Analytical Photogrammetry and Remote Sensing



The exercise is located at the laboratory of NCKU Dept. of Geomatics Engineering building which is located at the basement of the building. The images were taken at Thursday, March 26th 2016.


The images were taken by Sony NEX-5N. The detail description about parameters I used were camera focal length set to 16 mm, shutter speed to 1/50, aperture to F13 and the last ISO is set to 400. In order to calibrate the camera, we need about six photos minimum. But more is better. Then we organize the rotation about 45o clockwise at every capture. See on the figure 1. And as the result of the images captured, we can look at all direction where the location of camera is. See on figure 2 opened from Australis software.


Figure 1. Captured Image Direction


Figure 2. Captured Image in all direction


For the result of calibrated camera using Australis Software, I attached it on the “last page”. Based on the result, there are many kind of distortions in the lens, such as radial distortion and tangential distortion usually called decentering distortion. We can know the each of characters from the histogram (plotting) in the Australis result.

Radial Distortion

Based on the Australis software, the computation is using Gaussian computation for Radial distortion. See on the figure 3 reveal that, for the center of the image, there are no distortion. At least the distortion is almost zero. But, when the radius of its image center become larger, the distortion is starting appear. When the radius is enlarged to at least 11 millimeter from the center of image, the distortion is experiencing ‘saturation’ and going down until negative. For the increasing curve, we could see the distortion is outward, keep away from its center of image. But when the curve is decreasing, the distortion is inward, the arrows lead to its center of image. See on the figure 5.

Radial distortion causes the position of the image distortion along the radius from the main point. Distortion outwards is considered positive and distortion inward is negative. Positive radial distortion often called pincushion distortion, the distortion is that original image becomes distorted square shaped on its sides curved toward the center of the image. While conversely, negative radial distortion is called barrel distortion, which is illustrated in the figure 4.


Figure 3. Gaussian Radial Distortion (Australis Software)


Figure 4. Distortion character


Figure 5. Gaussian Radial Distortion (Using matlab coding)

The coding I used here is ‘’QUIVER function”, the center of an image is practically undistorted. When we move outward of the image to the blue rectangle, it has a two kind of distortions (inwards and outward) in which the inward is from increasing of curve and outward is decreasing of curve see on figure 3. And I attached the coding of this model below:

K1 = 2.35369*10^(-4);

K2 = -1.18567*10^(-6);

K3 = -1.41643*10^(-48);

K4 = -2.81086*10^(-56);

K5 = -6.34561*10^(-62);

 [x,y] = meshgrid(-12.8:1:12.8,-8.16:1:8.16);

r = sqrt((x.^2)+(y.^2));

dr = (K1.*r.^3)+(K2.*r.^5)+(K3.*r.^7)+(K4.*r.^9)+(K5.*r.^11);

[px,py] = gradient(dr,1,1);

 quiver(x,y,px,py,'r-'), axis image


xlabel ('milimeter');

ylabel ('milimeter');

And I also plot the Gaussian Radial Distortion as figure 3 using Matlab coding. By plotting the x-axis for radius and the y-axis for dr (distortion). The code is:

K1 = 2.35369*10^(-4);

K2 = -1.18567*10^(-6);

K3 = -1.41643*10^(-48);

K4 = -2.81086*10^(-56);

K5 = -6.34561*10^(-62);

[x,y] = meshgrid(-12.8:1:12.8,-8.16:1:8.16);

r = sqrt((x.^2)+(y.^2));

dr = (K1.*r.^3)+(K2.*r.^5)+(K3.*r.^7)+(K4.*r.^9)+(K5.*r.^11); 

Plot (r,dr, 'r-')


xlabel ('milimeter');

ylabel ('micrometer');

And the image is shown bellow:


Figure 6. Gaussian radial distortion (Using Matlab coding)

Decentering distortion

Non-metric camera lens is a combination of several lenses having different center of point. Improper adjustment of the focal point on the combined lens (centering) cause happening tangential distortion is also called decentric distortion. Centering error of the lens from combined lens on non-metric image is illustrated on figure 7:


Figure 7. (Up) perfectly combined lens, and (down) imperfectly combined lens.

Here, is the result from Australis software of Decentering distortion figure 8.


Figure 8. Decentering Distortion Plot

And here I also attach the coding of Matlab below:

P1 = 4.2055*10^(-5);

P2 = -6.0711*10^(-5);

[x,y] = meshgrid(-12.8:1:12.8,-8.16:1:8.16);

r = sqrt((x.^2)+(y.^2));

DD = ((sqrt(P1^2+P2^2)).*r.^2)*1000;

plot (r,Pr, 'g-'); axis image


xlabel ('Radial Distance (r) (in milimeter)');

ylabel ('P(r) (in micrometer)');

For decentering distortion, it uses the formula of  . We could rearrange the formula using Matlab language to be “DD”.

And for the image plotting, can see on figure 9. Here, the figure explains that when the radial distance is enlarged or far from the center of its image, the distortion is concurrently becoming large also.


Figure 9. Decentering Distortion Plot

Computation and program design and validated data.

As we know the images have many distortions which are caused by imperfectly camera lens shape and camera lens arrangement. So, after the distortion in the camera has calibrated, we could know the parameters. The parameters are shown on the table 1.

Tabel 1. Parameters of Calibration

Resolution 4912 x 3264 pixels
Pixel width 0.0050mm Pixel height 0.0050mm
Principal distance c 16.5229mm 0.003mm
Principal point offset in x-image coordinate xp -0.1561mm 0.003mm
Principal point offset in y-image coordinate yp 0.1847mm 0.003mm
3rd-order term of radial distortion correction K1 2.35369e-004 7.1991e-007
5th-order term of radial distortion correction K2 -1.18567e-006 5.5116e-009
7th-order term of radial distortion correction K3 -1.41643e-048 3.0777e-030
Coefficient of decentering distortion P1 4.2055e-005 1.951e-006
Coefficient of decentering distortion P2 -6.0711e-005 8.276e-007
No significant differential scaling present B1 0.0000e+000 3.078e-020
No significant non-orthogonality present B2 0.0000e+000 3.078e-020

xp and yp is the error of principal point in the image coordinate. Its value reveal that the center of the image was different than before calibration procedure. Again, we found the K1, K2 and K3 that state the radial distortion correction due to the camera lens imperfectly shaped. For the quality of the camera lens arrangement, we can know based on the parameters P1 and P2 or called decentering distortion.

Correcting the Image coordinate

I test some points of the data provided in Australis software. See on the table 1. Based on the theory, the corrected image coordinates  and  can be calculated from the measured coordinates  and  by using the formula. However, we need to transform the xp and yp in (mm) to pixel (Cp and Cr) in order to transform the Image Coordinate to Camera coordinate.


X and Y are now respect to the principal points,


%Distortion Parameters

xp = -0.1561*0.005;

yp = 0.1847*0.005;

K1 = 2.35369*10^(-4);

K2 = -1.18567*10^(-6);

K3 = -1.41643*10^(-48);

K4 = -2.81086*10^(-56);

K5 = -6.34561*10^(-62);

P1 = 4.2055*10^(-5);

P2 = -6.0711*10^(-5);

%Coordinate of Points

Cmeas = [962.301;










Rmeas = [2308.571;










%Point respect to the principal point

C = Cmeas - xp;

R = Rmeas - yp;

r = sqrt((C.^2)+(R.^2))*0.005;

dr = (K1.*r.^3)+(K2.*r.^5)+(K3.*r.^7);

%Corrected points

Ccorr = C + (C.*dr./r + P1.*(r.^2 + 2*C.^2) + 2*P2.*C.*R)*0.005

Rcorr = R + (R.*dr./r + P2.*(r.^2 + 2*R.^2) + 2*P1.*C.*R)*0.005

For the result could see on the table 2 C and R corrected.

Table 2. Measured and Corrected Coordinate Points of Image (in pixel)

Point Label Sample Points
CODE15 962.301 2308.571 961.38 2306.4
CODE17 1865.707 593.363 1866.6 593.65
CODE20 1544.668 2956.671 1545.2 2952.9
CODE30 2501.580 10.722 2504.3 10.733
CODE32 1197.849 2719.721 1196.4 2716.5
CODE35 2541.048 2933.785 2538.2 2930.5
CODE44 3274.365 247.508 3278 247.78
CODE52 2198.468 3209.994 2195.3 3205.4
CODE58 1363.712 1352.201 1363.5 1352.4
CODE59 1159.154 2030.220 1158.3 2028.8

Coordinate System Transformation

And we need to transform them (points) to the camera coordinate system. The formula to transform the Camera coordinate to Image coordinate system is shown below.


However, before we obtain the points, we need to know the value of principal point (, ). Because of the image size are 4912 in width and 3264 in height,  and  are a half of them, respectively. The formula can be drawn to the Matlab language.

%Points coordinate of image system

clear all; clc

C = [962.301









1159.154 ];

 R = [2308.571









2030.220 ];

 %Parameter of image coordinate system

ds = 0.005;

Cp = 2456;

Rp = 1632;

 %Transformation of image to camera

Xc = ds*C - ds*Cp

Yc = -ds*R + ds*Rp

 %Backward transformation

Ci = Xc/ds + Cp

Ri = -Yc/ds + Rp

The result is shown in Table 3.

Table 3. Points Coordinates of Image to be transformed to Camera Coordinate System

Point Label Sample Points (Computed in Matlab) Points from Australis Software
CODE15 962.301 2308.571 -7.4685 -3.3829      -7.4685      -3.3829
CODE17 1865.707 593.363 -2.9515 5.1918      -2.9515       5.1932
CODE20 1544.668 2956.671 -4.5567 -6.6045      -4.5567      -6.6234
CODE30 2501.580 10.722 0.2279 8.1063       0.2279       8.1064
CODE32 1197.849 2719.721 -6.2908 -5.4386      -6.2908      -5.4386
CODE35 2541.048 2933.785 0.42524 -6.5089       0.4252      -6.5089
CODE44 3274.365 247.508 4.0918 6.9225       4.0918       6.9225
CODE52 2198.468 3209.994 -1.2877 -7.89      -1.2877      -7.8900
CODE58 1363.712 1352.201 -5.4614 1.399      -5.4614       1.3990
CODE59 1159.154 2030.220 -6.4842 -1.9911      -6.4842      -1.9911

And the result after transforming from Image coordinate to Camera coordinate by Matlab computation, I compare the transformed coordinate, coordinate camera by my own computation, with camera coordinate by Australis. But there are no significant different instead most of the values are the same.

And after successfully transform to camera system. Then, would be done to backward transformation to image system in the same script.

%Backward Transformation

Ci = Xc/ds + Cp;

Ri = -Yc/ds + Rp;

Table 4. Points Camera coordinate to be transformed to Image Coordinate System

Point Label Sample Points (Backward Transformation)
CODE15 -7.4685 -3.3829 962.301 2308.571
CODE17 -2.9515 5.1918 1865.707 593.363
CODE20 -4.5567 -6.6045 1544.668 2956.671
CODE30 0.2279 8.1063 2501.58 10.7220
CODE32 -6.2908 -5.4386 1197.849 2719.721
CODE35 0.42524 -6.5089 2541.048 2933.785
CODE44 4.0918 6.9225 3274.365 247.508
CODE52 -1.2877 -7.89 2198.468 3209.994
CODE58 -5.4614 1.399 1363.712 1352.201
CODE59 -6.4842 -1.9911 1159.154 2030.22

And the result of the backward transformation are back to the initial values.