SELF-CALIBRATION CONSTRAINTS ON EUCLIDEAN

BUNDLE ADJUSTMENT PARAMETERIZATION

Application to the 2 Views Case

Guillaume Gelabert, Michel Devy and Frédéric Lerasle

CNRS; LAAS; 7 avenue du Colonel Roche, 31077, Toulouse, France

Université de Toulouse; UPS, INSA, INPT, ISAE; LAAS; Toulouse, France

Keywords: Self-calibration, Focal estimation, 3D reconstruction, Bundle adjustment.

Abstract: During the two last decades, many contributions have been proposed on 3D reconstruction from image

sequences. Nevertheless few practical applications exist, especially using vision. We are concerned by the

analysis of image sequences acquired during crash tests. In such tests, it is required to extract 3D

measurements about motions of objects, generally identified by specific markings. With numerical

cameras, it is quite simple to acquire video sequences, but it is very difficult to obtain from operators in

charge of these acquisitions, the camera parameters and their relative positions when using a multicamera

system. In this paper, we are interested on the simplest situation: two cameras observing the motion of an

object of interest: the challenge consists in reconstructing the 3D model of this object, estimating in the

same time, the intrinsic and extrinsic parameters of these cameras. So this paper copes with 3D Euclidean

reconstruction with uncalibrated cameras: we recall some theoretical results in order to evaluate what are the

possible estimations when using only two images acquired by two distinct perspective cameras. Typically it

will be the two first images of our sequences. It is presented several contributions of the state of the art on

these topics, and then results obtained from synthetic data, so that we could state on advantages and

drawbacks of several parameter estimation strategies, based on the Sparse Bundle Adjustment and on the

Levenberg-Marquardt optimization function.

1 INTRODUCTION

This paper proposes some simple comparative

results concerning the precision of the absolute 3D

Euclidean reconstruction we can expected from 2

different pinholes cameras and from matched points

between views. The cameras are supposed to have

very distinct relative orientations, so that it cannot be

considered as a stereovision head; cameras

parameters are unknown, except focal lengths that

are approximately known from the constructor’s

data sheet. We focus on the projective reconstruction

of 3D points from matched pixels on two views and

on the quality of the Euclidean structure estimated

without prior geometric information, but imposing

constraints/priors on the cameras parameters space.

Imposing priors on the parameters may give better

theoretical precision rather than fixing parameters.

Nevertheless one must take care about

parameterization imposed by the self-calibration

constraints. We will particularly insist on this point.

This work is motivated by an application about the

analysis of video sequences acquired by two or more

cameras, typically on a crash test experiments. Two

images are presented on Figure 1: they are acquired

by two uncalibrated cameras with very different

viewpoints. The challenge consists in extracting 3D

information from these images recovering in the

same estimation process, the intrinsic parameters of

the two cameras.

2 CAMERA MODEL

We are looking for the best Euclidean 3D

reconstruction we can obtain from m sets of n

corresponding images points x

coming from n 3D

point X

, projected onto the image planes by m

distinct pinhole cameras P

. To reach this goal, we

want to estimate a parameter vector p that contain

the camera parameters (P

projection matrix for the

camera j) and the 3D points

573

Gelabert G., Devy M. and Lerasle F. (2009).

SELF-CALIBRATION CONSTRAINTS ON EUCLIDEAN BUNDLE ADJUSTMENT PARAMETERIZATION - Application to the 2 Views Case.

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 573-579

DOI: 10.5220/0001806205730579

 SciTePress

p = (P

, X

)

(1)

Without considering optical distortions, we have a

total of (11m+3n) parameters. The parameter vector

p gives a predictive camera model

: R

x R

→R

(2)

The model is for the 3D point i seen by the camera j:

ij ij ij j i

x = u (p) = P X i=1...n, j=1...m

(3)

The model will impose implicit constraints,

c(x

, u

) = 0 (4)

between underlying feature x

from noisy measurements

of the feature x’

, and have to be consistent with the

feature.

2.0.1 Feature Error Model

Due to the very distinct viewpoints of the cameras,

the feature points are selected manually and suppose

free from outliers. The observation noise d(x

)

induced by this manual selection is assumed to have

Gaussian independent and identically distributed

terms, with variance s

x’

= x

+ d(x

)

(5)

2.0.2 Cost Function

We have chosen the Maximum Likelihood Estimator

(MLE) as decision criterion to estimate the

parameters that best fit to the feature error model.

MLE gives the global minimum of the inverted log

likelihood, taken as a function of the parameters p,

p = argmin J(p)

(6)

With

()

n,m

ij ij

i,j=1

J(p) = x -u p

2σ

∑

(7)

It is well known that this cost function does

generally not have a unique minimum and is very

dependent on the initial estimate of the parameters

. These problems occur if it exists a coordinate

transformation g of the parameter space P such that

J(p) = J(g.p)

(8)

The set of all such transformation form the group G,

called the group of gauge transformation. The set of

all parameters such that p = g.p

(p is geometrically

equivalent to p

) form what is called the leaf Pp

associated with p

, which is a sub manifold of P. So

some constraints have to be imposed on the

parameters set in order to have a unique solution,

which minimizes eq.(7), for each connected

component of the leaf. However we recall that this

will not be a global minimum of J. Moreover, these

constraints need to be linear.

2.0.3 Numerical Optimization

The Non Linear Least Square problem defined by

MLE eq.(7) will be solved by numerical

optimization via a Damped Levenberg-Marquardt

algorithm allowing simple bounds constraints on the

variables (Gill et al., 1981) , also called a Bundle

Adjustment procedure (Triggs et al., 2000) as the 3D

point coordinates are part of the parameters vector p.

2.1 Counting Argument

If we consider that the m cameras are uncalibrated,

the projection matrixes P

contain 11m independent

parameters, removing the projective scale factor.

These parameters are only defined up to 15 degrees

of freedom (noted dof) coordinates transformation T

of the projective space P

, defining a camera

parameter space with (11m-15) essential degrees of

freedom (noted edof).

That simply means that to have a unique solution on

the leaf Pp

, the parameters must be constrained by

15 algebraically independent gauge equations c

These equations define a sub manifold C in P of co

dimension 15 and ensure that there is a unique gauge

transformation g that maps a parameter p of Pp

another parameter pc of Pp

which respects the

constraints.

Now, let us consider the same problem with

calibrated cameras: every projection matrix has 6

dof, and are defined only up to a 7 dof similarity

transformation of the Euclidean coordinate space,

leaving (6m-7) edof for the parameter space. So

intuitively, if we want to move from the uncalibrated

projective space to the calibrated Euclidean space or

goes in the inverse way, we have (5m-8) edof left

and 8 constraints more to impose.

2.2 Parameterization

The classical way of representing a perspective

camera is to define the following model for a camera

projection matrix

= K

(10)

ij ij j i

x = (P T)(T X ) i=1...n, j=1...m

−

(9)

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

574

Figure 1: Example of a two-cameras application.

where K

is the 3X3 upper triangular intrinsic matrix (5

dof) that links 3D point coordinates in the camera

reference frame to images 2D pixel coordinates,

iu i0

iv i0

fsu

K= 0 f v

001

⎡⎤

⎢⎥

⎣⎦

(11)

is the extrinsic parameters matrix (6m dof), that

links the world coordinates to the cameras ones.

= [ R

| t

] (12)

However, this model will impose non linear constraints on

the feature instead of linear ones.

2.2.1 Calibrated Case

In the calibrated case, intrinsic parameters are

known, leaving a (6m) dof parameterization. So,

beginning from an initial parameter vector p

, 7

independent constraints have to be imposed on the

camera parameters space to have a unique solution

among C. Indeed without gauge, Euclidean

reconstruction is obtained up to a similarity. A

solution is to fix arbitrarily the world coordinate

frame to the first camera frame by imposing that

= [I

3X3

|0] (13)

and to fix the unknown scale of the Euclidean

reconstruction (Kanatani and Morris, 2000,

Grossman and Victor, 1998) by imposing

|| t

….t

|| = 1 (14)

This is the standard Euclidean gauge. We then

parameterize minimally the cameras relative

orientations by 6*m-7 free parameters, independent

from each other’s: 3(m-1) Euler angles via the

Rodrigues formula and (3m-4) parameters for the

normalized multi-camera translation. The

constrained parameter space C has (6m-7) edof.

2.2.2 From Calibrated Euclidean Space to

Uncalibrated Projective Space

Now, if we want to extent the calibrated Euclidean

parameterization defined above to the particular case

of uncalibrated cameras, the intrinsic parameters of

m cameras must be added into the system

parameterization. Beginning with totally unknown

intrinsic parameters, the projective cameras are

parameterized in a calibrated Euclidean way. The

counting argument allows (5m-8) edof to

parameterize the intrinsic parameters considering the

simple difference between the edof in the Projective

and Euclidean cases; equivalently it makes

mandatory to have 8 constraints on the (5m) dof of

free intrinsic parameters. If we denote f the fixed

intrinsic parameters and k the known ones among

the m views, we can derive the well known counting

process

mk+(m-1)f = 8 (15)

By the way, we recover the “self-calibration”

constraints, which explain the fact that “the intrinsic

parameters should be parameterized so that the self-

calibration constraints are satisfied” (Pollefeys at al.,

1998).

The Euclidean uncalibrated parameterization impose

implicit constraints on the projective parameter

space via

K [R | t] = P T (16)

which is directly related with the constraint

described by Triggs (Triggs, 1997)

P Q PT = KK

(17)

This constraint is applied on the absolute quadric

and is expressed algebraically by the above counting

argument.

Q = T Diag(1,1,1,0) T

(18)

SELF-CALIBRATION CONSTRAINTS ON EUCLIDEAN BUNDLE ADJUSTMENT PARAMETERIZATION -

Application to the 2 Views Case

575

We can visualize these 8 constraints, fixing the

unknown projective scale, by taking

proj

= [I

3X3

|0] (19)

so that T and so Q, are parameterized by 8

parameters (5 for intrinsic/absolute conic parameters

and 3 for the plane at infinity). It gives a local

parameterization of our gauge group G (Triggs,

1998).

⎡⎤

⎢⎥

⎣⎦

(20)

We have now (11m-15) dof for our set of

parameters; it is consistent both with the number of

edof of the projective space and with the counting

argument rule to well parameterize the intrinsic

parameters following the number of views and the

priors we have about them. It is also important to

mention that the counting argument is only valid for

non-critical configurations (configurations that do

not permit to locate exactly the absolute quadric in

the projective space). These configurations depend

either on 3D points parameters (critical surface

(Hartley and Zisserman, 2006)), or relative positions

between cameras (critical motion sequences as

described by Sturm (1997)), or both of them. Some

specific approaches have to be developed in such

cases.

2.3 Dealing with Parameters Inter

Correlations

However, using this natural approach, we are not

able to ensure that our essential parameters defined a

set of independent parameters. The calibrated

Euclidean parameterization provides an independent

set of parameters, but in the uncalibrated case, the

chosen parameterization gives intercorrelations, as

the intrinsic parameters are highly correlated with

the extrinsic ones (Shih et al., 1996). For instance,

the principal point position is correlated with the

camera orientation and the focal length with the

translation along the optical axis. In this case, the

camera parameters covariance matrix contains some

abnormally elevated values. As a consequence,

intrinsic gauge constraints imposed by eq.(10), if

perturbed by noisy measurements, will greatly

impact the Bundle Adjustment as the free parameters

will move to compensate this initial error induced by

the badly fixed/known ones.

So we could think that a free intrinsic gauge

approach, like the one proposed by Malis and Bartoli

(Malis and Bartoli, 2001) will greatly improve the

solution. Basically, the authors adopt an elegant

method, equivalent to obtain a reduced model of the

parameter set by the classical way. If we

differentiate eq. (7) with respect to the intrinsic

parameters, we set the result to zero, and we solve

the resulting equation, then, intrinsic parameters are

expressed in terms of the remaining parameters

(image point, extrinsic and 3D points). Substituting

it into eq. (3) we will obtain a function of the

remaining parameters. However, as the intrinsic

values are embedded in the free intrinsic gauge

parameterization, the self-calibration constraints still

have to be needed on remaining parameters leading

to the same problem; they are imposed either by

Lagrange multipliers or weighting methods.

A method to have a better Euclidean reconstruction

is to use some priors about the free parameters

during the optimisation, imposing the free

parameters to stay between some bounds. In the next

section, we propose some comparative results using

priors on focal lengths during a Bundle Adjustment

process, in the typical standard Euclidean gauge

(process noted EBA hereafter). For the 2 view case

with distinct cameras, EBA process is applied with

a weighting method, with bounds constraints, and

using Malis and Bartoli intrinsic free

parameterization with artificial weights. Results can

be found on Figure 2.

3 TWO VIEWS CASE

Our objective here consists in comparing several

ways to calibrate a two-cameras system, depending

on initial knowledge available on the intrinsic

parameters. Seven algorithms are compared.

3.1 Parameter Choice

The number of projective edof authorizes us to

parameterize the space parameters with 7 parameters

that must ideally be independent, being far from the

possible critical configurations for self-calibration

and 3D reconstruction: a theoretical study for these

ones has been performed by Sturm (Sturm, 1997) for

the specific two view case. We recover by the way,

the 7 dof of the fundamental matrix (scaled matrix of

rank 2), which encapsulates the 2 views epipolar

geometry. This is a classical way to show that 2

camera intrinsic parameters can be recovered from

images alone by a self-calibration procedure, as it

has been shown in a pioneered contribution by

Hartley for the focal length in 1993(Hartley and

Zisserman, 2006).

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576

Using the above Euclidean camera model with the

Euclidean standard gauge, we can add to our 5

Euclidean parameters space, 2 more parameters

chosen among the intrinsic ones to reach the 7

parameters allowed. If we add more parameters, we

will neither obtain a good 3D reconstruction nor

good estimate of the intrinsic parameters

(Bougnoux, 1998, Grossman and Victor, 1998) as

the parameterization will not respect the self-

calibration constraints and will be over-

parameterized.

Let us define two situations: (1) we can suppose that

we have estimated the camera calibration parameters

by some way, e.g. reading the camera data sheet: we

know approximately the focal lengths, and can make

the usual square pixel assumption (s=0, fu = fv = f)

with (u0, v0) at image centres. (2) Using a

calibration pattern, we can also consider focal

lengths as unknown parameters and fix the other

ones to their initial values. So that our set of

parameters p = (pC, pX) are the camera parameters

pC = (f1, f2, rx, ry, rz, tx, ty) plus the unknown 3D

points pX = (X1…XN), with the standard Euclidean

gauge and the self-calibration constraint imposed by

fixing all the intrinsic parameters apart from the 2

distinct focal lengths.

3.2 Initialisation

First we recover the relative camera orientation with

the Essential matrix, via the estimated intrinsic

parameters and the fundamental matrix calculated

with the Gold Standard Algorithm (projective BA)

as described in (Hartley and Zisserman, 2006).

The initial Euclidean reconstruction of the 3D points

is obtained with the 2 view optimal methods as

described in (Hartley and Zisserman, 2006).

If the true intrinsic are used, we call the obtained

reconstruction, the

initial calibrated reconstruction,

and the

initial pre-calibrated one if initial

parameters are only approximated. This provides

two initial parameters set (p

, p

) used as initial

guesses for the Bundle Adjustment function.

3.3 Algorithms

We recall that the goal is not to recover very precise

focal lengths, but to allow a better reconstruction of

the scene by adding the 2 focal lengths as parameters

during the optimisation process.

First of all, we run the optimization (BA) with the

standard Euclidean gauge for the initial calibrated

reconstruction and initial calibrated one. Only

Extrinsic and 3D points are free during the

optimisation process. We obtain respectively the

reconstruction called

EBA calibrated and EBA

pre-calibrated

. The first one being the best 3D

reconstruction we can obtain from noisy

measurements without constraint on the structure

parameters. These will be our reference data, as the

algorithms used to obtain it are well known.

Then, even if results do not appear here, we have

verified that if there is only noise on the focal

lengths, the free 7 parameters, exactly converge

through the exact focal lengths values. This means

that self-calibration constraint and Euclidean gauge

defined a unique minimum on the parameter set, and

not a sub manifold of the parameters set. This result

is also true for more cameras as long as the camera

parameterization and the Euclidean gauge impose 15

independent gauge constraints, which can easily, be

verified experimentally.

Next we investigate the results of our algorithms,

when the self-calibration constraints are badly

defined by coarse approximations of the intrinsic

parameters, so that the correlation between intrinsic

and extrinsic parameters (Shih et al., 1996) leads to a

set of parameters that have not the required edof.

To control the focal lengths during the BA process,

it is assumed that we have probabilistic priors f =

N(f

, s

) about them. A study of the focal length

variability versus optical centre can be found in

(Willson and Shafer, 1993).

In the intrinsic free Euclidean BA called

EBA

intrinsic free, these priors are added by imposing a

weight value to the cost function of the form ||f-f

where the focal is a function of remaining

parameters; other constraints coming from the fixed

intrinsic parameters are imposed by adding heavily

weighted artificial measurements as their variances

are supposed to be null.

The same procedure applied to the pre-calibrated

case, leads to the reconstruction called

EBA pre-

calibrated weighted

reconstruction.

We use then the numerical optimisation described in

(section 2.2.3) where focal lengths are subject to

linear bounds constraints (Gill et al., 1981) during

the non linear least square optimisation eq.(21),

leading to the

EBA pre-calibrated bounds

reconstruction.

0j j

7+3M

p= R

i=1..M

j=1,2

min J(p) subject to |f -f | 2

∈

≤

(21)

3.4 Synthetic Image Data

We model the object scene by 20 points randomly

created in a sphere of diameter 1000 of centre (0, 0,

SELF-CALIBRATION CONSTRAINTS ON EUCLIDEAN BUNDLE ADJUSTMENT PARAMETERIZATION -

Application to the 2 Views Case

577

0). The 20 points agree with the maximum pair of

points an operator can reasonably pick up in images

pair.

The two modelled cameras are of respective centres

C1(1866, 316, 3523) and C2(-3922, 358, 6963) with

2 respective optical axis pointing towards two

distinct points of the scene Z1(-0.35, -0.15, -0.92)

and Z2(0.51, -0.04, -0.85). The Y axis of each

camera is nearly parallel to the ground, in order to

model a realistic situation not critical for the self-

calibration of the 2 distinct focal lengths (Sturm,

1997). Ground truth is given by the respective

camera focal lengths in pixels f1 = 1000 and f2 =

2000 for 500X500 pixels camera images, and the

respective Principal points image coordinates

(260,240) and (230,220), with square pixel

assumptions.

Noise simulation on intrinsic parameters is imposed

by choosing the following values for respective

camera focal lengths (1100, 1800) and principal

points (290,200), (210,280).

For 10 values of the image noise variance, ranging

from 0 to 2 pixels, we generate 100 corrupted

images from the true one and run the distinct

algorithms with the 2 sets of estimated parameters.

To measure the 3D error E

on the reconstructed

scene, which may not be exactly Euclidean, we use

the average Horn reconstruction Error (Horn, 1987)

that gives the absolute position of the reconstructed

3D points from the true ones, eq.(22).

r reconstructed true

i=1

Ε = ||X - (sRX + t)||

∑

(22)

In eq. (23), s, R and t define a similarity of the

projective space estimated by linear minimization of

the following criterion

recons

i=1

(s,R,t) = Argmin

||X - (sRX + t)||

true

∑

Some insights on the true projective transformation

existing between the estimated reconstruction and

the true one, have been studied by Bougnoux

(Bougnoux, 1998).

3.5 Simulation Results

The best reconstruction, as guessed, is obtained by

the

EBA calibrated and the worst for small to

average values of the noise level for the

initial pre-

calibrated

case. As expected too, beginning from

the two sets of initial parameters, a better 3D

reconstruction is provided by the Euclidean BA

(with fixed intrinsic parameters).

Noise level (pixel)

Error

initial pre-calibrated

EBA pre-calibrated

EBA pre-calibrated weighted

EBA pre-calibrated bounds

EBA intrinsic free

EBA calibrated

initial calibrated

Noise level (pixel)

Error

initial pre-calibrated

EBA pre-calibrated

EBA pre-calibrated weighted

EBA pre-calibrated bounds

EBA intrinsic free

EBA calibrated

initial calibrated

Figure 2: These graphs show the average Horn

reconstruction errors for various algorithms applied on our

synthetic set of points, generated from 2 distinct views.

We now focus on the interesting case of noisy

intrinsic parameters. The

EBA pre-calibrated

weighted

gives the worst results. It is basically an

intermediate between

EBA pre-calibrated with

fixed focal length (heavy weight) and the one (which

is not represented) with totally free focal length

(weak weight). As pointed out by Hartley and Silpa-

Anan (Hartley, 2002), in their quasi-linear Bundle

Adjustment Approach (Bartoli, 2002), weights are

difficult to choose optimally, but if there is little

noise on intrinsic parameters and on images, then

imposing weak bounds will generate the better

results. However, for high value of the noise, it

performed badly, as the better approach will be to

fix the parameter or equivalently, imposed heavy

weights to the focal lengths terms, as the correlation

between the parameter set will be higher. The same

remarks apply to the

EBA intrinsic free, which

performed significantly well. Finally, the better

results are obtain with the propose optimization

scheme, where the focal length are well controlled

during the numerical optimization procedure. As the

image noise is increased, the priors approach

performs equally but asymptotically, we guess that

the better reconstruction will be obtained for the

EBA with fixed intrinsic.

4 CONCLUDING REMARKS

This paper points out some of the difficulties that

arise when intrinsic cameras parameters are

estimated in the same time as the structure and

motion parameters via the classical Bundle

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

578

Adjustment procedure (sequential quadratic

programming).

We have linked the famous self-calibration counting

argument to the number of degree of freedom in our

parameter set in order to have a minimal

parameterization of the projective dof derived from

the calibrated Euclidean one. The so defined model

implies non linear constraints on the parameters set

and leads to interdependencies on the parameters

that are difficult to deal with.

The comparative studies in the two views case show

that using artificial penalty on the cost function

gives good results. Moreover, imposing priors on the

focal lengths, even if the initial principal points are

far from the true values, leads to correct 3D

Euclidean reconstruction when the image noise is

quite low. We conclude that for very noisy images

with few points (20), the maximum likelihood

estimator (MLE) performed better when intrinsic

parameters are approximately fixed. To obtain even

better results, a search control approach during the

step damping of the BA may be helpful. However

we see that even with perfect intrinsic parameters,

the reconstruction is really dependant on the image

noise and quite imprecise. A solution will be to use

some constraints coming from the structure to

improve the quality of the Euclidean reconstruction.

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