A. M. Loghin, V. E. Oniga [616785]

A. M. Loghin, V. E. Oniga
A comparative study on camera calibration algorithm s

135 A COMPARATIVE STUDY ON CAMERA CALIBRATION
ALGORITHMS

AnabMaria LOGHIN – Ph.D Student: [anonimizat]., „Gheorghe As achi” Technical University of Iasi,
Faculty of Hydrotechnical Engineering, Geodesy and Environmental Engineering,
[anonimizat]
Valeria Ersilia ONIGA – Lecturer, Ph.D Eng., „Gheor ghe Asachi” Technical University of
Iasi, Faculty of Hydrotechnical Engineering, Geodes y and Environmental Engineering,
Department of Terrestrial Measurements and Cadastre , [anonimizat]

Abstract: In the recent decade, buildings 3D models are in a high demand by many
public and private organizations. The extraction pr ocedure of high accuracy measurements
from images is one of the principal tasks of closeb range photogrammetry. The particular
techniques used in buildings 3D models creation mos tly require an accurate calibration
process of metric or nonbmetric digital cameras. Ov er the years, there were developed many
calibration algorithms by several authors, such as: Tsai, Heikkilä & Silven, Bakstein & Halir,
Zhang, etc. This paper aims to present a comparison between the intrinsic calibration
parameters determined using the Tsai calibration al gorithm, respectively the Heikkilä &
Silven algorithm and their influence on building 3D model accuracy. In order to obtain the
results, the 3D model of the historical monument “D osoftei House” from IașibCity was
created, based on image – data acquired with the Ni kon Coolpix L810 digital camera. The
camera calibration process, was performed using a 3 D calibration object and the two
algorithms mentioned above.
Keywords: building, 3D model, image, accuracy, calibration

1. Introduction

Nowadays, there is a growing interest in the constr uction of 3D models, especially of
urban and build environment, being used in many sci entific domains of activity, such as:
architecture and preservation, engineering, archaeo logy, surveying, medical and chemical
industries, design projects, tourism, property sect or and also the emergence situations
institutes [1]. An important advantage of buildings 3D modelling is the capability of
preservation in time of the city image. The buildin gs 3D models are useful for many
applications such as: urban planning and environmen tal simulation, cartography, tourism and
mobile navigation. Automatically generating buildin gs 3D models, in the form of 3D CAD
representation, is the major part of city modelling and a challenge for many researchers.
In recent years, non-metric digital cameras have kn own a great technical development,
being used in extracting metric information from th e environment, in areas such as traffic
collisions and accident reconstruction, industrial inspection, preservation and cultural heritage
projects [2].
A non-metric digital camera is a camera with a comp letely or partially unknown
interior orientation and often unstable. All “off the shelf” or “amateur” cameras are integrated
in this category, being characterized by the absenc e of fiducial marks [3]. Thus, in order to
obtain the optical characteristics of a non-metric digital camera, also called intrinsic
parameters, a camera calibration process is require d.

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136 Camera calibration is a fundamental process, that h as always been an essential
component of photogrammetric measurement. The camer a calibration process is used to
obtain metric information of the three-dimensional (3D) world from two-dimensional (2D)
images. Many applications such as close range thre e dimensional measurement and other two
dimensional measurement tasks require a precise cam eras calibration process. Thus,
corrections of image distortion in cameras has been an important topic over time [4].
Camera calibration continues to be an area of activ e research within the CV
community, with a perhaps unfortunate characteristi c of much of the work being that it pays
too little heed to previous findings from photogram metry [5]. Over time, in photogrammetry
and CV literature there have been reported various camera calibration algorithms. These
algorithms are generally based on perspective or pr ojective camera models and they are
developed by several authors, such as: Tsai (Tsai, 1987), Heikkilä & Silven (Heikkilä &
Silven, 1997), Bakstein & Halir (Bakstein & Halir, 2000) or Zhang (Zhang, 2000).
In this paper are presented the Tsai and also the H eikkilä and Silven’s calibration
algorithms to determine the intrinsic parameters of the non-metric digital camera Nikon L810.
Tsai’s calibration model assumes that some paramete rs are provided by the manufacturer, to
reduce the initial guess of the estimation. It requ ires n features points (n > 8) per image and
solves the calibration problem with a set of n linear equations based on the radial alignment
constraint. A second order radial distortion model is used while no decentering distortion
terms are considered. The two-step method can cope with either a single image or multiple
images of a 3D or planar calibration grid, but grid point coordinates must be known [5].
The technique developed by Heikkilä & Silven first extracts initial estimates of the
camera parameters using a closed-form solution (DLT ) and then a nonlinear least-squares
estimation is applied to define the interior orient ation and compute the distortion parameters.
The model uses two coefficients for both radial and decentering distortion, and the method
works with single or multiple images and with 2D or 3D calibration grids [6].
The principal purpose of this article is to find ou t the intrinsic parameters of a non-
metric digital camera, using two different calibrat ion algorithms, Tsai’s and Heikkilä and
Silven’s calibration algorithm, in order to determ ine the degree of influence on a building 3D
model accuracy, created based on digital images.

2. Presentation of the Study Area, Materials and Eq uipment

2.1. Presentation of the Study Area

Museum of Old Moldavian Literature from 1970, the “Dosoftei House” also known
as “The House with Arcades”, located in Iasi city ( Romania), Anastasie Panu Avenue no.54,
was the Metropolitan of Moldavia between 1670 and 1 686. Built from stone, the historical
monument has a special architecture with a regular cubic shape (Fig. 1).

(a) (b)
Fig. 1. The study building – “Dosoftei House” Museum (a) p erspective view of the main
facade and (b) perspective view of the main side fa cade

A. M. Loghin, V. E. Oniga
A comparative study on camera calibration algorithm s

137 2.2. Materials and Equipment

The images of the historical monument were acquire d with a Nikon Coolpix L810
digital photo bridge camera (16.1 Mega pixel), equi pped with a 6,26 mm by 4,69 mm image
sensor (Fig. 2a). In this paper, there were used di gital images with the greatest resolution of
4608 x 3456 pixels and a 1,359 µm pixel size, taken with the minimum focal length.
The camera calibration process was performed by usi ng a 3D calibration object . This
target contains a number of 42 points, 36 placed in the corners of 9 wood cubes and 6 placed
at the middle of the distance between them, with di fferent heights. These 42 control points
have 18 mm in diameter and consist of metal parts m anufactured by means of a lathe (Fig.
2b). This target was attached to a room wall [6].
In order to place the 3D calibration grid target in the world coordinate system, it was
used a device produced by Aberlink, named coordinat e measuring machine (CMM), with an
uncertainty within the working space of 2 µm (Fig. 2c).

(a) (b) (c)
Fig. 2. (a) Nikon Coolpix L810 digital photo camera, (b) 3D calibration grid,
(c) coordinate measuring machine (CMM)

2.3. Data processing

In order to obtain the image coordinates for the 3 D calibration object control points,
Lisa software was used. This is a digital photogram metric software created by Dr.-Ing.
Wilfried Linder, Bad Pyrmont – Hagen from Germany, that allows the measuring in images
process. Therefore it has many fields of applicatio ns, like: agriculture and forestry,
archaeology, architecture, coastal protection, disp osal monitoring, geography and
environmental sciences, hydrology, material testing , monument preservation, urban and
regional planning [7].
The calibration process was made using the Matlab s oftware. This is a high
performance language for technical computing, devel oped by MathWorks, which integrates
numerical computation, visualization and programmin g. It is used to analyze data, to develop
algorithms, and create models and applications, all owing matrix manipulations, plotting of
functions and data, implementation of algorithms, c reation of user interfaces, and interfacing
with programs written in other languages, including C, C++, Java, Fortran and Python.
In order to obtain the building 3D model, the image s were processed with the
PhotoModeler Scanner 6 software developed by Eos Sy stems Inc. Company from Vancouver,
Canada. This software creates 3D models and allows accurate 3D measurements from
photographs taken with most standard cameras (eithe r digital or film), which represents a very
cost-effective way of doing accurate 3D scanning, m easurement and surveying. The 3D
models are created and exported with photographic t extures extracted from the original
images.

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138 3. Results and discussion

Usually, the calibration algorithms have traditiona lly employed reference grids, the
calibration matrix K being determined using one or more images of a known object point
array, such as checkerboard patterns. Commonly adop ted methods are those of Tsai (1987),
Heikkilä & Silven (1997) and Zhang (2000). These al gorithms are all based on the pinhole
camera model and include terms for radial distortio n modelling. The principal characteristic
of the pinhole camera model is the principle of col linearity, where each point in the object
space is projected by a straight line through the p rojection center into the image plane. This
model is only a approximation of the real camera pr ojection and it is not valid when high
accuracy is required. Therefore, a more comprehensi ve camera model must be used, which
includes corrections for both radial and tangential lens distortions.

3.1. Image observations for the Nikon Coolpix L810 digital camera calibration

The first step in the camera calibration process is represented by image observations of
the object. Having a 3D target, one image is enough to estimate the camera parameters
through the calibration process and for this experi ment it was used a single image, taken at 1
meter distance, using the minimum focal length (Fig . 3).

Fig. 3. Image observation of the 3D object, using t he Nikon Coolpix L810 digital camera

3.2. Tsai calibration algorithm

The camera calibration based on Tsai’s algorithm ( 1987) recovers the interior
orientation (intrinsic parameters), the exterior or ientation (extrinsic parameters), the distortion
coefficient and also an image scale factor. The alg orithm given by Tsai is a two-step method
that can cope with either a single image or multipl e images of a 3D or planar calibration grid,
but grid point coordinates must be known. Its imple mentation needs corresponding 3D point
coordinates and 2D pixels in the image.
For the present study, it was used a Matlab toolbox implementing the Tsai’s
calibration method, with the first term of radial d istortion correction, accessed via www-
cgi.cs.cmu.edu/afs/cs.cmu.edu/user/rgw/www/TsaiCode .html. There was used a single photo
of the calibration object, at one meter distance, w ith a minimum focal length. In order to do
the computations, there was used a number of 15 poi nts with known 3D world and image
coordinates.
The calibration process uses a two-stage technique . The first stage determines the
extrinsic parameters: focal length, rotation matrix , scale factor and the translation vector, by
solving a system of linear equations whose input is the coordinates of points in the calibration
pattern, both in the image and in real world. The s econd stage computes the radial distortion
factor, which cannot be determined from the calibra tion pattern [8].

A. M. Loghin, V. E. Oniga
A comparative study on camera calibration algorithm s

139 3.3 Heikkilä & Silven calibration algorithm

The camera calibration model proposed by Heikkilä & Silven (1997) determines a set
of camera parameters that describes the mapping bet ween 3-D reference coordinates and 2-D
image coordinates. These are the intrinsic paramete rs, such as focal distance (f), optical center
point (u o, v o), correction of radial distortion (k 1, k 2), correction of decentering distortion (p 1,
p2) and the image scale factor (s u), as well as the extrinsic parameters (r ij , X o, Y o, Z o).
For the present experiment a Matlab toolbox impleme nting the Heikkilä & Silven’s
method with the two terms for both radial and decen tering distorsion correction was used.
This Matlab toolbox is available at www.ee.oulu.fi/ ~jth/calibr/ and utilizes a new adjustement
procedure for circular control points and a recurse ve method for distorsion [9].
The entire calibration process is done in four step s. The first step is a linear parameter
estimation of the camera parameters using a closed- form solution (DLT). The DLT method is
based on the pinhole camera model, ignoring the rad ial and tangential distortion coefficients.
In this first step it is solved the linear transfor mation from object coordinates to image
coordinates.
The second step computes the distortion parameters by applying a nonlinear least
square estimation technique. The camera parameters are estimated by minimizing the
weighted sum of squared differences between the obs ervations and the model. This step
includes the transformation from the 3D camera coor dinate system to ideal (undistorted)
image coordinates.
The third step of the calibration procedure is repr esented by the correction for the
asymmetric projection. This correction is applied t o the center points of the circular control
points, due to the fact that the perspective projec tion of a circular feature on the image plane
will not remain circular, but an ellipse. In order to correct the projection error of the circular
control points, the camera parameters are computed recursively.
The fourth step, image correction, is used to solve the back-projection problem, in
order to determine the re-projected 3D coordinates recovering the line of sight from the
observed image coordinates. The unknown parameters for the inverse model were computed
by least-square method, using a generated grid of a bout 1000-2000 points, covering the whole
image area.

3.4 Comparative analysis of calibrating algorithms

The camera calibration model proposed by Tsai is a two-stage process that computes
the intrinsic and also the extrinsic parameters of the camera. This algorithm obtains the
intrinsic parameters, such as focal distance (f), o ptical center point ( u0, v0), correction of radial
distorsion ( k1) and also the image scale factor that minimizes th e measured image coordinates
corresponding to known target point coordinates (s x), as they are presented in Table 1.

Table 1. The Nikon Coolpix L810 digital camera intr insic parameters obtained with Tsai’s
calibration algorithm, for the minimum focal length f = 4 mm
Focal length f [mm] u0 [pixeli] v0 [pixeli] k1[mm] sx
f=4 mm 4.0557 2332.245 1734.104 5.2417•10 -4 1.0025

Regarding to the Heikkilä & Silven’s calibration al gorithm, in the first three steps,
there were computed the intrinsic parameters of the Nikon Coolpix digital camera, as shown
in Table 2.

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140 Table 2. The Nikon Coolpix L810 digital camera intr insic parameters obtained by Heikkilä &
Silven’s calibration algorithm, for the minimum foc al length f = 4 mm
Focal
length f
[mm] u0
[pixeli] v0
[pixeli] k1
[mm] k2
[mm] p1
[mm] p2
[mm]
f=4 mm 4.0958 2345.857 1751.143 7.2721 •10 -4 -3.3710 •10 -5 -1.2215 •10 -3 -2.5316 •10 -4

In the fourth step of the camera calibration proce ss, the unknown parameters for the
inverse model are solved, the resulting errors caus ed by the back-projection model being
represented as histograms in both horizontal and ve rtical directions [10].
After the calibration process of the Nikon Coolpix L810 digital camera, using Tsai and
Heikkilä & Silven’s calibration algorithms, a compa rative analysis of the two different sets of
intrinsic parameters was realized, the differences being presented in Table 3.

Table 3 – The comparative results of the calibration proces s of the Nikon Coolpix L810
digital camera, obtain by using Tsai and Heikkilä & Silven’s calibration algorithm

The distortions were computed at 4 mm in relation t o the image center. The two
profiles of the radial distortion, based on the com puted intrinsic parameters using the Tsai and
Heikkilä & Silven calibration algorithm, are repres ented in Fig.4.

Fig. 4. Radial distortion profiles for the Nikon Co olpix L810 camera, computed using the
calibration parameters resulted from Tsai and Heikk ilä & Silven calibration algorithms

3.5 The 3D model creation

The 3D model of the “Dosoftei House” Museum of Iasi city, was created using
“PhotoModeler Scanner” software. In a first phase, there were taken ten i mages around the
building, from ten different positions, at differen t angles. The photos have 25%- 60% overlap
and an angle of at least 20 o between them. All images were taken with the minim um focal
length of the Nikon Coolpix L810 camera lens (4 mm) , mounting the camera on a tripod at
each station point, for image stability. Parameter Calibration method
Tsai/ Heikkilä & Silven
[mm] Differences
[mm] Differences
[%]
Focal length 4.0557 / 4.0958 -0.0401 -0.9887
Principal point (x p) 3.1746 / 3.1873 -0.0127 -0.4001
Principal point (y p) 2.3779 / 2.3793 -0.0014 -0.0589

A. M. Loghin, V. E. Oniga
A comparative study on camera calibration algorithm s

141 Then, in a second stage, all photos were imported i n the software and in the correlation
process the method of manually match common feature s was used, every detail point or line
of the building being manually marked and reference d.
When the correlation process was finished, the buil ding 3D model was created, firstly
using the intrinsic parameters computed by Tsai’s c alibration algorithm. In order to place the
obtained 3D model in a real position in space, ther e were applied four transformations: a
scaling and three rotations. Therefore, the 3D mode l was scaled and rotated, using the 3D
known coordinates of three characteristic points, l ocated on different facades, with sensible
distance between them. The coordinates were determi ned by reflector less measurements
made with a total station.
In order to get a real appearance of the building, the 3D model was textured, using the
software’s high- quality option.

(a) (b)
Fig. 5. The “Dosoftei House” 3D model, created in “PhotoModeler Scanner” softwa re (a)
perspective view of the main facade, (b) perspectiv e view of the north-east facade

Secondly, it was obtained the 3D model of the “Casa Dosoftei” museum, using the
intrinsic parameters computed by Heikkilä & Silven’ s calibration algorithm. It was used the
same project, following the same steps, the only di fference being the replacement of the
calibration parameters.

3.6 Comparative analysis of 3D models

In order to analyze the accuracy of these two calib ration algorithms, an evaluation of
the building 3D digital models obtained using the “ PhotoModeler Scanner” software, was
made. Therefore, there were pointed out the differe nces between the values of two sets of
coordinates of 20 characteristic points of the buil ding, like: building corners, window edges,
doors, etc., as resulted from total station measure ments and by digital image processing, as
shown in Table 4.
The plane rectangular coordinates were determined i n the National Projection System,
named "Stereographical on unique secant plan-1970" and the normal altitudes in the „Black
Sea 1975” reference system for heights, through the GNNS technology using the South S82-V
GNSS Rover.
In order to make the computations, the real coordin ates rigorously obtained with the
Leica TCR 407 total station were used as a referenc e base.

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142 Table 4. The differences between the two sets of co ordinates
Point
no. Differences b PhotoModeler
(Tsai) – Total Station
∆X (m) ∆Y (m) ∆Z (m) Differences b PhotoModeler
(Heikkilä & Silven ) – Total Station
∆X (m) ∆Y (m) ∆Z (m) RMSE (m)
PhotoModeler b
Tsai PhotoModeler
Heikkilä &
Silven
1 -0.038 0.196 -0.170 0.035 0.020 0.020 0.263 0.045
2 0.166 -0.080 0.000 0.167 -0.060 0.000 0.184 0.177
3 -0.016 0.026 -0.018 -0.131 0.263 0.040 0.035 0.29 6
4 0.018 0.025 0.039 -0.061 0.091 -0.007 0.050 0.110
5 0.038 0.014 -0.016 -0.065 -0.033 -0.033 0.043 0.0 80
6 0.048 0.029 -0.097 0.049 0.025 0.101 0.112 0.115
7 0.041 0.216 -0.176 -0.034 -0.005 -0.025 0.282 0.0 42
8 0.018 0.031 -0.075 0.011 -0.029 0.083 0.083 0.089
9 0.036 0.146 -0.047 -0.070 -0.045 0.020 0.158 0.08 5
10 -0.019 0.078 0.049 0.070 0.069 -0.257 0.094 0.27 5
11 -0.035 -0.278 -0.245 0.107 -0.026 -0.103 0.342 0.151
12 -0.037 -0.015 -0.073 0.071 0.032 -0.261 0.083 0.272
13 -0.038 0.005 -0.181 -0.099 -0.063 -0.115 0.185 0 .164
14 -0.036 -0.156 -0.025 0.048 0.111 -0.290 0.162 0.314
15 -0.146 0.031 -0.029 -0.121 0.296 0.028 0.152 0.3 22
16 -0.087 0.028 -0.030 -0.011 0.011 0.029 0.096 0.0 33
17 -0.103 0.026 -0.017 -0.043 0.244 0.009 0.107 0.2 48
18 -0.188 0.038 -0.152 -0.193 0.250 -0.150 0.245 0.350
19 -0.282 0.037 -0.190 -0.291 0.010 -0.294 0.342 0.320
20 0.039 0.031 -0.234 -0.031 0.069 -0.038 0.240 0.0 84
0.163 0.179

The results show the maximum differences of 28.2 cm on the X coordinate, 27.8 cm
on the Y one and 24.5 cm on Z axis for the model cr eated using the parameters obtained with
Tsai calibration algorithm and 29.1 cm on X coordin ate, 29.6 cm on Y and 29.4 cm on Z axis,
for the second model, based on Heikkilä & Silven’s camera calibration algorithm.
The Root Mean Square Error was computed, using the following formula:

( ) ( ) ( )2 2 2
error r i r i r i RMS X X Y Y Z Z = − + − + − (1)

where: X r, Y r, Z r – the coordinates obtained with the total station TCR 407 Ultra,
Xi, Yi, Zi – the coordinates obtained by digital 3 D models interrogation.

From Table 4 it can be noticed that the greatest t otal error for the model created using
the Tsai’s calibration parameters is of 34 cm, whil e for the other one is of 35 cm. After the
results analysis, the Cumulative Root Mean Square Error for these two 3D models was
computed and it has the value of 16 cm for the firs t model and of 18 cm for the second one.

Fig. 6. The errors distribution histogram of
the measured detail points image coordinates Fig. 7. The error repartition of the building
detail points image coordinates

A. M. Loghin, V. E. Oniga
A comparative study on camera calibration algorithm s

143 For the resulted 3D model, obtained by using the ca libration parameters computed
with Tsai calibration algorithm, the errors distrib ution histogram of the image coordinates of
the detail points, was calculated as it can be seen in Fig. 6 and also the error repartition of the
building characteristic points image coordinates (F ig. 7).
Finally, the overall residual of the project using the parameters computed with Tsai’s
calibration algorithm, was of 1.312 pixels.
The angles between the projection rays of the deta il points range between 9o23’12”
and 89 o43’21”, with an average of 50o13’23” as they are shown in intervals of 20 o in Fig. 8.

Fig. 8. The repartition of the angles between the p rojection rays, used in points coordinates
computation, based on calibration parameters obtain ed with Tsai’s calibration algorithm

For the project using the calibration parameters co mputed with the Heikkilä &
Silven’s calibration algorithm, the overall residua l was of 1.401 pixels. Also, it was realised
the error distribution histogram of the detail poin ts image coordinates (Fig. 9) and their
repartition (Fig. 10).

Fig. 9. The errors distribution histogram of the
measured detail points image coordinates
Fig. 10. The error repartition of the building
detail points image coordinates

In the case of the distribution of medium angles be tween the projection rays,
corresponding to building characteristic points, a chart based on the angles values grouped in
intervals of 20 o is shown in Fig. 11. The angles range between 12 o06’71” and 89 o94’93”, with
an average of 52 o24’19”.

Fig. 11. The repartition of the angles between the projection rays, used in points coordinates
computation, based on calibration parameters obtain ed with Heikkilä & Silven’s calibration
algorithm

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144 4. Conclusions

The camera calibration process has always been an e ssential component of
photogrammetric measurement, especially used in clo se-range measurements. In order to
extract precise and reliable 3D metric information from images, an accurate camera
calibration and orientation procedures are necessar y. A camera is calibrated when parameters
like: the principal distance, the principal point o ffset and lens distortion parameters are
known. In many applications, especially in computer vision domain, only the focal length is
determined, but when high-accuracy photogrammetric measurements are needed, all
calibration parameters must be known.
This article presents a comparative study on two di fferent calibration algorithms
developed by Tsai and Heikkilä & Silven applied in the Nikon Coolpix digi tal camera
calibration process. A single image of the 3D calib ration object was used in both cases. In
order to estimate the camera parameters, the initia l data for these two algorithms was
represented by a set of 3D points and their corresp onding 2D projection on an image plane.
In order to compare the accuracy provided by each a lgorithm, the same set of test
points has been used and the 3D model of the same b uilding, “Dosoftei House” Museum of
Iasi city, fact which allows the results to be reli ably compared.
The RMSE has value of 16 cm for the 3D model create d using the Tsai calbration
parameters and of 18 cm for the one created using t he Heikkilä&Silven calbration parameters.

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