l>t(X,y) = P (r) cos (<1> - <1>.)

Tangential Profile

........---- .....~AxiS of Ma~dmum Tangentiol Distortion

FIG. 1. III ustra.ting .geom.etrical sigr:ificance of parameters definingtangential distortIOn according to thin prism model.

DUANE C. BROWN*

D. Brown Associates, Inc.Eau Gallie, Florida

Decentering Distortion of Lenses

The prism effect encountered in metric cameras can beovercome through analytic calibration.

INTRODUCTION

T HE DISTORTION of a perfectly centeredlens composed of individually flawless

elements is strictly symmetric about theoptical axis. A significant degree of decenter­ing will introduce both tangential distortionand asymmetric radial distortion. The physi­cal suppression of such distortion to a valuenot exceeding five microns over the plateformat of the typical mapping camera re­quires appreciable skill and patience on thepart of the optical technician in aligning thelens; its suppression to less than two micronscalls for luck in addition to skill and patience.In view of the increasingly stringent require­ments for calibration resulting from recentadvances in analytical photogrammetric tri-

* Presented at the Annual Convention of theAmerican Society of Photogrammetry, Washing­ton, D. c., March 1965, under the title "Decenter­ing Distortion and the Definitive Calibration ofMetric Cameras."

angulation, it is no longer tenable to ignore ordismiss the metric consequences of decenter­ing distortion of even exceptiorlally wellcentered lenses.

I n this paper we shall review tHe generalproblem of decentering distortion, (;otlsider itsanalytical representation, establish its pro­jective properties, and demonstrate thepracticality of its precise calibratiort. In doingso, we shall review and partially recortcile twoapparently conflicting theories of decehteringdistortion: the thin prism model and Con­rady's model. We shaH also provide experi­mental verification of our major findiitgs. Thepresent paper is in large measure a revisionand extension of a section of an earlier paper(Brown, 1964; Section 10: Calibration ofDistortion Caused by Lens Decentration).

THE THIN PRISM MODEL

According to the thin prism model, the dis­tortion introduced by a slightly decentered

444

DECENTERING DISTORTION OF LENSES 445

lens can very nearly be duplicated by placinga.n appropriately orien ted thin prism of ap­pI'opriate deviation in front of a perfectlycentered lens. In general, a single such prismis adequate to account for the compositeeffect of any number of decentered elements,for a group of individual prisms in objectspace (one associated with each decenteredelement) can be replaced by a single, equiva­lent prism. In reviewing both the photogram­metric and the optical literature, we havefound that with but one notable exception

ment with what was to be expected from thethin prism model. The thin prism model wasadopted by Sharp (1949) in his considerationof the consequences of tangential distortionoccuring in multiplex projectors and bySewell (1952) in his treatment of cameracalibration in the ASP Manual of Photogram­metry. Washer (1941) adopted the thin prismmodel in considering the effects of decentering(or of a bent optical axis) on the physical de­termination of the principal point. Later, in acomprehensive experimental investigation of

ABSTRACT: The thin prism model has been widely adopted in the photogram­metric literature to describe the effects of a sensibly decentered lens. Exact ex­pressions are derived for the radial and tangential components of the disto'rtionintroduced by a thin prism placed in front of a perfectly centered lens. Thismodel is compared with an alternative model (Conrady, 1919) based on rigorousanalytical ray tracing through a decentered lens. When the principal point ofautocollimation is adopted as the plate origin, the two models are found to be inprecise agreement regarding the tangential component of decentering distortion,but are found to be at variance by a factor of three regarding the radial compo­nent. However, when compensatory translation of the plate and tipping of thecamera are permitted to operate, the two models are found to be projectivelyequivalent to terms of leading order. Because this projective equivalence doesnot extend to higher order effects (which may assume prominence with wide anglecameras), Conrady's model is clearly to be preferred for general application.A n extended form of Conrady's model has been put to practical application inthe stellar calibration of ballistic cameras. Results of ltctlutl calibrations arepresented and discussed, and the implications of the present development toanalytical photogrammetry are examined.

(Conrady, 1919) the thin prism model ap­pears to have been almost universally adoptedto account for decentering distortion.

The few textbooks on optics according anyconsideration at all to decentered lenses in­voke the thin prism model with little or nospecial justification (Hardy and Perrin, 1932;Strong, 1958; Martin, 1948). The same is truethroughout the photogrammetric literature.Bennett (1927) was one of the earliest to re­sort to the thin prism model to explain thetangential distortion observed in a number oflenses. Pennington (1947) noted the systema­tic effects of tangential distortion on photo­gram metric extension of control and dis­cussed the practical determination of tan­gential distortion, pointing out that its ob­served characteristics are in general agree-

thin prism distortion Washer (1957 a, b)photographed target arrays both with andwithout a thin prism of known deviationplaced in front of a well centered lens. He wasthereby able to demonstrate J:!xplicitly thatsignificant asymmetric radial e~ects are to beexpected from thin prism distprtion in adcji­tion to the tangential effects :'considered byother investigators.

I t should be noted, however, that the radialeffects of decentering distortion had beenimplicitly taken into consideration, in aninvestigation performed by Carman (1948).In this investigation sets of rays from pointson a uniform grid in object space were nu­merically traced through a thin prism placedin front of a hypothetical mapping camera.Carman was thereby able to demonstrate

446 PHOTOGRAMMETRIC ENGINEERING

At the principal point the tangential profilePer) is zero and is tangen t to the axis ofmaximum tangential distortion.

Using the method suggested by Penning­ton, Livingston (1951) measured tangentialdistortion across both diagonals of the photo­graphic format of a total of 33 Metrogonlenses and one Topogon. As we shall presently~ee, Livingston's results are not generally instrict accord with the thin prism model forangles in excess of about 25° from the axis ofthe camera. However, they do substantiatethe "cosine variation" of tangential distortionin accordance with Equation 1.

in which

~t(x,y) = tangential distortion at x, y inimage plane (origin taken at prin­cipal point of autocollimation),

r = (X2+y2)1/2 = radial distance,Per) = tangential profile (tangential dis­

tortion at radial distance r alongaxis of maximum tangential dis­tortion) ,

cPo = angle between positive x-axis andaxis of maximum tangential dis­tortion (see Figure 1),

cP = angle between posi ti ve x-axis andradius vector from origin to x, y.

that a suitable choice of principal point cansignificantly reduce, though not eliminate, theeffects of thin prism distortion.

It is well to consider at this point the pre­cise behavior of tangential distortion accord­ing to the thin prism model. As described byPennington (1947) there exists in the imageplane an axis passing through the principalpoint along which the tangential distortion ismaximum (the term principal point is usedhere in a loose sense). There also exists in theimage plane an axis through the principalpoint along which the tangential distortion iszero. The axis of zero tangen tial distortion isorthogonal to the axis of maximum tangentialdistortion. The tangential distortion alongany other axis passing through the principalpoint and lying in the image plane is propor­tional to that along the axis of maximumtangential distortion, the constant of propor­tionality being the cosine of the angle betweenthe two axes. In analytical terms, the thinprism model for tangential distortion may beexpressed as

(2)

EXACT ANALYTICAL EXPRESSIONS

. FOR THIN PRISM DISTORTION'

Despite the widespread adoption of thethin prism model, we have been unable to findin the literature exact analytical expressionsdefining the precise relationship between theradial and tangential distortion induced by athin prism at any specified azimuth. 'vVasher's(1957 a, b) explicit results, for instance, arerestricted to the plane through the principalsection of the prism and to the plane normalto the principal section (both planes containthe optical axis); his discussion of azimuthalvariation is brief and is strictly qualitative,apparently being based on limited experimen­tal findings rather than on detailed analyticalresults. In order to shed light on this matter,we have performed exact analytical, three­dimensional ray tracing through a thin prism.The key resul ts of this investigation as re­ported in our earlier paper (Brown, 1964) areas follows. The radial and tangen tial com­ponents of thin prism distortion of an imageof plate coordinates x, yare given by

Ll,(X, y) = P cos (cf> - cf>o),

Ll,(X, y) = P sin (cf> - cf>o),

where

c = principal distance (same units as x, y),€ = angle of prism (radians),

f.J. = index of refraction of prism,cP = angle between radius vector to x, y

and positive x-axis,cPo = angle between positive x-axis and

image of edge of prism (if cP=oo, a linethat is normal to the image of theapex of the prism and is directedthrough the principal point coincideswith the positive y-axis; if cPo = 180°,such a line coincides with the negativey-axis) ,

80 = angle between undeviated principalaxis and ray to image point (undevi­ated principal axis is arbitrarily takento be normal to front surface of prism),

81 = angle between principal ray and imageray after refraction by first surface ofprism,

82 = angle between normal to second sur­face of prism and refracted ray withinprism,

83 = angle between emergent ray and nor­mal to rear surface of prism.

in which

P = c[ (cosO, COSJ1.f - cos (3) SinJ1.f

+ (1 - cos f cos J1.f) sin 00 sin (cf> - cf>o)] (3)

(1)= ( : coscf>o + : sin cf>o)P(r)

Ll,(X, y) = P(r) cos (cf> - cf>o)

DECENTERING DISTORTION OF LENSES 447

From these and Equation 2 it follows that thex, y components of thin prism distortion are

D.x = D.r cos </> - D., sin </>,

D.y = D.r sin </> + D., cos </>.

O2 and hence 03 are weakly dependent on theazimuth cP. This means that even in the sim­plified Expression 4, the profile function P isnot strictly a function of radial distance (or of( 0) but varies weakly with cP as well. However,for small ~ this dependence of P on cP isnegligible for all practical purposes.

The general relationships between the x, ycomponents (Ll x, Lly ) of distortion and theradial and tangential components (here de­noted more compactly by Llr , Llt) are given by

(8)

(7)

(6)

sin </>0 = - ~xlP, cos </>0 = ~,,/P.

~x = - P sin </>0,

D.y = P cos </>0,

and

I t follows that

In the strict thin prism model P is a positiveand monotonically increasing function ofradial distance. Accordingly, Equation 7 forP and Equation 8 for cPo are entirely unam­biguous. Later, we shall have occasion torelax the thin prism model by allowing P toassume both positive and negative values. Toavoid ambiguities of sign under such circum­stances, we shall invoke bilateral symmetryto restrict cPo to the range 0:;;. cPo :;;. 180°. Thismeans that sin cPo can assume only positivevalues and, hence, that the sign of P mustalways be taken opposite that of Llx .

The character of thin prism distortion isillustrated in Figure 2. Here, we have ana­lytically projected a grid with 20 mm. divis­ions through a thin prism (p. = v2, ~ = 10')onto a hypothetical camera of 600 mm. focallength and 17° X 17° field. The azimuth of theprism has been taken as cPo = 0°, and the trueelements of orientation have been specified asa=O, w=O, K=O, xp=yp=O, c=600. mm. Be­cause cPo=O, Equation 6 reduces to Llx=O,D.

y= P, and the distortion in Figure 2 is

entirely in the y direction. The results ofFigure 1 constitute an analytical parallel tothe experimental results of Washer (1957b) inwhich a target array was photographed bothwith and without a thin prism in front of thelens. In the present case, however, the objectpoints are at infinity, whereas in Washer'scase they are at a sensibly finite distance(namely, a distance of approximately three

For specified c, p., ~ and cPo the various quanti­ties appearing in Equation 2 can be com­puted from the plate coordinates x, y bymeans of the following sequence of equations:

l' = (x2 + y2)'/2

sin (Jo = 1'/(1'2 + C2)1/2

sin </> = xiI'cos </> = ylrsin (J, = (sin (Jo) I jL

cos (J2 = sin (</> - </>0) sin (J, sin <+ cos (J, cos <

sin (J, = jL sin (J2.

lt is to be emphasized that the above expres­sions for P and for Ll r (x, y), Llt(X, y) are exact;no approximations were resorted to in theirderivation.

PROPERTIES OF THIN PRISM DISTORTION

If ~ is regarded as a small angle and onlyfirst order terms are retained, P can be re­duced to the form

P = (CjL<)(cos(J, - COS(J3). (4)

This in turn can be expanded to either of thefollowing:

P = CI'~ (1-~) sin2 (Jo2 jL2

+ higher order terms in </> and powers of sin2(Jo

, 1'< ( 1)P = 2c 1 - jL2 1'2

+ higher order terms in </> and powers of 1'2. (5)

These equations demonstrate that to terms ofleading order neither p. nor ~ is individually ofconsequence, but rather that both combine toform the essential parameter of the prismwhich is expressed by the common factor inthe cof'ficients of sin2 00 and 1'2. Accordingly,insofar as the tangential profile is concerned,there exist families of projectively equivalentthin prisms and one is at liberty to assign anyval ue such that p. > 1 and ~ >°for either p. ort, but not for both simultaneously. In prac­tice, it is convenient to specify a fixed valuefor p. that is typical for glass and to let ~

assume the role of the free parameter. Thevalue p. = v2, though slightly low for normalglasses, has been used in our studies because itlends some simplification to the ray tracingequations. Washer (1957 a, b) adopts thevalue p. = 1.5.

The explicit formulation provided by Equa­tions 2 shows clearly that the radial andtangential components of thin prism distor­tion are of essentially equal magnitude atequal radial distances along orthogonal radii.From the ray tracing equations it is clear that

448 PHOTOGRAMMETRIC ENGINEERING

80 r-r-j-j-!'-i-j-r-It60 r i f f t t f j r40

j f t 6 6 6 t f j

20 i f • A • f ij t • A • f j

-20j f 6 A A • , j

-40 i f t • 6 • t t i-60 j i f f f f t i j-80 r i j i i j i

-80 mm -60 -40 -20 20 40 60 80mm

Residual Scale t--+--1o 10 20 microns

FIG. 2. Residual vectors of thin prism distortion when true elements of orientation are enforced(1-'= vi, <= 10',1>=0, c=600 mm.); mean error = 7.71-'.

focal lengths from lens which leads to a 2 Xreduction of the target array). We believethat this significant difference in object spacedistances accounts for the partial discrepan­cies between Washer's results and ours.

PROPAGATION OF THIN PRISM DISTORTION

THROUGH LEAST SQUARES PROJECTIVE

TRANSFORMAnON

During the course of the past decade wehave had the opportunity to study the re­sidual vectors from scores of stellar plates,each taken explicitly for the calibration ofsymmetric radial distortion and each contain­ing from 100 to 200 fairly uniformly dis­tributed images. In those cases where de­centering distortion was sufficiently large tobe obvious from visual inspection of theresiduals, the general nature of the systematicpattern bore little resemblance to that ofFigure 2, due allowance being made for arotation in <Po. This is not surprising, for in aleast squares stellar calibration the elementsof orientation will naturally adj ust in such amanner as to minimize the quadratic form ofthe residuals. Therefore, the calibrated ele-

ments of orientation will compensate in partfor the effects of decen tering distortion, andthe basic pattern of decentering residuals willbe altered by the compensative process.

In order to gain insight into the precisenature of the compensation resulting from aleast squares projective transformation, wesubjected the plate coordinates of the dis­torted grid of Figure 2 to a least squares ad­justment under the assumption that the gridcoordinates in object space were perfectlyknown. The resulting residual vectors areplotted in Figure 3 (the pair of curves plottedin the figure are for future reference). The'calibrated' elements of orientation turnedout to be a=O°, w=0°.0255, K=O°, xp=O,Yp=0.260 mm., c=600.000 mm. This demon­st~ates that the basic mechanism afforded byelements of orientation for compensation con­sists of:

a. A shift of the principal point away fromthe edge of the prism,

b. A tilt of the camera axis away from theedge of the prism (a corresponds to xtilt and w to y tilt),

DECENTERI G DISTORTION OF LENSES 449

I

I

!

•

..

~HYPOTHETICAL TRUE TRAJECTORY

~ ,,~ ."),,/ ~,

............t" ~ ·......

~ ... ", J ~ ~

f

40

20

60

80

-40

\

\

~\

-20

-80 \ .. .. 'fI

-80 mm --60 -40 -20 20 40 60

Residual Scole: 1---+---1o 5 10 microns

FIG. 3. Residual vectors of thin prism distortion when elements of orientation are obtained fromleast squares ~djllstment (same thin prism parameters as in Figure 2); mean error=3.6!,.

I t is noteworthy that the principal distancedoes not enter into the compensative processand that the compensative translation of theplate and til t of the camera are both confinedto the principal section of the prism. In com­paring Figures 2 and 3 one should note thatthe scales of the residual vectors differ by afactor of two. The rms error of the raw re­siduals of Figure 2 is 7.7 )1., whereas that of theresiduals of Figure 3 is 3.6)1., a more than two­fold improvement. The residual vectors afterthe least squares projective transformationare no longer strictly unidirectional as inFigure 2 but have sizeable components inboth x and y. As we shall illustrate presently,the general pattern of the residual vectors ofFigure 3 correlates well with the observedpattern of the systematic components ofresidual vectors resulting from stellar calibra­tions of symmetric radial distortion.

PROJECTIVE COMPENSATION OF THIN PRISM

DISTORTION FOR WIDE ANGLE CAMERAS

With narrow projective bundles (as inFigures 2 and 3) a small shift of the principalpoint is very nearly the projective equivalentof a small til t of the camera axis. This near

equivalence or translation and rotation doesnot hold for wide projective bundles. In orderto determine the nature of the compensativeprocess for wide projecti\"e bundles, we re­peated the computations leading to Figures2 and 3 for a lens of 115 mm. focal length and76° X 76° angular field. The prism angle waschanged from 10' to 2' so that the generalmagnitude of the distortion would be un­altered (as is clear from Equation 5, theeffects of thin prism distortion vary as adirect product of prism angle and focallength). Because the residual patterns beforeand after projective compensation turned outto be practically identical with those ofFigures 2 and 3 respectively, they are notreproduced here. The only essential change inthe over-all results was that the principalpoint underwent essentially no adjustment inthe case of the wide projective bundle (thevalue yv=O.l)1. was obtained); all compensa­tion resulted from a tilt of the camera axisaway from the edge of the prism. This cor­roborates Washer's (1957b) observation thatan appropriate til t of the camera can offset toan appreciable extent the effects of the asym­metric radial component of thin prism distor­tion.

450 PHOTOGRAMMETRIC ENGINEERING

AKALYTICAL EXPRESSIONS FOR

PROJECTIVE COMPE~SATION

From the fact that compensative transla­tion and tilt are confined to the plane of theprincipal section of the prism it can be demon­strated analytically that the equations forthin prism residuals resul ting from projectivecompensation (as in Figure 3) are of the fol­lowing form:

Llx = - (P + b - cy) sin <1>0

- 2- r' cos <I> si n (<I> - <1>0)c

Lly = (P + b - cy) cos <1>0

- 2-" sin <I> cos (<I> - <1>0) (9)c

in which b (same units as c) and "( (radians)denote the magnitudes of the compensativetranslation and tilt. The corresponding ex­pressions in terms of radial and tangentialcomponents are

Llr = (p + b - cy - : ,2) sin (<I> - <1>0) (lOa)

Ll l = (P + b - cy) cos (<I> - <1>0)' (lOb)

These relations explain fully the nature ofprojective compensation. In particular, theyclarify the relative roles of translation andtilt. As we have seen, the translation b isessentially unexercised in the compensativeprocess for wide angle cameras. On the otherhand, it performs an important function inthe compensative process for narrow anglecameras, for here tilt compensation by itselfis relatively ineffective. By largely, thoughnot quite completely, counteracting the effectof tilt, the translation b permits the applica­tion of a tilt which would otherwise be exces­sively large. As a result, the term ("(Ic)r' canbecome sufficiently large to be effective in theabove expression for D. r and yet not lead toovercompensation in the expression for D. t .

Equation lOb shows that for tjJ=tjJo thevalue of D. t becomes equal to b-c"( at x=y=O(here, P = 0). This demonstrates that tangen­tial distortion is not zero at the origin aftercompensation. Moreover, since P +b -c"(

passes through zero at a sufficiently largeradial distance, it follows that tangential dis­tortion can assume both positive and negativevalues across the format. The shape of thetangential profile is unaltered by the com­pensative process, which does nothing morethan to translate the profile by the amountb-c"(, thereby producing a positive and nega­tive balance of the profile across the format.

By contrast the compensative process for theradial component involves not only the trans­lation b-cy, but also the second order term-("(Ic)r', which serves to counteract much ofthe effect of the leading term of the expansionfor P given by Equation 5.

This explains why projective compensationis appreciably more effective in reducing theradial component of thin prism distortionthan in reducing the tangential component.We find, for example, that prior to projectivecompensation the rms values of the radial andtangential components in Figure 2 are bothequal to 5.4 }J., whereas after projective com­pensation (Figure 3) they become 2.5 and4.2}J. respectively. Perhaps the effectiveness ofprojective compensation explains why theradial effects of decentering have receivedrelatively little recognition in the literature.

Another possible reason is that the natureof the radial component is such that it has noeffect on the angle subtended by pairs ofradially symmetric points. This would renderimpossible the detection of the radial com­ponent in those procedures of lens calibrationwhich depend intrinsically on the determina­tion of the relative radial displacements ofopposing pairs of targets symmetricallyarrayed across the diagonals of the plate.Only when the measurements are made rela­tive to a central target imaged precisely at theprincipal point of autocollimation (as invVasher's investigation) can the existence ofthe radial component be detected and sepa­rated with certainty from possible effects ofcamera tilt. Indeed, it is to avoid such exact­ing alignment of the camera that some meth­ods for the calibration of radial distortiondeliberately avoid referring the measurementsto a central target and thereby implicitlyforego the possibility of measuring asym­metric radial distortion.

CONRADY'S MODEL FOR

DECENTERING DISTORTION

In our discussions above we have scrupu­lously avoided considering thin prism distor­tion to be equivalent to decentering distor­tion. This was done in anticipation of thepresent section in which we shall review a setof rigorous results derived by Conrady (1919)in an elegant but little known paper. Exceptfor Conrady, all of the references cited thusfar have uncritically adopted the thin prismmodel as accounting for decentering distor­tion. It seems that Conrady's paper is notwell known, for in all the literature we havereviewed, it is cited only by Livingston (1951)

DECE TERING DISTORTION OF LENSES 451

and there only in oblique justification for thethin prism model as it pertains to tangentialdistortion. As we shal1 see, Conrady's modelis in agreement with the thin prism modelregarding the tangential component of de­centering distortion but is substantial1y atvariance with the thin prism model regardingthe radial component. Conrady's results arebased on analytical ray tracing through a de­centered lens and are therefore exact throughthe order of the terms carried. They considernot only the effects of decentering on distor­tion but also its effects on the other aberra­tions as well. In particular, Conrady showsthat decentering has the fol1owing major pri­mary effects:

1. It in troduces a coma that (like normalcoma) varies with the square of apertureand (unlike normal coma) is uniformover the field both in magnitude and indirection;

2. It introduces an astigmatism character­ized by image patches in the form ofel1ipses varying in length, eccentricity'and orientation in different parts of thefield;

3. It introduces a distortion having radialand tangential components given by (inour notation)

t>r = 3P sin (", - "'0)

t>, = P cos (", - "'0) (11)

the origin on the plate being taken at theprincipal point of autoc01limation.

I t should be noted that Conrady's P3 V2 cor­responds to our P and that his angle X cor­responds to our angle of 90-(cP-cPo). Con­rady demonstrates that the' above propertieshold for a system having any number of de­centered surfaces.

The first and second consequences of de­centering affect the shape of the image patchbut not the position of its center relative toother images; moreover, in reasonably wel1centered lenses these effects are likely to bealmost imperceptibly smal1 in relation to theclassical aberrations. The third effect is thusthe only one of possible metric consequence.Comparing Conrady's expressions for ~" ~t

with the thin prism Expressions 2, we findthat the two models agree for ~l but differ bya factor of three for ~r. It f01l0ws that, as itstands, the thin prism model only partiallyaccounts for the effects of decentering distor­tion and would appear to be substantial1yinade.quate with regard to the radial compo­nent. As we shal1 presently see, this apparent

discrepancy is of no projective consequenceinsofar as first order effects are concerned.

In Figure 4 we have plotted residual vec­tors for decentering distortion according toConrady's model. Here, we have adopted thesame tangential profile, the same value of cPoand the same elements of orientation as inFigure 2. We see from Figure 4 that the pri­mary residual vectors of decentering distor­tion are not unidirectional as are the primaryresidual vectors thin prism distortion (notethat the scales of the residual vectors inFigures 2 and 4 differ by a factor of two). Thex, y components of decentering distortion aregiven by the following analytical expressions:

[( X2) xy Jt>% = - P 1 +2 -;;; sin "'0 - 2 1" cos "'0

t>y = - P[2 xy sin "'0 - (1 + 2 Y') cos "'oJ. (12)1" 1"

These expressions are appreciably more in­volved than their counterparts for thin prismdistortion (Equation 6).

PROPAGATIO:-\ OF DECENTERING DISTORTION

THROUGH LEAST SQUARES PROJECTIVE

TRANSFORMATION

As we did for the thin prism distortion ofFigure 2, we subjected the primary decenter­ing distortion of Figure 4 to a least squaresprojective transformation. The resultingtransformed pattern turned out to be identi­cal in every respect with Figure 3 and so is notseparately presented. From this, we may con­jecture that the thin prism model is projec­tively equivalent to Conrady's model andthat the thin prism model provides a suitablemodel for decentering distortion if it is modi­fied to embrace a suitable translation of theplate and tilt of the camera. The elements oforientation resulting from the least squaresprojective transformation of Figure 4 are:a=O, w=0.o1431, K=O, xp=O, yp=1.491mm., c= 600.000 mm. The elements undergo­ing adjustment (wand yp) are in this caseseveral times larger than the correspondingvalues for thin prism distortion, a result to beexpected from the threefold larger radial com­ponent of decentering distortion.

\Ve shall rtow examine the mechanism lead­ing to"the apparent projective equivalence ofthin prism distortion and decentering distor­tion. The radial and tangential components ofdecentering distortion resulting from the ap­plication to Equation 11 of a translation b'and a til t "(' can be shown to be given by

452 PHOTOGRAMMETRIC ENGINEERING

Ir

00 ~--\-rr-r-1---1-+/~60 \ \ \ , f f ! I I40 \ '\ ~ - • ~ ~ ! !

20 ~ ~ • ~ " ~ ~

• 10 A A 10 10 •~ ~ • • • .. .. ~-20

~ ! ! ~ ~ • - 1\ '\ \

-00 ! I ! f f , \ \ \-80 / I ! f \ \ \-80_ -00 -40 -20 20 .0 60 ao~

Residual Scale:~micrDns

FIG. 4. Residual pattern caused by decentering; distortion according to Conrady's model for lenshaving same tangential profile as thin pF'isltl of Figure 2. (Note that the scale of the residual vectors inthis figure differs by a factor of two from that of Figure 2.)

6, = (3P + b' - c-y' - -y; r2) sin (</> - </>0)

6, = (P + b' - c-y') cos (</>---~ (13)

According to Equation 5 the profile functioncan be expressed to the first order in r2 asP~Jlr2. If this substitution is made in Equa­ti6ns 10 and 13 and if the radial and tangen­tial components of transformed thin prismdistortion are equated to the correspondingtransformed components of decentering dis­tortion, the following results are obtainedupon reduction

b' = b + 2J,c2

-y' = , + VIC.

These relations between the two sets of com- ,pensative motions are independent o~ r, a rl;!­suit of pivotal importance stemmil1g <;lirectlyfrom the expression of P as equlvalef\t toJ,r2. This means that as long as P is of 'theform J,r2, the two models are projectivelyequivalent, for an appropriate translation andtilt can be found that will transform the onepattern of distortion into the other for allvalues of r. Indeed, it is to be noted that the

first order prajeet1ve equivalence of the twomodels is not dependent on the specific ratioof three between their respective radial com­ponents, bu t would hold as well for any ratio,say k, in which case the factors of two on theright hand sides of the above equations wouldbe replaced by k-l.

We shall shortly find that in certain in­stances at least one higher order term in r2

may be required in the profile function. Whenhigher order terms are significant, the strictprojective equivalence of thin prism anddecentering distortion no longer holds. Inview of this and in view of the special auxili­ary translation and til t required to render thethin prism distortion equivalent to decenter­ing distortion (a process, we would emphasize,that is possible only when higher order effectsare insignificant), we suggest that the thinprism model be abandoned entirely as amodel for decentering distortion. Conrady'smodel as expressed either by Equations 11 or12 provides the more suitable model fordecentering distortion the validity of which inno way depends on artificial compensativemotipns of the plate and camera.

DECENTERING DISTORTION OF LENSES 453

25u

r

•

10

15

20

. -120mm -00 -40 0 40 00 120mm

.....................

..............

... I I_40 0 _30 0 _200 _100 00 100 20° 30° 40°

FIG. 5. Average Profile function of tangential distortion obtained by Livingston from measurementsof 33 Metrogon lenses as compared with most nearly equivalent thin prism profile.

where p denotes the distance between the twopoints. In the thin prism model p is given by

p = c(p. - 1).

and ~, the prism angle, can in turn be com­puted from the leading coefficient J 1 of thetangential profile by means of

2cp.] 1• = -_-:--:-_-(p. - 1)(p. + 1)

Inasmuch as a decentered lens partiallyconforms to the thin prism model, it is clearthat, as a secondary effect, decentering willcause the image patches to be broken downinto minute spectra. In the strict sense, thismeans that the coefficients of the profile func­tions J 1, J 2, ••• are color dependent. How­ever, chromatic effects of decentering distor­tion are likely to assume practical significanceonly for objectives of very long focal lengthsuch as are used for astronomical observa­tions.

RELATIONSHIP BETWEEN PRINCIPAL POINT

OF PHOTOGRAMMETRY AND PRINCIPAL POINT

OF AUTOCOLLIMATION.

In the projective equations relating imageand object spaces, the fundamental point ofreference on the plate is the principal point ofphotogrammetry (xp, yp). In the analyticalprocess of calibration this point is recoveredrelative to the plate center as defined by asystem of fiducial marks. However, in our dis­cussions of decentering distortion the naturalorigin of the plate coordinate system becomesthe principal point of autocollimation (xp ."

Ypa)' Inasmuch as the displacement of the twoprincipal points must lie along the radius vec­tor of angle 90o +epo, the two points are inter­related by

(15)x". = X p + p sin <1>0

Y,>a = YP + p cos <1>0

HIGHER ORDER EFFECTS OF DECENTER1NG

As we have already noted, Livingston's(1951) results for wide angle lenses (Metro­gohs subtending 74°X74° fields) are notgenerally in strict accord with the thin prismmddel, for the typical tangential profile P(Figure 5) found by Livingston is not mono­tonic as required by the thin prism model butrather reverses its direction towards the edgeof the field. This observed reversal of tangen­tial distortion cannot be explained by errorsin camera set up as would have been the casehad such reversal been observed in theasymmetric radial profile; the tangential re­versal is clearly real and not a consequence ofexperimental procedure. It is noteworthythat Livingston's experimental results dem­onstrate the validity of the cosine variation oftangential di~tortion, reversal and all. Al­though revers!!l cannot be explained by thethin prism moclel, presumably it could by thehigher order t~rms of Conrady's model. Un­fortunately, Cqnrady does not develop theseterms and, in liE\\.l of supplementary analyticalresults, we must resort to conjecture concern­ing the nature Q,f the higher order effects ofdecentering. All \lvailable evidence suggeststhat higher order effects can adequately beaccounted for simply by adding powers of r2

to the profile function which then assumes theform:

P = J 1,2 + f2,. + J 3,6 + . . . (14)

As with higher order thin prism distortion, theexact profile function for decentering distor­tion will, we believe, ultimately be found to beweakly dependent on azimuth ep. Be this as itmay, experience to date indicates that Con­radY's model given by Equation 11 or 12, asmoqified by the incorporation of the expandedprofile Function 14, provides a working modelfor tlecen tering distortion that is sufficien t forall I:>ractical needs.

454 PHOTOGRAMMETRIC ENGINEERING

In Conrady's model the displacement in theprincipal section is three times greater than inthe thin prism model and the appropriateexpression for p accordingly becomes

p = 3c(!, - l)E.

Although, as we have already noted, the valueof I-' may be regarded as arbitrary insofar asthe tangential profile is concerned, this optionno longer holds when the separation of theprincipal points becomes a consideration, forthe value of p will clearly vary with eachchoice of 1-'. It follows that either p or I-' mustbe viewed as constituting an independentparameter when an attempt is made in thereduction to recover the principal point ofautocollimation relative to the principal pointof photogrammetry. By letting p rather thanI-' be the independent parameter, one makesno specific commitment to either the thinprism model or Conrady's model and accord­ingly one obtains a model valid for a combi­nation of decen tered optics and prismaticfilters.

To accomplish the calibration of p, oneneed merely replace x and y as consideredin foregoing development by the expressions

x - Xv - p sin </>0,

y - Yv - p cos </>0

in whichpnowconstitutesanadditional param­eter to be recovered in the calibration (atthis point we should again note that the co­ordinates of the principal point X p , YP are in­variably recovered in a definitive calibrationand hence do not constitute new parametersarising from a consideration of decentering).When decentering is small, the analytical re­covery of p is likely to be marginal becausethe variation in the partial derivatives of thex and y coordinates with respect to p is veryslight throughout the format, a consequenceof the fact that the variation in these de­rivatives depends primarily on the ordinarilysmall magnitudes of the coefficients of de­centering distortion Jl' J 2, . .. and coeffi­cients of symmetric radial distortion K"K 2, • •• (here it should be noted that theprincipal point of autocollimation serves alsoas the natural origin for computing the radialdistances entering the expressions for sym­metric radial distortion).

In most calibrations attempts to recover pto meaningful precision are likely to provefutile, for the coordinates Xp , YP can generallystrike a practical and effective compromisethat renders p projectively superfluous. Onthe other hand, with cameras of fairly lohgfocal length (on the order of several meters),

the value of p may amount to several milli­meters for moderate decentering. Here, theseparation between the principal points canconceivably assume projective significance,particularly when symmetric radial distor­tion is also large. In view of this, there may bemerit in carrying p provisionally through cer­tain reductions as an adjustable parameter,subsequently to be dropped in the event thatits inclusion fails according to an F-test toreduce significantly the quadratic form of theresiduals.

PRECISE ANALYTICAL CALIBRATION

OF DECENTERING DISTORTION

Decentering distortion can be calibratedanalytically in the same manner as has beenused successfully for years in the case ofsymmetric radial distortion (Brown, 1956),namely by incorporating the mathematicalmodel for decentering distortion into theoriginal projective equations generated by theplate measurements and by solving the result­ing system of normal equations for the de­centering parameters in addition to the otherparameters normally carried (elements oforientation, coefficients of symmetric radialdistortion, parameters of atmospheric refrac­tion, etc.). Inasmuch as the expressions for~x, ~y are nonlinear in cPo, the determinationof a suitable initial approximation for cPopresents a problem. Even when a 'suitableapproximation for cPo is available, it becomesnecessary to assign a discrete (nonzero) initialapproximation to J 1 in order to prevent thelinearized coefficients of the correction to theapproximation to cPo from being zero in all ofthe linearized observational equations. Oneway around this problem involves the exerciseof successively relaxed a priori constraints asdiscussed in Brown (1964). However, a farmore efficient and hitherto unpublished ap­proach is employed in our curren t version ofthe advanced plate reduction. It involves theintroduction of the new parameters PI, P 2,

P 3, ••• defined by

P, = -J,sin</>o

P. = J, cos </>0

P a = Jz!J,

P, = J 3/1,

which recasts the extended expressions forConrady's model into the form

L'lx=[P,(r2+2x2)+2P2xy][1+Par2+P.r'+··· ]

L'ly=[2P, xY+P2(r2+2y2)][1+Par2+P.r'+···],

DECENTERING DISTORTION OF LENSES 455

If the higher order coefficients h, J 3 werezero, the above expressions for t.x and t.y

would be reduced to linear functions of PI andP 2, and the initial approximations for theseparameters could be taken as zero. For themore general case where higher order coeffi­cien ts are to be considered, it suffices to adoptzero values for the initial approximations ofall the P's, the only precaution being thatarbitrary, but large, a priori variances beassigned to P 3 , p., ... to counteract (in theinitial adjustment) the indeterminacy other­wise resulting from the zero initial approxima­tions to PI and P 2• In subsequent iterations ofthe adjustment, the improved approxima­tions to PI and P 2 become discrete, thus ren­dering the solution for P 3 , p., ... determi­nate.

We shall not go further into the details ofthe extension of the plate reduction to em­brace the calibration of decentering distor­tion, for the computational mechanics of theprocedure are fully covered in Brown (1964).Rather, we would point out that when param­eters of decentering distortion are incorpo­rated into the adjustment, the covariancematrix of the adjusted values of these param­eters is contained in the inverse of the coeffi­cient matrix of the normal equations. Thismeans that the accuracy of the calibrationcan readily be computed for any specifiedpoints on the plate. I n stellar calibrations in­volving on the order of 200 well distributedstars, the error in calibrated decentering dis­tortion can generally be suppressed to onemicron or less at the extreme corners of theformat and to an rms value over the entireplate of abou t 0.4 microns.

EXPERIMENTAL RESULTS

To this point all results have been basedsolely on analytical considerations coupledwith supporting numerical simulations. Inorder to test the theoretical results we re­sorted to a special physical experiment. Incooperation with Space Systems Laboratoriesof Melbourne, Florida, we obtained stellarplates from a pair of Pth 60 Phototheodolitesmanufactured by SSL. The cameras havefocal lengths of nominally 600 mm., effectiveapertures of about 200 mm" and angularfields of 17° X 17°. They are designed to use6-mm. thick plates of dimensions 190X21Smm. One of the cameras (SSL 001) wasknown by inspection on an optical turntableto be out of alignment to the extent that smallfurther physical adj ustment would have beendistinctly worthwhile. The second camera

(SSL 002) was considered by optical tech­nicians to be aligned to the practical limit oftheir art and hence was deemed not to be sub­ject to further meaningful physical improve­ment.

The stellar exposures ",ere made simulta­neously for both cameras, the cameras beingside by side, and were of a common zenithalfield, Kodak Microflat glass plates coatedwith 103 F emulsion were employed. Theplates were photographically processed to­gether and were measured on a calibratedMann comparator by the same operator ondifferent days, A total of lSS well-distributedimages were measured on each plate, a pair ofsettings being made on each image. The platemeasurements \\'ere subjected to the ad­vanced plate reduction developed in Brown(1964), the appropriate parameters beingcarried for symmetric radial distortion butnone being carried for decentering distortion.No allowance was made for star catalog error,even though the General Catalog was em­ployed (typical GC error is equivalent toabout 2 microns on the plate of 600 mm.camera). This was deliberate and was done toprevent any possibility that the adjustment ofstellar positions might partially compensatefor locally significant systematic effects. Thex, y least squares residuals therefore reflectnot only random error in the plate coordi­nates, but also random error in the stellar co­ordinates as well as systematic error (such asdecentering error) not fully accountable bythe mathematical model of the reduction. Theresidual vectors for SSL 001 and 002 areplotted in Figures 6 and 7 respectively.

Very definite systematic tendencies of theresidual vectors are obvious from a visualinspection of Figure 6. \;Vhen due allowance ismade for the random component of theresidual vectors, we see that the general sys­tematic pattern of Figure 6 is in excellentcorrespondence with that of Figure 3 asrotated by about 70°. This correspondenceprovides solid experimental confirmation ofour theoretical results. It is noteworthy thateven though pronounced systematic effectsare evident in Figure 6, the rms error of theresidual vectors is only 3.9 microns, a resultdemonstrating the effectiveness of projectivecompensation.

The systematic effects so pronounced forSSL 001 are absent from 002 (Figure 7). Thisdoes not necessarily mean that decenteringdistortion is insignificant for 002, but ratherthat it is sufficiently small relative to the ran­dom error to elude casual visual detection. In

456 PHOTOGRAMMETRIC ENGINEERING

" .....so

~ ~ -...'\\'60 '- 't-• • ~/ ~. ~ •

r --- i-~, -\I "{ - _--t ~

" Ii20

--" i~-J' ---/.- ~ -..

"". '-.. - -.-20

"- --- •--- ~-.40 ~' -

-20-40-80 mm -60

~~_...- --., .....

,/ tI jl _~ -.:--~

"-20 4ojt/somm

Reliduot Scale: t-----t-----1o 5 10 microns

FIG. 6. Residual vectors from stellar calibration of SSL Camera 001 as obtained from least squaressolution for elements of orientation (no parameters carried for decentering distortion); meanerror=3.9J.l.

t -J-- ,--so .....

i~/..

\ ~\ ~, 4

60¥" :/' '\ /- ,

./ ..~ •

40

lif ...-----" "w\ t ~ ~• ,-~

~ -,20- '" ,

--+

r J :I •• J, ¥ • +)

~~-20 ,"

~ l/ I-40 , • .--A •

V I •.. •,A \ -- ..... ,(~~ --- t

'", ~ \ ~ •

" ~ r f /-80~ ""....

.... -~, •.-+-'-~-80 mm -{>o -.40 -20 20 40 60 80 rnm

Residual Scale: t-----t-----10 5 10 microns

FIG. 7. Residual vectors from stellar calibration of SSL Camera 002 as obtained from least squaressolution for elements of orientation (no parameters carried for decentering distortion); meanerror=3AJ.l.

DECENTERING DISTORTION OF LENSES

TABLE 1. MEAN ERRORS RESULTING FROM VARIOUS ADJUSTMENTS

457

Case I I Case II Case IIINumber of

Camera Control Mean Error of Mean Error of Mean Error of Mean Error ofPoints Plate Coord. Plate Coord. Plate Coord. Stellar Coord.

Residuals Residuals Residuals Residuals

SSL 001 155 3.91' 2.51' 2.11' 0."27SSL 002 155 3.41' 3.11' 2.81' 0."34

Case I. Star catalogue error and decentering distortion are not explicitly considered in the adjust­ment (residuals plotted in Figures 6 and 7).

Case II. Decentering distortion, but not star catalogue error, is rigorously treated in the adjustment.Case III. Star catalogue error and decentering distortion are both rigorously treated in the adjust­

ment (plate coordinate residuals for Camera 001 are plotted in Figure 8).

view of typical plate measuring accuracies of2 to 3 microns, it is altogether conceivablethat even after projective compensation,residual decentering distortion might amountto as much as 3 to 4 microns in some parts ofthe field and have an rms error of as much as2 microns. It is therefore clear that a needexists for a method of evaluating possibledecentering distortion that is more powerfuland less subjective than visual inspection ofleast squares residuals. As indicated earlier,an objective solution to this problem consists

of recovering the parameters of decenteringdistortion within the plate reduction itself.

RESULTS OF STELLAR CALIBRATION

OF DECENTERING DISTORTION

With the plate reduction extended to in­corporate parameters for decentering distor­tions, we repeated the stel1ar reductions forSSL Cameras 001 and 002. Results of threedifferent reductions of varying levels of re­finement are su mmarized for each camera inTable 1 and the final residual vectors for 001

~

180~ \ ..,.

~~

II ., ¥" • • ill60 ... .,.. - • • •,';4. " ~... - ........

40 .# .. '\• # •• -.....t ~ . ...... ~ •~ f I

20 A • • • - ~

" \ ~ ~

"' .....,~

.. ..- .. ~ •, - J - 1\ .. ".. •• ".. .. i" ......

-20 • \ • ~ .. -- .. .. -.. ~ f ..~-4 --;-/ - --~o

_.. .... .p' ~~ .# !A• j. /"~ • " ~

~ ... II ..- .-, - ""',,",. ""~ 11

"-80 ../ II .. ...l.-.r' • \ lo ......

-80 mm ~ -~o -20 20 ~O 60 80mm

Residual Scale: t----+------i0 5 10 microns

FIG. 8. Residual vectors from stellar calibratIOn of SSL Camera No. 001 as obtained from AdvancedPlate Reduction considering decentering distortion and random error in star catalogue (same originaldata as in Figure 6); mean error = 2.11'.

458 PHOTOGRAMMETRIC ENGINEERING

P(r)lOp~--...,....-----r---.,-------~-- ......

20

-sl----t----j--

-101---+-.,..---+----+----'-

-lSI----t----j----t---

" "" """ "" I" "-2Op L-__-'-__-----' -'- l. ..!-__\ • P+"

" p

\ :-"p

FIG. 9. Profile functions and associated one sigma confidence limits of decentering distortion resultingfrom stellar calibrations of SSL Cameras 001 and 002 by means of the Advanced Plate Reduction.

are plotted in Figure 8. The calibrated profilefunctions P(r) for both cameras are presentedtogether with their one sigma confidencelimits in Figure 9. We see from Figure 9 thatdecentering distortion for 001 is nearly threetimes as great as for 002 and amounts toabout 15 microns at a radial distance of 100mm. The one sigma error bounds of thecalibrated tangential profiles reach the level ofone micron only in the corners of the format.It follows that decentering distortion is amen­able to very precise calibration.

Although the calibrated profile function for002 grows to 5 microns at 100 mm., it shouldbe kept in mind that this is representative ofthe profile function without the benefit ofprojective compensation. ~Then such com­pensation is operative (as in Figure 7), themaximum value of the profile for 002 is re­duced to about 3 microns and its rms value ison the order of 1.5 microns. This provides a

good illustration of how decentering distor­tion can be significant (by ballistic camerastandards) and yet not be detectable fromvisual examination of residuals.

In comparing the residual vectors of Fig­ures 6 and 8 we note that the extended platereduction has been completely successful inremoving the systematic components of theresiduals in Figure 6. The randomness of theresidual vectors achieved in Figure 8 is en­tirely satisfactory. The mean error of 2.1microns achieved in the extended reduction isonly slightly more than half as great as themean error of 3.9 microns resulting when de­centering distortion was not explicitly takeninto account. The two parameters cPo and J 1

were found to be sufficient for the calibrationof both cameras; the coefficient J 2 was carriedinitially but was dropped for failing to lead toa statistically significant reduction of thequadratic form of the residuals. The cali­brated values of the decentering parametersare (for P and r in mm.) given in Table 2.

TABLE 2. CALIBRATED VALUES OF DECENTERING PARAMETERS.

Camera SSL 001

<1>0 = 71 0 .9 ± 30 .1J 1 =( -1.456 ± .075) X 10-6

Camera SSL 002

<1>0=6°.6 ± 10°.6J , =(0.502 ± .089)X10-6

DECENTERING DISTORTION OF LENSES 459

The appreciably greater mean error of cPo for002 is attributable to the fact that the de­centering distortion of 002 is appreciablysmaller than that of 001. It should be appreci­ated that if there were no decentering distor­tion, cPo would be indeterminate for therewould then be no axis of maximum tangentialdistortion. Hence, the greater the decenteringdistortion, the more sharply cPo is defined.

Inasmuch as random errors in the catalogedstellar positions were rigorously taken intoaccount in the final adjustment, residualswere also obtained for stellar right ascensionsand declinations. Each star was individuallyweighted according to the reciprocal of itsupdated variance as computed from appro­priate entries in the catalog. It will be notedthat the rms error of the stellar residuals forCamera 001 is only 0."27 which correspondsto 0.9 J.l on the plate. This relatively low valuereflects the fact that the particular star fieldemployed (Cygnus) is especially well deter­mined in the General Catalog (two thirds ofthe 42 different stars carried had updatedmean errors of less than 0."40 and only 4 hadmean errors in excess of 0."60).

We consider that our combined theoreticaland experimental results (including subse­quent calibrations not reported here) demon­strate conclusively the feasibility of the pre­cise analytical calibration of decentering dis­tortion. In sections to follow we shall reviewsome of the metric consequences of uncom­pensated decentering distortion.

EFFECTS OF DECENTERING DISTORTION ON

GEODETlC FLASH TRIANGULATION

Since the first demonstration of the feasibil­ity of employing ballistic camera observationsfor precise recovery of geodetic positions(Brown, 1958, 1959), full-scale global pro­grams have been or shortly will be inaugu­rated (ANNA, GEOS, PAGEOS) to exploitthis powerful geodetic tool. We shall considerhere how decentering distortion, if not ex­plicitly taken into account in the reduction,can be expected to lead to gross biases in re­covered geodetic heights in spite of all effortsto obtain improved accuracies through theexploitation of seemingly overwhelming re­dundancy of observations and plates.

The mechanism of the biasing process isbest explained through a reconsideration ofthe residual effects of decentering distortionfollowing least squares projective compensa­tion as illustrated in Figure 3. The heavy,solid line partially crossing Figure 3 repre­sents the trace of a hypothetical trajectory

as it would appear if there were no decenter­ing distortion. The broken curve representsthe displacement of the trace attributable tothe resid ual decentering distortion of thefigure (this displacement is magnified 2000times, as are the residual vectors). Vve seethat for the case under consideration, theeffect of residual decentering distortion is tointroduce a mean bias along the trace ofabout -5 microns, predominantly in thedirection of the y-axis.

We should particularly note that with theexception of the far corners of the format, theeffects of residual decentering distortion arelargest near the center of the plate and more­over are of almost constant direction withinthis region. This means that no matter fromwhat direction a reasonably well centeredtrace crosses the plate, the mean bias attribut­able to decentering will be of nearly the samedirection and of nearly the same magnitudeon plate-after-plate taken by a given camera.

Let us, for the moment, assume that thehypothetical camera giving rise to Figure 3were on an alt-azimuth mount with the+y-axis pointing in the direction of increasingelevation angle and the +x-axis pointing inthe direction of increasing azimuth. Then onplate after plate taken by the hypotheticalcamera, the mean bias of - 5 microns or sowould constitute a bias predominantly in areconstructed elevation angles of the rays. Ifthe camera had a 1000 mm. focal length, themean bias in elevation angle would amount toabout -1" of arc; if 300 mm. focal length, itwould amount to about -3" of arc. The pre­dominant effect of a more or less constant biasin the elevation angles of all of the rays con­verging on a given station would clearly be tointroduce a bias into the recovered height ofthe station. ·.1

By contrast, iLnhe various groups of raysfrom a moderate l17umber of plates were well­balanced in azimuth about the station, theresidual effects· bf decentering distortion inthe elevation angle would largely average outin the recovery of the horizontal coordinatesof the station. The effectiveness of this aver­aging process would hold equally well if therays were affected by significant azimuthaldecentering biases as, for example, would bethe case if the axis of maximum decenteringdistortion in Figure 3 were rotated by 45°.

It follows, then, that what need concern usin a well-balanced resection is the mean com­ponent of residual decentering distortion inthe direction of the elevation angle. If thiscomponent amounts to p microns for a camera

460 PHOTOGRAMMETRIC ENGINEERING

of focal length c microns, and if R, E denote,the mean slant range and elevation angle ofthe rays from the station to the points ob­served, the bias 13k to be expected in the re­covered height of the station is given roughlyby

The derivation of this approximation dependson the assumption that unbiased coordinatesof the flashes are independently available,thus reducing the geodetic problem to thelimiting case of independent resections. Itbears emphasizing that the above resultwould hold no matter how many plates wereemployed in the reduction for a given station;the bias in height attributable to uncom­pensated decentering distortion is not ame­nable to significant reduction through sheerexercise of redundancy.

We believe that residual decentering distor­tion provides the most likely explanation forthe untoward discrepancies encountered inthe recovery of heights of BC-4 camera sta­tions employed on a test triangle as reportedin Bulletin 24 of the U. S. Coast and GeodeticSurvey (1965). This test triangle involvedstations about 1500 km. apart in Maryland,Minnesota and Mississippi. BC-4 ballisticcameras equipped with synchronized shutterspassively observed several passes of Echo 1.The horizontal coordinates recovered in a six­pass reduction for the two stations treated asunknown (Minnesota and Mississippi) werefound to be in good agreement with the pre­existing first order survey, the discrepanciesamounting to only a few meters, and beingconsistent with the combined standard errorsof the two surveys. The ~iscrepancies in theheights of the two unknQW" stations, on theother hand, amounted to ~pput -20 to +31meters respectively. These d.iscrepancies com­pare unfavorably with the ~pected standarderrors of 6.3 and 3.9 meter ~ttributable torandom errors of the photog~~H\lmetric mea­surements and with the stan~ard errors ofabout 5 meters attributable '~o the astro­geodetic heights. If we take the mean dis­tance of the rays to Echo to be R''''' 2.5 X 106

m., and the mean elevation angle of the raysto be E = 45°, we find from Equation 16 thatfor cameras of 300 mm. focal length meandecentering biases in the direction of eleva­tion angle of p= -1.7 microns and p = 2.6microns, respectively, would suffice to explainthe observed discrepancies in the heights ofthe stations under the assumption that the

p- ­fh "" - R sec E.

c(16)

coordinates of the flashes were unbiased. Thisdegree of resid ual decenteri ng distortion atthe center of the plate is altogether plausible(indeed likely) for cameras of high metricquality.

I t should be noted that the ul ti mate effectsof residual decentering distortion are accen­tuated by the particular reduction employedby the USCGS. The critical factor in thisregard is that, following the least squares fit ofa high order polynomial to the measuredpoints on the trace, only a single rayon thefitted polynomial is retained for subsequenttriangulation and this, being a central ray, issubject (as we have seen) to greater residualdecentering distortion than any other possiblechoice wi th the exception of rays near thecorners of the format. Al though the smoothedplate coordinates corresponding to the centralray typically have standard deviations ofonly 0.2 fJ., their associated systematic errorsdue to residual decentering distortion are, onthe average, almost certain to range from fiveto fifteen times greater. Even so, the compo­nent of systematic error in elevation anglewould have relatively small effect on theclosures of triangulation and hence wouldlargely elude detection from least squaresresiduals. In our view, it would have beenbetter had USCGS evaluated the fitted poly­nomial at a pair of points having radial dis­tances of 60 to 70 mm., for here residual de­centering distortion approaches its minimumin those reductions employing stars over asubstantial portion of the plate. Better still,of course, would have been the explicit con­sideration of decentering distortion within theplate reduction employed by USCGS.

An alternative, though a less satisfactorymethod of counteracting decentering distor­tion, consists of carrying in the reduction setsof carefully paired traces such that the twotraces of a set cross the format in a similarmanner but are recorded on plates differingnominally by 180° in roll angle. To the extentthat decentering were independent of theforces of gravity acting on the lens, this pro­cess would allow appreciable cancellation ofthe systematic effects induced by decentering,for half of the rays would be biased positivelyand half negatively in elev(ltion angle. Theclosures of the adjustment would, of course,be rendered appreciably larger by this pro­cess, for they would fully reflec~ the biases inthe resecting rays. Clearly, this is a healthierand more desirable situation than one inwhich significant biases exist but are poorlyreflected by the residuals.

DECENTERING DISTORTION OF LENSES 461

The method of paired traces is readily ap­plied to cameras such as the PC-1000 whichcan be rolled about the optical axis; in fact, itis almost an automatic by-product of thepractice of rolling PC-1000's on each opera­tion so that the flashing light trace crosses theplate on one or the other of the diagonals.This together with the fact that triangulationis not limited to central rays perhaps partiallyexplains why it is that the recovered heightsof a six station PC-1000 test network ofdimensions comparable to those of theUSCGS Maryland-Minnesota-Mississippi nethave proven (Hadgigeorge, 1965) to be fullytenfold better in absolute agreement withastro-gravimetric heights than the corre­sponding BC-4 results reported in USCGSBulletin 24.

Even when the decentering distortion of agiven camera has been explicitly calibrated,we consider it prudent to employ the methodof matched traces routinely as a safeguardagainst the possibility of significant acciden­tal changes of decentering. Also, it is advis­able to evaluate decentering at moderatezenith distances with the camera in bothdirect and plunged orientations in order toestablish the influence, if any, of gravity onthe results.

EfFECTS Of DECENTERING ON PHOTOGRAM­

METRIC EXTENSION OF CONTROL

In the past, many of the metric shortcom­ings of mapping cameras could be toleratedby virtue of the compensation provided byfairly dense networks of pre-establishedground contro!' However, the establishmentof a high level of control constitutes a majorexpense of the mapping operation in bothtime and money. For this reason coupled withadvances in computer technology, the exten­sion of mapping control by means of ana­lytical aerotriangulation has aroused wide­spread interest. Recent breakthroughs(Brown, Davis, Johnson, 1964) have madefeasible the uncompromisingly rigorous si­multaneous adjustment of very large blocks ofphotography. However, the practical attain­ment of the full promise of analytical methodsdepends in great measure on the precisecalibration of the camera; accuracies of cali­bration four to five times greater than thosegenerally considered adequate in conventionalmapping are needed if the fullest benefits ofanalytical methods are to be realized. This isbecause residual systematic errors propagatethrough analytical aero triangulation in amost unfavorable manner. Thus, uncompen-

sated decentering distortion as small as 2 to 3microns has a rapid cumulative effect on theanalytical reconstruction of a strip andassu mes prominence relative to the effects ofrandom errors within a few models. Theadmissible length of analytical extension be­tween control is clearly a function of thegeometric fideli ty of the camera followi ngapplication of all known corrections. Up to acertain point, the more comprehensive andmore precise the calibration of the camera, thelower the requirements for absolute control inthe photogrammetric net. As the applicationof analytical methods widens, we foresee thegrowth of increasingly stringent demands forimproved accuracies of calibration of mappingcameras not only for symmetric radial distor­tion and elements of interior orientation, butalso for decentering distortion.

IMPLICATIONS OF DECENTERING DISTORTION

ON THE DESIG 1 OF METRIC CAMERAS

Because decentering distortion can beeffectively removed through calibration, ex­tremely precise centering need no longer beviewed as a stringent requirement for lensesto be employed in many photogrammetricapplications, especially in applications togeodetic flash triangulation. Indeed, decen­tering can now be tolerated to the extent thatit does not sensibly affect the quality ofimages. This means, in effect, that almost anywell-regarded commercial lens of suitablefocal length, aperture and angular field can beemployed for metric observations, for it isimage quality throughout the format thatnow becomes the overriding factor in the ul­timate determination of metric potentia!. Vilehave, for instance, been able to demonstratethat a specially modified commercial viewcamera employing a 480 mm. f/4.5 SchneiderXenar lens can produce stellar plates of su­perb quality over a cone angle of 20° and that,when properly reduced, can yield directionalaccuracies unsurpassed by any standard bal­listic camera of comparable or shorter focallength. This demonstration points out thattrifling technicalities and pedantic illusionshave been permitted to obscure the physicalessentials of the ballistic camera which, asidefrom focal length, aperture, and angular field,are merely twofold: (1) a high degree of short­term stability in any desired orientation(typically over a period of five to ten minutes)and (2) excellent image quality over thespectral range of interest. It is hoped thatone consequence of this paper will be to coun­teract the sophistry that unfortunately has

462 PHOTOGRAMMETRIC ENGINEERING

come to be superimposed on optomechanicaltechnology as it pertains to ballistic cameras.

CONCLUDING REMARKS

In our experience over the past decade wi ththe full scale stellar calibration of over 50different ballistic cameras for symmetricradial distortion, fully three quarters of thecalibrations have yielded mean errors in therange 3.5 to 5.0 microns, a range incom­patibly large relative to the 2 to 3 micronsnormally attributable to the combined effectof the plate measuring error corresponding todouble settings, the random instability of thephotographic emulsion, and all other sourcesof random error. Because of this it was usuallydeemed necessary in routine reductions fol­lowing the calibration of radially symmetricdistortion to resort to tedious piecewise pro­cedures wherein two or three compact butoverlapping groups of stars encircling differ­ent portions of long flashing light traces wereindividually reduced, the purpose being toallow the elements of orientation greaterfreedom for local compensation of unmodeledsystematic errors. Only through such indepen­dent reduction of limited regions of the platecould least squares residuals yielding a meanerror in the acceptable range of 2 to 3 micronsbe consistently obtained. In the light of ourpresent findings we are now convinced thatthe uncomfortably large mean errors fre­quently encountered in past full scale stellarcalibrations can largely be attributed to un­compensated decentering distortion. 'liVe arealso convinced that the problem of decenteringdistortion has now been overcome, and thatdecentering distortion is fully as subject toprecise analytical calibration as symmetricradial distortion. As we have indicated, thisfact bears consequences of fundamental im­portance to geodetic photogrammetry, toanalytical photogrammetry and to photo­grammetric instrumentation.

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--- (1965), Satellite Triangulation in the Coastand Geodetic Survey, USCGS Technical BulletinNo. 24.