The Effect of Prism on the Location of the Principal Point*· t FRANCIS E. WASHER, Chief, Optical Instruments Section, Optics and Metrology Division, National Bureau of Standards, Washington, D. C. ABSTRACT: The presence of prism in a lens-camera combination whether arising from a filter having non-parallel surfaces or from small decentra- tions of the lens elements is evidenced by a displacement of the principal point of autocollimation from the true princip;Ll point and the appearance of asymmetric values of the distortion. A method is described whereby the e.Dective prism angle and the displacement of the principal point can be determined from analysis of the asymmetric values of the distortion for the case of a camera properly aligned for calibration. Theoretical analysis of the problem is given and the results are confirmed by experiment. 1. INTRODUCTION T HE factors that affect the accuracy of calibration of airplane mapping cam- eras are of special interest to photogram- metrists inasmuch as the accuracy of the finished maps of an area is dependent upon the accuracy of the equipment used in the various steps intervening between the initial photography and the final map. In previous papers, the present author has discussed sources of error in camera calibration,1,2 effect of camera tipping,3 and location of the point of symmetry.4 In the present paper, a study is reported of the manner in which the values of the distor- tion are made asymmetric by the presence of a thin prism placed in front of the lens. In addition, the principal point of auto- collimation 5 is shifted from the principal point 6 ,7,8 in the presence of prism. The present paper analyzes theoretically the effect of a thin prism placed in front of an ideal lens and demonstrates the correctness of the theory by actual measurements. Asymmetric distortion patterns of the type that can be explained by prism-effect have been found frequently during camera calibrations in this laboratory.6,7 It was mainly because of this that this Bureau customarily locates the principal point which is an invariant point during its calibration. With the advance of knowl- edge, the centration of camera lenses has been improved markedly and more atten- tion is given to the parallelism of filters placed in front of the lens. 9 Consequently, the magnitude of the prism has decreased through the years and is now found to be negligible for most precision cameras. It is probable that the location of the point of autocollimation will replace the location of the principal point in laboratory camera calibrations. In the absence of prism-effect, the point of autocollimation usually coin- cides with the point of symmetrylO which is usually determined in field calibra- tions.u,12 520 * This is the paper of a series dealing with problems that relate to the calibration of pre cision airplane mapping cameras. The first paper was entitled "Sources of Error in Various Meth ods of Airplane Calibration and was published in this JOURNAL, Vol. XXII, no. 4. The second is in the March issue of this JOURNAL (Vol. XXIII, no. 1); the title is "A Simplified Method of Locating the ijpint of Symmetry." t This worki\vas sponsored in part by the U. S. Air Force. Approval for publication has been given by the Committee of the Standards Bureau and the sponsoring agency. ;"
Transcript
The Effect of Prism on the Location
of the Principal Point*· t
FRANCIS E. WASHER,
Chief, Optical Instruments Section, Optics and Metrology Division,National Bureau of Standards, Washington, D. C.
ABSTRACT: The presence of prism in a lens-camera combination whetherarising from a filter having non-parallel surfaces or from small decentrations of the lens elements is evidenced by a displacement of the principalpoint of autocollimation from the true princip;Ll point and the appearanceof asymmetric values of the distortion. A method is described whereby thee.Dective prism angle and the displacement of the principal point can bedetermined from analysis of the asymmetric values of the distortion forthe case of a camera properly aligned for calibration. Theoretical analysisof the problem is given and the results are confirmed by experiment.
1. INTRODUCTION
T HE factors that affect the accuracy ofcalibration of airplane mapping cam
eras are of special interest to photogrammetrists inasmuch as the accuracy of thefinished maps of an area is dependent uponthe accuracy of the equipment used in thevarious steps intervening between theinitial photography and the final map.
In previous papers, the present authorhas discussed sources of error in cameracalibration,1,2 effect of camera tipping,3 andlocation of the point of symmetry.4 In thepresent paper, a study is reported of themanner in which the values of the distortion are made asymmetric by the presenceof a thin prism placed in front of the lens.In addition, the principal point of autocollimation5 is shifted from the principalpoint6,7,8 in the presence of prism. Thepresent paper analyzes theoretically theeffect of a thin prism placed in front of anideal lens and demonstrates the correctness
of the theory by actual measurements.Asymmetric distortion patterns of the
type that can be explained by prism-effecthave been found frequently during cameracalibrations in this laboratory.6,7 It wasmainly because of this that this Bureaucustomarily locates the principal pointwhich is an invariant point during itscalibration. With the advance of knowledge, the centration of camera lenses hasbeen improved markedly and more attention is given to the parallelism of filtersplaced in front of the lens. 9 Consequently,the magnitude of the prism has decreasedthrough the years and is now found to benegligible for most precision cameras. I t isprobable that the location of the point ofautocollimation will replace the locationof the principal point in laboratory cameracalibrations. In the absence of prism-effect,the point of autocollimation usually coincides with the point of symmetrylO whichis usually determined in field calibrations.u,12
520
~
* This is the ~hird paper of a series dealing with problems that relate to the calibration of precision airplane mapping cameras. The first paper was entitled "Sources of Error in Various Methods of Airplane ~mera Calibration and was published in this JOURNAL, Vol. XXII, no. 4. Thesecond is in the March issue of this JOURNAL (Vol. XXIII, no. 1); the title is "A Simplified Methodof Locating the ijpint of Symmetry."
t This worki\vas sponsored in part by the U. S. Air Force. Approval for publication has beengiven by the Edi.~6rial Committee of the Standards Bureau and the sponsoring agency.
;"
EFFECT OF PRISM ON LOCATION OF PRINCIPAL POINT 521
c
FIG. 1. Schematic drawing showing the imageshift produced by placing a thin prism in frontof an ideal lens.
When the thin prism of refractive indexn and prism-angle a is placed in front of thelens, al1 rays are deviated in the directionof the prism-base and intersect the focalplane at points, X', 0 ' , and Y' . In the caseof an actual lens having prism-effect, thedistances 0'X' and 0' Y' are the only
(1)
x )t'
ox = OY =ftan(j
2.1 DETERMINATION OF THE EQUIVALENT
FOCAL LENGTH
For an ideal lens, the principal pointand the focal point (or center-cross) coincide. In practice, however, the center-crossis shifted away from the principal pointeither because of prism-effect in the lens ornonparal1elism of the surfaces of the filteron the front of the lens. This is il1ustratedin Figure 1 which is a schematic drawingof the displacement of the axial ray andrays inclined at anglefJ with the axis beforeand after placing a prism in front of anideal lens. In the figure, N is the rear nodalpoint of an ideal lens L. Initial1y, the axialray from an infinitely distant object isimaged at the principal point 0, consequently NO is equal to f the equivalentfocal length of the lens. The points of intersection of rays inclined at angles of +fJand -fJ with the optical axis are designatedX and Y. As L is an ideal lens having nodistortion,
2. EFFECT OF PRISM ON CAMERA
CALIBRATION
When a lens-camera combination isbeing calibrated under conditions suchthat the focal plane is truly normal to theline drawn to it from the central target, itsometimes happens that the values of themeasured distortions are asymmetric. Thisasymmetry has been found to be verysimilar to that produced by a combinationof a wel1-centered lens and a thin prism. 7
This effect of a thin prism in the opticalpath can be produced directly by a filterhaving non-paral1el surfaces placed infront of the lens, or indirectly by smal1amounts of decentration of the lens elements. The values of focal length anddistortion are affected in varying degreesby the presence of prism-effect on the lens.The effects have many similarities to thoseproduced by camera tipping.3 •4 However,while the asymmetric values of distortioncan be compensated in part by a smal1amount of camera tipping, they can becompensated in ful1 only by another prismof equal vertex angle and producing itseffect in the opposite direction. Perhapsthe most striking consequence is found inthe relation of prism-effect to the locationof the principal point. The term "principalpoint" in photogrammetry refers to thefoot of a perpendicular dropped from theinterior perspective center to the plane ofthe photograph. The same term in opticsis used to designate the intersection pointof either of the two principal planes withthe optical axis of the lens. In this opticalsense, the principal points are sometimesreferred to as the points of unit magnification or as two of the Gauss points. Theterm, "principal point," therefore, meansvery different things in photogrammetryand in optics. Consequently, confusionfrequently results when work is being described which properly belongs to bothphotogrammetry and optics. In this paper,however, the term "principal point" isused in its photogrammetric sense. I t mustbe mentioned that a third meaning of theprincipal point is sometimes found inphotogrammetric writings wherein thecenter of col1imation of a camera is referredto as the "indicated principal point."lo
In the succeeding sections, the effect ofprism upon the values of focal length anddistortion and upon the location of theprincipal point is analyzed.
522 PHOTOGRAMMETRIC ENGINEERING
(9)
DI = f[tan (fJ + E) - tan Eo - tan fJ] (11)
which can be written
f tan E(l + tan' (3)D1 = - f tan Eo (12)
1-tanfJtane
For small values of E, this can been written
fED1 = -- 0 +Etan fJ) - fEo (13)
cos' fJ
The distortion, D 2 on the other side is:
9 is used when E> 10 minutes, the valuesof I will be higher than the true value, theerror will run as high as +0.012 mm. forE=30 minutes. In such cases, it is necessaryto determine E by the method shown in alater portion of this paper and to use thisvalue in Equation 8. The value of I determined with Equation 8 agrees with thetrue value for values of E as high as 30minutes.
I t must be mentioned that these equa"tions are identical in form with thosefound for use in determining the focallength for a tipped camera. The chiefdifference is that here E is a deviation produced by prism-effect and will be shownlater to vary with {3, while in the tippingequation E is the angle of tip and does notvary with {3.
2.2 ASYMMETRIC DISTORTION PRODUCED BY
PRISM- EFFECT
When a camera has been properlyaligned but is equipped with a lens havingnoticeable prism-effect, it is found that anasymmetric distortion is produced. Thisdistortion affects the relative location ofimages in the image plane even as does thedistortion inherent in the lens arising fromlens-aberrations. There is, however, onepronounced difference in that it is asymmetric about the optical axis whereas distortion arising from lens-aberrations issymmetric. The asymmetric distortionproduced by prism resembles that produced by camera-tipping which was described in earlier papers3.\ but is sufficiently different to warrant distinguishingbetween the two types of asymmetricdistortion.
Referring to Figufe 1, the distortion, D I ,
on one side is:
(10)
(14D. = O'Y' - OY
DI = O'X' - oxor
O'X'+O'Y'f=----
2 tan (3
Thus, the equivalent focal length I can bedetermined to a good degree of approximation by using the measured values of 0'X',0' Y', and the known value of {3. The valueof I, so determined, will be correct to thenearest thousandths of a millimeter forE~ 10 minutes for values of the focallength of the order of 150 mm. If Equation
quantities directly measurable, and thepresent discussion is devoted to showinghow the focal length I, the prism-angle a,and the magnitude of 00' can be deducedfrom these measurements.
These determinations are made possiblebecause the angular deviation of the raysinclined at angle {3 to the axis is greaterthan the deviation EO of the central ray.By assuming that the ray passing throughthe prism and meeting the focal plane at0' is near the region of minimum deviation,it can be shown that El can be regarded asequal to E2, so for the balance of this discussion El and E2 are replaced by E. Since Eis always greater than Eo, th~ measurablequantities 0'X' and 0' Y' differ slightly inmagnitude, and it is this small differencethat permits the evaluation of I, a, and00' the shift of the center-cross from theprincipal point.
From Figure 1,
00' = f tan Eo (2)
O'X' = f[tan ((3 + E) - tan Eo] (3)
O'Y' = f[tan (fJ - E) + tan Eo] (4)
It is clear from Figure 1, that
O'X' + O'Y' = f[tan ((3 + E) + tan ((3 - E)] (5)
which may be written
2ftan (3O'X' + O'Y' = ---------- (6)
cos' EO - tan' (3 tan' E)
O'X'+O'Y'= cos'E(1-tan'ptan'E) (7)
2 tan (3
For small E, the term (1-E2) can besubstituted for cos E and Efor tan E; neglecting the term in E\ Equation 7 becomes
O'X'+O'Y'f = [1 - E'(1 + tan' fJ)] (8)
2 tan fJ
For very small values of E, the term in E2
can be dropped and the expression becomessimply
whence
EFFECT OF PRISM ON LOCATION OF PRINCIPAL POINT 523
(17)
which has been found to be useful are alsoincluded in the table.
TABLE 1
Values of EIEo for selected values of {3 forn = 1.5 and n = 1.6. In addition, values ofm=E/Eo cos'{3 are given for both cases
to simplify the determination of the prismangle Cl in the course of camera-calibration.Table 1 shows the values of ~/~o for n = 1.5for a series of value of {3; for comparisonpurposes a set of values .of ~/~o for n = 1.6is also included. In addition, values of thequantity
In analyzing the results of measurementmade on a camera where prism-effect ofunknown magnitude is present, the orientation of the effective prism and resultantdirection of the displacement of the centralimage 0' from the true position of theprincipal point 0 can be determined fromthe sign of the values of distortion. Theprincipal point lies on the same side of thecentral image 0 as those points showing
D2 = f[tan ({3 -~) + tan ~o - tan {3] (15)
which can for small values of ~ be written
-f~D, = -- (1 - ~ tan (3) +f~o (16)
cos2 {3
For small values of ~, D 1 is approximatelyequal to D 2 but opposite in sign. The effectof prism-defect in the lens is, therefore,an introduction of an asymmetric distortion whose magnitude is a function of {3and ~. To evaluate D1 and D 2 , it is firstnecessary to determine ~ and ~o. This canbe done when the prism-angle Cl is knownwith the aid of the following relation
(Vr.2 - sin2 {3 )
~='" -1cos {3
or
where ~ is the deviation of the ray incidentat angle {3 on the surface of the prism ofindex of refraction n and prism-angle Cl.
This is a sufficiently close approximationfor small angles of Cl when the deviation ~
takes place in the principal plane of theprism. Similar expressions can be derivedfor other planes in different azimuths. Forthe case of {3 = 0, Equation 17 becomes
~ = ~o = (n - 1)", (18)
As indicated in Equation 17, ~ is also afunction of {3. The manner in which ~
varies with {3 is shown graphically inFigure 2. These values of ~ were computedwith the aid of Equation 17. Because ofthis large change in ~ with {3 the interpretation of asymmetric distortion in termsof prism-effect is somewhat complex. Ithas been found convenient to postulate avalue of the index n and to determine theratio of ~ to ~o for a series of values of (3
6.0r---'I---,-'--,,---.-----'Ir------.
l/lZ 0::! 4.01-£:)<l:a::...J...J~ 2.0z
oo
o o o oo
o
o-
-
I I I-30· -IS· O· IS· 30·
ANGULAR SEPARATION FROM AXIS IN DEGREES
FIG. 2. Deviation E in the principal plane of a prism having a refractive index of 1.5 andprism-angle of 20 minutes (0.0058 radians) as a function of incident angle {J.
524 PHOTOGRAMMETRIC ENGINEERING
(21)
(20)
fEZ tan {3D=--·
cos' {3
averaging the values measured at the twocorresponding opposite angular separationsfrom the axis along a single diameter.This corresponds to the evaluation of
D1+D.D=--·
2
In the present case, the contribution oflens-aberrations to the distortion has beenassumed to be zero, so if D is not zero, thedeparture therefrom is a consequence ofthe prism effect alone. By using the valuesof D 1 and D 2 from Equations 13 and 16, itis clear that
I t is worthy of note that the above expression for residual symmetric distortioninduced by prism-effect is identical in formwith the relation found in the case ofcamera-tipping3•4• The difference is in E
alone which is a function of {3 for prismeffect and is invariant with respect to {3in camera-tipping.
Values of D for a specific case are shownin Table 2. The example shown is for aprism-angle ex of a magnitude unlikely tobe found for an actual lens. However, evenin this case, the value of D is practicallynegligible, even though D 1 and D 2 are verylarge. It is also of interest to note that thesign of the residual distortion D is negativein the case of prism-effect whereas it ispositive in the case of camera-tipping.
For the case of an actual lens havingmeasurable inherent symmetric distortion,it may be inferred that the value of thedistortion at a given angle {3 is given rea~nnably closely by Equation 20 providedthat ex is less than 30 minutes. If one isdealing with a distortion-free lens, an effective prism-angle of 30 minutes mayproduce an error of +- 0.003 mm. in thevalues of distortion referred to the calibrated focal length. I t is also likely thatsmall additional errors may raise in averaging for those val ues of {3 where the distortion characteristics are changing rapidly with {3. Even in such instances, theerrors in distortion will usually be smallcompared with the total distortion. If necessary, however, approximate correctionscan be made if the characteristic distortioncurve for the particular type lens is known.
maximum negative distortion. In somecases when symmetric distortion arisingfrom lens aberrations is present, it may benecessary to graph the observed values ofdistortion. Inspection of such a graph willreadily show which branch is the lower andwhich is higher because the asymmetriccontribution arising from prism-effect isadded to the symmetric distortion inherentin the lens. In such instances, the principalpoint lies on the side of the central imageindicated by the lower branch of the curve.To express it another way, the central image is displaced from the principal pointtoward the region of those points showingmaximum positive distortion or lying onthe upper branch of the curve.
To illustrate the manner in which thepresence of prism affects the values of distortion, computed values of D 1 and D 2 arelisted in Table 2 for a prism having anangle ex = 20 minutes and index n = 1.5 foran ideal lens having a focal length of 150mm. I t is clear from the table that evenfor this relatively large prism-angle, thevalues of D 1 are nearly equal to D 2 butopposite in sign. It is also noteworthy thatthe values of D 1 and D 2 are large and wouldbe still very appreciable for a prism-angleas low as 2 minutes.
a) Evaluation of the distortionThe values of the distortion assigned to
a camera-lens combination where prismeffect is present can still be obtained by
TABLE 2
Asymmetric distortion, D1 and D 2, induced inthe image-plane of a camera, equipped with alens having an equivalent focal length of 150mm. and having a prism-effect equivalent tothat produced by a prism of index 1.5 and anglea = 20 minutes. The shift of the central imagefrom the principal point isjEo=0.436 mm.
Distortion
{3 D,-D, D,+D,D1 D 22 2
degrees mm. mm. mm. mm.0 0.000 0.000 0.000 0.0007.5 - .014 .014 .014 .000
EFFECT OF PRISM ON LOCATION OF PRINCIPAL POINT 525
and
D, - D2 = 2f~ _ 2f(0 (32)2 cos2 fJ
The term ~/COS2 ,8 may be replaced by m~o
from Equation 19, and the above expression becomes
O'X' - 0'1" = 2f~o[m - 1] (29)
The values of 0'X' and 0' Y' are knownfrom the measurements, a sufficiently closeapproximation of f is known from Equation 9, and the proper value of m can betaken from Table 1. Accordingly
Actually one is usually more interested inthe value of f~o, which is the separation ofthe central image from the principal pointand can be obtained from Equation 29without the necessity of determining ~o
anda.b) Evaluation of a from the asymmetric
distortionIf the values of the asymmetric distor
tion, D1 and D 2, have been determined, itis a simple matter to obtain the value off~o and the prism-angle a. From Equations13 and 16,
f~ f~tanfJD, = --+-- - f~o (13)
cos2 fJ cos2 fJf~ f~ tanfJ
D2 = - -- +-- - f~o (16)cos2 fJ cos2 fJ
(31)
(30)
(18)
ex = 2~o.
~o
~=--.
n-l
O'X'- 0'1"~o=---
2f(m - 1)
For n assumed to be 1.5, then
and
or
2.3 EVALUATION OF THE EFFECTIVE PRISM
ANGLE a
When a camera is tested under conditions such that the test-targets are symmetrically located about the central targetand the focal plane is moreover perpendicular to the line joining the central targetand the front nodal point of the lens, it ispossible to evaluate the effective prismangle a in a relatively simple manner.There are two methods that can be usedleading to identical results. The firstmethod requires only the measured valuesof the distances separating image of thecentral target and the images of targets+,8 and -,8 respectively, plus the measured value of the focallengthj. The secondmethod uses the values of the asymmetricdistortion plus the value of f. Which of thetwo methods used in a calibration is determined by which one appears more convenient at the time.
a) Direct MethodThe evaluation of the prism-angle a di
rectly from the measured values of 0'X',0' y' and ,8 is called the direct method because it can be done without the necessityof determining the values of the distortionD1 and D 2• By referring to Figure 1, it isclear that
00' = f tan ~o (22)
O'X' = OX' - f tan ~o (23)
0'1" = 01" + f tan ~o (24)
On subtracting, Equation 24 from Equation 23, and substituting the proper functions of f,,8, and ~, it is clear that
O'X' - 0'1"
= f[tan (fJ +~) - tan (fJ - ~) - 2 tan ~o] (25)
which may be written
O'X' - 0'1"
(26)
2.4 LOCATION OF THE PRINCIPAL POINT
When a camera has been properlyaligned for test and no prism-effect is present, the central image and the principalpoint coincide. In such a case, the measured values of the distortion are symmetric about the central image. However,if prism-effect is present, the measuredvalues of the distortion are asymmetricand the central image is displaced from
= 2f[ tan ~ - tan~oJcos2fJ(l - tan2fJ tan2~)
For small ~, the above expression may besimplified to
[~ & tan2 fJ ]
O'X' - 0'1" = 2f --+--- - ~o (27)cos2 fJ cos2 fJ
For small values of ~, the second termwithin the brackets is negligibly small andEquation 27 becomes
O'X' - 0'1" = 2f[-~- - ~oJ (28)cos2 fJ
D, - D2 = 2f~o (m - 1)
which is identical to Equation 29.
(33)
526 PHOTOGRAMMETRIC ENGINEERING
the principal point by an amountfEo as indicated in Figure 1.
Therefore, if the magnitude of fEo is calculated in accordance with methods 2.3aor 2.3b, the distance separating the principal point and the central image is determined. The direction of the displacementis obtained from the rules given in section2.2. The location of the central image withrespect to the collimation index-markersof the camera can be measured directlyfrom the original test-negative and its coordinates obtained. Then, knowing thelocation of the principal point with respectto the central image and the location of thecentral image with respect to the center ofcollimation, it is a simple matter to determine the location of the principal pointwith respect to the center of collimation 7
3.0 EXPERIME TAL VERIFICATION FOR
A LENS AFFECTED WITH PRISM
To illustrate and verify the theory presented in the preceding pages, an experiment was performed using a precision
camera having a prism of known angleplaced in front of the lens. The camera selected was one having negligible prismeffect and for which the principal point andcenter of collimation had already beenbrought into coincidence. The camera wasplaced on the camera calibrator7 and soaligned that its focal plane was normal tothe optical axis of the central collimatorand so that the images from the variouscollimators would appear on the finalnegative in the manner shown in Figure 3.A thin prism having a dihedral angle, 0',
of approximately 12 minutes was placedon the front of the lens in the positionnormally occupied by the filter. The prismwas so oriented that its axis was parallelto the plane of the collimators that are thesources of the row of images marked I IIand IV in Figure 3. The maximum effectshould accordingly occur in the plane indicated by I and II.
I n making the first negative (designatedplate lA) the camera was so oriented inazimuth that the collimation marker A lay
: O"'~ 0 :I ii O~ 0 i:... :I !! O~ 0 II :, ,, ,] Dr l[ 1
io~:' ~ 0:L... ~------------------------..J
B
FIG. 3. Sch.ematic drawing of test negative obtained on camera calibrator showing the positionof the collimation markers A, E, C, and D, and the images of the various collimator targets. Thefour radial banks are indicated by the Roman numerals I, II, III and IV. This figure is to be considered in connection with table 3 and Figure 4.
EFFECT OF PRISM ON LOCATION OF PRINCIPAL POINT 527
midway between the radial rows of imagesformed by the lens of the targets containedin collimator banks II and IV. The base ofthe prism was positioned to give maximumeffect in direction 11. For the second negative (designated plate 2B), the camerawith prism still in place on the lens wasrotated 180 degrees in azimuth with allother conditions remaining unchanged.For this position, the maximum prismeffect should appear in direction 1. The location of all images on the two negativeswere measured with respect to the image ofthe central collimator-target (central image or center-cross).
The magnitude of the prism-effect expressed in terms of fEo, the shift of thecentral image, may be determined byeither of the two methods described in sections 2.3a and 2.3b. The method developedin section 2.3b will be described first underthe heading of the "one-plate method."The second method is based on the theorydeveloped in section 2.3a and is describedunder the heading of the "two-platemethod."
3.1 ONE-PLATE METHOD
The one-plate method makes use of themeasured values of the asymmetric distortion. To obtain accurate results, the valuesof {3 must be accurately known. In the present example, the measurements obtainedfrom plate 2B are used. The results ofmeasurement on the negative are listed inTable 3 for each value of {3 under the heading r. The symbols at the top and bottomof the column headed "ftan{3" indicate theorientation of the rows of images with respect to the collimator banks and t.hefiducial markers in the focal plane (see FIgures 3 and 4). The values of the equivalentfocal length were determined in the usualmanner and the value of f for each diameter appears at the top of the proper column. By using the proper f for each set ofmeasurements, the valuesof f tan{3 are computed and the value of the distortion, D,obtained from the relation
D = r - f tan {3 (34)
These values are shown under the headings "f tan {3" and "D." It is evident from
TABLE 3
Measured values of the distances separating images at angles {3 from the center-cross for Plate2B. A prism (O! = 12 minutes) is mounted in front of lens with its base facing the corner betweenA and D
Determination of the shift of the center-cross with respect to the principal point from consideration of the asymmetric distortion values given in Table 3
Average (22.5° to 45°).Jfo=.004 mm., fo=0.00003 radians, fo=O.0017° =0.10 min.
(35)
the measurements that the principaleffect has been obtained along the direction I and II as planned. It may be notedthat Dr (hereafter designated I) and Du(hereafter designa ted II), while opposite insign are of unequal magnitude. This difference in magnitude is a consequence ofthe large value of distortion inherent inthe lens itself.
Table 4 shows the values of the averageasymmetric distortion for each value of f3under the headings (I-II)j2 and (IIIIV)j2. The values of f~o are then computedfrom the relation
(D, - D,)f~o = -'c--'---_-:...
2(m - 1)
by using the values of m for n = 1.5 fromTable 1.
The direction of the displacement, f~o,
of the center-cross from the principal pointis given by the sign of f~o. I n the presentinstance, if f~o is positive, the displacementis along that radius where the targets incollimator bank I are imaged. The sketch,marked 2B in the upper part of Figure 4,
shows the coordinates of the displacementof the center-cross as determined from theasymmetric distortion. The lower part ofthe figure shows the coordinates of thecenter-cross directly measured with respect to the center of collimation. It isclear that excellent agreement exists between the two methods and that themethod of computing f~o from the asymmetric distortion is correct. It must beadded that too great reliance can not beplaced on values of f~o obtained for smallvalues of f3 because of the large effect produced by a 0.001 nim. error in rat f3= 7.5degrees and 15 degrees.
3.2 TWO-PLATE METHOD
When the prism-effect is small, as is usually the case, small errors arising fromsmall plate-curvatures and small inequalities in the absolute values of correspondingangles +f3 and -f3 may cause variation inthe magnitude and sign of the effect asdetermined by the one-plate method.These uncertainties can be reduced by using the two-plate method and the theory
EFFECT OF PRISM ON LOCATION OF PRINCIPAL POINT 529
IA
FIG. 4. Displacement of the central imagefrom the principal point by a thin prism havinga wedge angle of approximately 12 minutes.The upper sketches show the displacement computed from the distortion measurements. Thelower sketches show the actual displacementmeasured on the negatives.
shown in section 2.3a in determining thevalue of j(;o. In the two-plate method, twonegatives, plates 1A and 2B, are madewith the camera rotated 180 degrees inazimuth between exposures.
Tables 5 and 6 show the measuredvalues of the distances, r, separating thecenter-cross and the various images atangles {3 from the center, obtained fromplates 1A and 2B. The tables also show themanner in which these values are arrangedto facilitate computation. The two sets ofvalues are different and the differences forone bank of collimators are designated Vand the differences for the opposite bank ofcollimators are designated L.
The quantity V is essentially equivalentto the quantity
O'X' - O'Y' = 2j(;0(1II - 1) (29)
and L is the negative of the same quantity.Consequently, the average value is givenby
V-L-2- = 2j(;0(m - 1) (36)
R s O.246mm
AII lll:
O.158mm
0 O.l78mm C
1lI :r8
R"O.238mm
I",O.235mm '"
'"
,," ,
"IIR·O.235mm
Am
O.l43mm
D--=O""".I9'""O,--m-m+---- C
II8
R-O.238mm
TABLE 5
Determination of the shift of the center-cross from the principal point arising from prism-effect by the two-plate method for collimator banks I and I I. (Camera rotated 180° between plate lAand 2B)
fJ
degrees
4537.53022.515
7.5o7.5
1522.5
3037.545
lAj=131.551
r
mm.
130.748100.617
75.79854.40135.201
17.307o
17.31435.26254.564
76.133101.254131. 951
I j= 1~~.532
r
mm.
131.956101.247
76.12354.55035.256
17.319o
17.29735.20354.406
75.799100.630130.763
v
mm.
-1.208- .630- .325- .149- .055
- .012L
+.017.059.158
.334
.624+1.188
L
mm.
1.188.624.334.158.059
.017
V-L2
mm.
-1.198- .627- .330- .154- .057
- .14
mm.
.241
.239
.240
.241
.211
.221
IID A
IIB C
Average (22.5-45°)j'o=0.240 mm.
530 PHOTOGRAMMETRIC ENGINEERING
TABLE 6
Determination of the shift of the center-cross from the principal point, arising from prismeffect, by the two-plate method for collimator banks III and IV. (Can1€'ra rotated 180° betweenplates lA and 2B)
puted prior to determining f~o. The effectis twice that for the one-plate method.For these reasons, the two-plate method isused for all determinations of the prismeffect at the ational Bureau of Standards.
An additional advantage of the twoplate method is that it provides a simple,rapid means of determining whether or notplate-curvature is seriously affecting theaccuracy of the prism-effect and distortiondetermina tions. Considera tion of the quantity
leads to the conclusion that this quantityshould be zero if the two plates have identical figures and occupy the same positionwith respect to the lens, or should varydirectly with tan {3 if the two plates haveidentical figures and are displaced by asmall amount along the normal. The existence of values of (U+L) /2 tha t do not fallin these categories indicate the presence ofdiffering plate-curvatures or waviness ofthe test negatives. When the values of(U+L)/2 are near zero, one can havegreater confidence in the reliability of the
and
U-Lf~o = .
4(m-l)
The values of m for each value of {3 aregiven in Table 1. Values of (U - L) /2 andthe corresponding value of f~o are listed inTables 5 and 6. It may be noted that thevalues of f~o are essentially identical forboth the one-plate and two-plate methodsin this instance. The direction of the displacement, f~o, of the center-cross from theprincipal point is given by the sign of f~o inthe manner described in section 2.2. Itis also noteworthy that the magnitude of~o computed from the value of f~o is essentially identical with that predicted fromthe equation
~o = (n - l)a (18)
where n = 1.5 and a = 12 minutes.There are several advantages of the
two-plate method. The first is that it is notnecessary to know the values of {3 with ahigh degree of accuracy to get good valuesof f~o because the same angles are usedwith the prism-effect operative in oppositedirections. The distortion need not be com-
U+L2
(38)
EFFECT OF PRISM ON LOCATION OF PRINCIPAL POINT 531
TABLE 7Evaluation of the symmetric component of distortion for a lens affected by prism from the
measured asymmetric values of distortion. The initial values of distortion and f-o are taken fromTables 3 and 4
l( E. ) f_2 tan{32. ±f-o ---2-{3 -1 + 2 {3 (contribution from prism-effect, f_o=0.235 mm.).
fOCOS cos3. Values of distortion after compensation for prism-effect.4. Averaged values of distortion referred to the equivalent focal length (f=131.532 mm.).5. Departure of compensated distortion from average (3-4).6. t:.f tan {3 where t:.f+0.OO9 mm.7. Values of distortion referred to the calibrated focal length (fc=131.541 mm.).
measured values of distortion and prismeffect. Use was made of this quantity(U+L)/2 in deriving values of the probable error in distortion arising from pIa tecurvature that are shown in an earlierpaper (2).
3.3 EVALUATION OF THE DISTORTION
The values of the symmetric componentof the distortion can be obtained fairlyclosely by the use of Equation 20. It is ofinterest, however, to demonstrate that thesymmetric component can' be separatedfrom the asymmetric distortion by directcalculation. This has been done for one ofthe sets of distortion values contained inTables 3 and 4. The results are shown inTable 7. Column 1 shows the initial valuesof the distortion. Having evaluatedf~o, thecontribution from prism-effect is computedwith the aid of Equations 13 and 16 and islisted in column 2. The symmetric component is given by the difference of thevalues in columns 1 and 2 and is shown incolumn 3. The slight asymmetry remaining is indicated in columns 4 and 5, andfinally the computation of the distortionreferred to the calibrated focal length isshown in columns 6 and 7. The results arealso shown graphically in Figure 5.
The slight asymmetry that still existsarises in part from the difference in distortion at the angles ,B+~ and ,B-~. An additional correction can be made for this ifnecessary using procedures similar tothose described in a previous paper.
4. DISCUSSION
When a camera under calibration is soaligned that the focal plane of the camerais normal to the line joining the front nodalpoint of the lens and a distant object, thecentral image of the distant object islocated at the principal point of autocollimation of the camera. If asymmetricvalues of distortion are found under theforegoing conditions, it is evidence thatprism-effect is presen t and that the principal point does not coincide with theprincipal point of autocollimation. Moreover, the point of symmetry does not coincide with either of the other two points.The displacement of the central imagefrom the principal point can be evaluatedfrom the distortion-values, and the performance characteristics such as focallength and distortion can be determinedwith sufficient accuracy in the usual manner. However, a true point of symmetrycannot be located although, if the prism-
532 PHOTOGRAMMETRIC ENGINEERING
6f-
I
I-45° _30° _15° 0° 15° 30°
ANGULAR SEPARATION FROM AXIS IN DEGREES45°
FIG. 5. Evaluation of the true values of the lens distortion from the asymmetric measured values.The circles (curve 1) show the initial measured values of the asymmetric distortion. The X's(curve 2) show the values of distortion contributed by a prism of angle a = 12.3 minutes (fED=0.235). The crosses (curve 3) show the symmetrical distortion pattern remaining when curve 2is subtracted from curve 1. (See Table 7.)
effect is small (JED :::;0.010 mm.), it is possible to select a point of symmetry thatminimizes the asymmetric distortion to anextent that is tolerable.
5. ACKNOWLEDGMENTS
The author expresses his appreciation toother members of the staff for assistancerendered during the course of this work.The negatives used in the confirmatoryexperiments were made and measured byMr. W. P. Tayman. The drawings weremade by Mr. E. C. Watts.
6. REFERENCES
1. Washer, F. E., "Sources of Error in CameraCalibration." PHOTOGRAMMETRIC ENGINEERING, XX, 500 (1954).
2. Washer, F. E., "Sources of Error in VariousMethods of Airplane Camera Calibration."PHOTOGRAMMETRIC ENGINEERING, XXII,no. 4, pp. 722-740 (1956).
3. Washer, F. E., "The Effect of CameraTipping on the Location of the PrincipalPoint." J. Research NBS, 57, 31 (1956)RP 2691.
4. Washer, F. E., "A Simplified Method ofLocating the Point of Symmetry." PHOTOGRAMMETRIC ENGINEERING, XXIII, no. 1,pp. 75-88 (1957).
5. "Report of International Commission 1-
Proposal for International Photogram.metric Lens Tests." P hotogrammetria, I II,103 (1950-1951).
6. Washer, F. E. "Locating the PrincipalPoint of Precision Airplane MappingCameras." J. Research NBS, 27, 405 (1941)RP 1428
7. Washer, F. E. and Case, F. A., "Calibrationof Precision Airplane Mapping Camera."PHOTOGRAMMETRIC ENGINEERING, XVI,502 (1950); J. Research NBS, 45, 1 (1950)RP 2108
8. Roos, W., "Uber die Definition des Bildhauptpunktes und der Aufnahmeachse."Allgem. Vermess. Nachr. (Bildm. u. Luftbildwesen) P 235 (1950).
9. Sharp, J. V., "Basic Factors in Photogrammetric Instrument Performance." PHOTO·GRAMMETRIC ENGINEERING, XVI, 118(1950).
10. Sewell, E. D., "Field Calibration of AerialMapping Cameras," PHOTOGRAMMETRICENGINEERING, XIV, 363 (1948); MANUALOF PHOTOGRAMMETRY by the AmericanSociety of Photogrammetry, published1952, pp. 137-176.
11. Merritt, E. 1.., "Methods of Field CameraCalibration." PHOTOGRAMMETRIC ENGINEERING, XVII, 610 (1951).
12. Rock, D. 1.., "Field Determination of theCenter Cross." PHOTOGRAMMETRIC ENGINEERING, XVII, 596 (1951).
