Measurement of Vertical Heights from Single Oblique Aerial ... · Measurement at Vertical Heights...
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Measurement at Vertical Heights tram SingleOblique Aerial Photographs
SYLVESTER P. GAY, JR.,
Utah Construction Company, San Francisco, California
ABSTRACT: Nomographic methods for indirect measurement of verticalheights from images appearing on single-oblique aerial photographs arepresented in this paper. The application of the methods is quite elementary and may be learned by non-technical personnel in a matter ofminutes. A measurement of a height may then be performed in two orthree minutes with an accuracy of a very few per cent, depending primarily on the accuracy of photographic parameters and the skill of the operator.
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1. INTRODUCTION
I N A previous paper presented in thisjournal* the author described a method
for performing indirect measurements ofground objects from single-oblique aerialphotographs utilizing a set of nomographiccharts. The measurements presented inthat paper were limited to objects lying ina level plane---linear dimensions, areas, andangles. The same approach was subsequently applied to measurement of vertical heights, and it was soon found thatthe method was even more suitable for thisthan for level measurement. Image parameters are reduced to two, and the numberof nomographs required is reduced to one.The method is nearly, though not quite, assimple as the corresponding technique usedwith vertical photographs in which theobject height,
t .h = d X altitude,
where l is the image length of the heightand d is the distance from the top of theimage to the principal point of the photograph.
Early in the development of the nomographic method for measuring verticalheights, it became apparent that due tothe character of different height images ona photograph, two solutions would benecessary-one for heights projecting shortimages on the photograph, and anotherfor long images. Therefore, two differentmethods, equally simple, and each utiliz-
ing a separate nomograph are presentedin this paper. The difference between"long" and "short" images is determinedby the practicality of accurate measurement of the image and accurate manipulation of the image parameters on thenomograph.
The nomograph for long images mayalso be used to measure a component ofthe ground distance of a point from thenadir point of the aircraft, at the time ofexposure of the photograph.
Either height-measuring method maybe taught in ten or fifteen minutes to nontechnical personnel having little mathematical background. The principal skillrequired is ability to use a pair of dividersand to multiply two or three numbers together. An operator proficient in themethod should be able to measure aground height from a photo image inabout one or two minutes. Accuracy shouldbe within a very few per cent, dependingprincipally on accuracy of measurementof the image length.
The foregoing is based, of course, on thepremise that the operator is provided aphotograph with accurately known parameters of tilt, altitude, and focal-length,on which are marked accurate photographic axes. Methods of determiningthese parameters are found, among otherplaces, in the MANUAL OF PHOTOGRAMMETRY and in the aforementioned paperby the author, the latter containing severalshort-cuts, utilizing nomographs.
* Gay, S. P., Jr., "Nomographic Solution to Oblique Photo Mensuration," PHOTOGRAMMETRIC ENGINEERING, Vol. XXII, no. 4, pp. 674-692.
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MEASURING VERTICAL HEIGHTS FROM SINGLE OBLIQUE PHOTOS 901
One limitation of the nomographicmethods presented in this paper is that theheight to be measured must be a verticalheight, such as a flagpole, smoke-stack, orcorner of a building. I t will not measurerelative elevations of different objects aswill parallax methods utilizing stereoscopicpairs of photographs.
II. MEASURING HEIGHTS FROM SHORT
IMAGES
The height-factor chart (Figure 1) is thenomograph used in measuring verticalheights on an oblique photograph when theimage length is small, On this chart a number, or height-factor, is determined usingimage and photographic parameters. Theheight-factor multiplied by the imagelength and the quotient of the altitude andfocal-length gives the correct height of theobject being measured:
altitucteHeight of object = ----
focal lengthX image length X height factor (2)
To determine the proper height-factorfor a photographic image the photo interpreter simply measures the perpendiculardistance from the center of the image tothe x-axis of the photograph, i.e., thevertical displacement of the image. Thisdistance is then located on the scale forthe proper focal-length at the left edgeof the height-factor chart, and a line isextended horizontally until it intersectsthe curve for the correct depression-angleof the photograph (images above x-axis)or tilt-angle (images below x-axis). At thisintersection, the height-factor is read between vertical lines on the scales acrossthe top and bottom of the chart.
When the height-factor chart is reproduced at the scale corresponding to thecorrect focal-length for a photograph, theoperator may simply take the vertical displacement of the image with dividers andapply it to the chart in one step. For theother focal-lengths shown, a pair of proportional dividers may be used in scalingoff the distance. This method may also beused for photographs of any focal length.The ratio is an inverse one. Thus, thechart coordinate for an image point on aphotograph of 7" focal-length would be12/7 of its value for an image point of thesame vertical displacement on a photograph of 12" focal-length.
The height-factor chart, then, is universal in its use. It is not restricted to anyparticular depression-angle or focal-length.It requires no knowledge of mathematicsexcept as required by formula (2). Themanipulation of the chart is so elementarythat few advantages would accrue fromspecialization of the chart, such as adaption for one particular focal-length and depression-angle. The formula used with thechart, formula (2), may be recognized asthe same as that used for computinghorizontal distances on a map or on avertical photograph-multiplied by theheight-factor. This same formula is alsoused for computing horizontal distances onoblique photography with the authorsnomographic method, except that in thiscase, an "oblique" or level-line correctionfactor is used instead of the height-factor.
The height-factor chart in its presentform is restricted, however, to photographic images lying above, or beyond, thenadir point. On oblique photographs oflow tilt-angle, there is quite a sizeablephotographic area lying between the nadirparallel and the lower edge of the photograph, and it may occasionally be desirable to measure heights in this region.Large heights may be measured on thenomographic chart described in the following section, but it was not felt that theneed for measuring small images herejustified construction of additional curveson the height-factor chart. Furthermore,near the nadir-point where the cameraviewpoint is almost vertical, the imagelength is necessarily very small and theheight-factor large, and the accuracy ofthe nomographic method rapidly decreases.Parallax methods using stereo pairs ofphotographs are much more applicable tothis situation. Such methods for obliquephotography are presently under development by the author.
There is another important qualificationin the application of the nomographicmethod. It has been implied that a directmeasurement of the photographic image ismade. This is only true for those smallimages far from the nadir-point of the aircraft and near the principal line of thephotograph which are essentially parallelto the principal line. As images of verticallines on an aerial photograph always pointto the nadir, then many images on thesides of the photograph will be quite nonparallel with the principal line. For these
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"d::r:o..,oC)::tl:>=::is:tIl..,::tl>-<(')
tIlZC)
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FIG. 1. Height Factor Chart-for measuring heights from small photographic images.
FIG. 2. Height Nomograph for measuring heights from large photographic images. Also used for determining componentof nadir point distance parallel to principal line.
Nadir Point Distance~rt..... ..g~sL""", "t'!!9!1!•.~2m2graoh - Lcirqe Images
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904 PHOTOGRAMMETRIC ENGINEERING
I-ZW~WU<l..JQ.(/)
0 Step 3..J L<lU
i=0::W>0
r- L ---j ,r>-/ Height of Object
I! ! I ! I I . I I Io .2H"H .6H .8H .9H
FIG. 3a. Basic technique for determining heightswith height nomograph-large images.
tom of the chart, where the height of theobject being measured is given as a fractionof the al titude.
I t can readily be appreciated that measurements are more accurate the longerthe image-length, but even for smallimages careful work should give quiteaccurate results. However, heights of lessthan about one-tenth the altitude areprobably more accurately measured onthe height-factor chart'. This generallylimits measurements from the heightnomograph-large images to lower altitudephotography. For the sake of accuracy itshould be mentioned that the aircraft altitude used is always the height above thebase of the object being measured.
An alternate technique for applying thevertical-displacements of the top andbottom of the image to the chart is shownin Figure 3b. This technique provides ameasure of what the author has beenstriving to do with all his nomographicmethods-that is, to make the measurement of objects from oblique photographsas simple and as similar to measurementfrom vertical photographs as is possible,(without requiring the use of an electroniccomputer), thereby speeding up the measurements and limiting the amount of newinformation which a photo interpretermust learn-new information in most casessimply being the manipulation of thenomographs.
In this technique for applying the vertical-displacemen ts, the displacemen t tothe top of the image is first taken withdividers, or proportional dividers, and the
images the component of the image parallelto the principal line must be measuredrather than the image itself. This is nogreat problem and may be performed quiteaccurately with a tube magnifier by placing the scribed measuring lines parallel tothe x-axis.
As the height of the object being measured approaches an appreciable fractionof the aircraft altitude, more and moreerror results from use of the height-factorchart. Another nomograph was developedto overcome the error and to take fulladvantage of the high degree of accuracypossible with longer images.
III. MEASURING HEIGHTS FROM LONG
IMAGES
\i\'hile the height method for shortimages resembles methods for measuringlevel-lying dimensions, the height methodfor Joil"ii'images is quite similar to the measurement of. pe'rpendicular heights fromvertical photographs. Derivation of thetwo is, in fact, almost identical.
The nomograph used in measurement ofheights from long photographic images isshown in Figure 2. After performing thenecessary operations on the chart, theoperator arrives at the height as a certainfraction of the aircraft altitude. The onlymathematics necessary, then, is the multiplication of this fraction and the altitude.I n this respect the method is even simplerthan the corresponding technique forvertical photography, as a graphical stepreplaces a mathematical one.
To use the height-nomograph-largeimages, the vertical displacements to thetop and bottom of the image are determined on the photograph and are appliedto the chart in the same manner as previously explained for the height-factorchart, com men ts concerning other focallengths being equally applicable. However,on the height-nomograph-Iarge images,both positive and negative vertical-displacement is presented, and the photographic depression angle is always usednever the complementary tilt-angle. Horizontal Jines are extended from the twovertical-displacement values until theyboth intersect the correct depression-anglecurve for the photograph (see Figure 3a).The horizontal distance, L, between thesetwo intersections is then measured, or preferably taken with dividers, and is appliedto the left side of the scale at the very bot-
MEASURING VERTICAL HEIGHTS FROM SINGLE OBLIQUE PHOTOS 905
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Ien 1-1--Sle 3-----
0 f-L--J.J Yz<fL)
Y, ........ -e-~
a: Iw>0
FIG. 3b. Alternate technique for determiningdistance, L, on height nomograph-largeimages.
proper depression-angle intersection islocated on the chart and marked. Then,as with vertical photography, the imagelength, or in this case, the difference invertical-displacements, is taken with thedividers and is laid off vertically downward from the marked point. Dividers arenext rotated about the bottom point, andthe distance, L, to the depression-anglecurve is taken for application to the heightscale.
On the height-nomograph-large imagesdepression-angle curves have been plottedfor every degree from 0 to 90. I t can beseen that application to the chart of depression-angle values measured to withina fraction of a degree is less simple and notas accurate as for whole degrees. Therefore,it is suggested that the depression-anglevalue be rounded off to the nearest wholedegree; or if necessary, two measurementsmay be made for the curve either side ofthe correct depression angle, and the finalanswer weighted accordingly.
Although not specifically developed fordetermining nadir-point distances, theheight-nomograph-large images servesan important auxiliary function for thisuse. The nadir-point distance is the distance of a point on the ground from a pointdirectly underneath the camera at thetime of exposure. To fully define theposition of the ground point, it is necessary to know both the location of the nadirand the azimuth of the point from the nadir with respect to some known azimuth.The height-nomograph merely gives thecomponent of the nadir point distanceparallel to the camera axis, or on thephotograph, parallel to the principal line.
I t is determi ned on the chart at the properintersection of (1) vertical-displacementof the image point and (2) photographicdepression-angle by the scales at top andbottom of the chart in fractions and multiples of the altitude. There is an additionalconversion scale at the bottom for determining this component in nautical milesfor an aircraft at 10,000 feet altitude,which may be quickly converted for otheraltitudes by multiplying the ratio of actualaltitude to 10,000 feet.
The component of nadir-point distanceperpendicular to the principal line maybe determined nomographically by previously mentioned methods developed bythe writer. The ground point may then belocated, provided the position of the nadirand the azimuth of the photographic axisis known.
The ove:-rid!ng family of curves in thelower left of the height-nomograph-largeimages is used for measuring negativenadir-point distances and for determiningheights from images lying below the nadiron the photograph. The method used isexactly the same as for points and imageslying above the nadir.
IV. DERIVATION OF MATHEMATICS AND
CONSTRUCTION OF NOMOGRAPHS
Derivation of the mathematical formulas required for construction of the nomographic-charts used in measuring verticalheights from oblique photographs isnearly as simple as the use of the chartsthemselves. The derivation of the formulaon which is based the height-nomographlarge images will be explained first. Allmathematics occurs on the principalplane-the plane which includes the principal ray, or camera axis, and the nadirpoint (refer to Figure 4.).
The five independent parameters necessary to measure a vertical height, h, froma large image are as follows:
Photographic parameters
H-aircraft, or camera, altitude abovebase of object.
j-focal-length of camerat-tilt-angle of camera, the complec
ment of depression-angle, O.
Image parameters
Yl-vertical-displacement to bottom ofimage.
906 PHOTOGRAMMETRIC ENGINEERING
FIG. 4. Side view of oblique photo geometryfor measuring vertical heights.
where k is any point. Therefore, combining (3) and (4) and using fl's to top andbottom of image
This, very simply, is the formula utilizedin the measurement of vertical heightsfrom large images.
The height-factor chart for measuringheights from small images is based on aformula equivalent to (5) but derived by anindependent method utilizing a helpingplane through the base of the image perpendicular to the camera axis. The formula is:
Y2-vertical-displacement to top of image.
Dependent parameters used in the derivation are D 1 and D 2 , the nadir-pointdistances (on principal plane) for imagepoints representing the bottom and topof the object, and fll and fl2, the interiorcamera-angles for these two points. Thenadir-point distances aid in the derivationin the same manner as with the corresponding derivation for vertical photography.
From the diagram (Figure 4) it is seenby similar triangles that
11 D2 -D1
H=iJ-;-'
---0.
and therefore,
11= H (1- Dl).D2
Also it is seen that
Dk = H tan (t + (3k),
[tan (t +fJ1)]
1I=H 1 .tan (t + fJ2)
I 0.-0'=1
(3)
(4)
(5)
H cos fJl cos fJ211 = - I. ' (6)
j cos (t + fJ,) sin (t + (32)
where tv = Y2 - Yl, the component of theimage parallel to the principal line of thephotograph. Rather than present here thederivation for (6) it is simpler to merelyshow its equivalence to formula (5).
From (5), substituting sine and cosinevalues for the tangent, and simplifying:
r cos (t + (3,) sin (t + 82) lL' - sin (t + fJ,) cos (t + (32) J
1I=H .cos (t + (3,) sin (t + (32)
Using sum of angles reductions for thenumerator, multiplying, and gatheringterms:
. [COS #1 sin fJ2 - co, (J, si" ,'"1 , 11I=H ----------- .
cos (t + (3,) sin (t +#,)
Again changing the form of the numerator:
(tan (32 - tan fJ,) cos {3, cos tJ211 = H ------------ .
cos (t + (3,) sin (t + /3,)
From Figure 4 it may be seen that,
Y2 - y, I.tan {32 - tan {3, = -7- = 7 .
Therefore, substituting:
I. cos fJ1 cos fJ211 = H - ---------- .
j cos (t + fJ,) (sin t + fJ2)
This formula is identical to (6), thus proving the equivalence of (5) and (6).
The difference in the methods for measuring heights from long and sh~rt imagesis provided by a simplifying assumptionmade with formula (6) which cannot bemade with (5). This assumption is that asthe image becomes smaller and approacheszero length, fll and f3 2 approach a commonvalue flo. Therefore, for images of zerolength the common value f30 may be substituted in (6) to give
H cos2fJo11=-1. . (7)
j cos (t + flo) sin (t + fJo)
Substitution of the common value f30 in(5) gives the result, h = 0, but in formula(7), h = 0, only when tv = 0. Therefore, (7)may be used when tv ~ 0, by assuming thevalue of flo to occupy a point midwaybetween f31 and fl" i.e., the image midpoint. Detailed studies have not been performed as to the error involved in thisassumption, but preliminary calculationsshow that the error is very small whenmeasuring heights of less than about one
MEASURING VERTICAL HEIGHTS FROM SINGLE OBLIQUE PHOTOS 907
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(13)
are plotted for whole number values of tfrom 0° to 90° corresponding on the chartto depression-angle value of 90° to 0°.The yd! values are plotted directly; mkvalues, logarithmically.
The mk values plotted on the logarithmicscale act similarly to a slide rule. A givendistance along the scale represents thedifference in logarithms of the numbers atthe ends of the given distance. The number, or antilogarithm, corresponding to thegiven distance is the quotient of the twonumbers. Thus, the distance, L, in Figure3 actually represents the quotient:
1It1 tan (t + (31)
1It2 tan (t + (3,)
This distance applied to the proper scalewould give the value of (13) which, whensubtracted from unity (10), is the properaltitude fraction. However, by going onestep further, the values on the scale at thebottom of the height nomograph-largeimages have already been subtracted fromunity. Application of the distance L to thescale thus solves (10) completely, givingthe height-measurement directly as afraction of the altitude.
The mathematics for the component ofthe nadir-point distance lying on theprincipal-plane has already been derivedformula (4). And by comparing (4) and(11), the latter being the equation for thecurves on the height-nomograph-largeimages, it is seen that these curves givethe direct solution for nadir-point distance
tan (t+{3I)altitude fraction = 1 - . (10)
tan (t + (3,)
The formula for the family of curves onthe nomograph is simply
tenth the aircraft altitude, increasing,however, with increasing nadir-point distance of the height. Image parameters formeasuring heights from small images are,therefore, ly and Yo, rather than YI and Y2.
Formula (7), it may be seen, is equivalent to formula (2) where:
cos2 {3oheight factor = --,-- (8)
cos (t + (3o) sin (t + (3o)
From Figure 4,
{3o = tan- 1 ~f
The above two formulas, (8) and (9), arethe equations for the family of curves onthe height-factor chart. The vertical axisof the chart is the Yo/! axis; the horizontalaxis is the height-factor axis. The lattervalues are plotted logarithmically; theformer, directly for direct application ofvertical-displacement values with rule ordividers. Curves are plotted for values oft from 0° to 90°.
After construction of the chart was begun, it was discovered that the curves weremirror images about the height-factor axis,and that curves for values of depressionangle above the axis corresponded to thosefor tilt-angle below the axis. Therefore, thebottom half of the nomograph was eliminated, the top half sufficing for both.
Explanation of the construction of theheight-nomograph-large-images has beendeferred to the last to show the readertricks which are possible with nomographs,and which although at first appearingcomplicated, are actually quite elementary. The following solution, it may beadded, was not arrived at initially by deduction, but through trial and error.
The formula on which the heightnomograph-large images is based is (5)where,
1/tk = tan (±{3k), (11)
where
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k being any point. The vertical axis, aswith the height-factor chart, is the Yk/!axis; the horizontal, the mk axis. Curves
FIG. 5. Vertical displacements, Yl and Yz, arethe same for all boards in fence, proving validity of methods for performing measurements ofvertical heights from images located anywhereon the photograph.
908 PHOTOGRAMMETRIC ENGINEERING
in one operation. Thus, it was only necessary to add the proper scales for nadirpoint distance along the horizontal axis.
So far in this discussion, the mathematics has not departed from the principal plane, and explanations for the nomographs have blandly implied that themethod is valid for any point on theoblique photograph. A non-rigorous, butlucid, proof of the validity of the mathematics for any height'image on the photograph is here presented.
I n Figure 5 is shown a schematic diagram of an oblique photograph containinga board fence parallel to the x-axis of thephotograph. All the boards in the fence arevertical and of equal height, and on thephotograph they all project identical vertical displacements. Therefore, the fore-
SPECIAL CAMERAS ARE BEING TESTED
Special automatic cameras which make a record of a radar screen are being experimentallyput into operation by the Civil Aeronautics Administration to evaluate the Air Traffic ControlRadar Beacon System.
Use of the special cameras manufactured byGordon Enterprises, North Hollywood (Calif.)at C.A.A. radar sites in New York, Chicago,Washington and Norfolk has been announcedby D. M. Stuart, Director of the C.A.A. Technical Development Center, Indianapolis.
The move may presage use of such camerasto record almost all aircraft landings and takeoffs. Extension to marine use might in the future provide definite records of responsibilityfor such disasters as the Andrea Doria-Stock-
going mathematics, which was derivedonly for the center board, i.e., the boardlying on the principal line, will give anidentical height for all the boards, provided the vertical displacements, or component of the image parallel to the y-axis,is applied to the formulas or nomographs.
V. ACKNOWLEDGMENTS
The writer would like to acknowledgethe aid and encouragement given by MajorByron L. Schatzley, USAF, who was responsible for the establishment and supervision of this project at Wright Air Development Center, Wright Patterson AirForce Base, Ohio, and would also like tothank personnel of the Computer Laboratory, WADC, for computation of points forthe nomographs.
holm collision which occurred off Nantucket onJuly 25.
The special cameras, made to C.A.A. specifications, will record information on plan-positionradar indicators during special flight tests orduring periods in which it is desired to recordactual movement of air traffic along airways orin traffic patterns near airports, Stuart said.
"\iVhen the information is recorded on film, itis possible to make detailed studies after thetest period has been completed to be sure thatno significant data has been overlooked,"Stuart said.
The Gordon Enterprises cameras, using35 mm film in 100 foot magazines, are operatedautomatically by the sweep of the radar antenna. I n addition to photographing the radarscreen, the camera simultaneously photographs a clock face, so that the exact time isincluded as part of each picture. A data card focpertinent information is also photographed.
One feature of the cameras is that they canconstantly photograph the radar screen unattended by an operator. Except for changing offilm and winding of the timepiece, the cameraswill perform in excess of 500 hours without repair or adjustment.