Analytical Aerotriangulation by theDirect Geodetic Restraint Methods *

HUGH F. DODGE,

U. S. Geological Survey, Washington, D. C.

ABSTRACT: The paper courses the history of the U. S. Geological Survey's re­search project on analytical aerotriangulation. The objectives of the programare discussed along with the general plan for attacking the problem. The mainbody of the text is devoted to describing the basic portion of the geometry of theDirect Geodetic Restraint Method of analytical aerotriangulation. This de­scription is confined to terms familiar to the photogrammetrist and is givenwithout recourse to mathematical formulas. A brief resume of tests already con­ducted or in progress concludes the paper.

HISTORY AND CURRENT STATUS

I NSTRUMENTAL methods of photogrammetriccontrol extension have long been used in

the Geological Survey as an integral part ofits map-production procedure. It has alsobeen recognized for many years that ananalytical solution to the control-extensionproblem holds certain theoretical advantagesover instrumental methods. Developments inelectronic computers caused the GeologicalSurvey to activate a preliminary project in1955 for the purpose of collecting and review­ing available information on analytical aero­triangulation and for selecting a procedurethat could be adapted for electronic computa­tion. It was decided in August 1957 that effortshould be concentrated on a completelyanalytical method then being developed atCornell University under a contract from theU. S. Army Engineer Research and Develop­ment Laboratories. i .2

Subsequent investigations demonstratedthat the approach to the problem developedat Cornell required a certain amount of revi­sion. The system as refined by the GeologicalSurvey has been named "The Direct GeodeticRestraint Method." Tests of the method havebeen encouraging, but it must be emphasizedthat in the Geological Survey to date all workin analytical aerotriangulation has been ex­ploratory in nature; it has not gone beyondthe research stage.

1 Dodge, Hugh F., "Geometry of Aerotriangula­tion" (a thesis), Cornell University, June, 1957.

2 McNair, Prof. A. J., et at., "A Solution of theGeneral Analytical Aerotriangulation Problem,"Cornell University, May 1958.

HUGH F. DODGE

The Geological Survey is at present using afixed-decimal-point, single-precision, Data­tron 205 electronic computer program forthese investigations. This program can ac­commodate the simultaneous adjustment offrom one to nine single-camera photographs.In its present state, the program is not suit­able for production use because developmentof the error theory and weighting system isnot complete; also the program is inefficientand lacks many of the checks and peripheralcomponents that would be required in prac­tice. It is completely adequate, however, forthe purposes of the research program.

* Presented at the 25th Annual Meeting of the Society, l;Iotel Shoreham, Washington D. c., March,10, 1959. Publication approved by the Director, U. S. Geological Survey.

590

ANALYTICAL AEROTRIANGULATION 591

GEOMETRICAL BASIS

THE PHOTOGRAPH

The first item to be considered is the photo­graph itself. Figure 1 is a diagram of a"theoretical photographic system." The term"theoretical" is used in the sense that thisphotograph represents the simplest theoreti.calgeometry, with the object rays a~l co.nverglngat a point 0, the photograph lYing 111 a per­fect plane, and so on. Each photograph in agiven bridge has its own set of ~oordinate

axes with the origin at the perspective center.The z-axis is collinear with the camera axis;the x- and y-axes are parallel to lines joiningthe two sets of fiducial marks.

Coordinates of points on "theoreticalphotographs" are determined by making ap­propriate corrections to measurements ob-

weighting routines by electronic com­puter tests.

7. Appraisal of available physical com­ponents for the system (camera, film,glass plates, comparator) based uponthe error theory and operational con­siderations.

8. Development of calibration proceduresand computer routines for compensationof systematic errors in the physicalcomponents of the system.

9. Testing of the en tire system under actualproduction conditions.

The first two items in this list-the devel­opment of a workable geometry and testsverifying it-have been completed for single­camera photography. The remainder of thispaper will be devoted to an outline of themore important geometrical techniques usedand a short resu me of tests completed or inprogress.

~ ,~~>\L5 ~..... .Y" .-AXl

FIG. 1

PERSPECTIVECENTER

Fl GUR£ IIDEAL GEOMETRY

PLAN OF ATTACK

I t was necessary to formulate a general planof attack which would lead to the successfulrealization of the given objectives. This planemphasizes theoretical development and anal­ysis as a first step in every phase. This is fol­lowed by verification, first, in completelycontrolled tests involving fictitious photog­raphy and, second, in tests conducted undernormal operating conditions. Such a proce­dure lends itself to the following sequence:

1. Development of a geometrical method ofsolution suitable for electronic computa­tion and the preparation of an electroniccomputer program.

2. Verification of the geometrical develop­ment in a series of tests with fictitiousphotography.

3. Determination of the theoretical effectsof the propagation of errors by means ofa differential error analysis based onproven geometry.

4. Establishment of the validity of theerror theory by controlled tests.

5. Construction of data-weighting com­puter routines.

6. Demonstration of the correctness of the

PURPOSES OF RESEARCH

The development of analytical aerotri­angulation procedures in the Geological Sur­vey is governed by two major objectives:

1. To provide a general solution to the aero­triangulation problem for both stripsand blocks by a completely analyticalmeans.

2. To develop knowledge of the basic lawsand factors involved in aerotriangula­tion, with the expectation that suchknowledge can be applied to other photo­gram metric systems.

No attempt is being made to supplant hu­man judgment with an electronic computer.On the contrary, the aim is to design a systemthat will relieve the photogrammetric opera­tor of the tedious repetitive tasks that do notrequire judgment, and to provide him with amaximum amount of information so that hisexperience and intelligence may be appliedmore effectively. Since the operator can bestutilize his experience and skill in situationsthat are familiar to him, the system is beingdesigned so that the information supplied willnot be foreign to current photogrammetricthinking. With the system so designed, thebasic knowledge developed can be applieddirectly to other photogrammetric methods.

592 PHOTOGRAMMETRIC ENGINEERING

OPOSURE STATION I EXPOSURE STATION 2:

- -~ - - - - - -fliGHTPAfii - - - - - -~---

3 .~erget, Dr. Paul, "Interim Report on GroundPOSItIOn Computation From Shoran-ControlleclPhotography," OSURF Technical Paper No. 179,January 1954.

5. Formation of a new set of conditionequations based upon the new positionand orientation of each bundle of rays.

6. Formation of new normal equations andtheir simultaneous solution.

7. Repetition of the cycling (or iterating)until the photogrammetric requirementsare adequately satisfied.

While iterative procedures such as this arecommonplace, the form of the condition equa­tions used should be of interest.

Z?

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\ /~ '"00"' eo,",•FIG. 2

FIGURE 2PARALLAXCONDITION

RE~lOVAL OF PARALLAX

In taking up the photogrammetric require­ments and the equations associated with each,one by one, a logical place to start is with theremoval of parallax. Use is made here of themethod of Dr. Paul Herget3 whose work atOhio State University forms the foundationfor much of the work in this field at bothCornell University and the Geological Survey.

In Figure 2, two successive camera stationsare represented by the two aircraft silhouetteswith rays emanating from each. The longdashed lines represent the initially estimatedlocations of rays to a ground point whose truelocation is at G. When the bridge is absolutelyoriented these two rays will intersect at G,the correct posi tion in space, except for smallerrors. However, because the ini tial estimatesof position and orientation for the bundles ofrays are not exactly correct, these rays will,at first, neither intersect nor pass throughthe true position to the point.

Assuming that the elevation and horizontalposition of the point G are unknown, the onlycondition that can be legitimately enforcedis that the rays intersect. The correct posi­tioning of this~intersection is accomplished by

tained from actual photographs. These cor­rections are based on camera-calibrationdata, fil m-distortion measurements and atmos­pheric-refraction information.

The "theoretical photographs" resultingfrom these corrections will not be perfect be­cause residual errors in the photo-coordinatevalues will still exist after the application ofthe corrections. The theory is, however, thatthe resulting photographs may be representedby the simplest ideal geometry and will havegreater fidelity than the actual photographs.These theoretical photographs are used in allsubsequent calculations.

The coordinates of point P are shown inFigure 1 as x and y (as measured in a com­parator and corrected as described above) andz, which is equal to minus the focal-length.These dimensions are used to find the direc­tion cosines of each ray with respect to thephotographic coordinate system. These direc­tion cosines of all object rays are the digitalform that the electronic computer uses to rep­resent the bundle of rays associated with eachphotograph.

SATISFYING THE PHOTOGRAMMETRIC REQUIRE­

MENTS OF THE BRIDGE

The question now becomes: How can theestablished photogrammetric requirementsof the bridge be satisfied by using these digitalrepresentations of bundles of rays?

This is accomplished by executing thefollowing general sequence.

1. Estimation of the position and orienta­tion of the camera at each exposure sta­tion. These estimates comprise the fol­lowing elements: latitude, longitude,altitude, x-tilt, y-tilt and camera head­ing.

2. Formation of an appropriate equation,based upon the initial estimates, for eachphotogrammetric requirement that is tobe enforced. In other words, there willbe one condition equation to removeparallax at a given point, one to force afit to a bench mark, and so on.

3. Development of a set of normal equa­tions from the condition equations andthe simultaneous solution of these todetermine the corrections necessary tothe position and orientation of eachbundle of rays so that the photogram­metric requirements will be closer toful fill ment.

4. Application of these corrections to pro­duce a new position and orientation forthe camera at each exposure station.

A ALYTICAL AEROTRIANGULATION 593

EXPOSURE STATION

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FIG,4

\\ .'FIGURE 4

COMMON MODELSCALE PRINCIPLE

FIGURE :3DR HERGET'SMETHOD OF FITTING \~....A GROUND CONTROL POINT '\

FIG. 3

the simultaneous enforcement of all conditionsto be imposed on the photogrammetric bridge.To force these rays to intersect, or in otherwords, to remove the parallax, Dr. Hergetdeveloped an equation that reduces the short­est distance between the two rays (indicatedby q) with each iteration until this distancebecomes negligible. As already mentioned, thepositions and orientations of the various bun­dles of rays in the bridge are adjusted witheach iteration to achieve this.

An interesting sidelight is that Dr. Hergetwas able to use this same general form of equa­tion to fit a ground control point that had aknown elevation and horizontal position.Figure 3 shows such a ground-control pointwith two synthetic vectors (indicated by thesolid lines) emanating from it. By forcin~

the dashed line, which represents an objectray, to in tersect each of these two vectors, theray was forced through the ground point it­self. By thus forcing rays to intersect, Dr.Herget was able to resect the initial photo­graph of a strip and to add successive photo­graphs in a can tilevering process.

However, resecting single photographs im­poses far too severe ground-control require­ments for practical topographic mapping inmost areas of the world. The obvious need isfor an operational analytical system broadenough to encompass those photogrammetricprinciples)hat will permit a general solution.

COMMON MODEL SCALE

The next photogrammetric principle to beconsidered is the requirement that all modelsof a strip have a common scale. Figure 4illustrates three consecutive exposure stationswith rays emanating from each that shouldintersect at a common point. The q termsagain indicate the shortest distance betweenthese rays. One of Dr. Herget's parallax equa-

tions is used to force ray 1 to intersect ray 2and a second equation to force ray 3 to inter­sect ray 2. I t is now necessary to use an equa­tion that requiresjt to equalj2. This will bringthe three rays togethel' at a common point asthe iterative procedure takes place. By en­forcing such an equation for at least onepoint common to each three consecutive ex­posures of a strip, common model scalethroughout the strip is assured.

RELATIONSHIP BETWEEN AD} ACENT STRIPS

I t is also necessary to enforce the properrelationship between adjacent strips in ablock solution. In Figure 5 two flights areshown, The long rays all relate to the sameground point. The shortest distance bet\\'eenrays of the first fligh t is denoted by qt and forthe second flight by q2. The intersection of qlwith ray 1 is indicated by D, and of q2 withray 12 by E. Three equations are used to forceD and E to assume the same position in space.The first causes them to draw together in anX direction, the second in Y, and the third inZ. As the q dimensions go to zero in the itera­tive process, the effect is to bring all the raysthrough a common point, \Vhen two or moresuch ground points common to two adjacent

FIG. 5

594 PHOTOGRAMMETRIC ENG! NEERING

FIG. 7

fiGURE 7GEODETIC

~

this cone has certain limitations. When thelatitude of the point is close to 0° or 90°, it isnecessary to use the plane of the prime ver­tical to restrain the point. In this case, it isnecessary also to keep the point on the correctmeridian plane in order that the total effectwill keep the point on the normal (N2 inFigure 7).

To recapitulate on the matter of a hori­zontal station, assume a triangulation stationof unknown elevation on two photographs of astl-ip. Three equations are written for thispoint: first, the parallax equation; second, anequation to force the intersection onto themeridian plane; and third, an equation toforce the intersection onto the latitude cone(or prime vertical plane).

A similar situation exists for a point ofknown elevation. In Figure 7 a segment of asurface is delineated that represents all pointsof equal elevation. 'When the elevation of aground point or exposure station is known,the specific surface that point is to reach andremain on is defined. A spherical surface isused locally to approximate this actual surface.The poin t of known elevation is restrai ned byan equation so that it reaches and remains onthis spherical surface. Th us, in the case of abench mark appearing on two photographsof a strip, two equations are enforced: first,the parallax equation and, second, the eleva­tion restraint just described.

The utility of the geocentric coordinatesystem will be seen from the fact that theserestraining surfaces are most easily definedmathematically in terms of the basic geodeticreference system.

Other equations and restraint methods areincluded in the Geological Surveys currentcomputer program but the foregoing rep­resents the basic portion of the Direct Geo-

GEOCENTRICV-AXIS

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OF THEEARTH

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FIG. 6

FIGURE. 6CUT-AWAY vIEWOF EARTH'SREFERENCEE.LLlPSOID

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flights are used, the correct relationship willexist between the flights.

By enforcing these five equations (parallax,common model scale, and the three block­adjustment equations), wherever applicable,the internal integrity of a correctly plannedphotogrammetric bridge or block will be as­sured. All that remains for a completely gen­eral solution is a system for fitting all avail­able grou nd con trol.

GEOCENTRIC COORDINATE SYSTK\1

To understand the means of accommodat­ing the various ground-control conditions, itis necessary to consider a second basic co­ordinate system. Figure 6 depicts the refer­ence ellipsoid of the earth. Since the origin ofthe coordinate axes shown is at the center ofthe reference ellipsoid, the system is called ageocentric coordinate system. The positive Z­axis runs through the geographic North Poleand the positive X-axis is defined by the in­tersection of the plane of the meridianthrough Greenwich and the plane of the equa­tor. The Y-axis is then chosen so that an or­thogonal right-hand system will result.

Also shown in Figure 6 is a random merid­ian plane. All points in this plane have thesame longitude, and any point with thislongitude must lie in this plane.

The equation that is enforced for a point ofknown longitude in a photogrammetric bridgeis one that causes the point to reach and re­main on its meridian plane. A ground point orexposure station of known longi tude is re­strained in this way.

Similarly, if the latitude of a point is known,the point must be restricted to the surface of aknown cone. Figure 7 shows this situation.It should be mentioned here that the use of

ANALYTICAL AEROTRIANGULATION 595

detic Restraint Method and also indicateswhy the method was so named.

RESULTS OBTAINED

The Direct Geodetic Restraint Method wasfirst tested on a series of 26 problems. Thisinitial series was made up from the fictitiousdata developed by Earl Church and given inthe Harvard Compu tational Laboratory Re­port No. 25. Correct photographic-coordi­nates and ground-point positions were used inthese tests, each of which was a minimumtheoretical case. The tests ranged in complex­ity from the resection of a single photographto the sol ution of a block of eigh t photographs.In addition to some normal cases, a numberof bizarre problems were included, as well asa few that depended for solution upon expo­sure station restraints. The results of thisseries of tests are available at the U. S. Geo­logical Survey's Washington office.

A second series of tests is now being con­ducted to determine the practical range with-

in which the initial estimates must fall. A re­port on this series will soon be available.

A preliminary test using photography overthe Arizona test area is also under way. Thephotographic coordinate data was measuredby the U. S. Army Map Service on a Cam­bridge Stereocomparator. The results of thistest will be available in the near future.

The results obtained with the Direct Geo­detic Restraint Method to date have beensatisfactory. It is contemplated that thepresent computer program will be used foradditional testing, largely in relation toerrors in the physical components of the sys­tem, the propagation of errors, and dataweighting methods. It is expected that testsunder production conditions will be possiblein approximately one year.

I n closing, I would like to acknowledge theimportant contributions to this work of theother two members of the U. S. GeologicalSurvey's research team: M r. Robert C. Ellerand Mr. David Handwerker.

An Evaluation of Aerial Photography forDetecting Southern Pine Beetle Damage *

R. C. I-IELLER, R. C. ALDRICH AND W. F. BAILEY,

RESEARCH FORESTERS,

Div. of Forest Insect Research, Forest Service,U. S. Dept. of Agriculture

(A bstract is on next page)

INTRODUCTION

F OREST insects are insidious killers of ouration's timber. The most recent com­

pilation of statistics (5) showed that theywere responsible for killing seven times thesawtimber volume as that consumed by fire.In the South alone, during the past 5 years,the southern pine beetle (Dendroctonusfrontalis, Zimm) killed more than 400 millionboard-feet of pine timber annually (3). Howare we going to reduce these losses to tolerablelevels? How can we best detect outbreakswhile they are small and easy to control? Theold army maxim, "Those that get there fust­est with the mostest, win the battles!" ap-

plies equally well in the battle with thebeetles. Because bark-beetle epidemics fre­quently cover large areas of timberland andfluctuate widely in amount and degree ofdamage, an aerial method to discover andllo­cate outbreaks rapidly has more merit thanthe slow and costly ground methods used inthe past.

Two aerial approaches for speeding up de­tection are possible. The first involves visualobservation from the air by trained observerswho plot suspected infestations on maps. Thesecond involves taking special aerial photo­graphs on which interpreters try to locate theinfestations. The first method is inexpensive

* Presented at the Society's 25th Annual Meeting, Hotel Shoreham, vVashington, D. C. March 8to 11, 1959. This paper is a part of the Panel on The Application of Color Photography in Photo Inter­pretation.