unfortunately, due to image noise and cali-

bration errors, this process generally provides an esti-

mate only of the sought point, which depends on the

criterion chosen to match the available image points

with the image projections of the estimate on all the

cameras. The multiple-view triangulation problem

has numerous key applications, such as 3D object re-

construction, map estimation, and visual servo con-

trol, see for instance (Hartley and Zisserman, 2000;

Faugeras and Luong, 2001; Chesi and Vicino, 2004;

Chesi and Hung, 2007).

The multiple-viewtriangulation problemwith per-

spective cameras has been studied for a long time,

and numerous contributions can be found in the lit-

erature. Pioneering contributions have considered the

minimization of algebraic errors for deﬁning the esti-

mate of the sought point, since the resulting optimiza-

tion problems can be solved via linear least-squares,

while later contributions have proposed the minimiza-

tion of geometric errors since they can generally pro-

vide more accurate estimates, see for instance (Hart-

ley and Zisserman, 2000) about the deﬁnition of al-

gebraic and geometric errors. A commonly adopted

geometric error is the L2 norm of the reprojection er-

ror, for which several solutions have been proposed.

In (Hartley and Sturm, 1997; Hartley and Zisserman,

2000), the authors show how the exact solution of tri-

angulation with two views can be obtained by com-

puting the roots of a one-variable polynomial of de-

gree six. For triangulation with three views, the ex-

act solution is obtained in (Stewenius et al., 2005)

by solving a system of polynomial equations through

methods from computational commutative algebra,

and in (Byrod et al., 2007) through Groebner basis

techniques. Multiple-view triangulation is considered

also in (Lu and Hartley, 2007) via branch-and-bound

algorithms, and in (Chesi and Hung, 2011) via convex

programming. Other geometric errors include the in-

ﬁnity norm of the reprojection error, see for instance

(Hartley and Schaffalitzky, 2004).

This paper considers the multiple-view triangula-

tion problem in a vision system with perspective and

non-perspective cameras, hereafter simply denoted as

generalized cameras. In particular, cameras that can

be modeled through a spherical projection followed

by a perspective one, such as perspective cameras

and ﬁsheye cameras, are considered by exploiting a

uniﬁed camera model. An approach based on repro-

jecting the available image points onto virtual image

Chesi G..

On the Multiple-view Triangulation Problem with Perspective and Non-perspective Cameras - A Virtual Reprojection-based Approach.

DOI: 10.5220/0003981300050013

In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 5-13

ISBN: 978-989-8565-22-8

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

planes is hence proposed for the multiple-view trian-

gulation problem, which has the advantage of trans-

forming such a problem into a new one for which

the existing methods for multiple-view triangulation

with perspective cameras can be used. In particular,

algebraic and geometric errors of such methods are

now evaluated on the virtual image planes, and the

solution of the new problem exactly approaches the

sought scene point as image noise and calibration er-

rors tend to zero. The proposed approach is illustrated

by several numerical investigations with synthetic and

real data.

The paper is organized as follows. Section 2 pro-

vides some preliminaries and the problem formula-

tion. Section 3 describes the proposed approach. Sec-

tion 4 shows the results with synthetic and real data.

Lastly, Section 5 concludes the paper with some ﬁnal

remarks.

2 PRELIMINARIES

The notation adopted throughout the paper is as fol-

lows:

- M

: transpose of matrix M ∈ R

m×n

;

- I

: n × n identity matrix;

- 0

: n × 1 null vector;

- e

: i-th column of I

;

- SO(3): set of all 3× 3 rotation matrices;

- SE(3): SO(3) × R

;

- k vk: 2-norm of v ∈ R

;

- s.t.: subject to.

Let us denote the coordinate frame of the i-th gen-

eralized camera as

= (O

) ∈ SE(3) (1)

where the rotation matrix O

∈ SO(3) deﬁnes the ori-

entation and the vector c

∈ R

deﬁnes the position

expressed with respect to a common reference coor-

dinate frame F

ref

∈ SE(3). Each generalized camera

consists of a spherical projection followed by a per-

spective projection. The center of the sphere coin-

cides with c

while the center of the perspective cam-

era is given by

= c

− ξ

(2)

where ξ

∈ R is the distance between c

and d

. Let

X =









(3)

denote a generic scene point, where x,y,z ∈ R are ex-

pressed with respect to F

ref

. The projection of X onto

the image plane of the i-th generalized camera in pixel

coordinates is denoted by p

∈ R

3×3

and is given by

= K

(4)

where K

∈ R

3×3

is the upper triangular matrix con-

taining the intrinsic parameters of the i-th generalized

camera, and x

∈ R

3×3

is p

expressed in normalized

coordinates. The image point x

is the perspective

projection of the spherical projection of X. Speciﬁ-

cally, the spherical projection of X is given by

= A

(X) (5)

where

(X) =

(X− c

)



(X− c

)



, (6)

while the perspective projection of X

is given by

= B

) (7)

where

) =

+ ξ





+ ξ





. (8)

The solution for p

in (4) as a function of X is denoted

= Φ

(X). (9)

Figure 1 illustrates the spherical projection and the

perspective projection just described for the i-th gen-

eralized camera of the vision system.

i,s

i,c

Figure 1: A point X is ﬁrstly projected on the the point X

according to a spherical projection (frame F

i,s

). Then, the

point X

is projected on the image point x

(in normalized

coordinates) according to a perspective projection (frame

i,c

). The image point p

(in pixel coordinates) is hence

obtained as p

= K

. The distance between F

i,s

and F

i,c

, while the distance between F

i,c

and the plane where x

lies is 1.

Problem. The multiple-view triangulation problem

for generalized cameras consists of estimating the

scene point X from estimates of the image points

(denoted by

) and functions Φ

(·) (denoted by

(·)), i = 1,. .. ,N, where N is the number of gener-

alized cameras:

given



(·)



, i = 1, .. ., N



, estimate X. (10)

ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics

3 PROPOSED APPROACH

Let us start by observing that existing methods for tri-

angulation with perspective cameras cannot be used

to estimate X with the image points p

(clearly, un-

less ξ

= 0 for all generalized cameras, since in such

a case the cameras are perspective ones). This is due

to the fact that, as it can be seen from Figure 1, the

scene point X does not lie on the line connecting the

image point x

(i.e., p

expressed in normalized coor-

dinates instead of pixel coordinates) to the center of

its projection (i.e., F

i,c

The idea proposed in this paper consists of repro-

jecting the image points p

onto virtual image planes,

one per camera, in order to obtain new image points

for which this problem does not occur. This can be

done by determining the intersections of the lines con-

necting the scene point X to the centers of the spheri-

cal projections (i.e., F

i,s

) with virtual image planes V

Figure 2 illustrates this procedure for the i-th camera.

The virtual image plane V

is chosen for convenience

parallel to the true image plane of the camera and at

a unitary distance from the center of F

i,s

. The new

image point is denoted by y

. Let us observe that y

exists whenever X has a positive depth in the frame

i,s

, while y

tends to inﬁnity as this depth tends to

zero.

i,s

i,c

Figure 2: The image point x

(in normalized coordinates) is

reprojected onto the virtual image plane V

, ﬁrstly, by deter-

mining the intersection X

of the line connecting x

to F

i,c

with the sphere, and secondly, by determining the intersec-

tion of the line connecting X

to F

i,s

with V

. The new image

point is denoted by y

In order to derive the expression of the new image

point y

on the virtual image plane V

, let us proceed

as follows. First, let us recover the expression of p

normalized coordinates, i.e. x

. This is given by

= K

−1

. (11)

Second, let us express x









(12)

where u

∈ R. The line connecting x

to F

i,c

can be

parametrized with respect to the frame F

i,s

(α) =





αu

αv

−ξ

+ α





(13)

where α ∈ R. The spherical projection of X is hence

given by the intersection of l

with the sphere, i.e.

= l

(α

∗

) (14)

where α

∗

is the solution of

∗











(α)k = 1





1− ξ





(α) > 0.

(15)

This solution is given by

∗

+ δ

1+ u

+ v

(16)

where

1+ (1− ξ

)(u

+ v

). (17)

Then, the line connecting X

to F

i,s

can be

parametrized with respect to the frame F

i,s

(β) = βX

(18)

where β ∈ R. The intersection of this line with the

virtual image plane V

is hence given by

= m

(β

∗

) (19)

where β

∗

is the solution of

∗

: e

(β) = 1. (20)

This solution is given by

∗

1+ u

+ v

− ξ

+ v

)

. (21)

It is possible to verify that the overall expression of y

in normalized coordinates is given by









(22)

where γ

is deﬁned as

1+ ξ

1− ξ

+ v

)

. (23)

We denote the expression of y

as a function of p

ac-

cording to

= Ω

). (24)

Let us observe that y

exists whenever

+ v

) 6= 1 (25)

OntheMultiple-viewTriangulationProblemwithPerspectiveandNon-perspectiveCameras-AVirtual

Reprojection-basedApproach

i.e. whenever X has a positive depth in the frame F

i,s

The procedure just described assumes that p

and

Ω

(·) are known. However, in real situations this is

clearly not true due to the presence of uncertainties,

and hence the new image points have to be deﬁned

using the available data. In particular, p

is replaced

, while Ω

(·) is replaced by

Ω

(·) which is ob-

tained as in (22)–(24) by replacing u

, v

, γ

, δ

and ξ

with their available estimates ˆu

, ˆv

and

, re-

spectively. The estimates of the new image points are

given by

ˆy

Ω

(

). (26)

In order to estimate the scene point X, let us ob-

serve that the new image points satisfy the relation-

ship

= P

X (27)

where λ

∈ R and P

∈ R

3×4

is the projection matrix

given by





(28)

where the rotation matrix R

∈ SO(3) and the transla-

tion vector t

∈ R

are given by

= O

= −O

(29)

Hence, (27) can be rewritten as

X. (30)

The relationship (30) can be exploited to estimate X

from the available estimates of the new image points

ˆy

. In the sequel we discuss two criteria for this esti-

mation.

The ﬁrst method that we consider is based on the

estimation of X by minimizing the algebraic error in

the relationship (30). Speciﬁcally, according to this

method, the estimate of X is obtained through the lin-

ear least-squares problem

min

ˆc

alg

(Y) (31)

where Y ∈ R

and

ˆc

alg

(Y) =

∑

i=1



Y−

ˆu

Y−

ˆv



(32)

and

is the available estimate of P

. The solution of

linear least-squares problems can be obtained either

in closed form or through a singular value decompo-

sition (SVD). Indeed, let us deﬁne

A =













b =







ˆu

ˆv

ˆu

ˆv







. (33)

It follows that (31) can be rewritten as

min



AY−



. (34)

The minimizer of (34), denoted by

alg

, is given by

alg





−1

b. (35)

Alternatively, one can get this minimizer by introduc-

ing the SVD



A −



(36)

and by deﬁning

alg

ˆv

(37)

where ˆv

∈ R

is the vector with the ﬁrst three entries

of the last column of

V ∈ R

and ˆv

∈ R is the fourth

entry of such a column.

The second method that we consider is based on

the estimation of X by minimizing the L2 norm of

the reprojection error in the relationship (30). Specif-

ically, according to this method, the estimate of X is

obtained through the optimization problem

min

ˆc

(Y) (38)

where

ˆc

(Y) =

∑

i=1



(Y) −

ˆu

(Y) −

ˆv



(39)

and the function

(·) is the available estimate of the

function Ψ

(·) which deﬁnes the solution for y

in (30)

as a function of X, i.e.

= Ψ

(X). (40)

We denote the minimizer of (38) as

= argmin

ˆc

(Y). (41)

The computation of this minimizer can be addressed

in various ways. For instance, in (Chesi and Hung,

2011) a technique based on convex programming has

been proposed recently,which provides a candidate of

the sought solution and a simple test for establishing

its optimality. See also the other techniques described

in the introduction.

It is important to observe that the two methods just

described provide estimates of the sought scene point

by minimizing an error (either algebraic or geomet-

ric) deﬁned for the new image points ˆy

. This means

that such an error is evaluated on the virtual image

planes V

unless the cameras are perspective (in such

a case, in fact, the virtual image planes V

coincide

with the image planes of the cameras). Let us also

ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics

observe that the estimates provided by these methods

approach the sought scene point as image noise and

calibration errors tend to zero (clearly, if enough in-

formation is available for triangulation).

In the sequel we denote the 3D estimation errors

achieved by minimizing the algebraic error in (31)

and by minimizing the L2 norm of the reprojection

error in (38) as

alg

= k

alg

− Xk

= k

− Xk .

(42)

4 EXAMPLES

In this section we present some results obtained with

synthetic and real data. The minimization of the al-

gebraic error in (31) is solved through (35), while

the minimization of the L2 norm of the reprojec-

tion error in (38) is solved with the TFML method

described in (Chesi and Hung, 2011) available at

http://www.eee.hku.hk/∼chesi. In both cases, the

data are pre-elaborated in order to work with normal-

ized data.

4.1 Example 1

Let us consider a vision system composed by three

generalized cameras with 180 degrees-ﬁeld of view

deﬁned by

∀i = 1, 2,3















200 0 400

0 200 400

0 0 1





= 0.5

= e

[θ

]

and









, c





−9









−π/2





, c





−1

−7









−π/3

π/2





, c









The scene point is

X =









and the corresponding image points in pixel coordi-

nates are given by





677.926

344.415





, p





351.895

159.473









133.527

465.346





Figure 3 shows the three cameras and the scene point,

while Figures 4a–4c show the image points and the

boundary of the visible region in each camera.

−8

−6

−4

−2

−10

−5

−1

Figure 3: Example 1: the three cameras and the scene point

(“+” mark).

In this example we want to consider the presence

of image noise on the available image points. To this

end, we deﬁne the available image points as

= p

+ ηn

∀i = 1, 2,3

where η is a parameter deﬁning the image noise in-

tensity and









, n





−1





, n





−1





The problem consists of estimating X for η varying in

the interval [0,6] pixels.

We repeat the multiple-view triangulation proce-

dure described in the previous section for a grid of

values η in [0,6]. Figure 5 shows the obtained esti-

mates by minimizing the algebraic error and by mini-

mizing the L2 norm of the reprojection error. In par-

ticular, for η = 6, the 3D estimation errors achieved

OntheMultiple-viewTriangulationProblemwithPerspectiveandNon-perspectiveCameras-AVirtual

Reprojection-basedApproach

−100 0 100 200 300 400 500 600 700 800 900

100

200

300

400

500

600

700

[pixel]

(a)

−100 0 100 200 300 400 500 600 700 800 900

100

200

300

400

500

600

700

[pixel]

(b)

−100 0 100 200 300 400 500 600 700 800 900

100

200

300

400

500

600

700

[pixel]

(c)

Figure 4: Example 1: image points (“o” marks) and bound-

ary of the visible region (solid line) for each camera.

by the two methods are

alg

= 0.231, d

= 0.139.

As we can see, minimizing the algebraic error pro-

vides quite worse estimates than minimizing the L2

norm of the reprojection error in this example. Inter-

esting, the next examples will show that the situation

is generally different.

1.05

1.1

1.15

2.85

2.9

2.95

3.05

1.86

1.88

1.9

1.92

1.94

1.96

1.98

2.02

alg

Figure 5: Example 1: solutions

alg

and

for η ∈ [0, 6].

4.2 Example 2: Statistics with Synthetic

Data

Here we present some results obtained with synthetic

data. Speciﬁcally, we have generated 500 vision sys-

tems, each of them composed by a scene point to re-

construct (denoted hereafter as X) and 4 generalized

cameras with 180 degrees-ﬁeld of view, in particular

with intrinsic parameters given by

∀i = 1,..., 4















300 0 600

0 200 400

0 0 1





= 0.5.

For each vision system, X and the centers of the cam-

eras are randomly chosen in a sphere of radius 500

centered in the origin of the reference frame, while the

orientation matrices of the cameras are randomly cho-

sen under the constraint that X is visible by the cam-

eras. Figure 6a shows the scene points and the gener-

alized cameras for 10 of the 500 vision systems, while

Figure 6b shows the image points and the boundary of

the visible region in these cameras.

In order to generate the corrupted data, we have:

• added random variables in the interval [−η,η]

pixels to each coordinate of the image points,

where η ∈ R deﬁnes the noise intensity;

• multiplied ξ and each intrinsic parameter times

random variables in the interval [1 − η/100,1+

η/100];

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−50

100

−100

−50

100

−80

−60

−40

−20

(a)

0 200 400 600 800 1000 1200

100

200

300

400

500

600

700

800

[pixel]

(b)

Figure 6: Example 2 (synthetic data): (a) scene points (“+”

marks) and generalized cameras for 10 of the 500 vision

systems; (b) image projections of such scene points (“o”

marks) and boundary of the visible region (solid line).

• multiplied the camera centers and the angles of

the rotation matrices times random variables in

the interval [1− η/100,1+ η/100].

Hence, we have repeated the triangulation for 3

numbers of available cameras (i.e., 2, 3 and 4) and

for 4 values of noise intensity (i.e., η = 0.5,1,1.5,2),

hence solving a total number of 3 × 4 × 500 = 6000

triangulation problems. Table 1 shows the average

values of d

alg

and d

denoted by “alg” and “L2”, re-

spectively.

4.3 Example 3: Wadham College

Sequence

Lastly, we present some results obtained with almost

real data. In fact, we do not have real data for a non-

perspective camera, moreover with real data it is im-

Table 1: Example 2 (synthetic data): average 3D error for

different number of generalized cameras (N) and noise in-

tensity (η).

N = 2 (2000 points)

method \ η 0.5 1 1.5 2

alg 2.021 3.6212 6.1174 8.1519

L2 2.0181 4.0248 6.1692 8.42

N = 3 (2000 points)

method \ η 0.5 1 1.5 2

alg 1.0846 2.1688 3.0144 4.1615

L2 1.0817 2.0342 3.047 4.2966

N = 4 (2000 points)

method \ η 0.5 1 1.5 2

alg 0.9066 1.7793 2.4941 3.4809

L2 0.94721 1.7375 2.5739 3.8066

possible to know the true scene points that we would

like to use for evaluation. Hence, we have considered

the Wadham college sequence available at the web-

page of the Visual Geometry Group of Oxford Uni-

versity, http://www.robots. ox.ac.uk/∼vgg/data/data-

mview.html. This sequence consists of 5 views taken

with a perspective camera, the projection matrices of

such views, and 3019 image points corresponding to

1331 scene points visible in at least 2 of such views

(with known correspondence).

In particular:

• 1052 points are visible in 2 views;

• 215 points are visible in 3 views;

• 50 points are visible in 4 views;

• 14 points are visible in 5 views.

First, we have estimated the 1331 scene points using

standard triangulation for perspective cameras, which

are shown in Figure 9. Second, we have computed

the projections of these scene points onto generalized

cameras with same orientation, same center except

for a translation along the optical axis in order to en-

large the spanned image area, and intrinsic parameters

given by

∀i = 1,..., 5















256 0 512

0 192 384

0 0 1





= 0.5.

Figures 7–8 show the ﬁrst image and last one of

the 5 images, the corresponding extracted points, and

the same points after transforming and shifting the

cameras.

The data obtained so far will be used as “true”

data. Third, we have corrupted the true data as done in

the previous subsection for the case of synthetic data

with noise intensity η = 1. Fourth, we have repeated

OntheMultiple-viewTriangulationProblemwithPerspectiveandNon-perspectiveCameras-AVirtual

Reprojection-basedApproach

(a)

100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

800

[pixel]

(b)

0 100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

[pixel]

(c)

Figure 7: Example 3 (Wadham college sequence): (a) ﬁrst

image of the sequence; (b) points extracted in such an im-

age; (c) same points after transforming and shifting the

cameras.

the triangulation using for each scene point the max-

imum number of cameras where the point is visible.

Table 2 shows the average values of d

alg

and d

de-

noted by “alg” and “L2”, respectively.

(a)

100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

[pixel]

(b)

0 100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

[pixel]

(c)

Figure 8: Example 3 (Wadham college sequence): (a) last

image of the sequence; (b) points extracted in such an im-

age; (c) same points after transforming and shifting the

cameras.

5 CONCLUSIONS

We have addressed the multiple-view triangulation

problem in a vision system with perspective and

ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics

−16

−14

−12

−10

−8

−6

−4

−2

−15

−10

−5

Figure 9: Example 3 (Wadham college sequence): esti-

mated scene points.

Table 2: Example 3 (Wadham college sequence): average

3D error for different number of generalized cameras (N).

N alg L2

2 6.2038 8.4202

3 1.2768 1.3405

4 1.0436 0.93116

5 0.34739 0.3462

non-perspective cameras, and we have proposed an

approach based on reprojecting the available image

points onto virtual image planes. This approach has

the advantage of transforming the original problem

into a new one for which the existing methods for

multiple-view triangulation with perspective cameras

can be used. In particular, algebraic and geometric er-

rors of such methods are now evaluated on the virtual

image planes, and the solution of the new problem

exactly approaches the sought scene point as image

noise and calibration errors tend to zero.

The obtained numerical results suggest that mini-

mizing the simple algebraic error on the virtual image

planes can provide competitive estimates compared

with those provided by the minimization of the L2

norm of the reprojection error on such planes. This is

indeed interesting, and it is probably due to the dif-

ferent meaning that the L2 norm assumes when eval-

uated for the new image points. Future work will in-

vestigate this aspect.

REFERENCES

Byrod, M., Josephson, K., and Astrom, K. (2007). Fast

optimal three view triangulation. In Asian Conf. on

Computer Vision, volume 4844 of LNCS, pages 549–

559, Tokyo, Japan.

Chesi, G. and Hung, Y. S. (2007). Global path-planning for

constrained and optimal visual servoing. IEEE Trans.

on Robotics, 23(5):1050–1060.

Chesi, G. and Hung, Y. S. (2011). Fast multiple-view L2 tri-

angulation with occlusion handling. Computer Vision

and Image Understanding, 115(2):211–223.

Chesi, G. and Vicino, A. (2004). Visual servoing for

large camera displacements. IEEE Trans. on Robotics,

20(4):724–735.

Faugeras, O. and Luong, Q.-T. (2001). The Geometry of

Multiple Images. MIT Press, Cambridge (Mass.).

Hartley, R. and Schaffalitzky, F. (2004). l

∞

minimization

in geometric reconstruction problems. In IEEE Conf.

on Computer Vision and Pattern Recognition, pages

504–509, Washington, USA.

Hartley, R. and Sturm, P. (1997). Triangulation. Computer

Vision and Image Understanding, 68(2):146–157.

Hartley, R. and Zisserman, A. (2000). Multiple view in com-

puter vision. Cambridge University Press.

Lu, F. and Hartley, R. (2007). A fast optimal algorithm for

triangulation. In Asian Conf. on Computer Vision,

volume 4844 of LNCS, pages 279–288, Tokyo, Japan.

Stewenius, H., Schaffalitzky, F., and Nister, D. (2005). How

hard is 3-view triangulation really? In Int. Conf. on

Computer Vision, pages 686–693, Beijing, China.

OntheMultiple-viewTriangulationProblemwithPerspectiveandNon-perspectiveCameras-AVirtual

Reprojection-basedApproach