A review and analyses of plumb-line calibration

Shorter Contribution

A REVIEW AND ANALYSES OF PLUMB-LINECALIBRATION

Jose L. Lerma ([email protected])

Miriam Cabrelles ([email protected])

Polytechnic University of Valencia, Spain

Abstract

Radial and decentring distortion parameters have long been reported asfundamental elements in camera calibration, especially in digital close rangephotogrammetry. This paper reports on an investigation for the determination of bothradial and decentring distortion parameters using analytical plumb-line calibration.Straight line patterns, ranging from 1, 2, 4, 10 and up to 20 lines, not strictly plumb,are examined on a set of distorted imagery. Additionally, not only the number ofstraight lines but also the linear image distribution is analysed. For that purpose,linear features have been distributed in different positions and orientations over thewhole of the imagery. Furthermore, the relationship between the parametersmentioned and the location of the principal point is also analysed, and shows howsensitive they are to deviations from the centre of the sensor.

The motivation for this paper is the measurement of the minimum number of linearfeatures for proper calibration of non-metric digital cameras. Both radial anddecentring distortion parameters are examined. Finally, some line patterns arerecommended for making the plumb-line calibration technique a reliable, easy and fastprocedure. Photogrammetrists could use it as a first step in the procedure of cameracalibration; userswhoare not experts in photogrammetry could employ it as a final step.

Keywords: camera calibration, close range photogrammetry, lens distortion, straightlines

Introduction

Camera calibration is considered a necessary part of most photogrammetric processes.Knowledge of the internal geometry of the camera is essential when medium or high accuracywork is required. A correct model of the internal geometry of the camera guarantees thatmeasurements on the imagery will in theory be free of systematic errors, or at least that theywill be negligible. The (unknown) real behaviour of the camera system depends on thesuitability of the mathematical model used in the camera calibration. Nevertheless, andconsidering that the reliability of the calibration model is right for the relevant photogram-metric application, it means that neither the image measurements nor the derived object

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measurements in object space will be degraded in accuracy. Additionally, the image texturedraping of 2D or 3D models will be appropriately driven.

Digital cameras and digital images have gained wide acceptance in the field of close rangephotogrammetry and computer vision. Nowadays, digital cameras are widely required forreconstruction and modelling of static and dynamic features in both real and virtual worlds.Industrial, medical, architectural and archaeological applications, among others, are certainlyincluded in this. In fields such as the documentation of cultural heritage, off-the-shelf digitalcameras, still video cameras or other non-metric cameras (from a photogrammetric point ofview) are almost always used. Therefore, a camera calibration procedure is mandatory whetheror not the users require maximum geometric performance.

Camera calibration processes and definitions have changed greatly over recent years(Boland, 2004). There are now many different methods, from standard aerial calibrationprocesses based on goniometers, collimators and multicollimators for only one camera at atime, to bundle adjustment with self-calibration of multiple cameras, stellar methods,underwater methods and plumb-line methods, as well as multi-camera convergent methods.Most textbooks on photogrammetry (for example, Kraus et al., 1997; Wolf and Dewitt, 2000;Mikhail et al., 2001; Lerma, 2002; McGlone, 2004) review camera calibration methods appliedto aerial as well as close range optical cameras; only a few of them are particularly focused onclose range camera calibration (Faugeras, 1993; Fryer, 1996; Gruen and Huang, 2001; Hartleyand Zisserman, 2003). Clarke and Fryer (1998) also review the development of models forcamera calibration and the evolution of their methods, highlighting the driving forces behindeach improvement.

A calibration procedure in photogrammetry should produce numerical values representingspatial relationships of the measurement system. These include numerical estimates of camerainterior orientation represented by principal distance, location of the principal point andcoefficients of appropriate models representing lens distortion. If the measurement systemincludes additional sensors, their spatial position and orientation properties must also bedetermined by calibration (Boland, 2000). Additionally, image plane unflatness (out-of-planedistortion) and in-plane image distortion estimates should also be considered (Fraser, 1997).

Calibration is usually carried out in one of three settings: in the laboratory, on the job orby self-calibration, although the second and third of these are best suited for close range cameracalibration. In order to determine the six exterior orientation parameters defining the camera’sspatial location and attitude and the unfixed number of interior orientation parameters, thecollinearity equations with additional parameters (self-calibration) or the direct lineartransformation (DLT) are the basis for a least squares solution (Fryer, 1996). Many papersrelated to self-calibration have appeared in the literature (for example, Fraser and Edmundson,1996; Fraser, 1997; Lichti and Chapman, 1997; Ahmad and Chandler, 1999; Mills et al.,2003). Calibration requirements for both on-the-job calibration and self-calibration, such asmultiple images from several camera stations with a convergent geometric arrangement, thenumber and the relative positions of the camera stations, the network design, the number ofimages used in the adjustment, the camera’s attitude and, last but not least, the distribution ofcontrol points, are topics which have been well assessed, tested, understood and properlydocumented. The same comment is also applicable to DLT (Abdel-Aziz and Karara, 1971;Bopp and Krauss, 1978; Dermanis, 1994; Kraus et al., 1997; Kwon, 2001). On the other hand,as far as the authors are aware, analytical plumb-line calibration (Brown, 1971; Fryer andBrown, 1986), although considered ‘‘unique among analytical calibration methods’’ (Boland,2004), has produced modest results in close range photogrammetry (apart from somecomments and experience outlined in Fryer and Fraser, 1986; Fryer et al., 1994; Shortis et al.,1995; Clarke et al., 1998; Karras and Mavrommati, 2001).

Lerma and Cabrelles. A review and analyses of plumb-line calibration

136 � 2007 The Authors. Journal Compilation � 2007 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd.

This paper reports on an investigation for the determination of both radial and decentringdistortion parameters using analytical plumb-line calibration. Straight line patterns, rangingfrom 1, 2, 4, 10 and up to 20 lines, not strictly plumb, are examined on a set of distortedimagery. Additionally, not only the number of straight lines but also the linear imagedistribution is analysed. For that purpose, linear features have been distributed in differentpositions and orientations over the whole of the imagery. Furthermore, the relationshipbetween the parameters mentioned and the location of the principal point is also analysed, andshows how sensitive they are to deviations from the centre of the sensor.

Indoor calibration patterns of lines can either be easily fabricated from string or fishing line(over a contrasting background) or quickly plotted on a sheet by a computer. However, sometimesthere is no chance of measuring many straight lines because the target has only a small number ofthem. Sometimes, pictures were initially taken for other purposes than camera calibration. Thislast statement typically applies when photogrammetrists try to reconstruct architectural 3Dmodels from old photographs or postcards. The motivation behind this paper is the measurementof the minimum number of linear features for proper calibration of non-metric digital cameras.The decision to select the plumb-line calibrationmethod for testing is based on the fact that it onlyrequires the capture of image lines (straight lines in object space). A linear feature can also be astraight edge, and straight edges are typical in images of man-made objects (such as buildings,industrial parts, polyhedral shapes and so on) and environments. Additionally, there are someadvantages of using line rather than point features (Liu and Huang, 1991). Firstly, lines are ofteneasier to extract in a noisy image than are point features. Secondly, it is possible to determine theorientation of a line to sub-pixel accuracy. Thirdly, lines have simplemathematical representation.Fourthly, image lines can easily be found in architectural and industrial environments.

The next section describes the plumb-line calibration method. The mathematical modelthat relates straight lines in image space and object space is presented, followed by details ofresults obtained from the implementation of this approach, using synthetic data. Finally, someconclusions are presented.

Background: The Plumb-line Method

The analytical plumb-line method that accounts for the variation of distortion with objectdistance and throughout the photographic field was first attributed to Brown (1971). It wasparticularly suited to the task of pre-calibrating a camera for a specific focal length setting.

The principle of this technique lies in the truism that straight lines in object spaceshould project through a perfect lens as straight lines. Any variation from straightnessis attributed to radial or decentring distortion, and a least squares adjustment isperformed to determine the distortion parameters. (Fryer and Brown, 1986:52)

However, distortions which are not corrected prior to applying the plumb-line method, such asfocal plane unflatness, would slightly change the recovered lens distortion parameters. Theother way round, determined distortion parameters can model not only the lens but also someother elements of the camera system.

During the plumb-line calibration method, both the radial lens distortion and thedecentring lens distortion parameters are determined. In order to estimate the curvature ofstraight lines, the following straight line equation is considered:

ðxþ DxÞ sin hi þ ðy þ DyÞ cos hi ¼ qi ð1Þ

in which

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� 2007 The Authors. Journal Compilation � 2007 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. 137

Dx ¼ �xðK1r2 þ K2r4 þ . . .Þ þ P1ðr2 þ 2�x2Þ þ 2P2xy� �

. . .

Dy ¼ �yðK1r2 þ K2r4 þ . . .Þ þ 2P1xy þ P2ðr2 þ 2�y2Þ� �

. . .

�x ¼ ðx� xP Þ�y ¼ ðy � yP Þ

ð2Þ

where hi is the angle between the y axis and the normal to the ith line passing through theorigin and qi denotes the distance of the ith line from the origin, x and y are the coordinatesof the points on the image of the straight line, Dx and Dy are the image coordinatescorrection functions, K1, K2 . . . are the coefficients of radial distortion and P1, P2 . . . are thecoefficients of decentring distortion; both sets of global coefficients are only valid for theobject plane being considered.

Substituting equations (2) into equation (1) gives an observational equation that isfunctionally of the form

F ðx; y; xP ; yP ;K1;K2; P1; P2; hi; qiÞ ¼ 0: ð3Þ

After linearisation, a line of the form (1) or (3) is fitted to the given points such that the sumof the squares of the distances between the data points (x, y) and the ith lines is a minimum.

The simplicity of the mathematical model, the availability of linear features in mostscenes, the unnecessary knowledge of equations of straight lines and/or coordinates of anycontrol point, and, last but not least, the lack of correlation between the interior and the exteriororientation parameters make this method unique. Moreover, either the manual or the automaticmeasurement of points on linear features is an easy and quick stage in the camera calibrationprocedure, and it can be successfully carried out with only one picture for each camera settingor at least one picture for each refocusing.

On the other hand, one of the disadvantages of this method is that the principal distancecannot be determined. In general, offsets of the principal point also cannot be determinedbecause they become coupled with radial and decentring distortion coefficients; only in thecase of fisheye lenses where the distortion is very large can the offsets be computed withconfidence (Fryer, 1996).

The parameters determined with plumb-line calibration can be used in photogrammetry inmany ways: firstly, for the correction of image coordinates in cases where only nominal values(uncalibrated values) are available; secondly, as an independent check of lens distortionparameters obtained with another camera calibration method; and thirdly, for resamplingdistortion-free imagery. The latter means, for instance, that this calibration method can beapplied prior to the rectification of planar objects or features by means of a two-dimensionalprojective transformation.

Experimental Results and Analyses

Four experiments on synthetic images (also known as simulated images) were conducted totest whether or not the plumb-line method can provide good estimates of radial and decentringdistortions, in relation to the number and spatial distribution of line patterns, and last but notleast, the effect of small and large errors in the nominal centre of an image in a digital camera.

Initially, a set of 62 straight line patterns were examined. However, this paper summarisesand reports on the 14 most relevant ones (Fig. 1). The image resolution of each pattern was setto 3072 pixels wide and 2048 pixels high, the typical resolution of a digital camera with 6Æ1million effective pixels. Patterns were classified by the number of straight lines in five groups:



first group, 1 line; second group, 2 lines; third group, 4 lines; fourth group, 10 lines; and fifthgroup, 20 lines. The codification assigned to each straight line pattern is composed of twonumbers: the first code number represents the pattern and the second the corresponding numberof lines. Table I lists the number of groups, their features and the synthetic patterns assigned.All the straight lines were measured with 41 of the points, irrespective of their lengths.

Four patterns were assigned to three out of five groups. The idea behind this was to checkstraight line arrangements in order to generalise the distribution of straight lines. Thejustification for using only one pattern in the third and fifth groups, 1-4 and 1-20 patterns,respectively, was that the former is the straight line pattern recommended by Fryer et al. (1994)for fast and accurate estimation of the lens distortion parameters, both radial and decentring.The latter was an extension of the usual plumb-line pattern to 20 lines, gathering 10 verticaland 10 horizontal straight lines. Thus, with only one shot, the same result would be obtained asif the two sets of straight lines had been independently obtained by rolling the camera through90� to produce diagonal lines.

Straight line distributions were set considering not only horizontal and vertical lines butalso lines with arbitrary orientations on the imagery. The only constraint imposed on the sets ofstraight lines was qi ‡ 10 pixels; this value was empirically determined and set at an early stageof the experimentation in order to avoid misleading mathematical results. Thus, none of thestraight lines was allowed to pass through the principal point.

All the straight line patterns were distorted for each experiment with realistic values usingdifferent sets of additional parameters. Four experimental tests were carried out to checkwhether it is possible to recover those parameters after least squares adjustment (LSA) bymeans of the plumb-line method or not. The first set of additional parameters was defined bythe radial distortion coefficients K1 and K2; and the second set by the coefficients K1, K2, P1

and P2. These are suitable for most scientific and off-the-shelf digital cameras. Additionally,some errors in the principal point offset xP and yP were introduced to the sets mentioned. TableII reflects the values used for the generation of the pixel coordinates; they were taken from a

4-23-22-21-21-1

1-4 1-10 2-10 3-10 4-10 1-20

2-1 3-1 4-1

Fig. 1. Schematic images representing 14 straight line patterns.

Table I. List of experimental groups with their features and patternsassigned.

Group Number ofstraight lines

Number ofpatterns

Code patterns

1st 1 4 1-1, 2-1, 3-1, 4-12nd 2 4 1-2, 2-2, 3-2, 4-23rd 4 1 1-44th 10 4 1-10, 2-10, 3-10, 4-105th 20 1 1-20



camera calibration certificate. The first two experiments will allow the recovery of the radialand/or decentring distortion parameters to be analysed. The latter two experiments will allowthe principal point shifts and their effects on the estimation of the lens distortion parameters tobe investigated.

FotograUPV software developed at the Department of Cartographic Engineering,Geodesy and Photogrammetry, Polytechnic University of Valencia, has been used for allthese testing purposes.

First Experiment: Radial Distortion

All the synthetic images were affected only by radial distortion. Therefore, this firstexperiment was designed to test the effects of straight line patterns on the recovery of K1 andK2 radial distortion coefficients after LSA. The location of the principal point wasmathematically fitted to the centre of the digital image.

A summary of the results of LSA is given in Table III. These results indicate that allpatterns exactly recovered the K2 coefficient, although there were some slight differences forthe K1 coefficient. Furthermore, the standard deviations of the adjusted additional parameterswere not sensitive to the ‘‘real’’ deviations. For example, the estimation of the K1 parameterwas more accurately determined for pattern 1-1 than for pattern 3-1, but its standard deviationwas better for pattern 3-1 than for pattern 1-1. Fig. 2 shows the radial distortion profiles for theseven patterns selected in Table III and for the true one. This figure in combination with TableIII can help to analyse both the degree of magnitude of the distortion values and the degree ofvariation values between different patterns.

Regarding the patterns with only one straight line (first group), the predefined versus theestimated Gaussian radial distortion functions are plotted in Fig. 3. All the patterns offeredgraphical plots close to the true one up to a third of the horizontal resolution (768 pixels). Thepattern with the vertical line (pattern 3-1) showed maximum deviation not only for the firstgroup (Fig. 3) but also for the whole set of patterns (Figs. 2 and 4). Its root mean square (rms)

Table II. Simulation values applied to the patterns in each experiment.

Parameter Experiment

1st 2nd 3rd 4th

xP (pixels) 0 0 (1, 2, 5, 10, 50, 100) (1, 2, 5, 10, 50, 100)yP (pixels) 0 0 ) (1, 2, 5, 10, 50, 100) ) (1, 2, 5, 10, 50, 100)K1 1e–8 1e–8 1e–8 1e–8K2 )1e–15 )1e–15 )1e–15 )1e–15P1 0 1e–8 0 1e–8P2 0 )1e–7 0 )1e–7

Table III. Results of LSA for radial plumb-line calibration.

Additionalparameters

Patterns

1-1 3-1 2-2 4-2 1-4 4-10 1-20

K1 1Æ04e–8 1Æ11e08 1Æ03e08 1Æ04e–8 1Æ07e–8 1Æ03e–8 1Æ05e–8rK1

6Æ00e–11 7Æ10e–12 1Æ12e–11 5Æ83e–12 7Æ77e–12 4Æ59e–12 6Æ34e–12K2 )1Æ00e–15 )1Æ00e–15 )1Æ00e–15 )1Æ00e–15 )1Æ00e–15 )1Æ00e–15 )1Æ00e–15rk2 0 0 0 0 0 0 0



Fig. 2. Distortion profiles for selected patterns in Table III.

0

0·5

1

1·5

2

2·5

3

3·5

4

1-1 2-1 3-1 4-1 1-2 2-2 3-2 4-2 1-4 1-10 2-10 3-10 4-10 1-20Pattern number

Pixe

ls RmsMax. residuals

Fig. 4. Rms deviations and maximum radial distortion residuals.

Fig. 3. Radial distortion profiles for patterns with only one straight line (first group).



discrepancy between the true and the derived functions was 1Æ7 pixels, meanwhile the bestpatterns offered residuals around 1 pixel (independently of the group or number of straightlines). Particularly interesting is that patterns with neither horizontal nor vertical straight linesoffered better results than their counterparts: first group, patterns 1-1 and 4-1 versus patterns2-1 and 3-1; second group, patterns 1-2 and 4-2 versus patterns 2-2 and 3-2; fourth group,patterns 2-10 and 4-10 versus patterns 1-10 and 3-10. Additionally, the larger the number ofstraight lines measured, the smaller the rms and maximum residuals among patterns werefound to be. However, some of the aforementioned patterns with only one or two straight linesyielded better results than patterns with up to 4, 10 or 20 straight lines.

This experience illustrated that it is possible to mathematically recover the radialdistortion coefficients with only one or two appropriate lines (pattern 3-1 should be excluded).Nevertheless, it must be stressed in those cases that the parameters determined are mainly validfor the direction of those lines. Measurements of more straight lines (up to 10 or 20) could bejustified if the ‘‘real’’ lens distortion was rather heterogeneous throughout the whole imageformat.

Second Experiment: Radial and Decentring Distortion

All the synthetic images were affected by radial and decentring distortion. Therefore, thissecond experiment was designed to test the effects of straight line patterns on the recovery ofthe four most used lens distortion coefficients in camera calibration, K1, K2, P1 and P2. Thelocation of the principal point mathematically fitted to the centre of the digital image.

A summary of the results of LSA is given in Table IV. Unlike the previous experiment,none of the patterns belonging to the first group (patterns with only one straight line) properlyestimated the four lens distortion coefficients. This result can also be confirmed after analysingthe distortion profiles in Fig. 5. The same comment was also confirmed by patterns with allstraight lines passing near the centre (patterns 1-2 and 2-10). However, the standard deviationsof unit weight for all these patterns were always acceptable. Besides, it was not possible todetect wrong radial distortion parameter estimations for the whole set of patterns after theanalyses of the standard deviations of the adjusted additional parameters. On the other hand,this was possible for the decentring distortion parameters on 6 out of 14 patterns (1-1, 2-1, 2-2,4-2, 3-2 and 3-10). Fig. 5(b) shows the huge profile deviations in decentring distortion for thefirst four patterns selected in Table IV.

Fig. 6 shows the deviations obtained for all 14 patterns. It can be seen that the highestdeviations were obtained by patterns with only one line (first group), as well as by patterns withall straight lines passing near the centre of the imagery (patterns 1-2 and 2-10). In a similar way

Table IV. Results of LSA for radial and decentring plumb-line calibration.

Additionalparameters

Patterns

1-1 2-1 2-2 4-2 1-4 4-10 1-20

K1 1Æ49e–7 )3Æ12e–8 1Æ00e–8 1Æ04e–8 1Æ07e–8 1Æ03e–8 1Æ05e–8rk1 6Æ55e–8 7Æ07e–10 2Æ81e–10 5Æ98e–12 7Æ72e–12 4Æ62e–12 6Æ35e–12K2 5Æ00e–14 )1Æ00e–15 )1Æ00e–15 )1Æ00e–15 )1Æ00e–15 0 )1Æ00e–15rk2 3Æ00e–15 0 0 0 0 0 0P1 )1Æ02e–7 )6Æ06e–9 )2Æ02e–7 2Æ15e–7 1Æ14e–8 1Æ07e–8 1Æ11e–8rP1

2Æ60e–7 2Æ96e–8 2Æ20e–7 1Æ78e–7 3Æ92e–9 9Æ87e–10 3Æ14e–9P2 4Æ84e–7 4Æ42e–5 1Æ72e–8 2Æ57e–8 )1Æ14e–7 )1Æ07e–7 )1Æ09e–7rP2

2Æ91e–7 7Æ48e–7 1Æ45e–7 1Æ19e–7 1Æ70e–9 6Æ58e–10 1Æ39e–9



(a)

(b)

Fig. 5. Distortion profiles for selected patterns in Table IV: (a) radial distortion; (b) decentring distortion.

00·20·40·60·8

11·21·41·61·8

2

1-1 2-1 3-1 4-1 1-2 2-2 3-2 4-2 1-4 1-10 2-10 3-10 4-10 1-20

Pattern number

Pixe

ls

RmsMax. residuals

Fig. 6. Rms deviations and maximum radial and decentring distortion residuals.



to the previous experiment, patterns with neither horizontal nor vertical straight lines offeredbetter results than their counterparts: second group, pattern 4-2 versus pattern 3-2; fourthgroup, pattern 4-10 versus pattern 1-10. The pattern with a set of straight lines in a cross shapecentred in the middle of one quadrant (pattern 2-2) achieved a 0Æ42 pixel rms deviation value,twice as good as the second best (pattern 4-10).

Third Experiment: Radial Distortion versus Principal Point Error

All the synthetic images were affected by the same radial distortion coefficients but thelocation of the principal point was systematically shifted from the centre of the digital image.Therefore, this third experiment was designed to investigate the principal point offsets and theireffects on the estimation of the K1 and K2 radial distortion coefficients after LSA.

Regarding patterns with only one straight line, deviation results from patterns 1-1, 2-1 and4-1 can be checked in Table V. Meanwhile the pattern with a straight line passing near thecentre (pattern 1-1) was highly dependent on principal point offsets (for example, an offset ofonly 5 pixels in the x axis and )5 pixels in the y axis yielded an rms deviation of 8Æ61 pixels),the rest of the patterns were not very sensitive to shifts in the principal point; they admitted arelaxed tolerance (approximately up to 50 pixels) in the location of the principal point for notaltering the recovery of radial distortion coefficients.

Similar results as compared to the previous patterns were achieved by patterns with two ormore straight lines. Bad results were produced only for patterns with all the straight linespassing near the centre (patterns 1-2 and 2-10). Fig. 7 shows the deviations obtained for all thepatterns involving two straight lines.

Besides, the introduction of additional straight lines allowed the stabilisation of the radialdistortion error up to ±50 pixels in the x and y axes, and last but not least, a slight improvementin the estimation of radial distortion coefficients when the offsets were up to ±100 pixels(patterns 3-2, 4-2, 1-10 and 4-10). The best patterns for absorbing the image shifts were twopatterns with neither horizontal nor vertical straight lines (patterns 4-10 and 4-2), followed bythe two patterns with simultaneous horizontal and vertical straight lines (patterns 1-20 and1-10). Measurements of additional straight lines slightly reduced the deviation errors, forexample, pattern 1-20 only reduced it by 0Æ01 pixels when compared with its homologouspattern 1-10; and, similarly, fourth group patterns only reduced the deviation error by 10%when compared to second group patterns.

As a result of this experience, it can be concluded that patterns with all the straight linespassing near the centre should only be considered appropriate for recovering the radial

Table V. Radial distortion deviations introduced with various principal point offsets.

Principalpoint offsets

Pattern number

1-1 2-1 4-1

(x0,y0) Rms dev. Max. residualdeviation

Rms dev. Max. residualdeviation

Rms dev. Max. residualdeviation

(1, )1) 0Æ99 1Æ79 1Æ32 )2Æ72 1Æ08 )2Æ47(2, )2) 2Æ85 5Æ69 1Æ33 )2Æ74 1Æ10 )2Æ51(5, )5) 8Æ61 17Æ90 1Æ36 )2Æ79 1Æ15 )2Æ60(10, )10) 18Æ01 37Æ68 1Æ40 )2Æ87 1Æ24 )2Æ75(50, )50) 86Æ93 179Æ73 1Æ69 )3Æ33 1Æ90 )3Æ81(100, )100) 158Æ96 323Æ16 1Æ84 )3Æ46 2Æ53 )4Æ62



distortion coefficients when the knowledge of the location of the principal point is veryaccurate.

Fourth Experiment: Radial and Decentring Distortion versus Principal Point Error

All the synthetic images were affected by the same radial and decentring distortioncoefficients but the location of the principal point was systematically shifted from the centre of

050

100150200250300350400

(1,-1) (2,-2) (5,-5) (10,-10) (50,-50) (100,-100)

Principal point offsets

Pixe

ls RmsMax. residuals

0

2

4

6

8

10

(1,-1) (2,-2) (5,-5) (10,-10) (50,-50) (100,-100)


Pixe

ls

RmsMax. residuals

0

0·5

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(1,-1) (2,-2) (5,-5) (10,-10) (50,-50) (100,-100)


Pixe

ls Rms

Max. residuals

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Pixe

ls Rms

Max. residuals

(a)

(b)

(c)

(d)

Fig. 7. Rms deviations and maximum radial distortion residuals after introducing principal point offsets topatterns: (a) 1-2; (b) 2-2; (c) 3-2; (d) 4-2.



the digital image. Therefore, this fourth experiment was designed to investigate the principalpoint offsets and their effects on the estimation of the K1, K2, P1 and P2 distortion coefficientsafter LSA.

None of the patterns with only one straight line yielded acceptable deviation errors.The same comment was also true for all patterns with straight lines passing near the imagecentre (patterns 1-2 and 2-10). On the other hand, three out of four patterns with twostraight lines could be used to achieve profitable results, if and only if the principal pointoffsets were accurately known beforehand (up to 2 pixels, pattern 4-2; up to 1 pixel,patterns 3-2 and 2-2).

Besides, patterns with four or more straight lines admitted a relaxed tolerance in thelocation of the principal point for not altering the recovery of both radial and decentringdistortion coefficients. The only exception to this rule was the aforementioned pattern 2-10.Fig. 8 shows low deviation errors produced by the pattern with four straight lines (pattern1-4). This figure shows remarkably low values only for the first four situations, due to thestrong correlation between parameters for decentring distortion and the offsets of theprincipal point. However, the deviation substantially increased for large principal pointoffsets (Fig. 8(b)).

Final Analysis of Results

The experiments carried out should allow ideal patterns to be selected for recoveringeither radial distortion parameters alone or radial and decentring distortion parameters. For thispurpose, it was necessary to analyse patterns (the two former experiments) as well as theeffects of principal point shifts (the two latter experiments), mainly because the principal pointshifts are usually unknown for non-metric cameras (unless the camera was calibratedbeforehand).

0

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(1,-1) (2,-2) (5,-5) (10,-10) (50,-50) (100,-100)Principal point offsets

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Rms

Max. residuals

00·20·40·60·8

11·21·41·61·8

(1,-1) (2,-2) (5,-5) (10,-10) (50,-50) (100,-100)Principal point offsets

Pixe

ls

(a)

(b)

Fig. 8. Rms deviations and maximum radial and decentring distortion residuals after introducing principalpoint offsets to pattern 1-4: (a) radial; (b) decentring.



Regarding the plumb-line calibration of cameras only affected by radial distortion,patterns recommended for recovering the radial distortion coefficients with minimum effortwould be either one or two off-centre diagonal straight lines or a set of straight lines in a crossshape centred in the middle of one quadrant (patterns 4-1, 4-2 and 2-2, respectively). Ifmaximum reliability is required, then two parallel sets of five straight diagonal lines such aspattern 4-10 would be preferred.

Regarding the plumb-line calibration of cameras affected by both radial and decentringdistortion, patterns recommended for recovering all of these distortion coefficients withminimum effort would be a set of four straight lines close to the edges of the image format(pattern 1-4). If maximum reliability is required, then two parallel sets of five straight diagonalssuch as pattern 4-10 would be preferred. Unfortunately, the most accurate pattern afterdeviation analyses (pattern 2-2 with only two off-centre straight lines at right angles) shouldnot be proposed because of the negative effect of shifts of the principal point.

Conclusions

This paper has reviewed the plumb-line calibration method and has mathematically testedit from an unconventional point of view: considering different straight line geometries insteadof strictly plumb-lines in image space. For that purpose, the number of linear features as well asits spatial distribution in image space has been investigated in order to analyse which linearpatterns best allow the radial and decentring distortion parameters to be recovered.Furthermore, the principal point shifts and their effects on the estimation of the lens distortionparameters have also been investigated.

The analyses of multiple patterns with 1, 2, 4, 10 or 20 straight lines for estimating, on theone hand, the radial distortion parameters, and, on the other hand, both the radial and thedecentring lens distortion parameters makes these analyses unique. After the experimentation,it has been empirically demonstrated that the use of only one specific pattern (an off-centrelinear feature) could be used to properly estimate radial distortion if the lens was a perfectbody. Similarly, the determination of radial and decentring distortion would require a patternwith a set of four straight lines close to the edges of the image format. These two patternsdeliver the minimum number of linear features for proper calibration of non-metric digitalcameras from a mathematical point of view. However, in practice, a pattern with two parallelsets of five straight diagonal lines would be recommended.

Although plumb-line calibration is not a full camera calibration procedure comparable tothe self-calibration method, it can be accurate enough, easy to apply and fast even for non-experts; it only takes a few seconds to measure and adjust some linear features automatically,or a few minutes to measure them manually, determine the additional parameters and makecorrections. Furthermore, photogrammetrists could also make use of it, not only for purposessuch as checking lens distortion, but also for pre-calibration tasks in cases where a smallnumber of amateur images are available, for fast accurate determination of additionalparameters and, last but not least, for pre-processing (correction) of individual images prior toprocedures such as 2D image rectification or texture draping of 3D models.

Unfortunately, photogrammetrists sometimes have to deal with either past or presentdatabase imagery which lies outside the ideal case for fully automatic convergent self-calibration bundle adjustment. Situations such as that mentioned in the previous paragraph arenot unusual in cultural heritage documentation. In some scenarios the plumb-line calibrationmethod presented here is confirmed, not as a rival to the best camera calibration procedures,but as an optimum and cost-effective alternative method to correct both radial and decentringdistortion.



Acknowledgements

This research was partly funded by the Spanish Ministry of Education and Culturethrough the project HUM2005-03152. The valuable comments from anonymous reviewers aregratefully acknowledged.

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Resume

On a, de longue date, considere les parametres de decentrement et de distorsionradiale comme etant les caracteristiques fondamentales de l’etalonnage d’unecamera, particulierement lorsqu’il s’agit de photogrammetrie numerique a courtedistance. On presente dans cet article une recherche permettant de determiner a lafois les parametres de decentrement et ceux de distorsion radiale, en utilisant unetalonnage analytique sur fil a plomb. On constitue un jeu d’images empreintes deces distorsions sur un ensemble de lignes droites comprenant une, deux, quatre, dix etjusqu ’a vingt droites. On analyse nombre de ces lignes droites en tenant compte deleur repartition dans l’image. Pour cela on a reparti sur l’ensemble de l’image deselements lineaires selon diverses orientations et emplacements. On a mis en evidencela relation entre les parametres cites precedemment et la position du point principalet montre a quel point ils sont sensibles a tout ecart au centre du capteur. Le but de cetarticle est de determiner le nombre minimal d’elements lineaires necessaires a un bonetalonnage des cameras non metriques. On a etudie les parametres de decentrement etceux de distorsion radiale. Enfin on a degage des recommandations concernant laconstitution de ces ensembles de lignes droites pour que cette technique d’etalonnagesur fil a plomb soit fiable, facile et rapide. Les photogrammetres pourront l’employercomme premiere etape lors de l’etalonnage d’une camera ; les utilisateurs qui ne sontpas des photogrammetres confirmes pourront l’employer comme etape definitive.

Zusammenfassung

Radial-symmetrische Verzeichnungsparameter und radial-asymmetrische undtangentiale Verzeichnungsparameter sind fundamentale Elemente einer Kamerakali-brierung, insbesondere in der digitalen Nahbereichsphotogrammetrie. In diesemBeitrag wird uber eine Untersuchung zur Bestimmung der radial-symmetrischenParameter und der Dezentrierungsparameter der Verzeichnung berichtet, die mitHilfe einer analytischen Plumpline-Kalibrierung erfolgt. Muster von Testfeldgeraden,mit einer, zwei, vier, zehn und bis zu 20 Linien, die nicht notwendigerweise lotrechtsein mussen, werden mit Hilfe eines Satzes verzeichneter Bilder untersucht. Dabeiwird nicht nur die notwendige Zahl der Geraden analysiert, sondern auch derenVerteilung. Dazu wurden lineare Merkmale in verschiedenen Positionen undOrientierungen uber die Gesamtheit der Bilder verteilt. Der Bezug dieser Parameterzur Lage des Bildhauptpunkte wird untersucht, um ihre Abhangigkeit von der Lagezum Sensorzentrum zu bestimmen. Die Motivation fur diesen Beitrag liegt in derMessung der minimal moglichen Anzahl linearer Merkmale fur eine angemesseneKalibrierung nicht-metrischer digitaler Kameras. Dazu werden alle ermitteltenVerzeichnungsparameter analysiert. Abschließend werden einige Muster von



Testfeldgeraden vorgeschlagen, die eine zuverlassige, einfache und schnellePlumpline-Kalibrierung erlauben. Diese bietet sich dem photogrammetrischenExperten als erster Schritt einer Kamerakalibrierung an, fur Nicht-Experten kannes auch die endgultige Methode fur die Kalibrierung sein.

Resumen

Los parametros de distorsion radial y tangencial se han considerado desde hacemucho tiempo parametros fundamentales en la calibracion de camaras, especial-mente en fotogrametrıa de objeto cercano. Este artıculo expone los resultados de unainvestigacion que utiliza la calibracion analıtica mediante lıneas de plomada paradeterminar los parametros de distorsion radial y tangencial. Para conseguir dichoproposito se han examinado distintos patrones de lıneas rectas en imagenesdistorsionadas. Dichos patrones constan de una, dos, cuatro, diez y veinte rectas, nonecesariamente de plomada. Adicionalmente, se han analizado no solo el numero derectas sino tambien su distribucion en la imagen. Para dicha tarea, se handistribuido entidades lineales en diferentes posiciones y orientaciones sobre toda laimagen. Ademas, se ha estudiado la relacion entre los parametros mencionados y laposicion del punto principal, poniendo de manifiesto la sensibilidad de los mismos aldescentrado con respecto al centro del sensor.

El proposito de este artıculo consiste en medir el mınimo numero de entidadeslineales que aseguren una adecuada calibracion de camaras digitales no metricas.Para ello se determinan los parametros de distorsion radial y tangencial.Finalmente, se recomiendan patrones lineales que hagan fiable, facil de aplicary rapida la tecnica de calibracion mediante lıneas de plomada. Los expertos enfotogrametrıa podrıan usar este metodo como paso previo al procedimiento decalibracion de camaras; los expertos no fotogrametricos podrıan usarlo como unpaso final.



A review and analyses of plumb-line calibration

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