USER MANUAL

alibration

1.  CALIBRATIONS

1.1. Beginning with the program

1.2. Predefined configurations

1.3. Systems of units

Symbols of units

1.4. Input files

Photograph file

Control file

File extensions

1.5. Output files

Calibration information file

Configuration

Adjusted values and precisions

Importance of each distortion parameter

Photograph half diagonal

Residuals

Inner orientation file

Graphic file

1.6. Parametres to compute

Measured coordinates --> photo coordinates

Outer orientation

Focal length and principal point

Approximate/Known

Colour of the text boxes

1.7. Function of distortion

Decomposition

Different models

1.8. Modus operandi

2.  TOOLS

2.1. Saving and opening configurations

3.  TECHNICAL SUBJECTS

3.1. Method of adjustment

3.2. Rotation matrices

3.3. Dependencies among parameters

Transformation from measured coordinates to photo coordinates

Focal length

Ω, Φ and the principal point

3.4. Relation among the parameters α and β of the affinity and the scale ratio between axis and the angle contained by them

3.5. Formulation of each model

Polynomial models

Odd

Complete

Asymmetric models

Radial/tangential

Rotating vector

3.6. Graphs

Polynomial models

Complete

Odd

Asymmetric components

Radial/tangential

Rotating vector


1.  CALIBRATIONS

1.1. Beginning with the program

When Calibration is opened the program main window appears. There is a place to write the photograph file and a button next to it: . Click on that button and select the file imagen_px.ftm, in the folder Ejemplo_Calibracion. Proceed in the same manner for the control file and select coordenadas.prm. Near the top left corner there is a box displaying the text “Photo coordinates”. Click over there and select the configuration “Pixels / mm”. Right then the programs asks the user to input the pixel size.

Write 0.0078. After pressing the OK button it can be seen that , which was previously unchecked, has been checked on. Everything is ready now and by pressing the button the calibration is carried out.

Several result files will be generated as well as a graphic file, which can be viewed by clicking on the central and right buttons at the top: .

1.2. Predefined configurations

These are the ones offered at the listbox at the top left corner of the program window. In this example, the default configuration «Photo coordinates» was changed to «Pixels / mm». The firt one is used when the coordinates of the photograph file are expressed in milimeters or microns; i. e., when the transformation from pixel to photo coordinates has already been performed. The second one and those which follow it in the list are the ones suitable for a photograph file with the coordinates of the points within it written in pixels. In this case it is necessary to carry out a first transformation so that they are converted into photo coordinates, even if it is just to revert the orientation of the y axis. This transformation may also include, if it is desired, a change of units for transforming the pixels into milimeters, microns, inches or whatever system is chosen. It is therefore that the program asks for the pixel size when either the configuration «Pixels / mm» or «Pixels / microns» is selected.

1.3. Systems of units

Calibration ignores the units used by the user. It simply supposes everything is expressed in the same system. There are three systems of units: The measuring system (usually pixels), the photo coordinates one, and the terrain or ground one. All the quantities within each system must be expressed in the same units. The three systems are independent from each other.

If for example the photo coordinate uint system is millimetres, the photograph half diagonal inferred by the program will be expressed in millimetres, as well as the computed focal length and distortion values.

The measuring and the photo coordinate system bear a relation that has to be known, and is the value specified in the textbox “Scale” within the section “Measured coordinates --> Photo coordinates”. The principal point location can be computed either in the measuring units, either in the photo coordinate units. As for the former, the values “x translation” and “y translation”, within the section “Measured coordinates --> Photo coordinates”, shoud be selected for the adjustment, while on the other hand the values of “Principal point, x” and “Principal point, y”, within the section “Focal length and principal point”, should be left unchecked and with known value 0. This is the program's default option in the modes where the measured coordinates are pixels (“Pixel / mm” and “Pixel / micron”). For the second option, select these for parameters in exactly the opposite way.

The pixel size cannot be subject to calibration. It is chosen ad libitum, and according to the specified value thus will the result units (the photo coordinate system) be. If in the previous adjustment a value of 7.8 would have been written, the calibration results would have been expressed in microns.

When the measuring for the calibration is performed by some program and it is this program which generates the files, it usually applies a transformation from measured coordinates to photo coordinates, writing millimeters or microns to the file instead of pixels. The mode “Photo coordinates” should be used in these cases. This is the default mode, the one which is selected at program strart up. In this mode there is no transformations from measured coordinates to photo coordinates, i.e., the values of the parameters of this transformation equal 0 or 1 according to the parameter, and only two systems get involved in the calibration (photo coordinates and control points) instead of three.

If the measures are expresses in pixels the transformation «Measured coordinates --> photo coordinates» shoud be selected. By default the program supposes that the y coordinate grows downwards: ; it also supposes that pixels measure the same (approximately) horizontally and vertically: . If this is not the case and the x dimension of pixels is for instance double than the y dimensions, this would be told to the program thus: .

If metres are used for ground coordinates, so will be in metres the coordinates computed for the protection centre.

There is a fourth system of units: that of the angles. This one must be known to the program. It is chosen in the “Configuration” menu:

All the values referring tho angles must be in the selected system.

Simbols of units

Just for the purpose of output files, the simbols for the measuring, fiducial and ground units can be specified. Thus the number will be displayed followed by their unit. To do so, click at Configuration--> Output information--> Unit symbols:

The only difference between the default configurations Pixels / mm and Pixels / microns is the symbol of the fiducial system units.

1.4. Input files

Photograph file

Everything that appears before the string “-ff” that marks the beginning of the first photograph is ignored, so that any information can be written there. The names of photographs and points are literal strings that may contain letters or numbers. The format is free, provided the order of the fields is not altered.

The marks indicate whether the point enters the calculation or not. Thus, points can be eliminated from the computation without taking them out of the file. If the photograph is marked with 0, and so eliminated, the program continues reading the file till it finds a photograph marked with 1, which will become the calibrated photograph.

The marks are not necessary but must either be present in all photographs and points or not present at all. That is, the file may be marked or not marked, but not a mixtrure of both.

The points must always have both coordinates marked 1. If any of them is eliminated then the whole point is not included in the adjustment.


-ff      9052    153.668     1    
Normal photograph
-ff      9051    153.668     0    
Eliminated photograph
  
4143      69.09     13.079    11   
Normal point
4153     81.786    -66.747    01   
Eliminated point
5131     10.567     28.589    10   
Eliminated point
5141    -12.154     -100.7    00   
Eliminated point

If the input files aren’t marked or have PATB format the corresponding marked file is automatically generated.

Control file

The files can be marked in two different ways. Marked prm:

The points marked with 0 won't be calculated.

And marked pym:

File extensions

The extensions are not obligatory, but it is convenient to use them, for in that case Calibration recognises the files when they are selected pressing the button .

Photographs files
        not marked: .fot
             marked: .ftm
                 PATB: .f

Control files
       not marked: .apr, .apy
      marked prm: .prm, .ajs
      marked pym: .pym

If you wish to use a file with a different extension, after pressing the button it has to be chosen “All files (*.*)”. Right after, it has to be manually indicated the file type, clicking with the mouse at the place where that information is shown, and selecting the format wanted:


The format «Aerotri» of the phtograph file includes both marked and unmarked photographs files.

1.5. Output files

The program Calibration generates a few output files. The most important of them is the information file, with extension .inf, which is summarised in the pdf report sheet. A graphic file is also created, .gra, and a file containing the inner orientation data, .int, that can be read by Digi or other programs.

File of information of the calibration

At the beginning there is a header with the name of the input files and the number of points. Right after it it comes the configuration, that is composed of all the parameters that define the adjustment. When a parameter to be computed appears followed by a value, this is the approximate value indicated at the program window (or, for the focal length, in the file).

/------------------------------------------------------------\
|                       CONFIGURATION                        |
\------------------------------------------------------------/

  **Known values

     Measured coordinates --> Photo coordinates

        rotation      0
        esc       0.0078 mm/px
        esc x/y   1
        delta     0

        Inner orientation

           xp      0
           yp      0


  **Parametres to compute and approximate values

     Outer orientation

           X
           Y
           Z
           W       0
           PHI     0
           K

     Inner orientation

           f

     Function of distortion

        Computed for a maximum value of r of 12.5 mm (semidiagonal)

          Polynomial model: complete

                      s=r/12.5      cosA=x/r , sinA=y/r

          Symmetric, radial: 

            Dr= a2(3s^2-2s) + a3(9s^3-11.4s^2+3.4s) + a4(29.2s^4 -53.1s^3 +30.1s^2 -5.2s)

            Condition: Orthogonality

          Symmetric, tangential: 

            Dt= b2(3s^2-2s) + b3(9s^3-11.4s^2+3.4s)

          Asymmetric distortions.  Model: radial / tangential

            Dr= c3(4s^3-3s^2)cosA + c4(4s^3-3s^2)senA + c5·s·cos2A + c6·s·sen2A
            Dt= d1·s^2·cosA + d2·s^2·senA + d5·s·cos2A + d6·s·sen2A

After the configuration there come the adjusted values and their precisions for all the computed parameters. Each parameter of distortion is also followed by a value of importance, which is the mean quadratic distortion due to that parameter throughtout the phtograph.

For the function the formulas are displayed as a function of s, which is the value of is r divided by its maximum possible value. this value is compted by the program and it refers to it as the photograp half diagonal. For the example data its value is 12.5mm. s thus varies form 0 to 1.

Following the adjusted values are the residuals, and the estimated a posteriori standard deviation expressed in the units of the mesuring system. If there is no transformation from measured coordinates to photo coordinates it will coincide with the system of the later.

Inner orientation file

This file includes all the adjusted values in a format suitable for subsequent reading by Digi, provided the plug-in CamaraCalibra has been installed. Here is a complete example:

\begin Info
minx 	-23.693
maxx 	23.646
miny 	-13.566
maxy 	12.689
\end

\begin Orientacion interna media

f 	19.38626
xp 	0
yp 	0

\end

\begin Coordenadas medidas --> fotocoordenadas

Tx 	1491
Ty 	1007
a 	0.007800
b 	0.000000
c 	0.000000
d 	0.007800

\end

\begin Funcion de distorsion

semidiag 	12.5
Modelo polinomico 	Completo
Modelo asimetrico 	rad/tan

\begin Radial simetrica
a2 	-0.211513
a3 	-0.0282136
a4 	0.0191263
\end

\begin Tangencial simetrica
b2 	-0.000305508
b3 	0.000763013
\end

\begin Asimetrica serie1
c5 	-0.000716259
c6 	-0.000744935
\end

\begin Asimetrica serie2
d1 	0.00213269
d2 	0.000487395
d5 	-0.000347419
d6 	0.000841181
\end

\end Funcion de distorsion

The distortion components that correspond to each parameter are developed in Formulation of each model, as well as in the explanatory windows from each set of parameters. The transformation from measured coordinates to photo coordinates is the first transformation applied. The parameters Tx,Ty,a,b,c,d are those of an affine transformation:

The relation that exists among the parameters a,b,c,d and those that can be selected by the user at the program window is a tecnical subject which is explained at the end of this manual.

Graphic file

It includes a representation of the distortions as well as the residuals. both in the fiducial system (the photo coodinates one). The distortion is shown as a set of segments corresponding to points laying in a rectangular grid. The number of points in each grid axis can be chosen in Configuration--> Output information--> Graphic.

1.6. Parametres to compute

A calibration/orientation consists of the computation of a set of parameters selected from the collection of all possible parameters. These are grouped into three sets: measured coordinates --> photo coordinates, outer orientation and inner orientation or image geometry. This last group includes the parameters of distortion.

The first set of parameters is interpreted in different ways acording to the nature of the image: digital image or scanned analogic image.

Measured coordinates --> photo coordinates

These are the parameters that transform from the measuring system (comparator system or pixel coordinates) to the fiducial system. If the photograph was scanned the measuring system is the scanner, which assigns (x,y) pixel coordinates to each point. In this case this set of parameters is to be used when that engine is to be calibrated instead of the camera.

The parameters are the ones corresponding to an affine transformation. If the measuring engine were perfect there should only exist the first three, that correspond to a movement and parametrise the position of the measuring axes with respect to the fiducial system. Therefore the last three are distortions.

Even if you have digiatal images it has to be noted that the measuring software may already apply a transformation of this kind, generating files with coordinates expressed int millimeters or microns (or som other unit) as if it had been an analogic image. Therefore these parameters should not be selected, for if thew were we would be applying twice a transformation from measured coordinates to photo coordinates.

If on the other hand the image is directly a digital one, the most common practise is to use this transformation in order to transform from pixels to millimeters or microns and to place the origin of the coordinate system at the principal point:

In order to do so the pixel size is specified as the value of the scale parameter and the translation parameters are selected for the adjustment. These parameters are applied before the scale factor is, and therefore they will be expressed in the units of the measuring system, i.e., pixels. After applying the translation, the resulting coordinates x and y are multiplied by the scale factor, thereby finishing the transformation to the fiducial system: centered at the principal point and having millimeters as units (or microns, or any other value according to the value of the pixel size that was specified).

The “Pixel ratio x/y” is 1/1 if the pixel is square, 2/1 if it measures double in the x direction than in the y direction, 1/2 if double in y than in x, etc. It is the round, intended value. The parameter “scales x/y” is the small difference of the actual ratio x/y with respect to the intended one and is always close to 1. The angle of axis is the difference from 90º of the angle formed by them, and is positive if the positive directions of the axis form less than 90º. The formula that exactly defines the parameters is at technical subjects.

When the pixel is not square-shaped, the adequate value of “scales x/y” has to be specified as known value, and for the “scale” the geometric mean of the lengths of the x and y sides. If for example the pixel measure 6 microns along the horizontal direction and 3 microns along the vertical one, a value of 2 will be specified for “scales x/y” and a value of 4.24 at “scale”. Only few decimal figures should be written because, as is explained in Systems of units, the pixel size is not a calibrable magnitude and it is more important to use a simple value than an “exact” value, that is in any case conventional.

Outer orientation

They are the six parameters of the outer orientation of a camera.

Focal length and principal point

In the example here shown the principal point is not among the parameters to calibrate because the parameters “x translation” and “y translation” have already been selected within the first set of parameters. Let us suppose however that the pixel matrix measures 4000x3000 pixels and that the location of the principal point is wanted in the units of the photo coordinates (for example, milimeters) and with respect to the center of the pixel matrix. In this case the values of “x translation” and “y translation” withitn “Measured coord. --> Photo coord.” won't be selected for the calculation but rather they would be included as known values, 2000 and 1500 respectively (this is the transformation usually applied by measuring programs), while the two parameters of the principal point within this window would be selected. We are thus asserting that the origin of the photo coordinate system is located at the point (2000,1500) of the measuring system, and that to transform from the later to the former a scale factor has to be applied, after the translation, equal to the value written at “Scale”, for example 0.005. If the value arising for the principal point as a result of the adjustmen is for instance (0.052,-0.009), this means that the location of the principal point in the photo coordinate system is at the point (0.052,-0.009).

Summarizing, the order of application of the translations and the scale factor from the measuring system till the arrival at the the principal point is as follows:

x, y translations, Scale, Principal point x,y

The step «Scale» includes the changing of sign of the y coordinate.

It is recommended that the condition to define the radial symmetric distortion and the focal length be that of Orthogonality. With respect to this condition, look at Dependencies among parameters.

Approximate/Known

If a parameter is selected for the adjustment an approximate value can be optionally provided. For the parameters Ω and Φ the initial approximate value is mandatory. It is also for the focal length, but if in this case no one is given the value is taken from the file. Within the file, the value of the focal length must be expressed in measuring units (the one in which coordinates for measured points within that file are expressed), whereas if written at the program window its value must be expressed in the photo coordinate system.

On the contrary, if a parameter won't be computed then it must be given a known value, logically. It happens many times that we want a parameter “not to exist”, which is equivalent to say that it does exist and has a known value of 0 (or 1). For example, in the default configuration all the parameters that form the transformation from the measuring system to the fiducial system are known and have a value of 0 or 1, accordingly to which of them means “not to exist”.

Colour of the text boxes

The colour of the text boxes reflects the impossible, optional or obligatory nature of the value within them.


ObligatoryOptionalNot allowed

Each parameter in the main program window has two text boxes, the one corresponding to the known value, if the parameter will not be calculated, and the one of the initial approximate value, in case it enters the adjustment. In each case the other text box is locked. When the parameter takes part in the adjustment, in a few occasions it is obligatory to indicate an initial value, but in the majority of cases it is not.

The value of the photograph's half diagonal doesn't need either to be indicated. It is usually left to the program the task of computing it. If however it is wanted to use an exact value and not the one guessed by the program, that value should be written in the corresponding text box.

1.7. Function of distortion

The distortions are expressed as a function of the polar coordinates of the points: (θ,s), were s=r/rmax. The value of rmax is the photograph half diagonal.

Decomposition

The function of distortion is composed of symmetric radial distortion, symmetric tangential and two other series of asymmetric distortions.

The tangential distortion is a linear value t in the direction perpendicular to the radius, positive if the point in the image moves counterclockwise with respect to the theoretic position.

The symmetric radial distortion is, for each distance s, the mean radial distortion of the points (θ,s) for all values of θ, and likewise for the tangential distortion.

The asymmetric distortion is the total distortion minus the symmetric component.

Different models

The models of Calibration are based upon orthogonal functions, so that the parameters be independent among each other and each of them acquires a meaning. The program's default polynomial model is the complete one. The polynomials of the odd model have smaller coefficients but make use of higher powers of s.

Regarding the two asymmetric models offered by the program, none in particular is recomended, and experience beter than precept will tell which model suits better which camera. In the model of the rotating vector the parameters c5 and c6, from series 1, are the parameters of an affine distortion--a scale difference among the axes and the missperpendicularity of the angle contatined in between.

The parameters of both models are on the whole equivalents, in such a way that every four parameters from one row of the rotating vector model are, when taken together, are equivalent to the four parameters of the same row from the radial/tangential model. For example, the joint efect of the four distortions c5, c6, d5, d6, if all of them are selected, is the same in both models.

The formulas for each model can be viewed clicking on the text displaying the number of parameters: , etc., when the desired model is selected, or in technical subjects.

1.8. Modus operandi

There are obviously other possible ways of proceeding; the one proposed next is just a particual one.

• The images of the points with known coordinates are measured, in pixels. The photograph file is prepared with the coordinates in pixels as is shown above. The approximate focal length is expressed in pixels as well. The x coordinate must increase to the right; they coordinate upwards.

• In the program main window, at the top left corner, the configuration pixels / mm is chosen and the pixel size written, e.g., 0.008 (if we wanted the results to be expressed in microns the value to write would be 8.0).

• The adjustment is performed.

• It is checked whether some point has so high a residual that it would mean an error in its measurement. The residuals mat be viewed numericallt in the information file and graphically in the graphic. Since they are expressed in pixels, a very high residual can be, for instance, 5 px. If there is such a point it will be elliminated, by marking it with 00 in the photograph file, and the adjustment is carried out again. If there are several points with a very high residual they can all be elliminated, but from a set of several nearby points with a high error only the one of the largest residual should be elliminated. If in doubt, elliminate one at a time. The process is repeated till no blunder errors remain.

The elliminated points are re-measured and are included back in the adjustemnt. If there is not the possibility of measuring them again they shall be left elliminated.

• We now proceed by performing different trials selecting different sets of distortion parameters. The most common options are the selection of more parameters if there are thousands of measured points, or the removal of all of them save those from the simmetric radial distortion in case there are less than 50 measured points. To judge wheter a parameters should be included or removed from the adjustment, its value at the information file should be compared against its standard deviation, which is the precision with which the value was computed. The former should be at least double the later.

• A good value for the a posteriori standard deviation is one near 0.3 px.

2.  TOOLS

2.1. Saving and opening configurations

All the selected options from the program main window, and all the indicated values, except the names of the files, can be saved to a configuration file, at File-->Save configuration, and opened later.

The program includes four configurations: Photo coordinates, Pixels / mm, Pixels / microns and Pixels / pixels. The only difference between the second and third is the symbol of the units of the fiducial system, for the output files.

3.  TECHNICAL SUBJECTS

3.1. Adjustment method

The adjustment of the orientation/calibration is carried out by means of the method of least squares.

3.2. Rotation matrices

The form of the rotation matrix in function of the angles Ω, Φ and Κ is the following one:

This is the matrix that allows for the pass form the photo coordinates system to the system of the control points. The one of the reverse transformation, which is usually noted by M, is the transposed.

This figure exhibits the sign convention.

3.3. Dependencies among parameters

Among all possible parameters there are some of them which are equivalent, and so they cannot be included simultaneously in a calculation. There are some couples or groups of equivalent parameters. When any parameter is selected that is incompatible with any other that was already selected the program automatically deselects this last one.

The parameters to pass from measured coordinates to photo coordinates (fiducial coord.) are an alternative subset, to calibrate the measuring engine in stead of the camera or to move some parameters to pixel units. Hence, each of them is equivalent to some orientation parameter.

Tx    ---->  Principal point, x
Ty    ---->  Principal point, y
rotation ---->  Outer orientation, K
scale   ---->  focal length

The parameters “scales x/y” and “angle of axis” are equivalent to asymmetric distortions. If the model for “other distortions” is the rotating vector, these parameters are equivalent to c5 and c6 respectively. If the model is radial/tangential, each of them is related to one parameter of the asymmetric radial distortion and another analogous one of the tangential distortion.


If the rotation differs from zero the correspondences Tx<-->x and Ty<-->y no longer stand. In those cases (if it is known and different from zero or if it is unknown), (x,y) of the principal point must both be known or both be unknown.


The focal length is equivalent to the first term of the symmetric radial distortion (in any model, it is always the same). Thus, if the focal length is calculated that term vanishes from the radial distortion. Such is the condition of orthogonality. If it is desired that the function of distortion matches any other condition, once the calculation is finished the first term of the s.r.d. is modified in such way that the condition holds, and afterwards the focal lend is varied in the opposite way so that both changes cancel each other.

If any component is added to the distortion its mean square value increases (always), and it is for that that the condition of orthogonality is always recommended.

If the parameters Ω and Φ are selected, the selection of which is made jointly, then:

If the x coordinate of the principal point is selected (or its equivalent Tx) then the term s2cosθ of the asymmetric r.d. cannot be calculated.

If the y coordinate of the principal point is selected (or its equivalent Ty) then the term s2senθ of the asymmetric r.d. cannot be calculated.

3.4. Relation among the parameters α and β of an affinity and the ratio of scales between axis and the angle formed by them

The program uses for the parameters “scales x/y” and “angle of axes” of the transformation from measured coordinates to photo coordinates two parameters which are equal to c5 and c6 from the rotating vetor model of the asymmetric distortion. Noetheless, both for the input and the output, the parameters used are the ratio of scales between axis and the deviation from 90º of the angle contained by them. The relationship between the two couples of parameters is the following:

From internal parameters to user parameters:





From user parameters to internal parameters:





Like with the formulas of distortion, it is not necessary to know them, since the program, as well as the parameters K and E, displays the matrix to pass from measured coordinates to photo coordinates.

3.5. Formulation of each model

The knowledge of the formulas that appear next is not required in order to be able to apply the calculated parameters, because the program expands the polynomials and groups terms, and displays each component of the function of distortion as a single polynomial.

The polynomials are expressed as a function of the variable s, which is the normalized radius, i.e., divided by its maximum possible value = half-diagonal. The output file explicitely shows it. For example, s=r/160000.

Polynomial models

Each polynomial model is composed of two series of polynomials. The first one or series p is formed by the components of the symetric distortion, be it the radial or tangential one, for the components are the same for both. It is also applied to certain asymetric components: las even components. The second series, series q, is applied to the odd components of the asymetric distortions.

Odd

Series p:

Series q:


Complete

Series p:

Series q:


Asymmetric models

Each asymmetric component is the product of a polynomial, which only depends on s (that it to say, the radius), times a component depending only upon the angle θ. The polynomials are taken from the selected polynomial model, while the components depending on θ are determined by the asymmetric model.

Radial/tangential

In this model each asymmetric component has a contant direcction (taking the radius as the reference) and its module vaires periodically as a function of . In the first series that constant direction is the radial direction whereas in the second it is the tangential direction.

The components shown below are the first ones corresponding to the complete polynomial model. For the odd model just replace the polynomials shown here by those of that model.

Series 1:

The series 2 is identical to series 1 in its formulation, the difference being that there the direction of the components is the radial one whereas here it is the tangential one.

Rotating vector

In this model each component has a constant unity modulus and keeps “turning around” as θ is being varied. the vectors of the first series turn around in the sense opposite to the grouth of&nbps; , i.e., clockwise. In the second series they rotate in the opposite sense. Within each series, the role played in the previous model by the pairs sin/cos is now taken by pares of orthogonal vectors u/v. The u vector are components that for points withθ=0 follow the radial direction, while the the v vectors follow the tangential direction for those same points.

What was said in the preceeding paragraph applies to all components save the first four, two from series 1 and two from series 2. These four are replaced by the first four of the radial/tangencial model, because due to certain theoretic considerations the components c1 y c2 from the asymetric radial distortion must appear explicitely in any model.

As it was done above, in order to display the components the polynomials from the complete model are used.

u means: Vector of modulus one that for a point on the photograph with coordinates (r,θ) makes an angle cθ with the radial direction.

v means: Vector of modulus one that for a point on the photograph with coordinates (r,θ) makes an angle ctheta; with the tangential direction.

Series 1:

Series 2:

3.6. Graphs

Polynomial models

Complete

Series p:

Series q:


Odd

Series p:

Series q:


Asymmetric components

The graphs are very similar for the complete and odd polynomial models. The ones shown here correspond to the complete series for the radial/tangential model and the odd one for the rotating vector model.

Radial/tangencial

c1, c2. These components will normaly not exist because they vanish if the principal point is calibrated; d1, d2.



c3, c4; d3, d4.

c5, c6; d5, d6.



c7, c8; d7, d8.

c9, c10; d9, d10.

c11, c11; d12, d12.

Rotating vector

c1, c2. These components will normaly not exist because they vanish if the principal point is calibrated; d1, d2.



c3, c4; d3, d4.

c5, c6; d5, d6



c7, c8; d7, d8.

c9, c10; d9, d10

c11, c11; d12, d12