4. Theoretical overview

In this paragraph, the mathematical model of the central projection and the denition of additional parameters are reviewed. Fig.4.1 shows two image coordinate systems: the pixel coordinate system with the origin in the upper left corner of the image and the metrical image coordinate system with the origin in the center of the image.
nx (0,0) ymetrical xPixel

ny (0,0)

xmetrical

Psx yPixel

Fig.4.1: Pixel and image coordinate system
The transformation from pixel to image coordinates is dened as:
nx x = x' ----- p sx 2 ny y = y' + ----- p sy 2
with: x, y.image coordinates; x, y.pixel coordinates; nx, ny..image size in pixel; psx, psy.pixel space in x and y.
The geometrical model for processing a camera sensor is based on perspective projection and the associated procedure for the adjustment of the image coordinate measurements and the estimation of the parameters is the bundle method. The basis of the bundle adjustment for the calibration are the collinearity equations:
r 11 ( X X 0 ) + r 21 ( Y Y 0 ) + r 31 ( Z Z 0 ) x = x p c --------------------------------------------------------------------------------------------------------------------r 13 ( X X 0 ) + r 23 ( Y Y 0 ) + r 33 ( Z Z 0 ) r 12 ( X X 0 ) + r 22 ( Y Y 0 ) + r 32 ( Z Z 0 ) y = y p c --------------------------------------------------------------------------------------------------------------------r 13 ( X X 0 ) + r 23 ( Y Y 0 ) + r 33 ( Z Z 0 )
with: x, y.image coordinates; xP, yP.principle point coordinates; c..camera constant; X, Y, Z.object coordinates of point; X0, Y0, Z0.object coordinates of projection centre; rik..elements of R, orthogonal rotation matrix dened as:
r 11 r 12 R = r 21 r 22 r 13 r 23 =

r 31 r 32 r 33

cos cos cos sin sin cos sin + sin sin cos cos cos sin sin sen sin cos sin sin cos sin cos sin cos + cos sin sin cos cos
with ,,.angles of rotations of the camera. All the elements dened above are described in Fig. 4.2. The deviations of the physical reality from the ideal imaging geometry of the collinearity condition lead to some systematic errors. These are accounted for by extending the collinearity equations with functions of additional parameters. The extended collinearity equations have the following form:
x x x p = c f + x y y y p = c f + y
x = f (additional parameters) y = f (additional parameters)

Zi y y O X0 z c x Yi

Fig.4.2: Object coordinate system and image coordinate system. PP (xp, yp) is the principal point in the image.
Additional parameters are used in a camera calibration to model: interior orientation, inadequaciens of the sensor, lens distortion, deviations from a at surface, arbitrary deformations. The additional parameter set used in this work is provided by (Brown, 1971), containing three parameters of interior orientation (camera constant c and principle points coordinates xp and yp) and ve parameters describing symmetrical radial and tangential lens distortion (k1, k2, k3, p1, p2). The set was extended for the use in digital photogrammetry by two additional parameters of an afne transformation (scale factor in x Scx and shear fac-
tor A), modelling a horizontal scale factor and a non-orthogonality of the image coordinate axes (El-Hakim, 1986; Beyer, 1987). This set can be expressed as:
2 x x = x p --c xSc x + y A + xr k 1 + xr k 2 + xr k 3 + ( r + 2x )P 1 + 2xyP 2 c 2 y y = y p --c + x A + yr k 1 + yr k 2 + yr k 3 + 2xyP 1 + ( r + 2y )P 2 c

where:

r = x +y

x = x xp y = y yp

x, y.image coordinates; xP, yP.principle point coordinates. This model is a physical model, because all its components can directly be attributed to physical error sources. The individual parameters represent: Scx.scale factor in x (afnity); A.shear; k1, k2,k3.rst three parameters of radial symmetric lens distortion; P1, P2.rst two terms of decentering distortion. and c, xp, yp..change in interior orientation elements. In Appendix B there is further explanation of lens distortion.

5. Software used

For the purpose, these software have been used: SGAP, SGAPITOOL, ITOOL, RUDI, REVIDI, SVIS, DIFFRP, XV. SGAP is a bundle adjustment program for close-range photogrammetry offering batch processing. SGAP accepts, as input, data les in particular format. For each project exists a control le, typically denoted by the extension *.d. This le contains a list of les, each prexed by a code, which contains the different photogrammetric network entity data. The principle les used are listed below: project.par : Adjustment Parameter le project.ic : Image Coordinate le project.ca : Camera Denition le project.st : Station le project.co : Control Point le project.ch : Check Point le project.ob : Object Point approximations le project.ap : Additional Parameter le SGAPITOOL creates the starting le for itool, using an output le of sgap. REVIDI and ITOOL have been used for the determination of the tie points in the images using the least squares matching. RUDI has been used for the rst rough exterior orientation of the images, for the rst determination of few tie points with the testeld and for visualizations of the residuals. SVIS has been used for the visualization of the residuals of the points in the image space. DIFFRP has been used for the visualization of the residuals of the points in the object space.

6. Preliminary calculations

Given (Fig.6.1):

dmonitor = 0.5 inch number of sensor elements in horizontal direction = 1600 number of sensor elements in vertical direction = 1200

a=1600 pixel

dmonitor b=1200 pixel

dpixel

Fig6.1: Monitor and its elements
its possible to calculate: d monitor = ( a + b ) = 2000 pixel
Since 1inch = 25.4001 mm dpixel = 0.00635 mm and supposing square pixel: pixel size = 0.00449 mm = 4.49 m Here the starting values used as approximation for the calibration: image dimension: 1600x1200 camera constant: 7.1 mm principal point in x: 800 principal point in y: 600 pixel dimension in x and y: 4.49 micron
7. Tests for calibration and accuracy potential
A collection of images of a 3D testeld from a variety of viewpoints is necessary. All these images are then processed to measure the image coordinates (xi, yi) of common points (Xi,Yi,Zi) in each image. Then using Gauss-Markov model, the sum of the squares of the residuals of these observations are minimized by adjustment of some parameters, as shows in Fig. 7.1. The residuals are computed as the differences between the given coordinates of some points and the calculated coordinates of the points.
( x i, y i ) = f ( X 0, Y 0, Z 0, c, , , , x p, y p, p 1, p 2, k 1, k 2, k 3, Sc x, A, X i, Y i, Z i )

measurements

approximations which are adjusted
Fig.7.1: Parameters adjusted for the selfcalibration
with xi, yi.image coordinates; Xi,Yi,Zi.object coordinates; ,,.angles of rotations of the camera; k1, k2, k3 , p1 , p2.parameters of radial and decentering distortion; c.camera constant; xp, yp.principal point coordinates; X0, Y0, Z0.object coordinates of projection center. For the testeld used in the calibration, 3D coordinates of references points have to be known. These signalized points had been determinated with high accuracy using a theodolite system. These coordinates are taken as reference for all the tests. The reference eld used are: - Testeld nr.1 (Fig.7.2): located in room HILC57 of the ETH - Institute of Photogrammetry and Remote Sensing, it consists of targets placed on a wall and on an alluminium structure in front of the wall. The testeld spanned 4.2 m horizontaly, 2 m vertically and the targets on the front rods are 1 m from the wall. The coordinate system is located with the X axis parallel to the oor, the Y axis parallel to the wall and the Z axis pointed away from the wall toward the center of the room. There are 4 different types of targets: coded target on a black background with a small white dot in the centre, a with symbol and a white circle near the border (Fig.7.2 - A) black circles on a white background with a small white dot in the centre (Fig.7.2 - B) black circles on a white background with two different diameters (14 mm and 20 mm) (Fig.7.2 - C)
There are nearly 250 targets. The 20 coded targets and the 77 black targets with a diameter of 20 mm have been measured using a theodolite system. The coded targets have less accuracy than the black targets; the average precision of the measurements (sigma a posteriori) of the points of types B and C is 0.04 mm. The coded targets have been used only to get the rst approximations; all the other measured targets have been used as reference in the tests.

Fig.7.2: Camera calibration testel in the room HIL C57 with its targets
- Testeld nr.2 (Fig.7.3): it is a testeld with 100 coded targets. The targets have a diameter of 15 mm. The dimensions of the eld are 20x35x15 cm. The coordinate system is located with the X-Y axis parallel to the base and the Z axis pointed away from base.
Fig.7.3: Testeld nr.2 and coded targets
The two reference elds have been used in two different modes: 1. as explicit control: the known coordinates of the signalized points are used as control points and spatial resections were performed from each image. 2. as free net: a self-calibrating bundle adjustment is performed as a free network adjustment, including the 3D coordinates of the signalized points as unknown into the adjustment process. Three different image sets have been processed: Case1: with testeld nr.1; Case2: with testeld nr.1 and some perturbations of the camera; Case3: with testeld nr.2. A further test was performed to investigate the stability of the camera.

7.1 Case 1 Testeld

Testeld nr.1 was used.

Image acquisition

Images were acquired from 5 different camera positions: - in the left part of the testeld at two different heights; - in the right part at two different heights; - one centrally. To avoid correlations between parameters of interior orientation, exterior orienatation and object points coordinates, the camera was rotated in three different positions: unrotated, rotated left (+90), rotated right (-90). Totally 15 images were acquired. In Fig.7.4 are show the positions of camera.

left high

right high

-90 +90 +90 -90

-90 left low middle middle

right low

Fig.7.4: Test conguration in case 1
Two types of calibration were performed : 1. with external constraints using 25 control points; 2. a free net, without control points.

Procedure

For a processing with SGAP the following input les have to be created. stations le: every image represents a station; the coordinates of the projection center and the values of the 3 angles of rotation af the camera (exterior orientation) are necessary. A rst rough exterior orientation, adjusted with RUDI looking at the positions of coded targets projected in the image have been taken.

image coordinates le: for every image, its necessary to measure some points (tie points). Image coordinates of the coded targets (Fig.7.5) were measured manually with RUDI.
Fig.7.5: Coded target used as rst tie points
control points le: in this rst part of the orientation and calibration, only the coordinates of the coded targets were used. additional parameters for the calibration: were set to 0.0 in this rst step. camera le: it contains the denitions of the sensor; values from the reference manual (camera costant and principal point) and as in paragraph 6 (pixel size in x and y) were used. object points: in this rst part no object points were used. The program will compute a rst approximation of the object coordinates. After this editing work, its possible to run SGAP and get a more precise station and camera le and a rst set of object points. The image coordinates of the coded targets were taken with low accuracy (the center of the target in the image has been measeured manually). To get a more precise image coordinate le, ITOOL can be used: the coordinates of points from former calculations are back projected onto the images using the rst rough orientation of each station. The program will measure the image coordinates of the new points using least squares matching. The image coordinates le can be so updated with the more accurate data. Now the process continues in two different ways.
Case1A: 25 control points as external contraints
This case uses control points in the bundle adjustment: a control point is a contraint for the adjustment; the image coordinates and the 3D coordinates have to be provided. A set of 77 points (targets) on the testeld has been used: a part of these points as control points (25) and the others as check points (52). These check points are needed to calculate the RMS of object points differences. The data of the calabration are shown in Table 7.1
Testeld Number of images Contraints Number of Control Points Number of Check Points nr.external (CP) 25 52
Table 7.1: Data of the calibration
With the new set of les, SGAP creates more accurate positions of the cameras, their parameters, new object points coordinates and a rst set of additional parameters. Now
these results can be used to update the set of les (and run SGAP again) and with a better orientation it is possible to measure automatically new image coordinates of points using ITOOL again (and update the image coordinates le). The results of this process must be checked: infact, due to an unprecise exterior orientation of the image, its possible that some points are mismatched with others. Fig.7.6 shows a mismatched point. In the case of a right back projection, the point 4041 should stay on the target behind the slab (as in Fig.7.6.A); instead due to imprecise exterior orientation, the point has been matched in a wrong position (Fig.7.6.B). Also the big residual (visualized in RUDI with a long line) helps to recognize incorrect matching.

Fig.7.6: Mismatching of points. In A the two points are well recognized. In B there is a mistake.
With RUDI its possible to visualize all the object points and their respective image coordinates. This is useful for the elimination of large errors. After some iteration steps, all the additional parameters can be determinated. The results of the calibration are showed in Tables nr. 7.2 and 7.3.
Camera Constant [mm] Principal Point X [mm] Principal Point Y [mm] 1 - Scale in x Shear K1 K2 K3 P1 P2
8.423 0.022 -0.185 4.454e-05 5.918e-05 -3.122e-03 2.361e-05 1.544e-06 -2.980e-04 8.082e-05
Table 7.2: Camera parameters
X [mm] Y [mm] XY [mm] Z [mm] a posteriori of dimension x [m] dimension y [m] average dimension xy [m] dimension z [m]
0.109 0.094 0.101 0.208 unit weigh [mm] 4.3.1 5
RMSX [mm] RMSY [mm] RMSXY [mm] RMSZ [mm] 0.00043 Relative accuracy in X Relative accuracy in Y Relative accuracy in X-Y Relative accuracy in Z

0.140 0.099 0.121 0.428

1:30000 1:20000 1:25000 1:12000
Table 7.3: Theoretical () and empirical (RMS) precision (view Appendix C) of the test
Case1B: free net In this case no control point is used and all points (measured with the theodolite) are introduced as check points. No contrains to the point is given: the program should adjust all the positions af the camera, all the parameters and should compute the object coordinates in the best way. This case should give the best results. In Table 7.4 are shown the data of this calibration.
Testeld Number of images Contraints Number of Control Points Number of Check Points HIL Cinternal (free net) 77
Table 7.4: Data of the calibration
The procedure is the same as described before: compute new values of the unknowns with SGAP, update the set of les (camera, stations, additional parameters, object points), nd new image points with ITOOL (back projection), check for accidental mistakes and remove them (RUDI). In the following Tables (7.5 and 7.6) all the results are summarized.
Camera Constant [mm] Principal Point X [mm] Principal Point Y [mm] 1 - Scale in x 8.421 0.022 -0.185 6.681e-05

Shear K1 K2 K3 P1 P2

5.825e-05 -3.147e-03 2.767e-05 1.348e-06 -2.916e-04 8.098e-05
Table 7.5: Camera parameters
0.101 0.085 0.093 0.190 unit weight [mm] 4.3.1 5
RMSX [mm] RMSY [mm] RMSXY [mm] RMSZ [mm] 0.00040 Relative accuracy in X Relative accuracy in Y Relative accuracy in X-Y Relative accuracy in Z

0.244 0.143 0.200 0.528

1:18000 1:15000 1:16000 1:10000
Table 7.6: Theoretical () and empirical (RMS) precision (view Appendix C) of the test
A comparison between the two different modes of calibration shows that the values of the camera parameters resulting from the free net adjustment and the adjustment with control points do not differ signicantly and can be considered as identical. Instead the RMS of the free net are bigger than the rst case: this is an unexpected effect. In the next chapter a thorougher analyse of the result will be presented.

7.2 Case 2 Testeld

The testeld used is the same of case1 (testeld nr.1).
Again 15 images are acquired, more or less from the same positions of case 1. This time (15 days later than the rst calibration), two perturbation to the camera have been introduced: 1. zoom in and zoom out before every capture; 2. turn off and turn on the camera between two different camera positions. These perturbations have been done in order to investigate the stability of the parameters of the camera. Again two adjustment have been performed: 1. with external constrains using 25 control points; 2. with internal constrains (free net), without control points.
The image data sets were processed in the same way as case 1, for both adjustments. In the following Tables (7.7, 7.8 and 7.9), the results of both cases are showed.
CASE 2A Testeld Number of images Constraints Number of Control Points Number of Ceck Points HIL Cexternal (CP) 25 52
CASE 2B HIL Cinternal (free net) 0 77
Table 7.7: Data of the calibration
Camera Constant [mm] Principal Point X [mm] Principal Point Y [mm] 1 - Scale in x Shear K1
8.424 0.020 -0.185 < 1e-07 7.457e-05 -3.151e-03
8.423 0.021 -0.184 2.227e-05 7.125e-05 -3.171e-03

K2 K3 P1 P2

3.279e-05 8.874e-07 -3.079e-04 7.632e-05
3.365e-05 9.819e-07 -2.936e-04 8.019e-05
Table 7.8: Camera parameters
CASE 2A X [mm] RMSX [mm] Y [mm] RMSY [mm] XY [mm] RMSXY [mm] Z [mm] RMSZ [mm] a posteriori [mm] Relative accuracy in X Relative accuracy in Y Relative accuracy in X-Y Relative accuracy in Z 0.105 0.150 0.099 0.104 0.102 0.129 0.185 0.403 0.00043 1:30000 1:20000 1:24000 1:12000
CASE 2B 0.090 0.240 0.077 0.116 0.084 0.189 0.158 0.480 0.00038 1:18000 1:18000 1:16000 1:10000
Table 7.9: Theoretical () and empirical (RMS) precision (view Appendix C)
Also in this case the camera parameters of the camera do not differ significantly. So the perturbations seems not to influence the stability of the camera. Instead the RMS of the free net are still bigger than the other case.

7.3 Case3 Testeld

Testeld nr.2 has been used.
A third set of calibration images was acquired. To achive good illumination conditions the images have been taken with the ash. The set is composed of: - 4 images above the testeld with four different rotations angles (0, +90, +180, -90); - 4 images from every corner of the testeld; - 4 images from every border of the testeld. The chosen camera arrangement with 12 camera positions and the rotated images should allow a reliable determination of the camera parameters.
Again the calibration (bundle adjustment) is realized with SGAP. The following input data has to be created: stations le: approximative values of the 6 parameters of exterior orientation, which can be rened with RUDI. image coordinates le: for every image, image coordiFig.7.7: different nates of the targets are measured. For the purpose REVtypes of targets in the used testeld IDI has been used: the program recognizes every coded target on the testeld, assignes it its proper code-number and image coordinates of the recognized targets are computed with least squares matching. control points le: no control points were used, due to the fact that accurated 3D coordinates of the coded points are not avaible. Only a free net calibartion has been performed. check points le: as check points all the coded targets measured in other works (measurement with Kodak DCS200 + two scale bares) have been used. The coordinates of these points come from a former photogrammetric measurement and their precision cannot be compared with a theodolite measurement; for this reason, the computed RMS are not very indicative of the precision of the measurement. additional parameters for the calibration: all values were set to 0.0. camera le: the starting value are the ones found in the reference manual (camera costant and principal point) and the one derived from the preliminary calculations (pixel size). The data were processed in the same manner as before. All the results of this calibration are shown in Tables 7.10, 7.11 and 7.12.
Testeld Number of images Contraints Number of Control Points Number of Ceck Points
Box 12 internal (free net) 0 86
Table 7.10: Data of the calibration
8.424 0.030 -0.188 < 1e-07 9.207e-05 -3.087e-03 1.958e-05 9.663e-07 -2.784e-04 3.326e-05
Table 7.11: Camera parameters
X [mm] Y [mm] XY [mm] Z [mm]

0.006 0.006 0.006 0.010

4.454e-05 5.918e-05 -3.122e-03 2.361e-05 1.544e-06 -2.980e-04 8.082e-05
1.2e-05 6.3e-06 2.1e-05 3.6e-06 1.8e-07 5.5e-06 5.3e-06
Table 8.2: Values of the additional parameters and their standard deviation in case 1A used as reference
If an additional parameteters belongs to the respective range can be consider stable for the camera. Additional parameters k2, k3, and Shear show the largest differences and they are not within the given range. For case 3, the causes of the differences could be the closeness of the testled and in the use of the ash; instead in case 2 the perturbations of the camera might have produce these differences. But, as it can be seen in Appendix D - Fig.1, 2, 3 - for case 1A, 2A and 3, the APs inuence the global distortion with the same behavior.

- Accuracy results

In general, the accuracy of the camera can be considered quite high. In all tested congurations, the standard deviation of unit weight is smaller than 1/10 pixel (Table 8.3).
CASE 1A a posteriori [m] 0.43 CASE 1B 0.40 CASE 2A 0.43 CASE 2B 0.38 CASE 3 0.21
Table 8.3: Sigma a posteriori of unit weight for all tested congurations
The comparison with the reference coordinates of the testeld is consistent only for the rst two cases. The inconsistency in the third cases is due to the not high precision of the reference coordinates (measurement with Kodak DCS200 + two scale bares).
CASE 1A X [mm] RMSX [mm] Y [mm] RMSY [mm] XY [mm] RMSXY [mm] Z [mm] RMSZ [mm] Relative accuracy in X Relative accuracy in Y Relative accuracy in X-Y Relative accuracy in Z 0.109 0.140 0.094 0.099 0.101 0.121 0.208 0.428 1:30000 1:20000 1:25000 1:12000 CASE 1B 0.101 0.244 0.085 0.143 0.093 0.200 0.190 0.528 1:18000 1:15000 1:16000 1:10000 CASE 2A 0.105 0.150 0.099 0.104 0.102 0.129 0.185 0.403 1:30000 1:20000 1:25000 1:12000 CASE 2B 0.090 0.240 0.077 0.116 0.084 0.189 0.158 0.480 1:18000 1:18000 1:16000 1:10000 0.010 0.006 0.006 CASE 3 0.006
Table 8.4: Theoretical and Empirical precision for all tested congurations
The results of the rst two cases must also be handled with care: observing the residuals in the object space, a systematic error can be observed in all the cases (Fig.8.1 and 8.2). Instead in the image space no systematic behavior of the residuals can be recognized (Fig.8.3 and 8.4).
Fig. 8.1: Case of calibration with 25 control points: residual in object space (for case 1A and 2A), between the object points and the check points. A systematic error is quite clear for the points near the border. (Top: case 1A; bottom: case 2A)
Fig. 8.2: Case of calibration in free net: residual in object space (for case 1B and 2B), between the object points and the check points. Its easy to see the systematic error on the border of the images. (Top: case 1B; bottom: case 2B)
Fig. 8.3: Residuals in image space (for case 1A and 1B). Only the central unrotated image is considered. It seems that no systematic error is present in the images as in the object space. (Top: case 1A; bottom: case 1B)

Fig. 8.4: Residuals in image space (for case 2A and 2B). Only the central unrotated image is considered. It would seem that no systematic error is present in the images as in the object space. (Top: case 2A; bottom: case 2B)
This behavior in the object space draws to conclude that the systematic errors could derive from the instability of testeld. All points on the wall (left and right part of the images) show the same behavior, while the residuals of the points on the testeld (cetral part) show different behavior (g. 8.5). The RMS of the free net cases show also unexpected large values; this fact suggest again to think a variation of the reference coordinates of the testled. Therefore also in these two cases, the RMS values of object points must be interpreted and considered with care due to possible insecure coordinates of the testeld. For all these reasons, the suitability of the camera for photogrammetric purpose cannot be drawn and its accuracy potential cannot be estimated exactly.
Fig. 8.5: Residuals in object space overlapped on the teseld. Scale: 1 cm (in the image): 0.15 mm (in the object space). The back projections of the errors do not correspond exactly in the image because the residuals are plotted without considering the z coordinate. So points that should stay on the slabs are on the wall.
9. The inuence of camera rotation on the additional parameters
In the precedent calibration, a constant value of the interior orientation and of the additional parameters in all the images had been assumed. But as the images have been taken rotating the camera, its quite interesting consider three different groups of camera: 1. all the images rotated left; 2. all the images rotated right; 3. all the images unrotated. Three independent set of additional parameters and three camera les are dened. The goal is to see if the parameters have the same value as the cases before. Procedure and results All the data of case 1 have been considered and the bundle adjustment (SGAP) has been performed with 25 control points. So a comparison with the results of case 1A must be done. The results are listed in Table 9.1.
group LEFT Number of images Camera Constant [mm] Principal Point X [mm] Principal Point Y [mm] 1 - Scale in x Shear K1 K2 K3 P1 P8.423 0.025 -0.188 -6.681e-5 8.269e-05 -3.140e-03 2.439e-05 1.407e-06 -2.646e-04 6.557e-05 group RIGHT 5 8.424 0.017 -0.181 <1e-7 8.377e-05 -3.122e-03 1.997e-05 2.041e-06 -3.103e-04 1.057e-04 group UNROTATED 5 8.421 0.025 -0.187 6.668e-5 4.776e-05 -3.144e-03 2.828e-05 1.243e-06 -3.055e-04 6.634e-05 case1A 15 8.423 0.022 -0.185 4.454e-05 5.918e-05 -3.122e-03 2.361e-05 1.544e-06 -2.980e-04 8.082e-05
Table 9.1: Comparison of the parameters with case 1A
Parameters K3 and P2 show the largest differences; also the principal point changes its location. Although this the values do not differ signicantly and can be considered as identical in both cases. The sensor properties remain unaffected to rotations.

10. Conclusions

A number of practical tests with different sets of images and testelds have been done. The calibration of the camera and its results have shown that the additional parameters of the camera to model the lens distortion can be considered well determined and the sensor is stable. Instead the accuracy potential cannot be well derived because of uncertain stability of the testeld and so the results are limited interpretable. A new measurement of the reference points to investigate if the coordinates are changed might be done.
APPENDIX A Camera SONY DSC-F505 CYBERSHOT Highlights
Sensor Resolution Image dimensions Monitor type Lens Type Lens - Focal Length (35mm equivalents) Lens - Aperture Range Zoom Filter Threads Exposure Focus Storage Image Capacity File Format Camera Size Usable Battery Types 2,022,096 pixel, 1/2 inch 1600x1200, 1024x768, 640x480 2" Hybrid LCD Glass, Carl Zeiss Vario-Sonnar 38-190mm 2.8-f/3.3 to f/8.0 5X Optical Zoom Lens 10X Precision Digital Zoom 52mm TTL, Automatic TTL Automatic, or manual with focus indicator Removable Memory Stick, 4 up to 64 MB 5-50 JPEG 4.25 x 2.5 x 5.4 inches (107.2 x 62.2 x 135.9 mm) InfoLITHIUM(tm) only
Table 1: Feautures of the Sony DSC-F505 Cybershot
APPENDIX B Lens distortion
All the camera systems whether metric, non-metric, or semi-metric do not possess a perfect lens system. They have measurable distortions and other optical defects (aberrations). The aberrations are spherical, chromatic, coma, astigmatism and curvature of eld. Aberrations degrade the quality or sharpness of the image. Instead lens distortion causes the displacement of the image. The distortion may be radial and tangential (Fig.1). Thus, in order to achieve high measurement accuracy, the lens distortions have to be corrected.
DISTORTION symmetric radial decentering tangential

ansymmetric radial

Fig1: Lens distortion
Lens distortion can be computed as: D = r + t

where (g.2):

t = tangential distortion and r = r + r is the radial distortion composed of a symmetric part (r) and of ansymmetric part (r). r + t is called decentering.
Fig.2: Two components of distortion
The symmetric part of the radial distortion can be modelled using a polynomial expression as: r = f(r) = a1r + a2r + a3r +. with r = radial distance. The decentering distortion can be modelled as (Conrady): r = 3f(r) sin ( - 0) t = f(r) cos ( - 0) with f(r) = b1r2 + b2r4 + b3r6 +. = angle of the point in the image 0 = angle where the tangential distortion has the maximum value
Fig.3: Decentering distortion
The maximum value of tangential distortion is in = 0. The ansymmetric radial distortion has its maximum in = 45.
APPENDIX C Theoretical and Empirical precision of object coordinates and Relative accuracy
Theoretical precision Its the average precision of the object coordinates. xi ------------nx

-------------ny

x + y ---------------------2

------------nz

with nx, ny, nz = number of points coordinates in x,y and z

x = 0 q xx i i

standard deviation of object space coordinates
with qxxi = diagonal element of the inverse of the normal equation matrix at the position of the corresponding unknown and:

0 = T v Pv -----------r

standard deviation a posteriori of the unit weight a posteriori
with: v = residual of the observations
Empirical precision (o RMS) Its computed from the comparison of the check point coordinates with the calculated coordinates of the object points. Its the empirical accuracy in object space. x i

---------------nx 2

2 xy =

---------------ny

x + y -----------------2

---------------nz

with: x = xrif - xcalc y = yrif - ycalc z = zrif - zcalc nx, ny, nz = number of check points coordinates and xrif, yrif, zrif = reference coordinates of check points The empirical precision gives the real accuracy of the measurement. If the points of referent are not well accurated, this value is inconsistent.
Relative accuracy Its calculated as: RAx = dimensionx / RMSx RAy = dimensiony / RMSy RAz = dimensionz / RMSz
APPENDIX D Global distortion in the image
Using the model equations and the values of the additional parameters, it is possible to visualized the global distortion of the image due the systematic errors of the camera. In the next images case 1A (Fig.1), case 2A (Fig.2) and case 3 (Fig.3) are showed.
Fig.1: Global distortion in case 1A:
1 - Scale in x Shear K1 K2 K3 P1 P2 4.454e-05 5.918e-05 -3.122e-03 2.361e-05 1.544e-06 -2.980e-04 8.082e-05
Fig.2: Global distortion in case 2A:
1 - Scale in x Shear K1 K2 K3 P1 P2 < 1e-07 7.457e-05 -3.151e-03 3.279e-05 8.874e-07 -3.079e-04 7.632e-05
Fig.3: Global distortion in case 3:
1 - Scale in x Shear K1 K2 K3 P1 P2 < 1e-07 9.207e-05 -3.087e-03 1.958e-05 9.663e-07 -2.784e-04 3.326e-05

References

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