DR. SANJIB K. GHOSH

Ass't. Proj., Ohio State Univ.Columbus, Ohio 43210

x- and Y-Parallax

Observations

ABSTRACT: A n experimental study determines the comparative preczswn ofX- and V-parallax observations in stereo-photogrammetric models. Supportingtheories are presented. It was determined empirically that the standard error ofX-parallax observations is about 0.4 times the standard error of V-parallax ob­servations. The effect of nsing dove prisms to change V-parallax to X-parallaxis also studied.

The failure of obtaining an intersectionmeans the existence of parallaxes (X or Y),which are defined by:

Such parallaxes are due to the errors in theorientation elements. The parallaxes existingat a model-point expressed as functions of theerrors in the orientation elements have beenderived and established by various photo­gram metrists earlier (e.g., Branden berger).Any text book may be referred to for suchderivations (e.g., Hallert, Schweidefsky, orZeller). Considering a right-handed coordi­nate system with clockwise rotations, withthe origin of the system at the left side projec­tion center and with regard to the use ofnegatives (and not diapositives) the func­tions are:

INTRODUCTION

T HE FORMATION of a three dimensionalstereo-model involves the intersection

of the optical rays, originating from the twocorresponding picture points, at each point inthe surface of the model. The intersection isperformed when at the model-point (in themodel space) both the rays have the sameX-coordinate (model coordinate) and also thesame V-coordinate. The mathematical ex­pression for this is that at a point P in themodel-space, XPX=XPII' where X PI is theX-coordinate with respect to the ray fromthe left-camera of the stereo-plotting instru­ment and X pu is the X-coordinate withrespect to the ray from the right-camera.Similarly, for the V-coordinates, Y pl = YpIl'

With a view to studying the feasibility ofusing X-parallaxes for the relative orienta-

(1)

(3)b

-Px = 5Z­Z

\ (X - b)'(- Z/1 + Z' Idq,1l

X XY-Px = db~'1 - - dbz, - Y ·dK' - -- dWI

Z Z

r (X') (X - b)+ Z 1 +-- dq" - db~'11 +--- dbz llZ' Z

(X - b) Y+ Y·dKIl + Z dWIl

-Py = dbYI - ~ dbz l +X·dKI - Z (1 +~) dWIZ Z'

XY Y+ - dq" - db)'f1 + - dbz!1 - (X - b)dKIlZ Z

( Y') (X - b)Y+ Z 1 + Z' dWIl - Z dq,f1 (2)

rtis apparen t that the use of ei ther or bothof the parallaxes would result in solving theerrors in the elements of relative orientation.The author was curious about it and wentdeeper into these aspects in a researchsponsored by the U. S. Army Engineer(GIMRADA, Fort Belvoir, Virginia).

The X-parallax at a model point is relatedto the corresponding elevation error accord­ing to an expression -Px=oZ(bx/Z); whereOZ is the elevation error. For all practicalpurposes, one can assume that the modelbase (b) is identical with the X-component(bx) of the base. I n practice, this is almostalways true. Then

Px = Xp! - X PIl and,Py = Y PI - Y PIl .

X-parallax (error):Y-parallax (error):

667

668 PHOTOGRAMMETRIC ENGINEERING

tion of a stereo-model and to analyze theaccuracy obtained thereby as compared tothe situation with the' use of V-parallaxes, aseries of experimental observations weremade in the Photogrammetric laboratory ofthe Department of Geodetic Science, TheOhio State University. Below is given an ac­count of the studies.

DESCRIPTION OF THE EXPERIMENT

For these investigations a \Vild Auto­gra ph A7 was used. The photographs weretaken wi th a Wild RC5 Aerial Camera;1=152.36 mm.; format: 23 cm.X23 cm.;standard photography. Picture scale was1 :8000 approximately. Model Scale was1 :4000.

Observations were made at or very neareach of three different points of the model\\·ith undulating terrain. One of the pointswas on a Aat side-walk that was runningEast-West (parallel to the X-axis), the secondpoint was on another side-walk that wasrunning North-South (parallel to the Y-axis)and the third point was on a grassy tennislawn with lines crossing each other at rightangles. Three hundred repeat-observationswere made at each point, thus making thetotal n'umber of observations 900. The 300observations at each point were broken downas follows:

Case AI: 50 observations of V-parallaxelimination with the element by" and cor­responding readings of the by" counter bymoving the right side measuring mark fromfront to rear;

Case A2: 50 observations of V-parallaxelimination with the element by" and cor­responding readings of the by" counter bymoving the right side measuring mark fromrear to front. lext, after changing the Y­parallax into X-parallax with Dove-prisms,

Case Bl: 50 observations of V-parallaxelimination with the element by" and cor­responding readings of the by" counter byapproaching the model surface with themeasuring mark from above;

Case B2: 50 observations of V-parallaxelimination with the element by" and cor­responding readings of the by" counter byapproaching the model surface with themeasuring mark from below. Next, aftercoming back to the normal situation, bysetti ng the Dove-prisms back to their orig­inal situations,

Case Cl: 50 observations of X-parallaxelimination (i.e., reading of spot elevations)

with the foot-disk and the correspondingZ-counter readings by approaching themodel surface with the measuring markfrom above;

Case C2: 50 observations of X -parallaxelimination (i.e., reading of spot eleva­tions) with the foot-disk and the corre­sponding Z-counter readings by approach­ing the model surface with the measuringmark from below.

The relevant elements were set at the in­strument to the following values: j= 152.36mm.; Z~304.7 mm.; bx=157,20 mm.; by'=by"=bz'=bz"=100.00 (i.e., zero value)m m.

Assuming the arithmetic mean to be thecorrect value for each case (separately forAI, A2, B1, B2, C1 and C2), the standarderror for each case was computed. These are,as averages obtained from all the threepoints:

Case AI: 15.7 microns, and Case A2:13.5 microns; their average gives the stand­ard error for Case A : 14.6 microns.

Case B1: 5.1 microns and Case B2: 5.7microns; their average gives the standarderror for Case B: 5.4 microns.

Case Cl: 10.9 microns and Case C2:18.8 microns; their average gives thestandard error for Case C: 14.8 microns.

However this Case C gives the standarderror of elevation determination in the model,which is related to the X-parallax accordingto the expression (3). This gives the standarderror of X-parallax determination from caseC to be, with proper substitutions in the ex­pression (3):

157.214.8-- = 7.5 microns.

304.7

All the above are expressed as in the model.In these studies the model scale was approxi­mately twice the picture scale. From thisconsideration, for the sake of standardiza­tion, it may finally be expressed that at thepicture scale (in this particular case):

The standard error of V-parallax deter­mination is 7.3 microns.

The standard error of V-parallax deter­mination, by changing them to X-par­allax with Dove-prisms, is 2.7 microns.

The standard error of X -parallax deter­fl1ination (from elevation observations)is 3.8 microns.

x- AND Y-PARALL.\X OBSERV.\TIONS 669

The above shows that the standard errorof X-parallax determination is between 0.37and 0.52 (i.e., between 2.7/7.3 and 3.8/7.3)~0.44, depending on the element used for thepurpose) times the standard error of Y-paral­lax determination. This means, generallyspeaking, the co-factors are related to eachother as follows:

Qpz,Pz = (O.44)2Qpy,py = O.2Qpy,py.

In case, however, Dove-prisms are uscd forchanging the Y-parallax to X-parallax, gen­erally speaking, QPx,Px~Qpy,py.

The author wishes to thankfully acknowl­edge the guidance and support he received

from his adviser Dr. A. ]. Brandenberger inthis study, which was a part of a researchsponsored and supported by the U. S. ArmyEngineer, Geodesy, Intelligence and Map­ping Research and Developmen t AgencyFort Belvoir, Virginia.

BIBLIOGRAPHY

1. Brandenberger, A. ]. "Fehlertheorie der aus­seren Orientierung "on Steilaufnahlllcn. Diss.,E.T.H., ZUrich, J9117.

2. Ghosh, S. K. "J J1\'estigations into the problemof relati"e orientation." Diss., Ohio State Uni­versity, 1964.

3. Jerie, H. G. Vorzeichenfragen an rallmlichenAliswertegeraten. Photogralllllletria, 1955-56,vol. 1.

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