The Spatial Model Conceptof Photogrammetry*

c. L. MILLER, Assistant Professor of Surveying,

Director, Photogrammetry Laboratory,

Massachusetts Institute of Technology

ABSTRACT: Although photogrammetry is a well established field, photo­grammetrists are constantly faced with the task of explaining the subjectto other engineers and scientists. This paper presents some simple con­cepts of photogrammetry which the author has found useful in describingthe subject to other engineers. A spatial model concept is developed andcomparisons are made with conventional surveying.

INTRODUCTION

D URING the past ten years, photogram­metry has developed into a field of

considerable technical and professional im­portance. As a result of this growth, photo­grammetrists are constantly faced with thetask of explaining the subject to a growingaudience of engineers and scientists. Inmany cases, the only explanation requiredis one sufficient to give the newcomer ageneral understanding and basic appreci­ation of the subject without creating anatmosphere of complexity and mystery.The purpose of this paper is to presentsome simple concepts of photogrammetryto serve as an aid in explaining the subjectto others.

A BASIC DEFINITION

Photogrammetry is defined by theAmerican Society of Photogrammetry as"the science or art of obtaining reliablemeasurements by means of photography."It is noted that this definition does notlimit us to anyone type of photographysuch as aerial photography nor to anyonefield of application, such as map making.Photogrammetry is essentially a method ofmaking spatial measurements. By spatialwe mean location, length, direction, size,shape, area, volume and similar dimen­sions, as contrasted with such dimensionsas pressure, voltage and velocity. How­ever, spatial measurements are often re­lated to the determination of other

variables, such as determining stress bymeasuring the spatial increment strain, orvelocity by measuring time and spatialdistance.

A BASIC CONCEPT

We may consider photogrammetry as ameans of creating a spatial model of theolJject photographed-a precision spatialmodel which can be measured. This simpleconcept removes much of the mysteryabout photogrammetry. We merely createa model and measure it instead of theactual object of interest. Although ourmodel may be mathematical, usually wecreate a three dimensional optical or opti­cal-mechanical model of the object.

Essentially, photogrammetry is an analogprocess. By analog we mean the objectbeing investigated is simulated by someother physical system. The use of electrical,mechanical, and hydraulic analogs is quitecommon in engineering practice for study­ing and measuring objects or conditions ofin terest.

ApPLICATIONS

In general, we can say that photogram­metry can be used to measure any objector phenomenon which can be photo­graphed. We usually find photogrammetryto be advantageous whenever it would bedifficult or uneconomical to directly meas­ure the actual object. For example, photo­grammetry might be applicable if the

* AUTHOR'S NOTE: This paper was developed from material presented in addresses before theBoston Society of Civil Engineers, April 4, 1956, and the New York Section, American Society ofPhotogrammetry, December 4, 1956.

31

32 PHOTOGRAMMETRIC ENGINEERING

object falls in one of the following cate­gories:

(1) Very large-such as a portion of thesurface of the earth

(2) Very small-such as a sand particleor metal surface roughness

(3) Moving-such as a structure underdynamic loading or a missile

(4) Changing-such as a hydraulic phe­nomenon or a glacier

(5) Inaccessible-such as an atomicparticle in a cloud chamber

(6) Complicated-such as a deformedairplane wing or the human body

Our photogrammetric spatial modelactually has four dimensions. In additionto the three linear dimensions of x, y, andz, the model has the important fourthdimension of time. The model is a truereplica of the object photographed at aspecific instance of time. If the object ofinterest is moving or changing, the timedimension is of paramount importance.Photogrammetry permits us to freeze theexact shape, size, and spatial location of theobject at any selected time or time inter­val.

One of the important advantages ofmeasuring with a model is that our workis conducted in the convenience of thelaboratory instead of in the environmentof the actual object. However, the photo­graphs must be taken in the environmentof the object under natural conditions.Therefore, the limitation on applying pho­togrammetric methods to a measurementproblem often causes some difficulty inobtaining suitable photographs and not abreakdown in the photogrammetricmethod.

In summary, if we have an object whichwe would find difficult to measure directlybut which can be photographed, we createa model of the object and measure themodel. With this concept of where photo­grammetry can be applied, the many novelapplications of photogrammetry periodi­cally reported are not so amazing, after all.In most cases, the object of investigationmeets one or more of the six general objectclassifications listed above. As an applica­tion example, the problem of making spa­tial measurements of clouds in the highaltitude jet stream was recently presentedto D. R. Schurz of the M.LT. Photogram­metry Laboratory by a meteorologicalresearch group. In this case, we have anobject which is very large, moving, chang-

ing, inaccessible, and complicated all at thesame time.

THE METRIC PHOTOGRAPH

In explaining the "how" of the photo­grammetric method of making measure­ments, our first consideration is the natureof the photograph. Geometrically, thephotograph is a perspective projection ofthe object photographed. In a perspective,all of the rays from the object converge toone point and the perspective projection isthe trace of the intersection of the con­verging rays with a plane. The function ofthe lens of the camera is to cause all of therays from the object to converge to onepoint. The plane of the photographic filmor plate is the intersecting plane. The func­tion of the photographic emulsion is tofreeze the trace of the intersecting rays.

The metric photograph enables us tomathematically and physically recover theeffective geometric point of convergence ofthe rays which formed the photograph. Byrecovering this point, we can recover andreproduce these rays, and hence the rela­tive directions of and angles between allof the rays to every point on the object.We therefore say that the photograph is agraphical record of a set of directions orangles, and that the photogrammetriccamera is an angle recording instrument.

A single photograph gives us the direc­tion to each point in object space from thecamera position, but not the location ofpoints in space. A point in object spacecould be located by the intersection of twodirection rays. This could be easily achievedby having two photographs of the sameobject from different camera positions.

I t is observed that human vision geo­metrically operates in much the samemanner. Each eye is similar to a camera,so our brain receives two separate anddifferent "photographs" from two differentcamera positions. In the case of the twocameras and the two eyes, we are essentiallytriangulating distances. The distance be­tween camera stations or eyes is the base­line, points different distances away sub­tending different angles opposite the base­line.

We can duplicate natural human visionby placing before each eye the correspond­ing photograph from two camera positions.This duplication is called stereovision andis an important part of the photogram­metric system since it enables us to see themodel we have created. The average hu-

THE SPATIAL MODEL CONCEPT 33

man eyebase is about 0.21 feet. This shortbaseline sets limits on our distance ordepth perception ability. However:, if weplace photographs before the eyes whichwere taken 2,100 feet apart, we effectivelyexpand our eyebase and extend our depthperception abilities by a factor of2,100/0.21 or 10,000 times. This amplifi­cation of our human vision is an importantfactor in our ability to make very precisemeasurements with photogrammetry.

CREATING THE MODEL

When the photograph was formed, rayscame from the object to the photograph.If we return to our camera position with aprojector (which we may consider to begeometrically similar to the camera andwhich merely reverses the direction of therays), the rays will return to their point oforigin on the object. Similarly, if we have asecond projector projecting the photographfrom a second camera position, the raysfrom a given image point appearing inboth photographs will intersect in space attheir point of common origin. Under theseconditions we will project a three dimen­sional model coinciding in space with theoriginal object.

If we maintain the same relative orien­tation between the two projectors, butmove them closer together along the base­line connecting the two camera stations,the rays from common image points willcontinue to intersect in space but will forma model smaller than the object. If wemove the projectors apart, the modelformed will be larger than the object. Theratio of the projector spacing to the cameraspacing is the scale of the model. If thephotographs were taken 3,000 feet apartand placed in projectors 1.5 feet apart, theprojected model would have a scale of1/2,000.

The spatial model formed by the pro­jectors cannot be perceived by the unaidedeyes. As mentioned before, each eye mustbe presented with a separate and differentpicture. Therefore, one eye is only allowedto see the picture from the first projector,and the other eye only the picture from thesecond projector. When this is done, themind will fuse the two sets of images intoa single spatial model.

THE NEED FOR CONTROL

We have seen how a spatial model of anobject can be created with photographs. Wecannot use the model for measuring the

object unless we know the scale of themodel and its location in an establishedreference framework. This is achieved byhaving dimensional control in the model.The need for control can be illustrated bycomparison with conventional surveying.It was previously stated that a photographis only a source of angular information.But angles alone do not define size. In tri­angulation, we know that we require thelength of at least one side of a triangle (abaseline) in a network of triangles beforewe can compute the triangles. The same istrue in photogrammetry. We have tomeasure at least one distance in our systemin orde: to determine size or scale of themodel.

A second basic requirement in any sur­veying system is horizontal orientation.What are the directions of our lines withrespect to our spatial framework referenceaxes? Here again, as in ground triangula-

. tion, we need the horizontal orientation,or bearing, of at least one line in our model.These first two requirements, scale andorientation, can be satisfied by having twopoints in the model of known horizontalposition. The distance between the twopoints establishes scale, and the bearing ofthe line between the two points establishesorientation.

A third consideration is the establish­ment of a datum plane in order that hori­zontal planes in the object will be hori­zontal in the model. Since it takes threepoints to determine a plane, we need thedifferences in elevation between threepoints in the model to define the datumplane. If we are compiling a topographicmap and would like our elevations refer­enced to a standard datum such as meansea level, we would need the mean sealevel elevation of one of the three points.In summary, to locate our model withinsome established spatial framework, wewould need at least two horizontal andthree vertical control points within themodel and referenced to the desired co­ordinate system.

We have spoken so far about an indi­vidual model formed by two photos, sayA and B. However, photos Band C, C andD, D and E, etc, may form additionalindividual models which collectively forma continuous model of a large area. Theindividual models can be compared to indi­vidual triangles in a triangulation network.If we know everything about one triangle,we can compute the other triangles. Simi-

34 PHOTOGRAMMETRIC ENGINEERING

larIy, if we have oriented, scaled, andleveled one of the individual models, wecan determine the spatial location and allpoints therein of the other models. Just aswith ground triangulation, we can onlycontinue this process of aerial or photo­grammetric triangulation until our erroraccumulation becomes too large to tolerate.The important point is that we do notnecessarily need "outside control" in everyindividual model.

MEASURING THE MODEL

The ultimate goal of most of our con­ventional surveying operations is the xyzcoordinates of selected points, the graphicalrecord of the xy position of selected lines,and the representation of relief by contourlines. Once we have created and controlledour photogrammetric model, we canachieve these goals directly without anyintermediate measurements, computations,adjustments, or plotting. How this is donemay be understood if you visualize thatwithin the spatial framework of the modelwe insert a visible reference mark whichcan be freely moved in model space. Thexyz position of the reference mark is socalibrated that the value of its position isalways known. With this movable andcalibrated reference mark, we have the per­fect surveying tool. This may be comparedto the surveyor having a little black boxwith three dials which always read thexyz spatial position of the box as it is car­ried from point to point.

When measuring the model, to locate apoint on its surface, we merely place thereference mark thereon and directly readits value. To trace the plan position of aline, we merely move the reference markalong the line in the model. To run out acon tour, we set the z value of OUI referencemark to equal the elevation of the desiredcontour. The mark is then moved until ittouches the surface of the model. This isone point on the contour. Now if the markis moved but constantly kept in contactwith the surface of the model, we are trac­ing out the desired contour. A pencilpoint is coupled with the reference markso that our point, plan line, or contourline is automatically plotted.

THE STEREOPLOTTER

There are three common approaches tosolving engineering problems: (1) analyti­car or mathematical, (2) graphical, and

(3) mechanical. By mechanical we meanwith a machine, instrument, or analog.Most photogrammetric problems may besolved by anyone or a combination of theseapproaches. For carrying out the conceptspreviously presented and in actual prac­tice, we find the third approach to be themost feasible and use an instru men t calledthe stereoplotter.

The stereoplotter is an instrument forcreating and measuring the spatial modeland recording the results. The functionalcomponents of the stereoplotter include (a)two or more projectors or other means offorming the spatial intersections of thereconstructed rays, (b) facility for movingthese projectors along and around threemutually perpendicular axes to reconstructrelative and absolute orientation of theprojectors, (c) means of separating theprojections for the eyes, (d) a movable andcalibrated reference mark, and (e) a plot­ting system. Although they all performbasicly the same function, there are a largenumber of different types and makes ofstereoplotters available to the engineerand scientist.

PHOTOGRAMMETRIC ACCURACY

All measurement systems and their in­strument components contain sources oferror. vVe therefore design our measure­ment systems in such a way that we can(1) reduce the errors to a tolerable level,(2) correct for the errors by instrumentdesign or measurement technique, or (3)make secondary measurements to enableus to compute the errors and correct theprimary measurement. Of course we nevercompletely eliminate all errors, but striveto reduce the total residual error until weobtain the desired precision and accuracy.In this respect, photogrammetry is no dif­ferent from any other measurement systemin that we are concerned with many sourcesof error. However, the photogrammetricengineer can reduce, correct, or computethese errors and so design any measure­ment project to obtain the desired ac­curacy.

Previously we presented a concept ofphotogrammetry as the creation of aspatial model of the object photographedand the measurement of the model. Theaccuracy of the method is therefore essen­tially concerned with our ability to createand measure the model. The accuracy withwhich we can create and measure the model

THE ~PATIAL MODEL CONCEPT 35

We Need a ModelWith a Scale Of

5/11/11/21/201/2001/2,0001/4,0001/10,0001/20,0001/40,0001/200,000

depends on many factors such as thecamera, lens, photographic materials, flightheight, control, stereoplotter, human ele­ment, nature of the object, and the designof the over-all system. Errors originatefrom these and many other sources.

What is important is the value of all ofthe combined individual errors at the scaleof the model. We therefore present a simpleconcept that the accuracy of our photo­grammetric measurements is a function ofthe size of the model for a given systemand set of conditions. As a hypotheticalexample, suppose at the scale of the modelthe total net residual error from all sourcesis 0.15 millimeters, 0.006 inches, or 0.0005feet. If the object being investigated is1,000 times larger (model scale 1/1,000),an error of 0.0005 feet tn the model co;­responds to an error a thousand timeslarger or 0.5 feet in terms of the object.For the case of our example, we could setup a table as follows:

If the AllowableError Is

0.0001 feet0.00050.0010.010.11251020100

If measurements of 1/10,000 of a foot arerequired, we would need a model five timesthe size of the object. If measure men ts to10 feet are required, we could use a model20,000 times smaller than the object photo­graphed. The error in running a contourline is about twice that of a discreet point.Since the allowable error in a contour isone-half the contour interval, the modelsizes given in the table must be multipliedby four if the numbers in the first columnare contour intervals. Two foot contourswould therefore require a model scale ofapproximately 1/1,000 for the hypotheticalexample.

It is obvious that there is no inherentlimit to the accuracy of photogrammetricmeasurements in terms of object accuracy.In fact with microphotogrammetry, meas­urements to the order of microns at thescale of the object are quite possible ifsuitable photographs can be obtained.

When people speak of limits for the accu­racy of aerial photogrammetric mapping,they really mean there are minimum prac­tical operating limits on the altitude of theairplane and associated problems in ob­taining suitable photography.

PHOTO ANALYSIS AND I TERPRETATION

In this paper we have dealt entirely withbasic concepts of the measurement ormetric aspect of photogrammetry. Inaddition to recording angles, the photo­graph records qualitative informationabout the object photographed which isidentified and interpreted by the photoanalyst. Therefore, photo analysis andphotogrammetry are closely allied butsomewhat separate fields of endeavor, sincethe areas of knowledge involved are quitedifferent. Mathematics and physics arebasic to the photogrammetric engineerwhereas geology, soils, and forestry aremore important to the air photo analyst.

The nature and extent of the informa­tion and data which the highly trainedprofessional photo analyst can obtain fromaerial photographs borders on the fan­tastic. Soils, geology, drainage, land classi­fication, vegetation, human activity areonly a few of the many areas of data thatcan be obtained. It is not within the scopeof this paper to discuss the qualitativeaspects of photography, but it is importantto note the distinction between the work ofthe photogrammetric engineer and thephoto analyst even though there is someoverlap between the two fields.

PROFESSIONAL PHOTOGRAMMETRY

Although the emphasis in this paper hasbeen on simple concepts which give thenewcomer an insight into photogrammetry,in theory and practice we are concernedwith a subject which has extensive techni­cal complexities. The photogrammetrist iscalled upon to have a working knowledgeof many scientific and technical fields.Therefore, we classify photogrammetry asa narrow and highly specialized field butone involving broad professional considera­tions. A major step in the direction of pro­fessional recognition will occur when weremove some of the mystery surroundingthe subject. It is hoped that the simpleconcepts presented in this paper will assistin obtaining a wider appreciation andunderstanding of the work of the photo­grammetrists.