U. S. Department of CommerceNational Bureau of Standards

Research Paper RP2108Volume 45, July 1950

Part of the Journal of Research of the National Bureau of Standards

Calibration of Precision Airplane Mapping CamerasBy Francis E. Washer and Frank A. Case

An instrument is described that permits the registration of all the information necessaryto calibrate a precision airplane camera on a single negative. Twenty-five collimatorsarranged at 7.5° intervals along two meridians at right angles provide optically distant tar-gets. These targets are photographed by means of the camera to be tested. Measurementsmade on the negative yield information on the equivalent focal length, distortion, resolvingpower, prism effect, orientation of the lines joining opposite pairs of collimation indexmarkers, and location of the principal point. This instrument was designed and built tomake the calibration of precision cameras required for all precision cameras used in Govern-ment mapping projects. A brief account of the calibration of a typical camera and a dis-cussion of the physical significance of calibrated focal length are given.

I. Introduction

The number of precision airplane mappingcameras submitted to the National Bureau ofStandards for calibration has increased steadilysince the first formal specification covering thistype of camera was promulgated by the UnitedStates Department of Agriculture in March 1940.[I].1 The demand for calibration of precisionmapping cameras has increased so rapidly since1945, that new equipment [2] for this work hasbeen developed and built to supplement the pre-cision lens testing camera that has been in usesince 1935 [3]. The precision lens testing camerawas primarily developed for use in photographi-cally determining the equivalent focal length,distortion, and resolving power of lenses mountedin barrels or shutters. The instrument was lateradapted to measuring these optical quantities forlenses mounted in cameras and further to locatethe principal point in airplane mapping cameras[4].

With increased volume of work, it was evidentthat the method of calibration involving use ofthe precision lens testing camera possesses severaldeficiencies. First, four negatives are required,the making of which is a time consuming opera-tion. Second, different sizes and shapes of the

* Figures in brackets indicate the literature references at the end of thispaper.

various makes of cameras cause difficulties inmounting them for test. Third, the entire fieldof the camera cannot be covered with a singlephotograph. The new camera calibrator wastherefore designed to remedy these difficulties.

Inasmuch as the optical characteristics of lensesintended for use in airplane mapping cameras canbe measured on the old precision-testing cameraprior to their installation in cameras, the newcalibrator was designed primarily as a pointinginstrument. It provides 25 beams of parallellight from known directions. The informationderived from it is therefore mainly of a directionalnature and includes data on the uniformity orlack of uniformity in the distortion characteristicsof lens-cone combinations. It permits morerapid and accurate location of the principal point,together with a determination of the equivalentfocal length of the lens as mounted in the camera.It provides quantitative information on themagnitude of the prism effect and tangential dis-tortion. Moreover it provides all this informationon a single negative, thus avoiding possible changesthat result from small movements between suc-cessive exposures. It provides a more stablesupport for the camera during test and alsogreater versatility in the types of cameras thatcan be tested.

The scope of the present paper is limited to adescription of the instrument and the instruments

Precision Camera Calibration 1

Hi I «

FIGURE 1. Collimator bank of camera calibrator.

Collimator bank before installation in the instrument. It consists of 25 col-limators arranged in the form of a cross. When mounted in the cameracalibrator, the central collimator points vertically upward with the remaining24 collimators spaced at 7.5° intervals along the two meridians.

developed for use in its calibration. In addition,the calibration of a precision airplane mappingcamera as conducted on the new calibrator isdescribed. The manner of interpreting the nega-tive is given for completeness.

II. Description of the Camera CalibratorThe camera calibrator and the details of its

construction are shown in figures 1 and 2. Figure Ishows the heart of the new instrument, which isa bank of 25 collimators arranged in the form of across. In use, it is mounted beneath a table asshown in figure 2. The central collimator pointsvertically upward, and the remaining 24 collim-ators arc spaced at 7.5° intervals along the fourarms of the cross from 0° to 45°. Departure fromthe 5° interval used in the Bureau's lens testingcamera [3] is necessitated by space limitations.The collimators are of the fixed-focus type withthe tubes cut to proper length so that each reticlelies in the focal plane of the corresponding objec-tive. The collimators are mounted in a specialcasting, whose inner and outer envelopes arespherical. The collimator axes are normal to thespherical surfaces and correspondingly pointtoward the center of curvature. Accuracy ofpointing of the collimators depends to a largeextent upon the accuracy of counlerbored holesand flanges by which the collimators are mounted,

but the reticles may be moved about a little inthe focal planes of the collimators so as to correctsmall deviations in the pointings. These adjust-ments are made with capstan screws, so arrangedas to afford complete translational freedom of thereticle. After adjustments, clamping by appropri-ate screws guards against any subsequent move-ment. The clear apertures of the collimators are18 mm in diameter, which is somewhat smallerthan those of the lenses commonly used on pre-cision mapping cameras. This, however, does notimpair the instrument so far as its main purpose isconcerned. To guard the collimators againstdisturbance the lamp housings, mounted inde-pendently on the arms of the cross, completelysurround the target end of the collimators butdo not touch them at any point. A 6-v. frostedflashlight bulb serves to illuminate the reticle.Provision is made for the insertion of a filter andglass diffusing screen between the lamp and thetarget.

The complete instrument is shown in figure 2.The table top is a flat steel plate having a large

FIGURE 2. Camera calibrator.

Instrument as arranged for use. The oamera under test points verticallydownward. Its proper orientation is effected by the autooollimatlngtelescope and the plane-parallel piece of glass placed on the focal plane ofthe oamera. Images on a finished negative appear every "..r>° along eachdiagonal, with the oollimatlon index markers registered on each side of thenegative.

Journal of Research

circular opening in its center. The collimatorbank is mounted beneath the table and centeredwith respect to this opening in the table top.The camera holder consists of a tripod placedover the central opening in the table. A circularopening in the center of the tripod permits thelight from the 25 collimators to reach the lensof any camera placed on the holder. The focalplane of the camera can be adjusted to normalitywith the axis of the central collimator with theaid of the adjusting screws on the tripod and theautocollimating telescope, whose axis is bent90° by the prism mounted in front of its objective.The adjusting screws also provide a limited amountof vertical adjustment to compensate differencesin distance between the front surface of the cameralens and the front end of the camera of the varioustypes of camera. Four flashlight bulbs mounted onadjustable arms reaching out into the circularopening of the table top serve to illuminate thecollimation index markers of the camera under test.

I l l Test Chart

The test charts of the new instrument includeresolving power patterns, although the limited sizeof the collimator objectives precludes the evalua-tion of the resolving power of the camera lens atits maximum aperture. However, the inclusionof the chart does permit study of the resolvingpower at reduced aperture. Since a new testchart for use in the precision lens testing camerahad been under consideration for some time, itwas decided to make a new chart at this time andto use a reduced version of it as the target reticleof the camera calibrator. This new test chart isshown in figure 3. The chart was first made on alarge scale and then photographed to the desiredsize, using Eastman high resolution plates, type649GH. The size of the target in the 0° collimatoris 8 mm square. Large scale charts are also madeand subsequently reduced for each of the variousangular separations from the axis, so that thecosine correction in the vertical dimensions andcosine-squared correction in the horizontal dimen-sion of each target pattern can be made. Thesecorrections are necessary in order that the corre-sponding patterns will all be imaged at the samesize regardless of angular separation from theaxis for a given lens-camera combination. Thethree-line patterns reduce in size in a geometricalprogression, the ratio being \^2, or approximately

1.189. Closer steps in the series are admittedlydesirable, for example \/2, but this smaller ratiocould only have been achieved at the expense ofcompressing the range of permissible values be-cause of the small size of the reticle. The spacingspresent in the patterns in the reticle range from2.3 to 161 lines per millimeter in 26 steps. Thesepatterns, when imaged by the lens under test, arechanged by the ratio of the focal length of collima-tor lens and lens under test. Table 1 shows therange of resolution in the image plane for the threefocal lengths most frequently encountered. Thenominal maximum /-number of the lenses aregiven. Since the diameter of the collimator lensreduces the effective aperture of the lens undertest, the effective operating /-number is given foreach lens. The line showing limit of resolutionshows the maximum theoretical resolving power*for each lens at its effective /-number. With thischart therefore, the upper limit of resolution isdetermined by the performance of the lens andlimitations of the emulsion of the photographicplate and not by insufficient range of the resolvingpower patterns. A considerable amount of workhas been done in other laboratories, wherein acircular target is used instead of a line target.

o o=•111o o=•111O O

= 111o o

o o

o o

o o= 111o o= 111o o

O O

= 111

FIGURE 3. Test chart.

o o

The test chart that is used in preparing the reticle for use in each collimatorconsists of patterns of parallel lines in two orientations with spaces varyingin geometric progression by steps equal to - ^ . The circle targets are includedfor comparative study of line and circle targets.

Precision Camera Calibration

TABLE 1. Limits placed on effective f-number by the diam-"• eter of the collimator objectives of the camera calibrator forthree lenses of different focal lengths; the maximumtheoretical resolution of these lenses operating at thesef-numbers, and the range of the resolving powers obtain-able for these different focal lengths provided by the testcharts of the camera calibrator

E quivalent focal length, mmUsual nominal /-numberE ffective /-numberMaximum theoretical resolution in lines

per mm, as determined by effective/-number

Minimum resolution in lines per mmprovided by test chart

Maximum resolution in lines per mmprovided by test chart

1306.37.2

198

4.7

332

1506.38.3

172

4.1

289

2106.8

11.7

122

2.9

205

A series of circular patterns is therefore includedin the test chart for comparison of results obtainedwith it with the results obtained with the 3-linepatterns.

The cross in the center of the chart is the fiducialmark with reference to which the angles separatingthe collimators are measured. When imaged ona camera test negative, the measured separationsof the crosses for various collimators serve for thedetermination of equivalent focal length anddistortion.

The National Bureau of Standards is at presentusing a high-contrast target and high-contrastfine-grained plates in the testing of photographicobjectives and airplane mapping cameras. Thisis being done because existing Government speci-fications require that lenses for use in aerial map-ping projects be tested under these conditions.It is recognized that there may be merit in theprocedures followed in other laboratories thatrequire the use of low-contrast targets and fastemulsions comparable to those used in actualaerial photography. Investigations are now inprogress at this Bureau on the effect of contrastof target on performance of lenses. When thisinvestigation is concluded, it is probable thai adifferent type of tar-get will be recommended foruse in the testing of photographic objectives andcameras with corresponding changes in the re-quired values of the resolving power for certifica-tion of lenses for Government use.

IV. Calibration of the Instrument

Adjustment of the angular relationships andmeasurement of the angles separating the colli-

mators in each meridian is necessary before theinstrument can be used in the calibration of pre-cision cameras.

1. Adjustment of Angular Relationships

It is required that the angles between corre-sponding collimators on opposite sides of the centerbe as nearly equal as possible, and moreover thatthe axes of the beams of parallel light emergingfrom the collimator in a given meridian be parallelto and in near coincidence with a common plane.This adjustment is made before the collimatorbank is mounted under the table. The collimatorbank is placed face downward over the openingin the table top, as shown in figure 4. The lamphousings are removed to expose the target reticles.A front surface mirror is placed at the center ofcurvature of the spherical casting. The mirror,shown at A, is mounted in a special device thatpermits easy leveling and rotation about a hori-zontal axis.

A microscope equipped with a vertical illumina-tor is mounted over the reticle of the centralcollimator, and the mirror surface is brought intoa position normal to the axis of the centralcollimator by means of the leveling screws, withthe collimator, mirror, and microscope serving asan auto collimator. If now the —7.5° reticle onone side is illuminated with the desk lamp, the

FIGURE 4. Arrangement of apparatus for adjusting the<i ngular relationsh ii>*.

Journal of Research

second microscope may be used to view the reticleof the +7.5° collimator on the opposite side whileadjusting the reticle to bring it into coincidencewith the reflected image of the —7.5° reticle.The two 7.5° angles are then equal, and the axesof the +7.5°, 0°, and —7.5° collimators fall inone plane. The microscope with the verticalilluminator is then brought over the +7.5°reticle and the mirror rotated and made normalto the axis of this collimator. The +15° and+22.5° collimators are then adjusted. The proc-ess is repeated by using the +22.5° collimator asthe auto collimator to adjust the +30°, +37.5°,and +45° collimators. On returning and usingthe 0° collimator as autocollimator, all of theremaining collimators on the same meridian canbe adjusted. Following this adjustment it canbe stated that the axes of all collimators in agiven meridian lie in the same plane, oppositeangles are equal, and moreover angles betweenadjacent collimators are equal. The entire processis repeated to adjust the collimators on the meri-dian at right angles to the first meridian. Itmust be mentioned that, although the anglesseparating collimators along a given meridianare equal, the corresponding angles betweencollimators in different meridians are not exactlyequal. The accuracy of the initial boring of thecasting is attested by the fact that no difficultywas encountered in equalizing the angles andmaking the collimators coplanar, although theamount of lateral adjustment available was lessthan 1 mm.

2. Measurement of the Angles

The angles between adjacent coUimators aredetermined by a comparison method using areflecting biprism and telescope, as shown infigure 5. The standard of comparison is a biprismhaving a known angle between the two surfacesforming the obtuse angle. The two surfaces area l u m i n i z e d a n d f o r m t w o m i r r o r s t h a t m a i n t a i n aconstant angle with respect to one another. Theangle formed between one Surface and the exten-sion of the o ther is 3.7600°. Ideally this angleshould be one-half of the 7.5° angle between thecoll imators, but its present value is sufficientlyclose, and moreover (he a m o u n t of depart ure from;>.7.r)° is known. The biprism is mounted in a,special device that permits rotation of the biprisma b o u t a h o r i z o n t a l a x i s c o i n c i d e n t w i t h t h e e d g e

separating the two aluminized surfaces. As shownin figure 5, this device is mounted on the cameraholder in such a manner that the edge of thebiprism coincides with the center of curvatureand is normal to the plane in which the collimatoraxes for a given meridian lie. The error thatarises from unsymmetrical use of the collimatorobjectives is made negligibly small by the use ofcollimator objectives that have very small longi-tudinal spherical aberration and making certainthat the reticles lie in the focal plane of theobjectives.

To measure an angle, the telescope, equippedwith a micrometer for varying its pointing, issighted at the center line of the biprism. Thebiprism is rotated about its edge until the image

7

FIGURE 5. Reflecting biprism for measurement of angles.

reflected by one face of the biprism of the reticlein one collimator coincides with the cross-hairsof the viewing telescope. T h e lower telescopemicrometer reading is recorded. T h e reticle ofthe first collimator is darkened and the second oneilluminated. T h e image reflected by the secondface of the biprism then appears in the field ofview of the telescope. If the angle betweencollimators is exactly twice the biprism angle, noshift- is evident . If the second image is shifted,the Lower telescope micrometer is used to correct

ihe pointing and the difference in micrometerreading noted. The difference in micrometerreading for the two pointings when translatedinto angular shift gives the difference in angleb e t w e e n , t h e t e l e s c o p e a n d t w i c e t h e b i p r i s m a n g l eB y rotating the biprism, the angles separating

Precision C a m e r a Calibration

TABKL 2. Values of the individual angles separating adjacent collimators and their deviation from 7.5° for each collimator bank

Collimator bank. _ _

Nominal angle

0 to 7.57.5 to 1515 to 22.522.5 to 3030 to 37.537.5 to 45

I

Angle

7.49637.49727.49857.49377.49407.4981

Deviation

-0.0037-.0028-.0015-.0063-.0060-.0019

II

Angle

7.49467.49787.49607.49557.49307.4992

Deviation

-0.0054-.0022-.0040-.0045- . 0070-.0008

III

Angle

7.50327.50387.50617.50217. 50697.5070

Deviation

+0.0032+.0038+.0061+.0021+ . 0069+.0070

IV

Angle

7.50417.50387.50967.51387.50807.5082

Deviation

+0.0041+.0038+.0096+ . 0138+.0080+.0082

each successive collimator can be determined forall the collimators lying in the meridian. Thebiprism holder and viewing telescope are thenrotated 90° about a vertical axis, and the aboveprocess is repeated to obtain the angular separa-tions of the collimators in the second meridian.The measured values of the angles are given intable 2 to show the agreement between the variousangles, together with their departure from 7.5°.The tolerance on angle for boring the casting was± 1 min. (±0.0167°); the table shows that theresults obtained indicate the accuracy of boringwas amply close.

V. Operation of the Camera CalibratorTo calibrate a camera on the instrument, one

must be sure that the focal plane of the camera isnormal to the axis of the central collimator. Todo this the operator adjusts the micrometers onthe viewing telescope until the image of thecenter cross on the 0° collimator coincides withthe cross-hairs of the viewing telescope. Thecamera cone is then placed on the camera holder,adjusted laterally until the lens of the camera iscoaxial with the 0° collimator. The camera isthen adjusted vertically until the images formedby the outer collimators show a minimum ofvignetting, the camera cone is rotated until thetwo diameters along which the two perpendicularrows of images fall coincide with the diagonals ofthe focal plane. The camera is then clamped toprevent its further motion. An optically flatplane-parallel glass plate aluminized on its uppersurface, is placed on the focal plane of the camera.Then using the viewing telescope as an auto-collimating telescope, the focal plane of the cam-era is adjusted to normality with the axis of theautocollimating telescope by small adjustments ofthe screws supporting the camera holder. Flash-light bulbs attached to the ends of small rods

supported by the table top are then moved aboutuntil the light from the bulbs illuminate each'iofthe four collimation index markers mounted in thecamera cone. The rods are then clamped. Theplane parallel is removed; the room is darkened,and a photographic plate is placed on the focalplane of the camera, and an exposure is made.It is customary to place a heavy piece of flat glassover the photographic plate during exposure tohold the plate in place and help to flatten it.

The finished negative shows the images of thetargets every 7.5° from the center to the corneralong each diagonal. A schematic drawing of afinished negative is shown in figure 6. Thecollimation markers are registered on the sides ofthe negative, so that there is little likelihood of thetarget images being fogged by the light thatregisters the markers.

FIGURE 6. Schematic drawing of test negative obtained withcamera calibrator.

6 Journal of Research

Eastman V-G spectroscopic plates are generallyused. In those instances, where a precisioncamera is to be used for infrared photography,Eastman IV—R spectroscopic plates are used.

VI. Calibration of a Precision Camera

Calibration of a precision camera consists ofthe location of the principal point with respectto the collimation index markers, determinationof the angle formed by the intersection of linesjoining opposite pairs of collimation index markers,determination of the equivalent focal length ofthe lens as mounted in the camera, determinationof the distortion, and resolving power from thecenter to the corner of the image plane. Detailrequirements relating to all these quantities arecontained in specifications set up by variousgovernmental mapping agencies concerning cam-eras that are to be used in Government mappingprojects [5]. It is further stipulated that precisiontype cameras intended for use in these projectsbe tested for compliance with specifications bythis Bureau.

It frequently happens that a camera on finalexamination does not comply with the require-ments, but that it can be brought into compliancewith the requirements by small lateral movementsof the lens with respect to the collimation indexmarkers or by small movements of the markersthemselves. Following these adjustments, a re-check of the camera shows that it complies. Toensure that nothing can then happen to changethis adjustment, holes are drilled and dowelsplaced so that the movable members remain fixedwith respect to one another.

VII. Interpretation of the Negative

With the new camera calibrator, all of theinformation necessary lo evaluate performanceof the lens-cone combination is contained in asingle negative. However, it is customary tomake two negatives and combine the results ofmeasurements from these, to minimize any pos-sible error. A schematic drawing of a typicalnegative is shown in figure (». The markers aredesignated A, li, C} and I) for convenience. TheLetter A is reserved for the markers bearing thearrow indicating direction of Might. The Romannumerals / , / / , / / / , and IV serve lo designatethe TOWS of images formed by the four collimalor

FIGURE 7. Equipment used in the determination of the 90°condition.

banks radiating out from the center. The centralimage formed by the 0° collimator is designatedthe center cross (C. C). The intersection oflines joining opposite pairs of collimation indexmarkers is designated the center of collimation,(CoiC).

1. Determination of the 90° Condition

The first measurements on a given negative aremade to determine whether or not the lines AB andCD are perpendicular within ± 1 min. This del er-mination is made on a special device shown infigure 7.2 The central feature of this device con-sists of a piece of plate glass bearing four shortradial diamond lines cut in the surface near the endof a 4.5-in. radius. The error in the positioningof the lines does not exceed ± 5 sec. This plate isembedded with the lines uppermost in the metalframe that bears the plate clamps, so that the sur-face of the metal frame and the upper surface ofthe glass lie in the same plane. The negative isplaced, emulsion side down, over the glass plate insuch a manner that the index lines of the collima-tion markers and the four diamond lines are in nearcoincidence. The negative is then clamped down,so that no relative movement can occur diningmeasurement. The metal frame is mounted on aspindle, so thai a given marker can be broughtunder the viewing microscope. The Separation of

• The authors wish to express appreciation to William r. Tayman, whodeveloped this device for holding the 90s standard plate and the negative inoontaot ciiiiint; measurement, which represents ;t definite advanoe over theearlier equipment used in making this determination.

Precision Camera Calibration886334r- 50 2

diamond line and index line on the negative arethen measured. This process is repeated for theremaining three markers. From these four meas-urements, the departure and the direction of de-parture of the collimation index markers from the90° condition can be determined.

2. Location of the Center Cross with Respect tothe Center of Collimation

The center cross, as used herein, is the pointwhere an infinitely distant object point lying on aline perpendicular to the focal plane is imaged bythe camera lens. The coordinates of the centercross are determined with respect to the rectan-gular coordinate system formed by the linesjoining opposite pairs of collimation index markershaving the center of collimation as origin. Thesemeasurements are made with the aid of the deviceshown in figure 8. The negative is placed,emulsion side uppermost, upon the flat metalplate and so oriented that the line AB is ap-proximately parallel to the bench ways uponwhich the slide rests. The negative is thenclamped to prevent any movement with respect tothe metal plate. By moving the slide along thebench ways, the measuring microscope can be setin turn on marker A, center cross, and marker Band readings taken. From this data the lateraldisplacement of the center cross from the line ABcan be obtained. The negative is then rotated

C.ofC '

C.C.V

ab

FIGURE 8. Equipment used in locating the center cross withrespect to the center of coll / mill ion..

FIGURE 9. Schematic drawing showing the center cross withrespect to the center of the collimation.

90° and similar settings made to determine thelateral displacement from the line CD. Theresults of measurement on a typical negative areshown schematically in figure 9. The magnitudesof the displacement are exaggerated for purposesof clarity in the figure.

3. Location of the Principal Point with Respect tothe Center Cross

The principal point of precision-type aerialmapping camera is defined as that point where aperpendicular dropped from the near nodal pointof the lens meets the focal plane [6]. For an ideallens, the principal point and the center crosscoincide. In practice, however, the center crossis shifted away from the principal point eitherbecause of prism effect in the Jens or because ofn o n p a r a l l e l i s m of the sur faces of t h e filter o n t h ef r o n t o f l l i e l e n s , w h i c h f o r m s a p a i l o f I l i e o p t i c a lsystem of the camera. Th i s is illustrated infigure 10, which is a schematic drawing showingthe displacement of the axial ray and rays inclined;il jingle jtf wit li the axis before and after placing aprism in front of an ideal lens.

In figure 10, 0 is the principal point and is thepoint where light from an infinitely distant objectpoint lying on the axis of an ideal lens would heimaged if no p r i s m effect is present . O n i n t e r -posing a prism having a refractive index of 1.5and angle a the axial ray is hen I away from (he

Journal of Research

normal by an amount eo=a/2 and now cuts thefocal plane at 0', which is now referred to as thecenter cross. To locate the position of theprincipal point with respect to the center cross thedistance O'O must be determined.

The determination of O'O is made possible be-cause the rays inclined at angle 0 are also displacedby amounts ex and e2, and cut the focal plane atpoints X[ and X2 instead of Xx and X2. Byassuming that the ray passing through the prismand meeting the focal plane at 0' is near the regionof minimum deviation, it follows that we mayregard ei as equal to e2 and can compute the rela-tive magnitude of 1 in terms of e0. The magni-tude of i is the average value of €i and e2 and al-ways greater than e0 and X'2 X2=X[ X^>0'0.None of these quantities can be measured directly,but their magnitudes may be obtained as follows:From the figure,

] = OX'2-O'OO'X[=j[t*n. (0-€i)+tan eol = [

or O'X2=OX2+X2X2-O'OO'X'^OX.-X', X1+0/0,

and0' X2-0'Xl=2(X2 X2-O'O) =

sinceX'2 X2=X[ X, and OXX=OX2.

In the case of an actual lens having prism effect,X[, 0', and X2 are the observed images and theseparations 0' X[ and O'X2 can be measured.Consequently, the value of AD can be determinedfor each pair of equal known angular separationsfrom the axis from measurements on the negative.This information is used most conveniently bycomparing the measured values of AD with valuesof AD computed for a definite set of values ofprism angle, a, focal length / , and separationangle /3. The computations can be readily per-formed and the observed values of AD can beinterpreted in items of a sufficiently accuratelyby this process, because AD is linear in a for afixed angle /3 for small value of prism angle a.Table 3 gives the computed values of i, XiX2,O'O, and AD for the case of a lens having a focallength of 150 mm combined with a prism of index1.5 having an angle a of 0.05°. It may be notedfrom the table that AD increases rapidly withincreasing 0. For the case shown, AD is negli-gible at 7.5° but increases to 0.3 mm at 45°. Itis evident from the table that the value of O'O

FIGURE 10. Schematic drawing showing the image shiftproduced by interposing a thin prism in front of an ideallens.

deduced at the wider angles is more sensitive thanthe value determined at the narrow angles. It isalso clear that the presence of appreciably largeprism effect may cause marked asymmetric dis-tortion.

TABLE 3. Image displacements of various points in thefocal plane for a prism of index 1.5 and angle a=0.05°combined with a lens whose focal length is 150 mm

Degrees07.5

1522.53037.545

e

Degrees0.0250.0254.0265.0283.0316.0364.0435

X2'X2=Xx'Xx

mm0.065.068.074.086.110.151.228

O'O

mm0.065.065.065.065.065.065.065

AD =O'XS-O'Xx'

mm0.000.006.018.042.090.172.326

Although it is feasible to determine the prismeffect from a single negative from the cameracalibrator by this process, it is customary tocombine the results from two negatives with thecamera being rotated 180° about its axis ofsymmetry between the marking of each negative.This procedure increases the accuracy and simpli-

Precision Camera Calibration

fies some of the computations. The coordinatesof the principal point P. P., with respect to thecenter cross, are measured along the diagonalsof the focal plane, as indicated in figure 11,because the calibrator is so constructed that theimages from the various collimations lie alongthe diagonals.

4. Location of Principal Point with Respect toCenter of CoUimation

To locate the principal point with respect tothe center of coUimation, it is only necessary tocombine the results obtained in section VII,2 and 3. Care must be taken to avoid errorsin sign and to make allowance for the fact thatthe measurements are made in two coordinatesystems rotated 45° with respect to one another.Figure 12 shows the resultant obtained by com-bining the measurements illustrated in figures9 and 11.

The usual requirement in precision mappingcameras is that the principal point must not bedistant from the center of collimation by morethan 0.03 mm. In the event that this require-ment is not met, it is customary to adjust themarkers with respect to the lens until the separa-tion of principal point and center of collimationis less than 0.03 mm. When this has been done,dowels are placed so that no relative movementof principal point and center of collimation canoccur.

c.aar — - * P . P .

B

FIGURE 12. Schematic drawing showing the principalpoint with respect to the center of collimation.

5. Determination of Equivalent Focal Length andDistortion

The same negative from which the location ofprincipal point is made is used to determine theequivalent focal length and distortion. In factthe same measurements, described in section VII,3, for use in locating the principal point withrespect to the center cross are used.

The equivalent focal length [6] is definedtheoretically by the equation

y1

tan i (1)

FIGURE 11. Schematic drawing showing the principalpoint with respect to the center cross.

where y' is the distance from the principal focusto the center of the image in the image spacefocal plane of an infinitely distant object pointthat lies in a direction making an angle /3 with theaxis of the objective. If a photographic objectivewere free from distortion, the quotient would beinvariant with respect to $. For many photo-graphic objectives, the distortion is negligible forpoints distant from the center of the useful fieldby not more than one-fifth of its radius. Con-sequently, it is often possible to obtain a satis-factorily accurate value of / by a single deter-mination of /3 and yf for a point lying near the axis.

The negative made for an airplane camera coneon the camera calibrator has images at the centerof the field and at six known angles proceeding in7.5° steps from the center to the edge of the useful

10 Journal of Research

field. Consequently, on substituting the meas-ured separation from the 0° to 7.5° image and themeasured value of tan 0 into eq 1., the value of theequivalent focal length can be determined. The0° to 7.5° separation occurs four times on the nega-tive along radii separated by 90° of azimuth, sothat four independent determinations of/ are pos-sible from a single negative. The accuracy of thisdetermination is further increased by the fact thatthe measurements made on opposite sides alongthe same diameter automatically compensate forany error of plate tipping. It must be mentionedthat practically the term equivalent focal lengthas used for a lens cone combination is actually thescale factor for use in interpreting distances meas-ured on our aerial photograph in the central area.Inasmuch as photographic objectives are mountedin the aerial camera in such manner as to yieldbest over-all definition, it is clear that the equiva-lent focal length as above determined is the scalefactor for the plane of best average definition.

Having established the value of the equivalentfocal length for the lens camera combination, thedistortion referred to the equivalent focal lengthcan be readily found. To evaluate the distortionlet y'i, y'21 VZJ • • • be the separation of theimages on the negative from the central imagecorresponding to the angles in the object spaceft, ft, ft. Let products/ tan ft,/ tan ft,/ tan ft,be computed. Then the values of the distortionDu D2, D3 . . ., are given by the relation

D1=y[—-fta,n ft

A=2/2—/tan ft

A=yi—/tan ft

focal length determined for the axial region, themeasurements made on the negative in the axialregion are free from distortion, whereas in theextra-axial regions the errors from distortion maybecome serious.

The customary procedure for alleviating thiscondition is to use a new scale factor that providesa better over-all accuracy in interpreting distancesmeasured on the negative, although it does so atthe expense of introducing negligibly small errorsin the axial region. This new scale factor is calledthe calibrated focal length [6]. The term cali-brated focal length, its significance, and the man-ner of its evaluation have been sources of con-siderable misunderstanding in photogrammetriccircles since the inception of the term. It isproposed in the following paragraphs to presenta study of the variation of equivalent focal lengthand distortion that occur for a lens affected withdistortion when the equivalent focal length isbased on the expression

J t a n j87

and j8 is allowed to take any value at finiteintervals between 7.5° and 45°. It is hoped thatcareful consideration of this analysis will lead tobetter understanding of the physical significance ofthe calibrated focal length.

In table 4, the row marked y' lists the measuredseparations from the axial image of the imagesoccurring at 7.5° intervals out to 45°. Thesemeasurements are from an actual negative madeon a lens-cone combination. In the second row,values of/, the equivalent focal length, are givenwhere each value of/ is obtained from the equation

Positive values of the distortion indicates adisplacement of the image away from the centerof the negative.

6. Determination of Calibrated Focal Length

The lenses used in aerial mapping are not freefrom distortion. Consequently the value of theequivalent focal length determined for the vicinityof the axis is not necessarily the best scale factor foruse in interpreting measurements made on thenegative at points well removed from the axialarea. If one holds fast to the value of equivalent

r

tan

It is clear that a different value of/ is obtained foreach value of ft This variation in / appears be-cause of the distortion in the lens. It is possibleto determine the values of the distortion for eachof the values of/ at each angular separation fromthe axis. This has been done, and the results areshown in the table. One of the natural conse-quences of this procedure is that, for whatevervalue of the equivalent focal length is determined,the distortion is necessarily zero for that value ofj3 when the distortion is evaluated with respect

Precision Camera Calibration 11

to that particular focal length. The location andthe magnitude of the positions of maximumpositive a ad negative values also change for eachvalue of/. These effects are shown graphically infigure 13. It may be noted in figure 13 that thecurves of distortion are very similar, and that the

-O.I

+0.1

o

-0.1

+0.1

0

-0.1

+0.1

U 15

I I I

i I I I I

7.5' 22.5° 30° 375°

FIGURE 13. Variation of distortion in the same image planewith equivalent focal length computed from the relationf=y'/tan p for varying angles of £.

The lowest curve showing the distortion referred to the calibrated focallength indicates that the calibrated focal length is equal to the equivalent focallength for values of 0 of approximately 16° and 42.5°.

TABLE 4. Variation of the values of equivalent focal lengthand distortion with angle upon which the computations arebased for a typical lens

7.5°

»;

mm20.064

/ i

152.400

15°

yf2

mm40.487

h

152.443

22.5°

r

mm62.182

ft

152.535

30°

r

mm88.112

U

152.614

37.50°

i

mm117.086

n152. 589

45°

H

mm152.345

/6

152.345

Distortion referred to the equivalent focal length f=y'/tan p

ftf2 ..

U - - ---/6

7.5°

0.000-.006—.018-.028-.025

.007

15°

0.012.000

-.025-.046-.039

.026

22.5°

0.056.038.000

-.033-.022

.079

30°

0.124.099.046.000.015.156

37.50°

0.145.112.042

-.019.000.188

45°

-0.055-.098-.190-.269-.244

.000

Distortion referred to the calibrated focal length fe=152.451 mm

0.006 -0.002 0.035 0.094 0.106 -0.106

variation of the curves is very like that whichwould be produced by rotating the entire curveabove the 0° point.

The map maker is primarily interested in thescale factor that introduces the least error whereverused throughout the entire area of the photograph.Thus, while use of the scale factor equal to theequivalent focal length at 7.5° gives zero errorwithin the 7.5° zone, it is at the expense of largeerror in the 37.5° zone. Inasmuch as there isusually no preferred area in a given photograph,it is obvious that it would be preferable to use ascale factor that would reduce error in the regionof large error even if a small error were introducedin the region of zero error. Considering thecurves in figure 13, it is clear that use of j2 insteadof/i as a scale factor will give reduced distortionin the 37.5° zone with the introduction of a smallamount of distortion at 7.5°. Selection of one ofthe other values of / changes the error patterneither for the better or for the worse.

It is, at present, standard practice to considerthe preferred error pattern as that one for whichthe absolute values of maximum positive and maxi-mum negative distortion are equal. The pre-ferred value of equivalent focal length for which

12 Journal of Research

this condition prevails is called the CalibratedFocal Length. It seldom happens that the cali-brated focal length coincides with the equivalentfocal length at one of the fixed angles (i. e. one ofthe 7.5° intervals) so that it is necessary to com-pute the increment A/ that must be added to theequivalent focal length determined at one of thefixed angles (usually the 7.5° angle) to yield thecalibrated focal length. This may be done withthe aid of the following formula:

aj tan ftn+tan ft/

where Dm and Dn are the values of distortionexisting at the angles pm and f$n referred to theequivalent focal length / . The calibrated focallength j c is then obtained from the relation

The above formula is a general one, and the newvalues of distortion referred to the calibratedfocal length at angles (3m and pn are equal in mag-nitude and opposite in sign. Usually ($m and pn

are the angles at which maximum positive andmaximum negative distortion occur. These equa-tions are essentially the same as those reported ina paper by Sewell [8].

The values of the distortion referred to thecalibrated focal length are shown in table 4 andfigure 13. The computation is made for,flw=37.5°and j3w=45°. It may be noted that the samevalue of j c will be obtained regardless of whichvalue of/ is selected as a base of computation, solong as the corresponding values of distortion areused.

It must be emphasized that the determinationof the calibrated focal length involves no shiftwhatever of the position of the focal plane of thecamera cone with respect to the lens. The cali-brated focal length is simply that value of theequivalent focal length that serves as the pre-ferred scale factor in interpreting distancesmeasured on the photograph.

7. Determination of the Resolving Power

The camera calibrator provides an excellentmeans of checking whether or not the camera lensas mounted in the camera is capable of producingusable definition from the center to each cornerof the field. If, as sometimes happens, the resolu-

tion along the four radii from the center to thecorners of the field is not uniformly good this willbe detected on the negative and thus minimize thepossibility of a camera being certified as satis-factory when the resolution satisfies minimumdefinition requirements along one diameter of thefield but does not do so at some points along thediameter at right angles to the first. Table 5 liststhe observed values of the resolving power for atypical 6-in. lens in the focal plane of the camerain which it is mounted. These resolvings powersare measured at effective aperture j/8.3 and maybe slightly higher than the values that would befound at maximum aperature.

It is clear from this table that some variationexists in the values of the resolving power for thesame angular separation from the axis on differentmeridians. For the lens shown, the variationsare not such as to bring the resolving power downat any point to values lower than the usuallyspecified minimum of 15 lines per millimeter forthis type of lens.

TABLE 5. Observed values of resolving power for a typical6-in. lens as determined from a negative made with thecamera calibrator

Collimator bank

III -III. .IV

Average

I . .IIIII

IV

Average

Resolving power for tangential lines in linesper millimeter with angular separation fromthe axis of—

0°

53535353

53

7.5°

53273239

38

15°

32272727

28

22.5°

27232719

24

30°

32323227

31

37.5°

27232723

25

Resolving power for radial lines in linesmillimeter with angular separationthe axis of—

0°

53535353

53

7.5°

53273239

38

15°

39323232

34

22.5°

46273932

36

30°

39393939

39

37.5°

32273232

31

45°

23231923

22

perrom

45°

*27272327

26

8. Flatness of Platen

The specifications for precision airplane mappingcameras [5] usually contain the requirement thatthe surface of the platen shall not depart from a

Precision Camera Calibration 13

plane by more than ±0.0005 in. The platen isthe flat surface against which the film is pressed toensure its planeness during the instant of ex-posure. This requirement is included in thespecifications, because small departures from flat-ness in this locating surface would be imparted tothe film and would in turn affect the location ofimages on the final negative.

It is therefore customary in the course ofcalibration of a precision camera to check theplaten for flatness. It has not been consideredexpedient at this laboratory to make a detailedcontour map of the entire platen surface. Insteadthe practice here has been to check the platen forflatness along selected lines or diameters of theplaten and to conclude that the platen is satis-factorily plane if no departure from planes inexcess of the specified tolerance occurs along theselines.

A simple device, shown in figure 14, has beenconstructed that permits a rapid measurement ofthe departure from flatness along a selected linewith an accuracy of ±0.0001 in. The deviceconsists of a metal beam supported by two legsset 8 in. apart. Midway between these two legs,a sensitive dial indicator gage is mounted with thelower end of the plunger, which motivates thegage lying in the line common to the two mainouter supports. The two outer supports and the

FIGURE M. Dial indicator device used for checking flatnessof platen.

The Instrument is graduated In microns and permits the rapid meas-urement of departures from flatness with an accuracy of d 0.0001 In.

central plunger have convex spherical surfaces atthe lower portions, which touch the surface to betested. The two smaller supports serve only tohold the instrument in a vertical position and toprevent the instrument from tipping over. Theinstrument may be rocked on the two outer legs.In use the small front leg is in contact with thesurface being tested, and the movement of theplunger is in a direction closely perpendicular tothe surface.

The instrument is graduated in microns.Before each use, it is placed on an optical flat toensure that the gage reads zero for a truly flatsurface. It is then placed on the platen undertest and the reading noted. The instrument ismoved across the platen, rotated 90°, and movedacross again. It is also moved about the platenin other orientations. If the pointer on the gagedoes not depart from zero by more than ±13^during the course of this operation, the platen isadjudged satisfactory.

VIII. Discussion

The method used at this Bureau for determiningthe location of the principal point of airplanecameras has been in use for some years. This typeof work was first performed on the precision lenstesting camera and it is now being performed onthe new camera calibrator. In recent years, thequestion has been raised whether the principalpoint as herein defined and determined is theproper point to use in the photogrammetric sensein the interpretation of aerial photographs. Inthe absence of lens distortion, particularly of theasymmetric type induced by prism effect, therecan be no question of the adequacy of the methodand its interpretation.

However, in the presence of unbalanced distor-tion, there may be some merit in the criticismsthat have been directed at this concept of theprincipal point. I t is not the aim of the presentauthors cither to refute or concur in the variousother suggested concepts of what constitutes thebest definition of principal point in cameras in thepresence of asymmetric distortion. Rather, thesematters are included in the present paper to showthat cognizance <>l" these alternative procedures isbeing taken; and it is hoped that ultimately somecommon basis may be round wherein these differ-ences can be reconciled.

14 Journal of Research

The most recent publication dealing with oneof the alternative definitions of principal point isthat of P. D. Carman of the National ResearchCouncil of Canada [9]. In that paper a proof ispresented that indicates that a point havingproperties identical with the center cross as de-fined in section VII, 2 of the present paper, oughtto be used as the principal point. This definitionof principal point is accepted in Canada as evi-denced by a paper by R. H. Field [10], who de-scribes the method of locating the principal pointused at the National Research Laboratories atOttawa.

In the September 1948 issue of PhotogrammetricEngineering, E. D. Sewell [8] presents the conceptof a "Point of symmetry/' which is a point aboutwhich all radial distortions are to be symmetrical.This point does not coincide with either the centercross or principal point as defined in the presentpaper, although all three points coincide in theabsence of asymmetrical distortion.

To illustrate the differing locations of thesepoints, consider the following case. A camerawhose lens showed negligible prism effect was cali-brated at this laboratory and the principal pointand center of collimation brought into coincidence.Following this, a thin prism was placed in frontof the lens with its base facing the direction A,and a test negative was made. The center crosswas found to be displaced in the direction A by0.239 mm. The principal point, as determined bythe method described in this paper, was found tocoincide still with the center of collimation within±0.002 mm. The point of symmetry was foundto be displaced in the direction A by 0.496 mm(the computation being based on targets separated37.5° from the axis and the 0° target following themethod described by Sewell). The values hereingiven are far greater than one likely to be found inpractice, as the angle of the prism used wasapproximately 12 min.

It is probable that in lens-cone combinationswhere little asymmetric distortion exists there islittle to be said in favor of any one of these con-cepts over the others. When appreciable asym-metric distortion is present, the authors are not ina position to state which of these points ought tobe used in photo-interpretatioa. This must besaid in favor of the principal point as located bythis Bureau: It is an invariant point for a givenlens-cone combination; its location is not appre-

ciably affected by placing filters of differing pris-matic power in front of the lens; it is possible thatour present procedure of reporting, however,should be amended to include magnitude of theprism deviation and its direction for a given lens-cone-filter combination so that the center crosscan be located by those users who prefer it intheir interpretation processes.

In the opinion of the authors, there are threecourses that may be followed to eliminate thesedifficulties. The first course is to set an upperlimit to the amount of prism deviation permissibleon a lens-cone combination, possibly 0.015 mmfor the axial ray. A lens-cone combination thathas excessive prism deviation would then be un-acceptable unless it were reworked to reduce theeffect. That such a course is practicable is evi-denced by the discussion contained in a paper [11]by J. V. Sharp and H. H. Hayes.

The second course of action is to make accuratemeasurements of the prism effect, either on thelens as mounted in its camera cone or on the lensalone. Having accurate knowledge of the magnitudeand direction of the prism deviation, it should thenbe possible to neutralize the effect by using a filteron the front of the lens, which is itself a thin prisminstead of the usual plane parallel. Other re-searches in progress at this laboratory indicate thatsuch a course is feasible. It is possible that smallresidual effects may still remain, but it is believedthat such residual effects will be negligible com-pared to the asymmetric distortions known to beproduced by the prism effect.

The third course of action resembles the first, ex-cept that an upper limit on " tangential distortion''rather than prism deviation is set. This is pre-ferred by J. V. Sharp, because it allows for thepossibility3 of the existence of a form of "tan-gential distortion" that deviates from the patternpredicted by the hypothesis that tangential dis-tortion is produced by prism effect. Even in thiscase it is possible that a compromise involvingpartial neutralization of the prism effect may beadvantageous.

The authors express their appreciation to othermembers of the staff of the National Bureau ofStandards for assistance rendered during the vari-

3 This possibility was brought to the senior author's attention in the courseof an informal discussion.

Precision Camera Calibration 15

ous phases of development of this equipment.Kobert E. Ward of the Instrument Shop did amajor portion of the machine work and in par-ticular performed the highly accurate boring oper-ation on the casting that holds the coUimators inalinement. George L. Buzas constructed thetarget holders and the devices used in calibratingthe instrument. Edgar C. Watts assisted in thedesign of testing devices and made the drawingsused in this paper. William P. Tayman madethe reticles used as targets in the coUimators andis the principal operator of the completed instru-ment. Arthur A. Magill, Roland V. Shack, IrvingMalitsky, and William P. Tayman assisted in thecalibration of the instrument. Wilbur W. Bran-non made the biprism used for calibration of theinstrument.

IX. References

[1] United States Department of Agriculture Specifica-tion No. A-APC-1102 (as approved March 1940).

[2] F. E. Washer and F. A. Case, New precision cameracalibrator, NBS Technical News Bulletin 33, 8(1949).

[3] I. C. Gardner and F. A. Case, Precision camera fortesting lenses, J. Research NBS 18, 449 (1937)RP984.

[4] F. E. Washer, Locating the principal point of precisionairplane mapping cameras, J. Research NBS 27,405 (1941) RP1428.

[5] Manual of Photogrammetry by the American Societyof Photogrammetry, p. 130 and 136 (PittmanPublishing Corp., New York and Chicago, 1940).

[6] ASA Standard No. Z38.4.21-1948. American Stand-ard methods of designating and measuring focallengths and focal distances of photographic lenses.

[7] I. C. Gardner, Relation of camera error to photogram-metric mapping, J. Research NBS 22, 209 (1939)RP1177.

[8] E. D. Sewell, Field calibration of aerial mappingcameras. Photogrammetric Engineering, XIV, 363(1948).

[9] P. D. Carman, Photogrammetric errors from cameralens decentering, J. Opt. Soc. Am. 39, 951 (1949).

[10] R. H. Field, A device for locating the principal pointmarkers of air cameras, The Canadian Surveyor(July 1949). (N. R. C. No. 1979.)

[11] J. V. Sharp and H. H. Hayes, Effect on map produc-tion of distortions in photogrammetric systems,Photogrammetric Eng. XV, 159 (1949).

WASHINGTON, February 7, 1950.

16 Journal of Research