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The Precise Evaluation of Lens Distortion* FRA CIS E. WASHER, National Bureau of Standards, Washington, D. C. ABSTRACT: Four methods of radial distortion measurement are described. Com- parison of results obtained on the same lens using each of the four methods is used to locate systematic errors and to evaluate the reliability of each method. Several sources of error are discussed. It is concluded that a precision of ± 2 microns can be achieved using anyone of the four methods provided adequate attention is given to the reduction of all potential errors to negligible proportions: that is, the P E 8 of D{3 does not exceed 2 microns where D{3 is the value of distor- tion obtained at angle fJ and PE 8 is the probable error of a single determination. 1. INTRODUCTION T HE measurement of radial distortion in the focal-plane of photographic objec- tives has been the subject of intensive study since the advent of mapping from aerial photographs. This particular aberration is of prime interest to photogrammetrists as its magnitude determines the accuracy with which the final negative maintains the correct relationships among the array of point images making up the photography of the cor- responding array of points in the area photo- graphed. It was early realized that the relia- bility of the quantitative information ob- tained from a photograph is increased as the distortion in the camera lens is decreased. Consequently, the development of improved lenses was encouraged with the result that succeeding series of new lenses were character- ized ever lower values of distortion. During this period of change, diverse methods of evaluating the distortion of lenses came into use at various laboratories. The reason for such diversity was primarily the availability of given types of measuring in- struments in various laboratories. There are now three principal methods of measuring distortion, plus numerous additional methods that are either the inverse of one of the prin- cipal methods or a variation thereof. The nodal slide bench is one of the oldest methods; this is a visual method capable of high accu- racy and is perhaps the most widely used. The photographic method is more recent and arose out of the desire to make measurements under conditions approximating those of use. The third principal method is the goniometric, which is used to considerable extent In Europe. Because of the diversity of methods used in evaluation of distortion, it seemed worth- while to investigate the results of measure- ments made in a single laboratory on a single lens by a variety of methods, to determine whether or not the values so obtained varied appreciably \\"ith the method. This has been done using four different methods and a com- parison of the results is reported in this paper. 2.0 METHODS OF MEASUREMENT The methods are as follows: 1. Photographic-Precision lens testing camera. 2. Visual-Nodal slide optical bench. 3. Visual-Inverse nodal slide on T-bench with small aperture telescope. 4. Visual-Modified goniometric using Wild theodolite. Methods 1, 2, and 4 have been described at some length in the literature,1,2,3 while Method 3 4 has been recently developed at this Bureau and is a modification of a very early methodS of measuring lens distortion. In the following sections, a brief description of each method is given. 2.1 PRECISION LENS TESTING CAMERA. METHOD 1 The precision lens testing camera,! shown in Figure 1, was developed at this Bureau by 1. C. Gardner and F. A. Case. It is one of the earliest successful devices developed to meas- ure the performance of lenses by photographic means. It consists of a bank of collimators * This paper was presented at the 1962 St. Louis ACSM-ASP Convention. 327
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Page 1: The Precise Evaluation of Lens Distortion...evaluation of distortion, it seemed worth while to investigate the results of measure ments made in a single laboratory on a single lens

The Precise Evaluation of Lens Distortion*

FRA CIS E. WASHER,

National Bureau of Standards,Washington, D. C.

ABSTRACT: Four methods of radial distortion measurement are described. Com­parison of results obtained on the same lens using each of the four methodsis used to locate systematic errors and to evaluate the reliability of each method.Several sources of error are discussed. It is concluded that a precision of ±2microns can be achieved using anyone of the four methods provided adequateattention is given to the reduction of all potential errors to negligible proportions:that is, the P E 8 of D{3 does not exceed 2 microns where D{3 is the value of distor­tion obtained at angle fJ and P E 8 is the probable error of a single determination.

1. INTRODUCTION

T HE measurement of radial distortion inthe focal-plane of photographic objec­

tives has been the subject of intensive studysince the advent of mapping from aerialphotographs. This particular aberration is ofprime interest to photogrammetrists as itsmagnitude determines the accuracy withwhich the final negative maintains the correctrelationships among the array of point imagesmaking up the photography of the cor­responding array of points in the area photo­graphed. I t was early realized that the relia­bility of the quantitative information ob­tained from a photograph is increased as thedistortion in the camera lens is decreased.Consequently, the development of improvedlenses was encouraged with the result thatsucceeding series of new lenses were character­ized ever lower values of distortion.

During this period of change, diversemethods of evaluating the distortion of lensescame into use at various laboratories. Thereason for such diversity was primarily theavailability of given types of measuring in­struments in various laboratories. There arenow three principal methods of measuringdistortion, plus numerous additional methodsthat are either the inverse of one of the prin­cipal methods or a variation thereof. Thenodal slide bench is one of the oldest methods;this is a visual method capable of high accu­racy and is perhaps the most widely used. Thephotographic method is more recent and aroseout of the desire to make measurements underconditions approximating those of use. Thethird principal method is the goniometric,

which is used to considerable extent In

Europe.Because of the diversity of methods used in

evaluation of distortion, it seemed worth­while to investigate the results of measure­ments made in a single laboratory on a singlelens by a variety of methods, to determinewhether or not the values so obtained variedappreciably \\"ith the method. This has beendone using four different methods and a com­parison of the results is reported in this paper.

2.0 METHODS OF MEASUREMENT

The methods are as follows:1. Photographic-Precision lens testing

camera.2. Visual-Nodal slide optical bench.3. Visual-Inverse nodal slide on T-bench

with small aperture telescope.4. Visual-Modified goniometric using

Wild theodolite.Methods 1, 2, and 4 have been described at

some length in the literature,1,2,3 whileMethod 34 has been recently developed at thisBureau and is a modification of a very earlymethodS of measuring lens distortion. In thefollowing sections, a brief description of eachmethod is given.

2.1 PRECISION LENS TESTING CAMERA.

METHOD 1

The precision lens testing camera,! shownin Figure 1, was developed at this Bureau by1. C. Gardner and F. A. Case. I t is one of theearliest successful devices developed to meas­ure the performance of lenses by photographicmeans. I t consists of a bank of collimators

* This paper was presented at the 1962 St. Louis ACSM-ASP Convention.

327

Page 2: The Precise Evaluation of Lens Distortion...evaluation of distortion, it seemed worth while to investigate the results of measure ments made in a single laboratory on a single lens

328 PHOTOGRAMMETRIC ENGINEERING

FIG.!. Precision lens testing camera (Method 1).

in evaluating the distortion at a given anglef3 is

where Y{3 is the measured distance separatingthe images produced by the lens under test inthe 0° collimator and that in the collimatorinclined at angle f3, and f is the value of theequivalent focal-length based on the separa­tion of the 0° and 5° images and that of the0° and 10° images.

2.2 VISUAL OPTICAL BENCH; DIRECT NODAL

SLIDE. METHOD 2

The visual optical bench has long been thebasic tool for evaluating the constants oflenses. The one used at this Bureau2 is shownin Figure 2. For measuring distortion, a col­limator, nodal slide lens holder, and microm­eter microscope are used. The lens is carefullyaligned in the holder and the axial-imageformed by the lens under test of the illumi­nated reticle of the target is brought into co­incidence with the object-plane of the viewing

spaced at 5° intervals having resolution testcharts as reticles. The lens under test ismounted at the center of convergence of thecolli mator fan, and can be aimed at anyone ofthem by rotation of a carriage which carriesthe lens holder and camera back whereon thephotographic recording plate is mounted. Aspresently constituted, the lens testing camerahas 10 collimators covering a total angle of45°. When used to test wide-angle lenses, thecamera is aimed at one of the extreme collim­ators (Position I) and the test made. In orderto cover a complete diameter, a second test ismade with the camera aimed at the collimatorat the opposite extreme (Position II). It hasbeen found by experience that the resultsobtained from two negatives made in thismanner are quite reliable.

From the measured separation of theimages recorded on the test negative madewith a given lens, it is possible to determineboth the equivalent focal-length f and thedistortion D{3. The determining equation used

D{3 = Y{3 - f tan {j, (1)

FTG. 2. Visual optical bench. Direct nodal-slide (Method 2).

Page 3: The Precise Evaluation of Lens Distortion...evaluation of distortion, it seemed worth while to investigate the results of measure ments made in a single laboratory on a single lens

PRECISE EVALUATION OF LENS DISTORTION 329

FIG. 3. Inverse nodal-slide (Method 3).

2.3 INVERSE NODAL SLIDE. METHOD 3

In the inverse nodal-slide method4, shownin Figure 3, an illuminated target, preferablya point source, is placed at the rear focal-pointof the lens under test. Light from the point­source emerges from the lens in a collimatedbeam which enters the viewing telescope. Bya series of adjustments, a condition is foundfor which a smal1 rotation of the lens about avertical axis does not produce an angular dis­placement of the collimated beam observablethrough the viewing telescope. The rearnodal-point of the lens is then in close co-

microscope. By a series of successive adjust­ments, a condition is found for which a smal1rotation of the lens about a vertical axis doesnot produce a displacement of the axial­image viewed. The rear nodal-point of thelens is then considered to coincide with thecenter of vertical rotation of the nodal-slide.

Assuming the equivalent focal-length f tobe known, the nodal-slide is rotated byamoun t fJ about the vertical axis using thecalibrated circle of the nodal-slide to positionit exactly. The entire saddle carrying thenodal-slide and lens is then moved away fromthe microscope toward the col1i mator by anamountf(sec fJ-l). The viewing microscope isshifted laterally to the new position of theimage and its dial read and recorded as read­ing RfJ. The nodal-slide is then rotated toposi tion -fJ and a second setting of the micro­scope, R-fJ , is made. These displacemen ts aremeasured in a plane normal to the chief raysinclined to the axis at angles +fJ and -fJ,respectively. The distortion DfJ is obtainedfrom the relation

where LHfJ=f+fJ-LfJ, and f is the equivalentfocal-length of the lens.

(4)

(3)t::.EfJ

Dfj = -fsec2 {32

DfJ = rfj - f tan {3.

3.0 RESULTS OF MEASUREMENT

An extensive series of measurements in asingle wide-angle lens using each of these fourmethods was made and values of distortiondetermined with each method. The results of

2.4 MODIFIED GONIOMETER. METHOD 4

The goniometer method, il1ustrated inFigure 4, used in this laboratory does notemploy a specially constructed camera goni­ometer such as is used in Europe, but morenearly approximates the method used byMerritt3• A calibrated linear-scale is placed inthe focal-plane of the lens under test and thescale is viewed through the front of the lenswith a precision theodolite. The angular dis­placements fJ of selected points on the scaledistant rfJ from the center line on the cali­brated scale are measured using the precisioncircle of the theodolite. The distortion DfJ isobtained from the relation

incidence with the center of rotation of thenodal-slide. The nodal-slide bearing the lensunder test is then rotated by amoun t fJ aboutthe vertical axis, the target reticle is movedaway from the zero position by amountf(secfJ-l), the poin tingoftheviewingtelescopeis adjusted to compensate for change in direc­tion in the collimated beam emergent fromthe lens under test, and the setting of thetransverse micrometer is recorded. This read­ing is 4fj. The process is repeated with thelens rotated to position -fJ, and the readingLfJ is recorded. The distortion Dfj is obtainedfrom the relation

(2)t::.Rfj

DfJ = -- sec (32

where t:J.RfJ = ·RtfJ - R_fj.

Page 4: The Precise Evaluation of Lens Distortion...evaluation of distortion, it seemed worth while to investigate the results of measure ments made in a single laboratory on a single lens

330 PHOTOGRAMMETRIC ENGINEERING

FIG. 4. Modified goniometer (Method 4).

In addition, the referral of the values of D{3to the calibrated focal-length removes fromconsideration errors in distortion arising fromerrors in equivalent focal-length. Hence,throughout the paper values of distortion arereferred to the calibrated focal-length for eachof the four methods.

3.2 PRECISION OF MEASUREMENT

The accuracy of the final accepted values ofD{3 obtained by any of the four methods isdependen t upon the magni tude of random andsystematic errors9 that may affect the meas­urement. For the most part, the magnitude ofrandom errors may be estimated either on thebasis of past experience or from analysis of theoriginal data taken during the course ofmeasurement. The systematic errors are lessamenable to evaluation as they arise in manyinstances from defects of various nature inthe equipment itself or from errors in calibra­tion of the measuring devices. It sometimespossible to detect this type of error fromanalysis of measurements of the same quan­tity using different methods.

It is clear that the values of the distortion

O~--..-=-c::.-----------+---l

,---,."./ \

/ \/ \

\1\

2 \\\k

30

{3 IN DEGREES

Dp BASED ON E.F.L

Oil BASED ON CFL.

o

100

-100

O~::::;t;:::==--=-_::::r...:::.--------l(\_-j---X lil--

(f)

zo0::(,)

:z~

0.... -100'------'------'------'---1

Zof=~ 100I-(f)

a

FIG. 5. Comparison of values of distortion, D{3as a function of (3 by Method 1 (Precision lens test­ing camera) and Method 2 (Direct nodal-slidebench).

In the upper frame, values of D{3 are referred tothe equivalent focal-length (EFL) while in thelower frame the values of D{3 are referred to thecalibrated focal-length (CFL).

these measurements (for a lens having a focal­length of 150 mm.), were reported in a seriesof papers'·6,7 which give a detailed discussionof each. The primary purpose of this paper isto presen t an overall picture of the problemsencoun tered and the sol utions thereof.

3.1 THE CALIBRATED FOCAL LENGTH

In comparing values of distortion obtainedby two methods such as Methods 1 and 2, thedistortion must be evaluated in such a mannerthat corresponding values are readily compar­able. For example, values of distortion ob­tained by Methods 1 and 2 are shown inFigure 5. In the upper frame, the values arebased upon the equivalen t focal-length (EFL)in each case. However, the equivalent focal­length used in Method 2 approximates theparaxial value while that used in Method 1 isthat value which makes D{3=O at {3=approx­imately 7.5°. It seems more reasonable there­fore to refer the values in both cases to acalibrated focal-length (CFL)8 such thatD3So= -D,so; this has the advantage of en­suring that the maximum positive valueequals minus the maximum negative value ineach case. The effect of this transformation isshown in the lower frame of Figure 5. It isevident that this transformation improvedthe si tua tion for the 35° to 45° region al thoughthe disparity of values of D{3 was increased inthe 5° to 30° region.

Page 5: The Precise Evaluation of Lens Distortion...evaluation of distortion, it seemed worth while to investigate the results of measure ments made in a single laboratory on a single lens

PRECISE EVALUATION OF LENS DISTORTION 331

obtained with the aid of Equations 1 through4 will be affected by errors in r, {3, I::>.R, and I::>.~.

In general, values of the probable error of asingle determination PEs do not exceed ± 1micron for measurements of rand I::>.R, and itis believed the systema tic error in thesevalues arising from calibration errors does notexceed ± 1 micron. In addition, values ofPE. do not exceed ± 1 second for measure­ments of {3 in methods 1 and 4; and values ofPE. do not exceed ±1 second for measure­men ts of I::>.~ in Method 3. Hence for anyone ofthese methods, the precision of measure men tis such that values of PEs of DfJ should bewi thin ± 2 microns throughou t the range of{3 = 0° to {3= 45°.

T

- D, + D. + Da + IJ,DfJ = ~----4---

where the subscripts indicate the method usedin determining this particular value of DfJ. The

3.3 COMPARISON OF RESULTS FOR THE FOUR

METHODS

Following evaluation of the distortion DfJfor a preliminary series of measurements madeby each of the four methods, an average Dfjwas obtained for each value of {3 using therelation

-I

_0.20

departure from the average I::>.DfJ" was deter­mined for each method using the relation

!!.D~n = DfJ - DfJn

where n indicates the method number andranges from 1 to 4. The magnitudes of thesedepartures from the average are shown foreach method in Figure 6. It is obvious that theobserved departures are considerably in excessof ± 2 microns although well within the toler­ance of ± 20 microns claimed as the accuracyof measured val ues of distortion prior to theadvent of nearly distortion-free lenses.

These measure men ts were all made in onelaboratory so that it was easier to isolatepossible causes of systema tic error and deter­mine their magnitude. After careful study andanalysis, it was concluded that the values ofDfJ obtained by Method 2, the direct nodal­slide method, were perhaps least affected bysystematic errors. The departures of themagnitude of the values of I::>.DfJ for Method 1can be substantially reduced by correctingfor a small amoun t of plate curvature or

O· IS'" 30· 45'"

ANGULAR SEPARATION FROM AXIS IN O~GREES

FIG. 7. Variation of distortion for incorrect andcorrect relative locations of the lens-aperture withrespect to the aperture of the viewing telescope asa function of angular separation {3 from the axis.

The values of the distortion referred to theequivalent focal-length are shown in box A; thevalues of the distortion referred to the calibratedfocal-length are shown in box B. The curvesmarked 1 show the maximum incorrect values ob­tained; the curves marked 2 show the minimumincorrect values; while the curves, marked 3, showthe normal or correct values of the distortion. Thevalues are for a wide-angle lens having a nominalfocal-length of 150 mm and maximum aperturef/6.3.10

0

10

.,Cg"e0- 10 3<J

0

-10

-20

0

FIG. 6. Variation of the departures !!.lJfJ from theaverage value DfJ for the four methods as a func­tion of {3 for each method.

The numbers in the upper left of each framespecify_the method for which the departures!!.DfJ = DfJ - Dpn are shown. Thus, Frame 1 gives thecomparison for the precision lens testing camera;Frame 2 for the direct nodal-slide; Frame 3 for th'~

inverse nodal-slide; and Frame 4 for the modifiedgoniometer. This figure shows the magnitudes ofthe values of !!.DfJ prior to the adjustment of vari­ous errors.

Page 6: The Precise Evaluation of Lens Distortion...evaluation of distortion, it seemed worth while to investigate the results of measure ments made in a single laboratory on a single lens

332 PHOTOGRAMMETRIC ENGINEERING

4. COKCLUSIONS

On the basis of the findings made in thisstudy, it seems probable that a precision,

differential plate tipping induced by warpingof the camera back of the precision lens test­ing camera.

Analysis of values of distortion by Method3 showed that substantial changes in distor­tion values are produced by unsymmetricalapertures. Some idea of the magnitude of thischange is gi ven in Figure 7. I n fact, the chiefvalue of Method 3 is its usefulness in evaluat­ing errors produced by asymmetrical use ofthe lens aperture.

The values of DfJ determined by usingMethod 4 were found to be affected by curva­ture of the scale used in the focal-plane and byasymmetrical use of the aperture.

Additional measurements were made by allmethods with special emphasis placed on re­ducing these causes of error. In addition, ad­justment of the final values was made to com­pensate for residual errors of plate curvaturewhere evident.

Values of departures from the average,I1DfJn, for each of the four methods deter­mined after making these corrections areshown in Figure 8. It is clear that a substan­tial improvement was effected. The averageprecision index, given as probable error of asingle determination PE., has a value of ±2microns for each method. I t is worthy of notethat 25 of the 36 values of I1DfJn do not exceed± 2 microns and only one value of I1DfJn is asgreat as 6 microns.

5. REFERENCES

1. Gardner, I. C. and F. A. Case, "Precision Cam­era for Testing Lenses. J. Research NBS, 18,449 (1937) RP984.

2. Washer, F. E. and W. R. Darling, "FactorsAffecting the Accuracy of Distortion Measure­ments made on the Nodal Slide Optical Bench,"J. Opt. Soc. Am., 49 #6, 517-534, June (1959).

3. Merritt, E. L., "Methods of Field Camera Cali-bration, PHOTOGRAMMETRIC ENGINEERING,XVII, 610 (1951).

4. Washer, F. E., W. R. Darling, "Evaluation ofLens Distortion by the Inverse Nodal Slide."J. Res. NBS, Vo!. 63C, No.2 (1959).

5. Bennett, A. H., "Aberrations of Long FocusAnastigmatic Photographic Objectives: Sci.Pap. BS, 19,587 (1923-24) S494.

6. Washer, F. E., W. P. Tayman, and W. R.Darling, "Evaluation of Lens Distortion byVisual and Photographic Methods." J. Re­search NBS, 61, No.6 (1958) RP 2920.

7. Washer, F. E., W. R. Darling, "Evaluation ofLens Distortion by the Modified GoniometricMethod. J. Res. NBS, Vo!. 63C, 10 . 2 (1959).

8. Washer, F. E., and F. A. Case, "Calibration ofPrecision Airplane Mapping Cameras: PHOTO­GRAMMETRIC ENGINEERING, XVI, 619 (1950);J. Research NBS 45, 1-16 (1950) RP 2108.

9. \,yasher, F. E., "Sources of Error in VariousMethods of Airplane Camera Calibration,"PHOTOGRAMMETRIC ENGINEERING, XXII, p.727 (1956).

(PE.), of ± 2 microns in the measured val uesof lens distortion can be achieved with any ofthe four methods provided adequate attentionis given to reducing all potential errors tonegligible proportions. For all methods it isdesirable to measure distances to ± 1 micronand angle to ± 1 second of arc. Errors arisingfrom plate or scale warpage should be reducedto the exten t that they do not exceed ± 1micron. In addition, if the measured values ofdistortion are to be accurate as well as preciseto the nearest 2 microns; it is desirable thatthe measurements be made by not less thantwo independent methods and that the cor­responding values for each of the two methodsagree within ± 2 microns.

Finally, while it may be possible to achievethis level of accuracy in the measuremen t ofdistortion along a single radius of a lens, itfrequently happens that the value of distor­tion, DfJ' for a given value of (3 may vary withazimuth to an extent far in excess of the pre­cison of measurement. Thus it would appearthat if it be necessary to know the distortionwithin ± 2 microns, it would be necessary tomake measurements along many radii. Fromsuch measurements, one could then make achart similar to a map which would show themagnitude of the distortion at any selectedpoint in the entire picture area.

I

v

I

2

4

30 45 0/3, DEGREES

15

3

o

FIG. 8. Variation of tiDfJ with fJ for eachof the four methods after corrections.

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10

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