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ANALYZE OF MATHEMATICAL MODELS FOR DIGITAL CAMERA CALIBRATION 
. * i 
M. Haen , S. M. Ayazi 
NCC of Iran, Iran-Mashhad-Vakilabad Blvd.-Ghadir Ave. NCC of Khorasan, - morihaeri@yahoo.com, sma@ayazi.com 
Commission V, WG-V-5 
KEY WORDS: Photogrammetry, Close range, Calibration, DLT 
ABSTRACT: 
By development of digital camera technologies and some capabilities of these cameras, development of models to contribute these 
capabilities for close range photogrammetry needed. With these high resolution images and many images that can be derived from 
an object we can get more accurate and early 3D map of object. This benefit could be implementing in industrial photogrammetry to 
control quality of this products by low cost and equipment. First step in close range photogrammetry routine is camera calibration. 
All recent digital cameras do not have any complete information from distortion of camera lens and unfitness of CCDs. Therefore 
obtaining some information about digital camera parameters is necessary. After calibration of cameras we can get accurate 
information from images and combine image observations to obtain 3D-models based on analytical photogrammetry methods. The 
accurate 3D-model could help manufacturers to control quality of industrial products. In this paper we design a test field for 
extracting accurate parameters of digital camera. We select a 3D part of building and labelling it with some targets. Precise 
coordinates of these points derived by accurate device. In some calibration projects sometimes physical parameters of camera derive. 
But in our project after obtaining physical parameters of camera by additional parameters method we obtain some mathematical 
parameters by rational functions. Wherever collinearity equation requires suitable initial values then parameter definition by rational 
function may be very useful in this matter. Presenting a linear model for camera calibration could be easier and comfortable. 
Therefore we can evolve this method for self calibration. In this case, without obtaining feasible initial value, we can compute both 
camera parameters and exterior parameters of images. 
1. INTRODUCTION 
Camera calibration is a very important stage in photogrammetry 
and it should be applied to each image before to could extract 
exterior orientation parameters of images. In such amateur 
camera and professional ones the calibration parameters are 
unknown. For this matter some calibration methods were 
applied to set of images derived from object or calibration 
pattern and then interior parameters determined. 
Calibration process could be solved with previous studies about 
it such as Brown and Ebner parameters. These parameters 
should be added to collinearity equation or DLT (Direct Linear 
Transformation) and solved by least square estimation. This 
nonlinear equation must linearized and for computing, feasible 
initial value required. 
DLT method often used for calibrated coordinates of two 
images but for calibration of camera we need additional 
parameters. In most cases additional parameters added to left 
part of equations. We added some similar parameters to 
numerator and denominator of DLT equations and make it a 
rational function with special additional parameters. We add 
third degree, fifth degree and seventh degree of X, Y and Z to 
each numerator and denominator. For testing the behaviour of 
each parameter we apply each parameter separately. In each 
stage residuals of errors in image points plotted. 
2. CASE STUDY 
The images derived for test captured with NIKON D80 camera 
from targets those assembled on the conjugate of walls. In this 
research the 3D based formula is used. For this reason we 
should have a set of 3D object points. Assembling of targets in 
three conjugate faces is for enabling 3D coordinate for 
calibration points. Coordinates of each point was surveyed with 
Leica total station with 2 PPM accuracy. Images captured in 
some location around this field. Coordinate of target points of 
each image determined. Now we have a set of image measured 
points and object coordinates of these points. In figure 1 a 3D 
plot of measured point are displayed. 
3. EXPERIMENTAL RESULT 
We applied DLT method without any additional parameter in 
the first step (equation 1). In three images that taken from these 
points some residuals derived in points. One sample of these 
images is shown in Figure 2. For the second stage three third 
order parameters added to numerator and denominator of DLT 
equation (equation 2) and then residuals plotted. In the stage 
three five order parameter added to equations and in the fourth 
stage seven degree parameters added. The RMSE of image 
points in these four stages listed in Table 1 and one plot of 
residuals is shown in Figure 3. 
* Corresponding author