Errors In Photogrammetry - ASPRS
PROF.JAMESP. SCHERZUniversity of WisconsinMadison, Wis. 53706Errors in PhotogrammetryIt is important that one understand the errors, their sources,characteristics, and relative magnitudes in order to applyphotogrammetric materials effectively.BASIC RELATIONSHIPS IN ANALYSISeffectively withphotogrammetric or remote sensing images, a basic understanding is necessary ofsources of photogrammetric errors and theirrelative and approximate magnitudes. Oftenthe subject of errors is covered i n suchmathematical detail that it leaves the user soconfused that he simply overlooks these errors entirely. What is really needed is a simple approach for analyzing errors and understanding their effects for (1) applicationsTOTEACHORWORKIn dealing with the subject of errors of anykind, it is important to realize that there aredifferent rules governing the propagation oferrors depending on whether the processused is addition and subtraction, or is multiplication and division.SIGNIFICANT FIGURES I N ADDITION A N D SUBTRACTIONIf one adds the number 1.2345 (five significant figures) to the number 123.4 (foursignificant figures), the answer is 124.6, accu-ABSTRACT:TO teach and work effectively with photogrammetry, oneshould have a basic understanding of the sources and relative magnitude of errors inherent in aerial photographs. Exact calculus approaches are often so complicated that they cause one to want toforget about errors entirely and pretend they do not exist. The approach described herein equates all source error effects t o a percentage, and as long as the mathematical manipulations are multiplication and division, the same percentages can be applied to the finalanswer to ascertain its probable error. The method described provides estimates of errors identical t o those obtained using calculus,b u t the described method is much easier. The method provides students and users with a ready and quick method for analyzing errorsand of obtaining a feeling for the relative magnitudes of errors i nphotogrammetry work.where aerial photos are used as map substitutes, (2) where photos are used in conjunction with stereoscopes and parallax bars, or(3)where photos are used with stereo plottersor analytical photogrammetry. This paperpresents a simple, comprehensive, and practical method ofthese errorsand ascertaining their approximate magnitudes. The techniques presented here havebeen developed by the author in six years ofteaching photogrammetry and have provenvery effective in analyzing and understanding errors and dealing with them in variousaspects of photogrammetry and remote sensing work.rate only one place to the right of the decimalpoint because the second number added isno more accurate than that.This same approach is used if one subtractsrather than adds.SIGNIFICANT FIGURES I N MULTIPLICATION A N D DIVI-SIONWith multiplication, if one takes t h enumber 1.234 (four significant figures) andmultiplys it by the number 1.23 (three significant figures), t h e answer 1.51782 isrounded off to 1.52 (three significant figures)because the product can have no more significant figures than the least in the number493
494PHOTOGRAMMETRIC ENGINEERING,1974being multiplied. The same general ap-CALCULUS METHOD OF DETERMINING ERROR PROPA-proach also holds for division. The erroranalysis for photogrammetric work hereindescribed is derived from the rules governing propagation of errors for multiplicationand division. This is important to remembereven though almost all photogrammetricequations to be analyzed involve multiplication or division in their solution.GATIONPhotogrammetric equations are based onthe fact that all light rays pass through thefront and rear nodal points of the camera lenswith direction unchanged. If one neglects thethickness of the lens, the geometry becomesthat of the pinhole camera. (See Figure 1).In Figure 1, by similar trianglesZ1X flI orZ (fIZ)X. Let us assume that the followingvalues and probable errors in their measurement are known (in inch, ft foot):f 10.000 inX 1000 ftI 5.00 in. 0.001 in%%1 ft0.01 in.The problem presented here is that Z is to becomputed along with the error in Z due to theerrors in f, X, and I. Two approaches will bepresented, the first is an exact calculus approach. The second is herein called theGoverning Percentage Method which is usedin the rest of the paper for analyzing morecomplex error situations.From Figure 1and the given values off, x,and I,Z 2000 ft some error.One can calculate the total probable value ofthis error by taking derivatives of Z with respect tof, I, andX and combining the effects.The error in Z due to f by derivatives is:dZ (XII) df (1000 ft15 in) x (0.001 in)dZW % 0.2 ftThe error in Z due to X is:dZ (flZ) dX (10 in15 in) x 1ftd& 2 f tThe error in Z due to I is derived from:Z (f/I)X I-1fX and (dZldI) (-fXII2)or:dZx 1000ft15 in x5 in) x .01 inch (-fX/12)d l (-10 inAccording to the theory of probability, thecombination of these various errors resultsin:Total Probable Error fl0.2)2 (2)2 (-4)2] 4.4 f torapproximately 4 ft.As any error can be either positive or negative, accumulative or compensating, one cansay that the total resulting error in Z due tof,X, and I may be as high as 0.2 2 4 6.2 ft.Assuming by some coincidence that some ofthe errors inf, X, and I really approach zero,then the total error on Z will perhaps lie between 0 to 6 ft; perhaps about 4 feet.DETERMINING ERROR PROPAGATION BY GOVERNINGPERCENTAGE METHODFIG.1.Simple photogrammetric relationship; thepinhole camera geometry. ZIX flI or Z fill.Legend: f, focal length; I, image; X, object; Z,flying height above terrain.Rather than using calculus, a simpler approach using percentages of errors will givethe same results. This method is hereincalled the Governing Percentage Method, andis extensively elaborated on later in this paper.It is possible to express the errors i n f , X,and I in terms of percentages of the numbersthemselves:
ERRORS I N PHOTOGRAMMETRYf 10.000 in.; error inf 0.001;percent error inf 0.001 inl10.000 in 0.01%X 1000 ft; error inX 1 ft; percent error inI 5.00 in; error in I 0.01 in;percent error inIf we sum the percentages of .Ol%,0.1%and 0.2%caused byf, X, andl, we have .31%.This same percent error will carry through tothe calculated value of Z. Assuming accumulating error, the error in Z is calculatedas follows:Error in Z 31% of 2000 ft 6.2 ftwhich is the same as the error obtained bycalculus. By analyzing the percentages of errors on the input figures we can see that thegoverning or largest percentage is 0.2%andthe final propagated error it causes is approximately 0.2% x Z, or 4 ft. This same governing percentage approach can be applied tovarious and complex photogrammetric calculations, involving multiplication and division.Whether or not a particular error is of concern depends on the photogrammetric technique being used. Many aerial photos areused simply as map substitutes. Where this isdone, certain errors should be understood.AERIAL PHOTOS USED AS MAP SUBSTITUTESIn Figure 1, we assume that the objectx ison flat ground and that there is no relief displacement, i.e., difference in scale over thephoto caused by difference in distance between the ground and the photo. We assumethat there is no relief-displacement error.Also, in Figure 1, we assume that the film isparallel to the flat ground, or that I is parallelto X. We assume that there are no tilt errors.We also assume that the light rays passstraight through the lens-directionunaltered. We assume, therefore, no lens distortion errors.We assume that the distance I measured onthe photo is the true distance projected by thecamera. We, therefore, assume no error dueto the film or paper-print shrinkage. In someinstances, we can enlarge images by projectors which take care of film shrinkage. Wethen assume that the shrinkage is uniformacross the photo. We assume that there is nodifferential film shrinkage.495Likewise, we assume that the plane uponwhich the image was projected was truly flat,that it had no bumps on it caused by dustbehind the film or caused by uneven thickness of the film emulsion. w e , therefore, assume no focal-plane flatness errors.One other important error in any photogrammetric work is the error caused in measurement. For analysis of Figure 1, we alreadystated that the error in measurement of I was .O1 in, so the measurement error has already been accounted for in this example.Other errors such as atmospheric refractionmay be present, but are usually insignificantcompared to the errors already listed.Generally speaking, if a photo is used as amap substitute, we assume that the followingerrors are zero:Relief displacement.Tilt.Paper or film shrinkage.Differential shrinkage.Lens distortion.Focal-plane flatness.PHOTO PRINTS USED I N CONJUNCTION WITH STEREO-SCOPES AND PARALLAX BARSIf one views two overlapping photos sideby side and measures parallax with a parallaxbar, all of the errors listed for the single photoexist and may be doubled except the error ofrelief displacement which is really the parallax being measured*. Several very significantadditional errors must also be considered inthis situation. In Figure 2 by similar trianglesZA/B f/Pa where ZA,f, and B are as shown inFigure 2, and Pa is the parallax of point a. Thisis the equation that is commonly used to calculate the difference in elevation betweenthe aircraft and any point on the ground. Ifwetake the parallax of the top and bottom of thetree in Figure 2 we have:Pa fBEAand PC fBIZcThe difference in parallax between the topand bottom of the tree is:z,-z* also equals dh and fromPa .PEA,ZA .PIPa*Accordingto the Theory of Probability ifwe use2 photos the resulting error in the combination canbe expected most probably to have a magnitude ofd2 1.414times the errors in a single photo. It isalso conceivable to have a maximum error 2 timesthe magnitude of the errors of a single photo.
496PHOTOGRAMMETRIC ENGINEERING,1974FIG.2. Geometry for using overlapping photos and a parallax bar.Therefore,dh (dpZJfB) (WPJ dpZJP,.Ifdh is small compared toZ,, Po approachesb, the photo base, anddh dpZ,lb.This is the equation that is often used to relate difference in parallax to difference inelevation.In these equations we assume a commonflying height for both photos, but there maybe up to 100 feet difference in flying height.Also, significant measuring errors are usuallycaused by transferring principal points andcalculating b or B. In summary, for work withparallax bars, we have all of the errors associated with a photograph used as a mapsubstitute except the relief displacementerror; this is reflected in the parallax which ismeasured. The remaining error effects are allincreased and, possibly, doubled becausetwo photographs are used. Additional errorsare due to unequal flying heights and errorsi n measuring and transferring principalpoints.ter is of no concern. The errors that still existwith the stereoplotter are errors caused bydifferential film and plate shrinkage, errors offocal-plane flatness and any stereoplottermeasuring errors.The same error analysis applies for analytical photogrammetry as well as for stereoplotting. With special cameras, it is possible withreseau grids etched on the focal plane tohandle the errors due to differential filmshrinkage. Very special cameras using glassplates rather than film can almost eliminateboth the effects of differential film shrinkageand focal-plane flatness. Such cameras arevery special indeed and are not normallyused for operational photogrammetry. Foroperational photogrammetry, be it stereoplotting or analytical work, the limiting errorswill normally be (1)errors due to differentialfilm shrinkage, (2) errors due to focal-planeflatness, and (3) errors due to measurements.Any other less-precise photogrammetricoperation will be limited by some combination ofthe errors previously listed. Followingis a detailed investigation of each errorsource and an attempt to ascertain the magnitude of each.ERRORS I N STEREGPLOTTING AND ANALYTICALPHOTOGRAMMETRYWith stereoplotting we measure the reliefdisplacement as with the parallax bar. However, each projector is also adjusted to takeout the tilt effects and, therefore, the errorsdue to tilt. The projectors are adjusted to takeout any errors caused by unequal flyingheights. If we use the proper projection lensor projection techniques, we can handle thelens-distortion errors. We adjust the projec-.tors until the projected image matches theplotted ground control distances so that uniform film or plate shrinkage is of no concern.The errors that still exist with the stereoplot-I n Figure 1, the scale of the photograph isZIX which is also equal to flZ or (focallength)l(flying height). Of course, as Zchanges, so does the scale. As a point ismoved up or down in elevation, its image isdisplaced on the photograph.In Figure 3 the general equation thatexpresses this displacement is:drlr dzlz*.*If this relationship is not readily apparent fromFigure 3, any good photogrammetric text will showits derivation.
497ERRORS I N PHOTOGRAMMETRYPhotof 1 0 . 0 0 inchesa p a 4 . 0 0 inches b 4 . 0 0 i n c h e sFIG.5. Calculating the ground distance AB whereu p , pb, f, a n d 2 are given and the photo is tilted 3".IAPFIG.3. Displacements on an aerial photo due todifferences in elevation.The distance r is really a photo representation of the ground distance AP. As point A ismoved upward by dz, a' is moved on thephoto by the distance dr. The resulting errorin scale of the line r is dr; the percentageerror in line r is drlr, which is equal to dzlz.Therefore, the percent error due to relief isequal to dzlz. The absolute magnitude of thiserror varies depending on the ruggedness ofthe terrain and flying height. Ifz is 1000 ft anddz is 100 ft, the percent error due to uneventerrain is 100/1000 10%.The most significant error in a single photoused as a map substitute is the error causedby uneven terrain. The next most significant77APPhotoBFIG. . Calculating the ground distanceAB whereu p , pb, f, and Z are given and the photo is assumed to be vertical.error is the error caused by tilt. For the photosof Figures 1and 3, the assumption is that thephotographs are vertical. Generally speaking, due to air turbulence, etc., such photosare likely to have up to 3" tilt. Following is ananalysis to arrive at the magnitudes ofcomputational error which might be caused by 3"tilt.Let us assume in Figure 4 that the imageson the photo are used to calculate the grounddistance AB, first we assume a vertical photo.From Figure 4,tan a 4.00110.00; a 21'48' andAP PB tan a x 10,000 ft 4000 ftAB AP PB 8000 ft.Now let us assume that there was really 3"tilt in the photo at the time of exposure asshown in Figure 5. From Figure 5,a 21'8' (as in Figure 4) andB a-3" 18'48'A a 3O 24'48'AN 10,000 x tan B 3404 ftNB 10,000 x tan A 4620 ftAB AN NB 8024 ft.If we assume a vertical photo and it was, infact, tilted 3' as shown, there is an error of8024-8000 24 ft in the calculated length ofAB due to the 3" tilt. The relative error due tothe 3" tilt in this instance is 24 ftl8000 ft or0.3%. Other methods of analyzing the effectsof the 3"tilt produce errors of the same general magnitude. As a general rule, the errorsdue to normal tilt can be expected to be between 0 and 0.3%.ERRORSDUETO SHRINKAGETo calculate the errors due to paper printshrinkage, one has to measure the distance
498PHOTOGRAMMETRIC ENGINEERING, 1974I(- d distortionExample :fdm,,- 152 mm0.15 mmFIG.6. Effects of lens distortion.between fiducial marks on a finished printand compare that to the distance on the negative. The resulting difference over the average distance is the relative error. Goodquality papers will produce shrinkage errorsfrom 0 to 0.2%, whereas poorer-quality papers will produce shrinkage errors as high as0.5%.To obtain the film shrinkage, one comparesthe negative to the camera opening. Theshrinkage errors of most aerial films can beexpected to be less than 0.1%. Differentialfilm shrinkage or non-uniformity of thisshrinkage in any one direction will perhapsbe about 1/10to 11100ofthis or about 0.005%.Figure 6 shows a sketch of lens distortionand a typical distortion curve for an oldercamera lens. In this example, if a 4S0, thedistance r is equal to f 152 mm. The lensdistortion at this angle is dm,, 0.15 mm. Therelative error in the distance r due to the lenserror is d,,,h 0.15 mml152 mm 0.1%. Asa general rule, the errors on a photo due tolens distortion will be less than 0.1% andconsiderably less on higher-quality cameras.In photogrammetric calculations, we assume a flat focal plane. Aerial cameras havevacuum systems to flatten out the film for thispurpose. However, pieces of dust may catchbetween the film and the vacuum platen, orthe thickness of the film itself may vary. Figure 7 shows a sketch of such errors.In Figure 7, the ray striking a truly flat focalplane would be imaged at a. However, becau.;e of the deviation from the flat focalplane due to distance t, the ray is really imaged at b. The error on the flattened image isthe distanced. The relative error in distance ris dh. Typical values fort are about 10 pm or0.01 mm. For a 4S0,d t. The value ofr fora normal camera will be about 150 mm. Therelative error then is 0.01 mmll50 mm or lessthan 0.01%.The magnitude of errors caused by lack offocal plane flatness will be in the magnitudeof less than 0.01%.Measuring errors are always present anddepend entirely on the technique used. Thepercentage of this error is, of course, calculated by forming a ratio of the probable errorAd and the distance measured d,Probable errorPercent errorin measurement a ddue to measurementmeasureddlengthCOMBINEDEFFECTSOF ERRORSFrom the foregoing analysis, the magnitudes of the different errors can be summarized as follows:USING A SINGLE PHOTO AS A MAP SUBSTITUTETable 1summarizes the errors that exist if asingle photo is used as a map substitute. Assuming flat terrain one can see from Table 1that the expected error will still be up toabout 0.5% which corresponds to a precisionof 0.51100 11200.It is clear that unless one corrects for papershrinkage and tilt effects, there is no use worrying about lens distortion, film shrinkage, orfocal-plane flatness in this example.Assuming terrain differences of 400 ft and aflying height above average terrain of 4,0001IIFocal PlaneTI -IFIG.7. Errors due to lack of focal-plane flatness.1
ERRORS I N PHOTOGRAMMETRYUneven terrain (&/z) (varies, depends on terrain)Tilt0 to 0.3%Paper Shrinkage0 to 0.5%Film Shrinkage0 to 0.1%Differential Film Shrinkage0 to 0.005%Lens Distortion0 to 0.1% %Lack of Focal Plane FlatnessO to 0.01%(varies with technique)Measuring (Adld)ft, the terrain error becomes 40014000 lo%,and terrain is clearly the governing error inthis application and will limit the expectedprecision.In this example, if one scales distances directly from the photo used as a map, the errorsare about 10% which gives a precision of101100 111 0 . Calculated ground distancesfrom such a photo can have errors as large as10% of the calculated length.Terrain effects are almost always governing in such instances and there is little pointin worrying about tilt, lens distortion, or filmshrinkage if there are significant differencesin terrain.PHOTO PRINTS USED WITH STEREOSCOPES A N D PARALLAX BARSIf two overlapping photos are fasteneddown side by side and used with a parallaxbar to obtain elevations, the uneven terrainerror ofTable 1 drops out because this is whatis being measured.All the other errors still exist and are increased and may be doubled because of thetwo photographs. Also, significant errors areintroduced due to transfer ofprincipal points,due to measuring, and due to unequal flyingheights, since the basic equations for suchwork assume common flying heights.In any event, assuming perfec
Errors in Photogrammetry It is important that one understand the errors, their sources, characteristics, and relative magnitudes in order to apply photogrammetric materials effectively. TO TEACH OR WORK effectively with photogrammetric or remote sensing im- ages, a basic understanding is necessary of sources of photogrammetric errors and their
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