AJethods Of Computing Cantilever

Control Extension*DR. C. S. SHU,

Photogrammetric and Highway Design Engineer,Robert E. Smith, Consulting Engineers.

Rochester, New York

ABSTRACT: Two methgds are described for p3rformi'7,g cantilever exten­sion of ground-control analytically. O'te m~thod is similar tg the familiarinstrumental method of control-ext3nsi;m employing a procedztre to per­form relative orientation of a model first a YJ,d then to perform the absoluteorientation or sealing of the model. Th~ other m~thod combines the relativeand absolute orientation into a simztltaneozts solzttion for each stereomodel.The second method is shown to be more favorable. For cantilever extensiongroztnd-survey control is given only in the initial photograph of the strip.Investigations have been condztcted using both a mi·timztm and a redun­dant nztmber of PJ,ss-p,ints. E'Lch method employs an iterative processwith the electronic-ealcztlator performing th3 t3digus comput'Ltions.

INTRODUCTIOl

A NYONE who has had the experience ofsolving a photogrammetric problem,

such as space resection and orientation bythe analytical method using a desk cal­culator, understands how tedious andtime-consuming the work is. Yet, com­pared to this simple problem, the compu­tation of control-extension in a strip ofphotographs is indeed a monstrous task.

Up to now, control-extension in photo­grammetry has been performed by eitherradial triangulation or stereo triangulation.Radial triangulat'on usually ut'lizes slottedtemplets as a means of horizontally con­trolling maps, charts, and mosaics. Stereotriangulation as performed on the precisestereoscopic mapping instruments pro­vides horizontal and vertical control fortopographic mapping. These two methodshave dominated the arena of control-ex­tension in photogrammetric mapping forapproximately 30 years. It was not untilthe introduction of the high-speed elec­tronic-computer that the concept of apply~ing analytical methods to photogram­metric control-extension received warmattention.

Under the sponsorship of the U. S. ArmyEngineer Research and Development Lab­oratory, Cornell University has engaged inthe study of a number of methods for solv-

ing the problems of space resection andorientation, extension of control, andbridging. These studies have been con­centrated largely upon existing methodswith some necessary modifications.

COORDINATE SYSTEMS AND

EARTH CURVATURE

To eliminate the problem of earth curva­ture and to obtain the true azimuth of theprincipal plane of each photograph, threerectangular coordinate-systems are intro­duced into the computations. These are:(1) geocentric, (2) local, and (3) photo­graphic.

The geocentric coordi na te-systemt hasits origin at the center of the referenceellipsoid of the earth. The positive Z-axisis toward the north pole. The X and Yaxes lie in the plane of the equator with thepositive X-axis in the meridian planethrough Greenwich.

The local coordinate-system, whichvaries for each photograph, is consideredto have its M and N axes in a plane tan­gent to the reference ellipsoid with thepositive N axis toward true orth. The

t Shu, C. S., and McNair, A. .T., "Determi­nation of Geographic Coordinates, FlightHeights, and True Orientations for Extensionsof Strips of Aerial Photographs," PHOTOGRAM­METRIC ENGINEERING, September 1956, p. 637.

* Presented at Society's 23rd Annual Meeting, Hotel Shoreham, Washington, D. c., March 5,1957.

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718 PHOTOGRAMMETRIC ENGINEERING

positive K axis is toward the Zenith andcontains the exposure station. The photo­graphic coordinate-system has its originat the perspective center with the negativez axis toward the principal point. The xand y axes lie in a plane parallel to theplane of the photograph with the positivex axis, as projected on the plane of thephotograph, pointed toward that fiducialmark which allows this axis to best describethe direction of flight.

The conversion from geographic coor­dinates of latitude, longitude, and eleva­tion above the reference ellipsoid of theearth to geocentric coordinates of X, Y,and Z or vice versa is straightforward.With an electronic-computer the conver­sion requires only a second or two. In thiscomputation, for a limited area, the sur­face of the reference ellipsoid of the earthcan be regarded as coinciding with sealevel. A matrix multiplication makes itpossible to convert between the photo­graphic coordinate-system and the localcoordinate-system. Thus, for any photo­graph in a strip the true azimuth of theprincipal plane is readily established inde­pendent of convergence of the meridians.

The method of computing a cantilevercontrol-extension is to solve for the geo­centric coordinates of the exposure stationsand of the pass-points, and also to solvefor the relations between the photographiccoordinates and the local coordinate-sys­tem of each photograph.

The relations between the photographicand the local coordinate-system providetilt, swing, and azimuth, or pitch, roll,and crab of each photograph.

COMPUTATIONAL PROCEDURES

Two methods, each of which is repre­sentative of a general approach to theproblems of computing cantilever control­extension will now be described: (1) Amethod which is very similar to the in­strumental approach; (2) A method whichsolves relative and absolute orientationsimultaneously.

(1) THE MODIFIED BRITISH ORDNANCE SUR­

VEY METHOD

The first method consists of first com­puting the relative orientation for eachpair of overlapping photographs in thestrip. Each is called a stereogram. Con­venient values are first assumed for therelative orientation elements. Then cor-

rections to these assumed values aresought. The formulas applied to this com­putation are a set of first-order linear equa­tions. These equations are derived fromthe concept that, if a pair of overlappingphotographs are in relative orientation,the rays emitted from both exposure sta­tions to any common image point in theiroverlap area must intersect. This is ac­complished by reducing the volume of afour-sided figure to zero by an iterativeprocess. The edges of this figure are thestereo base, the two rays from the exposurestations to the common image point, andthe line representing the shortest distanceor discrepancy between those two rays.

For cantilever control-extension, knownground-controls are provided only in thearea covered by the initial aerial photo­graph of the strip. A minimum of thereground-control points is required to es­tablish the correct space resection andorientation of this photograph with re­spect to the geocentric coordinate-system.This is computed by some other methodsuch as the Church or Herget method.Since the relative rotational elements foreach model have been obtained by theprevious computation, the correct spaceorientation for the second photograph canbe computed by means of a simple trans­formation process or matrix multiplica­tion. The space resection for the forwardphotograph in any stereogram can be ob­tained if at least two ground controlpoints appear in it to fix the scale of themodel.

The control-extension can now proceedby selecting a minimum number of twopass-points which are in the triple overlapof the first three photographs. The geo­centric coordinates of these pass pointscan now be computed. At this point thesecond photograph is completely orientedand resected, and two points, the geo­centric coordinates of which are known,appear in the second model. These for­ward points are now treated as ground­control and the resection and orientationof photo three is accomplished in the samemanner as photo two. Thus each exposurestation in turn is located out to the endof the strip.

(2) THE MODIFIED HERGET METHOD

The second method differs from the firstin that the relative and absolute orienta­tions for each model are completed in a

COMPUTING CANTILEVER CONTROL EXTENSION 719

single step. The space-resection and space­orientation of the initial photograph areaccomplished as in the first method. Thenfor the second photograph, six differentialerror equations with six unknowns can beestablished. These are composed of equa­tions for the two ground-control pointswhich are in the overlap area of the initialand the second photograph and of equa­tions for the two pass-points, which are inthe common area of the first and secondmodels. Three of these six unknowns repre­sent the three corrections to the approxi­mate geocentric coordinates of the ex­posure station of the second photograph.The other three unknowns represent therotational corrections for the camera axisof the same photograph with respect tothe geocentric coordinate-system.

Convenient values are assumed forthese elements and then corrected by aniterative process. As soon as the computa­tions for the space resection and orienta­tion of the second photograph are com­pleted, the geocentric coordinates of thetwo pass-points can be computed by es­tablished formulae. Again these two pass­points are used as ground-control pointsfor the orientation and resection of thethird photograph in the strip. By repeatingthe process, the computation of the con­trol-extension can be continued to the lastphotograph in the strip.

TESTS AND ERROR STUDIES

To test these two methods, data from aset of fictitious photographs with an aver­age scale of 1; 80,000 were processed. Thecomputations were programmed to suittwo types of high-speed electronic com­puters.

After the methods had been successfullytested using a minimum amount of data,tests were run for redundant data usingsix, ten, and fifteen pass-points per model.These tests correspond to an actual case inwhich every systematic error had beeneliminated for each pass-point. Each pass­point selected for computation was dis­placed from its true position on the photo­graph. This was done in a random mannerto simulate accidental errors in measure­ment. The standard error for the measure­ment was assumed to be 0.050 mm.

Seven sets of fictitious photographicstrips, ranging from four to nine photo­graphs per strip were tested with variousrandom photo-image measurement errors

and with different numbers of pass-pointsper model. The results indicate that thereis no assurance that fifteen pass-pointswould give better results than six pass­points per model. It can also be concludedthat the modified Herget method givesbetter accuracy than the modified Britishmethod.

The results of these sample sets fail toprovide a strong confidence limit for anyprediction of the range of the accumulatederrors from model to model. It is apparentthat after three models, the position of theground points does not maintain an ac­curacy of 1; 5,000. Thus the results ofcantilever extension will probably not meetthe required accuracy for controlling top­ographic mapping so long as the standarderrors in photographic image measurementare 0.050 mm. or a maximum error of0.150 mm. However, it should be remem­bered that similar accidental errors of thismagnitude are present in any instrumentalextension and would not permit theachievement of National map accuracystancards by any cantilever method.

ANALYTICAL ADVANTAGES

Although the methods mentioned wererestricted to the cantilever extension, someadvantages by analytical control-extensionversus the instrumental control-extensioncan be ascertained from this study;

(1) The analytical method can intro­duce any number of pass-points or anyexisting control-points into the computa­tion without complicating the problem,except that a redundant number of equa­tions requires a least squares solution.As we all understand, since the advanceof the electronic computer, the leastsquares solution by the electronic-computerhas become the least troublesome routine.Six points is the minimum number ofpass-points required in the control-exten­sion to solve the space-resection andspace-orientation of each new photographin the strip. Our study made evident thatusing 10 or 15 random selected pass­points may not always give a better resultthan by using 6 points. However, it canbe expected that when a redundant num­ber of random selected pass-points areused in the solution, the resulted modelwill have its error more uniformly distrib­uted over the entire area of the model,which will be the basis for mapping. Inother words the model determined by a

720 PHOTOGRAMMETRIC ENGINEERING

mllllmum number of pass-points may re­sult in a uneven distribution of errors;some locations in the model may tend toexaggerate the errors while in other loca­tions the degree of errors may be less.The flexibility of the analytical method foradopting any number of pass·points orground-control points in the control-ex­tension was lacking in the instrumentalmethod.

(2) When the minimum number of pass­points are used in the analytical control­extension, the pass-point distribution willbe confined to certain patterns as in theconventional instrument methods. Thiswill give the well defined pass-points onthe photograph a chance to be selectedin the control-extension, hence better ac­curacy can be expected than when thepoorly defined pass-points must be usedas in the instrumental method, when thereis restriction in choosing the pass-pointsin certain restricted areas on the photo­graph.

(3) Due to the limitation of the plottinginstruments, it is not natural that theorientations of a strip of photographs haveto be divided into two separate steps­relative-orientation and absolute-orienta­tion. By means of the analytical method,the space-resection and orientation of eachphotograph can be solved simultaneously.Furthermore such methods offer not onlythe solution of the resection and orienta­tion of each photograph in the strip one byone, but also the simultaneous solutionof the entire strip, that is, in a strip withknown controls at the two ends of thestrip or with known controls scatteredover the entire strip. Theoretically asimultaneous solution of a strip or severaladjacent strips, pooling all the availablecontrol data over the strip or strips in aleast square solution, will give the bestfit of the models in tJ-.e strip or strips andthe most probable solution for the entirescheme of the photo strip or strips. How­ever, the amount of computations in­volved in solving the simultaneous equa­tions is in proportion to the cube of thenumber of unknowns in the simultaneous

equations. \iVhether the accuracy achievedby the simultaneous solution of the stripor strips will warrant the great many com­putations should be a subject for furtherstudy, since the quality of photographicimage-points and the accuracy of the meas­uring devices have their own limitations.

(4) Since the measured photographic­coordinates are the initial data in the com­putation of analytical aerotriangulation,any known systematic errors such as lens­distortion, fil m-shrinkage, refraction oflight rays, etc., which can be made ascorrections to the photographic-coordi­nates, can be eliminated. In addition, theplotting instrument errors, optical ormechanical, will not exist in the analyticalmethod, because no plotting instrumentsare used at this stage.

(5) If the analytical method can takethe place of instrumental method inaerotriangulation, only simple stereo­plotting instruments will be needed toserve the purpose of making the contouror topographic maps.

(6) It has been proved that the analyt­ical method of control-extension bymeans of the electronic··computer is fasterthan the instrumental method. It couldbe expected in the future improvements ofcomputation speed in the electronic-com­puter and by the simplification of theformula and process of the analyticalmethods, the aero-triangulation for a stripor strips of photographs can be completedin a small fraction of the time required bythe instrumental method. Also in analyt­ical control-extension no skilled andhighly trained personnel are necessary asare highly demanded in the instrumentalmethod. Theoretically, the accuracyachieved by the analytical method shouldbe higher than by the instrumental method,because of the certain advantages of theformer that have been mentioned in thisraper.

Therefore, it can be concluded that inthe future, the control extension by meansof the analytical methods will be muchmore widely accepted in the field of photo­grammetry.