765 
PRECISE METHOD OF FISHEYE LENS CALIBRATION 
Michal Kedzierski, Anna Fryskowska 
Dept, of Remote Sensing and Geoinformation, Military University of Technology, Kaliskiego 2, Str. 00-908 
Warsaw, Poland - (mkedzierski, afryskowska)@wat.edu.pl 
Theme Session C2 
KEYWORDS: Digital Camera, Calibration, Close-range Photogrammetry, Fisheye Lens, Accuracy 
ABSTRACT: 
In close-range photogrammetry productivity and accuracy of optical systems is very important. Fisheye lens camera can provide data 
in hard to reach places or in very small distance to the object. Nevertheless such lenses have a very large distortion. Paper presents 
new approach to the digital camera calibration process. Proposed method bases on differential geometry and on determination of arc 
curvature in every segment of photographed test. Making regression of this curve, we obtain straight line and point laying on the 
straight without distortion. Researches were made using calibration test field containing about 250 reference points and two cameras 
Kodak DCS 14nPro with 10.5 mm and Nikon 65 with 16 mm fisheye lenses. In a paper there is described conception and results of 
precise fisheye lens calibration method. 
1. INTRODUCTION 
Fisheye lenses provide imaging a large area of the surrounding 
space by a single photo, sometimes more than 180 deg. They 
make possible to realize photo on very small distance, what in 
some engineering elaboration aspects may be particularly useful. 
Close range photogrammetry (central perspective) does not 
comply with fisheye image processing. The fundamental 
difference between a fisheye lens and classical lens is that the 
projection from 3D ray to a 2D image position in the fisheye 
lens is intrinsically non perspective. One fact has to be taken 
into consideration - that not all fisheye lenses give 
hemispherical image. In our experiment there were used fisheye 
lens with focal lens 10.5 mm, which image is not 
hemispherical. Application of such type of fisheye lens gives 
more possibilities of usage of images in close range 
photogrammetry eliminating from the image everything above 
FOV of 170°, and preventing simultaneously retrieval of image 
radius. Images were taken with digital camera Kodak DCS 14n 
Pro f= 10.5mm with matrix 4500x3000 pixels. 
In order to making our method more reliable, we repeated 
measurements for second camera with 16 mm lens, mounted in 
classic camera Nikon 65. In this second case, photos were taken 
in the B & W film with 50 ISO film speed, and scanned with 
2500 dpi resolution. 
Fisheye lens has a very large distortion, for which the distortion 
polynomial used here would not converge. For such a lens the 
image coordinates should be represented as being ideally 
proportional to the off-axis angle, instead of the tangent of this 
angle as it is in the perspective projection. Then, a small 
distortion could be added on the top of this. Furthermore, the 
position of the entrance pupil of a fisheye lens varies greatly 
with the off-axis angle to the object, therefore this variation 
would be modeled unless all viewed objects are very far away. 
The calibration of dioptric camera involves the estimation of an 
intrinsic matrix from projection model. The intrinsic matrix, 
which maps the camera coordinates to the image coordinates, is 
parameterized by principal points, focal length, aspect ratio and 
skewness. 
The projection from 3D rays to 2D image positions in a fisheye 
lens can be approximated by the imaginary equidistance model. 
Let a 3D ray from pp of the lens is specified by two angles. 
Together with the angle (p between the light ray reprojected to 
xy plane and the x axis of the camera centered coordinate 
system, the distance r is sufficient to calculate the pixel 
coordinates: u’ = (u\ v’, 1) and in some orthogonal image 
coordinate system , as u' = r ■ cosqp; v' = r • sinqp. The complete 
camera model parameters including extrinsic and intrinsic 
parameters can be recovered from measured coordinates of 
calibration points by minimizing an objective function with 
denotes the Euclidean norm. 
A circular fisheye camera is a result of the size of the image 
plane charged coupled device (CCD) being larger than the 
image produced by the fisheye lens. 
In the experiment, the calibration points (230) were located on 
3D test, with an error m X yz= ± 0.0007 m. The test is painted 
special super matt paint, precluding light reflections. While 
lighting the test does not cause any shadows on the test and its 
background. The picture used in this experiment was taken in 
the distance of 0.5 m (a depth of the test is 1.5 m). Points of the 
test are located on the simple metallic elements forming in the 
space straight segment. Image of these points in the photo is a 
circular sector on the plane. Because the lens elements of real 
fisheye lens may deviate from precise radial symmetry and they 
may be inaccurately positioned causing the fact, that the 
projection is not exactly radially symmetric, Kannala and 
Brandt propose adding two distortion terms, in the radial and 
tangential direction. In our investigations we propose 
determination of distortion on the basis of proper mathematical 
relations: between this segment in the space and the arc on the 
plane. 
Using differential geometry we can determine very precisely 
distortion value in radial and tangential direction. This relations 
base on determination of arc curvature in every segment of 
photographed test. In comparison with previous presented by us 
methods of fisheye lens calibration, application of such a 
method in determination of radial and tangential distortion, gave