The Effect of the Fixation Disparityon Photogrammetric Processes

SANDOR A. VERES,

Asst. Proj., Surveying and Mapping,Purdue University

ABSTRACT: This paper reviews the function oj the human eyes in photogram­metry, and discusses its limitations. The correction of the observation error ispresented by mathematical derivations and practical examples. The paperpoints out the need for continued development of techniques in view of con­stantly increasing requirements.

INTRODUCTION

PHOTOGRAMMETRIC instrumentation hasundergone a revolutionary development

since the second World War. A precisionphotogrammetric instrument today is able toprovide measurement to within a precision of± 3 microns. By using these precision in­struments the compensation of errors in­volved in the measurements becomes of pri­mary importance. The compensation of errorsis based upon the geometrical knowledge ofthe source of errors. The human eye is in­volved in every photogrammetric measure­ment; consequently the geometrical knowl­edge of the errors due to the limi tation of thehuman eye is very important. Because theorientation procedure is one of the most im­portant parts in the reproduction of thephotographic bundle of rays, in which thehuman eyes are directly or indirectly in­volved, the limitation of the human eyesdefines the accuracy obtainable in photo­grammetric data.

LIMITATION OF HUMAN EYES

The function of the human eyes in photo­grammetry can be classified into two catego­ries; a, observation and b, orientation.

a. [n photogrammetric observation both eyesare directed to a point on the photogram­metric model, which can be called the pointof fixation. The image of this point shouldfall on a corresponding retinal element. Themeasuring mark of a photogrammetric in­strument has to be adjusted until it coin­cides with the point of fixation or, physiologi­cally, the images of the measuring mark fallon the same retinal elements as the imagesof the fixation point.

The error of observation is represented asthe non-coincidence of the measuring markand the point of fixation. Consequently the

study of coincidence is most important fromthe photogrammetric point of view. For astudy of this kind an instrument called ahoropter has been used.

The horopter designed by Tschermak con­sists of thirteen steel channels mounted so asto converge to the midpoint of the inter­pupillary base line of the two eyes of anobserver. The central channel lies in a medianplane, perpendicular to the eyebase of theobserver, and the others make angles of 1, 2.4, 8, 12, and 16 degrees on each side of thecentral channel. A small vertical steel rod canslide smoothly in each channel and the rodsare mounted for use at visual distances of 20,40, and 75 cm. from the interpupillary baseline. The rods in the channels are movabletoward and away from the observer exceptfor the central one which is representing thepoint of fixation.

The motion is controlled by handles and isadjusted by the observer so that all the rodsshould be placed in a line parallel to the eyebase at the distance of the point of fixation.The discrepancies of the individual rods fromthe correct line can be measured on a measur­ing device. A screen located in front of theobserver's eyes permits him to see only thelower part of the left-side rods with the lefteye, and only the upper part of the right-siderods by the right eye. (This is the case inphotogrammetry, when the left eye is ob­serving the left picture and the right eye theright picture. The images are not the samesince the pictures are taken at different airstations.) The results published by Dr. K. N.Ogle are given in Figure 1.

It can be seen from the Figure 1 that theconnecting line of the corresponding rodsyield a hyperbolic line which is called the

148

THE EFFECT OF THE FIXATION DfSPARITY 149

FIG. 1. Spatial representation oflongitudinal horopter.

71 is the disparitya is one-half of interpupillarity distance

andb is the observation dista nee.

FIG. 2. Region of binocular single vision onthe longitudinal horopter.

be general agreement that the training cannotreduce the area beyond a certain limit.

This phenomenon can be represented on ahoropter diagram. The diagram is given inFigure 2 after Fischer (5).

In his experiment the observation distancewas 40 cm. It can be seen that the limits arenear the point of fixation and they are spacedfurther apart on both sides away from thefixation point.

Experiment has also shown that the size ofthe Punum area is larger in the case of iden­tical target patterns and smaller in the case ofdissimilar target patterns.

These limits are of primary importancefrom the photogrammetric point of view,because if the limits could be determined thecorrection and the point of fixation could b~

computed.

CORRECTION OF PHOl'OGRAMMETRIC

COORDINATES FOR FIXATION DISPARITY

The limits of the region are given by thesurface of a normal-stereo and pseudo-stereomodel. The situation is presented in Figure 3,where 0 1 and O2 are the left and right eyerespectively, and O. is the perspective stereocenter. P is a point on the surface of thenormal-stereo model and the correspondingpoint on the pseudo-stereo model is P'. Thecorresponding rays intersect the referenceplane R at PI and P 2• An approximate par­allelogram is given by P, Ph P 2 , pI, whichcan be called the error parallelogram. Thisparallelogram is developed into an errordouble cone or more precisely into an ellipsoidwhen taking into account the X and Y direc­tional errors or the longitudinal and lateraldisparity. In a three dimensional observationinside this error ellipsoid, it is possible toobtain a parallax-free orientation althoughthe corresponding rays do not intersect eachother.

(1)

'1_.-16° .-0

-<.rom

---I_

R. Channels

Llb7J = - 2a­

b2

~ ""L.E. R.E.

L. Channels

where

longitudinal horopter. Therefore, the imagesof the fixation point must be disparate. Themagnitude of the angular disparity at thepoint of fixation as determined by Dr. K.Ogle (1) is as follows:

This disparity is the observation error inphotogrammetry due to the limitation of thehuman eyes. Similar results have been ob­tained by Fischer, Ames, and others (2; 3).

b. The second and more important function ofthe human eyes in photogrammetry is inorientation. It was pointed out in the analy­sis of photogrammetric observation abovethat the images of a point should fall on thecorresponding point of the retinas. If thiscondition is fulfilled the images will fuse andwill be seen as a single image. Singleness wasthe criterion for determining the horoptercurve. Punum (4) studied this problem andshowed that an image on a retinal point ofone eye would fuse with an image fallingwithin a small area of other retina. Thismeans a certain region exists where only oneimage is seen; that is, within this region thetwo images are fused, although disparate.Photogrammetrically speaking, the parallax­free orientation of a model is possible withinthis region although the corresponding lightrays are not necesiarily intersecting eachother. The limit of this region determines theaccuracy of the orientation. Experiment hasshown that the size of this region,-or some­times called Punum area-can be decreasedsomewhat with training, but there seems to

150 PHOTOGRAMMETRIC ENGINEERING

28 29 )0 31 33 34 :3 5 381---1---1-_'---1---1-_1---1-- Points.

(2)

28 29 30 31 33 34 35 38

,'.............., ./ \.

\\" ! \" ,~, ~/' A \,

---\,/ v ~+--_---'_4.-_':--'---'_4.---+----'_4.- Polntso

20 20

30 30

10 10

PX(rnico) eX(CIll.)

40 J.O

30 30

20 20

10 10

fh",c.) ·Y(",.)

40 40

The model coordinates of the control pointswere transformed into the ground coordinatesystem by using the following well knowncoordinate transformation formulas.

y = - bx + ay + elf

X = ax+by+e'

FIG. 4. Relation between the residual error ofcoordinates (da~hed lines) and the standard errorof observations (solid lines). A is normal stereo andB normal plus pseudo stereo.

The Figure 4-B concerns the second group,and a high si milari ty exists between thestandard error of observations and the re­sidual error of coordinates. This result clearlyindicates the two surfaces, namely the nor­mal-stereo and the pseudo-stereo model,coincide with the limits of the region offusion. The results also indicate that, if thelimits are known, the corrections needed toobtain the correct position of a point offixation can be computed.

Three points out of the given eleven wereused for coordinate transformation, as well asfor the numerical absolute orientation. Theresidual coordinate error for the remainingeight points were computed and compared tothe standard error of observation. The resul tis given in Figure 4, where No. 4-A presentsthe result of the first group. On the abscissathe poi nts are gi ven and the errors are plottedas ordinates. The standard errors of observa­tions (J.l.y) are given in microns and presentedas the solid line. The residual errors (e y ) aregiven in centimeters and plotted as thedashed line. As can be seen, no relation existsbetween the two kinds of errors because thestandard errors of observation refer to the sur­face of the normal-stereo model.

7/

//

//

//

//

/~ /_ Normal Stereo Model

The camera testing area of U.S.c.&G.S.in northern Ohio was photographed by theOhio State Highway Department with aFairchild F-501 (j=6") camera, at the scaleof 1/12,000. The control points were pre­signalized and their coordinate values wereobtained from U.s.c.&G.s. in the StatePlane Coordinate System of Ohio. The obser­vation of control points were made with theWild A-7 Autograph of the Ohio StateUniversity. The orientation of the models wasaccomplished using the optical-mechanicalorien tation method in the usual man ner.

FIG. 3. Correction of fixation disparity_

pI

/ Pl

--~-4-~-'lL-...:!o...----EQO

\ /\ /

........ \ / -_ _ _ _ -- Pseudo Stereo Model

Two groups of observations were made.One consisted of four series of observations onthe normal-stereo model, the second wascomposed of two series of observations on thenormal-stereo model and two series of observa­tions on the pseudo-stereo model. From theseobservations the standard error of observa­tions has been compu ted in the usual man ner.

In order to prove that the limits of thefixation disparity are given by the surface of anormal-stereo and pseudo-stereo model anexperiment has been executed. The experi­ment was based upon the assumption that ifan area is photographed in such a way thatthe coordi na te axes of the grou nd controlpoints are parallel to the model coordinateaxes, the standard error of observation (If apoin t is related to the distance betweenpoints P and P' (Figure 3). It should show ahigh similarity to the residual error of the samepoint, because the point has been observed atthe upper and lower limits of the region.

THE EFFECT or THE FIXATION DISPARITY 151

+-----...!......'---.!-----_ X

(6)

(4)

(3)

t::uYP' = )(p - ...X pI

errors due to the fixation disparity in the leftand in the right photograph can be called LiXland LiX2 respecti\'ely. The incorrect positionpI of the examined point P can be constructedby using the picture coordinates of x, +Lixiand X2+Lix2 and is given by dashed lines inFigure 5. This is the location of the poin t inthe case of normal-stereo obsen·ation. If thephotographic plates are interchanged, in­cluding the picture coordinates, the pseudo­stereoscopic position P" of the P point can beobtained. as is presented hy dotted lines inFigure 5.

It'D.1t = bj (D.X, - D.X,)

The correction for XI" coordinate can beex pressed as

in \\'hich

Substituting the (3) formula into the aboveequation

ItXp' = - (x, + D.X,)

fIt'- bj2 (x,D.X, + D.~·,D.X, - ~',D.XI - D.X,') (5)

and/l

X,> = -~',

fSubstituting (5) and (6) into Equation (4),the correction becomes

The model coordinates of the point are XI'and Y1" in the case of normal-stereo observa­tion and XI''' and Yp" in the case of pseudo­stereo observations. As can be seen fromFigure 5, a tJ.h elevation error exists betweenpoints P and pI which can be computed inthe following manner:

or

It'D.Xp' = bj2 (X,D.X, +D.x,D.:r, - XID.X, - D.XI 2)

It- - 11,1', (7)

fA similar equation can be derived for thecorrection of XI''' coordinate such as

It'D.Xp" = ~f' (x,D.X, - x,l1x, - D.X,D.X, +D.X,2)

II+ --:- !h, (8)J

b,,,

//

/ ./,

". ,,/"'. "

"'. "-"" ,,"

\ "'."\ )<\pt//px

h

Ah

1

1 .,-,.:;--"'n--­

f

FIG. 5. Correction of X Moclel coorclina te.

This correction method for the Z coordi­nate is certainly correct with the assumptionthat the two limits are located symmetricallywith respect to the point of fixation or refer­ence plane. Since the X and Y model coordi­nates are also affected by this fixation dispar­ity, further investigation was undertaken(10).

A correction method involving the Z co­ordinate only has been published by Mr. H.Yzerman (6) as follows: "'vVe see that (Figure3) 0 1, Oz, PI, P z are points of a completequadrangle of which the t\\·o sides Ot, Oz andPI, P 2 are parallel. \Ve know therefore thatthe cross ratio 01020,E~= -1 and thus that0,0,= -0,02", This means that the inter­change of stereoscopies results in equal in\'er­sion of the model with respect to the refer­ence plane. Or in other \\'ords, if, due toFixation disparity or Fertch-effect. theparallax of Jf (measuring mark) with themodel point P seems to be nil, then the inter­change of stereoscopies \\'ould make the laten tparallax visible in the order of t\\'ice of itsvalue. This could not only be compared \\'ithan increase of B/H (Base-Height) ratio ormagnification of two times. Still more chal­lenging is that a true zero-method is availableinstead of an approximation method with allits su bjecti ve aberrations."

The correction for the X model coordinatecan be deri"ed from the follo\\'ing conception.In a stereoscopic model an arbitrary point"P" is given (Figure 5). The correspondingpicture coordinates are XI for the left and X2for the right camera. If one assumes thatthese picture coordinates are theoreticallycorrect, then the model coordinates XI' andY I' are also correct. The X componen ts of the

152 PHOTOGRAMMETRIC ENGINEERING

A further approximation can be introducedthat the

(16)

(15)[XP' (XP" - XP')+0.5 -

b 2

[Y, (Y "- Y')+0.5 ~ p P

b 2

YP" (YP" - YP') YP" - YP'J+ b 2 + 2

Yp = 0.5(Yp' + YP")

In the Equations (15) and (16) every termis known. Therefore these formulas can nowbe used for a practical purpose.

The test of these formulas was performedby using the model described earlier. Theground coordinates were computed frommodel coordinates with and without correc­tion and compared to the given coordinatesof corresponding points. Eleven control pointswere given on the model and three were usedfor absolute orientation and coordinate trans­formation. The residual standard positionerrors have been computed and the result isgiven in Figure 6. The dashed line representsthe residual standard position error obtainedfrom the uncorrected coordinates and thesolid line represents the residual standardposition error computed from the correctedcoordinates. As can be seen, the correctionprovides about 40 per cent increase in accu­racy of position. Eight different models wereexamined (10) about 45 per cent in X and Ycoordinates as well as in Z. The averageresidual elevation error was found to be1/7,000 of the flying height without correc-

and

Substituting equations (12), (13) and (14)into equation (11), the final formulas are:

XP = 0.5(Xp' + X p")

(14)

(13)

(12)

It X '+X"- (!lx - !lx) = p p - X'

2

and if t::.h=O compared to hand b.XI=O com­pared to Xl it can be written:

iX' d i x "x, = h p an x, = h p

By knowing the Xp', b.Xp', Xp" and b.Xp"the correct model coordinate of the point canbe expressed with the following formula

X= XP' + XP" + !lXp' + !lXp"

p 2 2 (9)

Substituting the corresponding equations forEquation (9), the final formula is

XP = 0.5(Xp' + Xp")

+ 0.5 [:' (:rl!lx, - XI!lXI - !lXI' + x2!lx, (10)

- X,!lXI + !lx,') + ; (!lx, - !lXI) ]

or simply

XP = 0.5(Xp' + Xp") +!lX (11)

Four unknowns exist in Equation (10);consequently the equation in this form can­not be used for any practical purpose inorder to solve the unknowns, three moreequations would be required. The equationscannot be established; consequently it isnecessary to introduce certain approxima­tions in order to obtain a useful correction. Itcan be assumed that:

h' h'- !lXI' = - !lx,' = 0bf' bI'

f'p (em.)

50

40

30

20

10

R

", ,/ \

I \I \

: \I \/ ,I ,I \I ,/ \/ \I \I/

1-----+--1---1---"1-----1---1--+---"1-----1--.+---1--- Points.28 29 30 31 32 33 34 35 38 39 46

FIG. 6. Comparison of standard residual position errors. For uncorrectedcoordinates (dashed line) and corrected coordinate (solid line).

THE EFFECT OF THE FIXATION DISPARITY 153

tion and 1/13,000 using the correction for­mulas.

CONCLUSION

The analysis of the above correctionmethod led to the conclusion that the correc­tion formulas have a very significant effect onthe improvement of accuracy. But one has torealize that the correction formulas werederived only to improve the observations anddo not improve the orientation. It is alsoemphasized that a close relation exists be­tween the orientation of a stereo-model andthe fixation disparity and also between themeasured Y parallaxes and the fixation dis­parity. Therefore these correction formulasare effective only as far as the observation isconcerned. Consequently, a large amount ofthe residual errors after the correction are dueto the uncorrected orientation. Further in­vestigation must be undertaken in order toobtain correction method or methods fororientation, which is highly important in anyaerial triangulation

ACKNOWLEDGMENT

The author wishes to express his gratitudeto Dr. A. ]. Brandenberger for his valuableadvice and of making possible the use of theinstrumentation of the Ohio State University,and also to Mr. L. O. Herd, Chief Engineerof Aero Survey Section of the Ohio StateHighway Department for providing thematerial needs of this study.

REFERENCES

1. Dr. K. N. Ogle: Binocular Vision, \Y. B.Saunders Co., Philadelphia and London, 1950.

2. A. Ames, Jr., K. N. Ogle and G. H. Gildon,Size and Shape of Ocular Images. Method ofDetermination and Physiologic Significance.Arch. Opth. (April) 1932.

3. A. Ames, Jr., K. N. Ogle and G. H. Gildon:"Corresponding Retinal Points, the Horopterand Size and Shape of Ocular Images." Optic.Soc. America, Vol. 22, 1932.

4. P. L. P. Punum: Uber die Einheitliche Ver­schmelzung Versch iedenartiger Netzhou tein­drUcke beim Sehen mit zwei Augen," Arch. f­A nat. ·und. Physiol. 1861.

5. F. D. Fischer: "Fortgesetzte St\l.dien UberBinokularsehen (Tschermak): 11 Uber Asyi­metrien des Gesichtssinnes Speziell des Raum­sinner beider Augen." Arch. f. d. des Physiol.1924.

6. H. Yzerman: "The S. D. I. A New Method forStereoscopic Measurements and Plotting,"IntemationalArch. of Photogrammetry. Vol. XII,1956.

7. S. A. Veres: "Physical Aspects of Aerial Photo­grammetry," Unpublished PhD Dissertation,The Ohio State University, 1962.

Additional ReferencesDr. A. J. Brandenberger: Fehlertheorie der Aus­

seren, Orientierung von Steilaufnahmen." Tech.Hochschule Zurich, 1947.

Dr. A. ]. Brandenberger: "Aerial Triangulationby Least Squares." Mapping and ChartingResearch Laboratory. The Ohio State Univer­sity, 1956.

Dr. A. J. Brandenberger: "Theorie und Praxis derGegenseitigen Orientierung von Steilaufnah­men." Zeitschrift fUr Vermessung und Kultur­technik. Heft 4,5, 1948.

D. R. Dwyer: "Visual Factors in StereoscopicPlotting." PHOTOGRAMMETRIC EKGINEERING,Vol. XXVI, No.4, September 1960.

Book Review

Proceedings of the Fifth National SurveyingTeachers Conference-August 1962. Spon­sored by Committee VIII, Surveying &Mapping of the American Society Engi­neering Education. Conference held atGeorgia Institute of Technology, Dah­lonega, Georgia. Published by CommitteeVII, Surveying & Mapping, AmericanSociety Engineering Education. 98 pageswith illustrations. Obtainable from Mr. D.H. Harkness, Secretary. Address W. &L. E. Gurley, Troy, New York.

The Publications Committee has reviewedthe Proceedings at the request of Mr. Hark­ness. Particular emphasis of the review wasplaced on the portion titled "A Course inBasic Photogrammetry, pp 13-32, presented

by Professor Wilfred H. Baker. I n the reviewit was noted that the discussion was primarilyconcerned with methods and procedures usedfor teaching students in the field of photo­grammetry, the fundamental principles ratherthan technical aspects. To most of our mem­bers, information of this nature is morereadily available from other sources in varioustextbooks, manuals and publications. In mostinstances the reference books will explain thesubject matter in greater detail than is dis­cussed in the Proceedings. However, for thoseinstructors who have little or no acquaintancewith the field of photogrammetry, this publi­cation will be of some assistance.

ABRAHAM ANSON, ChairmanPublications Committee