- Email: [email protected]
Basic investigations on behalf of the application of the theory of errors in photogrammetry
49 Basic investigations on behalf of the application of the Theory of Errors in Photogranunetry by A. Ji van der Weele, Delft, Holland.
I. Introduction. The application of the theory of errors to photogrammetry has not yet reached the same intensity as in the other branches of geodesy n o t w i t h s t a n d i n g the f a c t t h a t there is g r e a t need for the results of such an application. The f i r s t purpose of such a study is to find a numerical expression for the accuracy of a certain quantity. This expression may then be used in d i f f e r e n t ways e.g.: a) for a comparison of d i f f e r e n t methods (radial-triangulation versus spatial-triangu: lation, etc.) ; b) for an examination of the influence of d i f f e r e n t circumstances (scale of~ the photographs, type of camera, etc.) ; c) for an examination of the lines along which an improvement of instruments or methods can be obtained in the most efficient way; d) for a prediction of the accuracy t h a t might be expected in later applications; e) etc. etc. These considerations are naturally of g r e a t importance for the development of photogrammetry, and t h e i r practical value is increased if they are combined with economical studies. A t present, what has been done in this field in the Netherlands and abroad is not much more t h a n a modest beginning. There are several reasons for this fact: 1. the p r o b l e m s are very complicated and therefore require an elaborate quantity of work; 2. there are many d i f f e r e n t methods and instruments, s o - t h a t each theoretical investigation is only applicable in a limited number of cases. An example of this fact is the excellent work of Prof. W. K. Bachmann [(1)]. In this book the author applies the theory of errors to a method of relative orientation t h a t is based on theoretical considerations but which will probably be rarely used in practice. The resulting tables and formulae have therefore a v e r y limited practical value; 3. as f a r as I know, none of the authors of the present time have taken into account the real properties of the observations. The consequence is t h a t the results of the theoretical considerations do not correspond with those of practical experience. The weakest point of all theoretical treatise~ in this field lies in t h e f a c t t h a t until now no one has taken the trouble to prove t h a t his hypotheses and idealisations about the character of the observations sufficiently approach reality. There are three kinds of problems to be solved for the application of the theory of errors : A. in the f i r s t place there is the need for computation-rules of general validity. This need is completely filled by the publications of Prof. J. M. Tienstra [(2)]. Based on his theory it is 0nly a question of computation-technics to be able to execute the required work in the most efficient way. In this respect it is possible t h a t in future the electronic computation machines and such-like may render useful service; B. in the second place it is necessary to obtain a sufficiently accurate mathematical d e s c r i p t i o n of the problem a t hand. This may lead to considerable work but there is no principal difficulty in a precise and careful analysis of all observations and manipulations executed a t a given instrument or for a given method;
4,
50 C. at the base of the problem is finally the tensor of cofactors t h a t determines the values of the mean errors and correlations of t h e observations. 1) To be able to set up this tensor it is necessary to go back to experiments. Series of observations must be produced, varying the circumstances as f a r as these may have influence in practice, to conclude with a statistical examination for the determination of the cofactors. This phase of the problem offers a lot of practical difficulties. A f i r s t examination of the photogrammetric problems shows t h a t we may separate the influences o n ' t h e observational accuracy into three groups: (1) the influence of the properties of the photograph; (2) ,, ,, ,, . . . . of the i n s t r u m e n t ; (3) ,. . . . . . . . . of the observer. In which order of succession these groups are i m p o r t a n t cannot be said a priori. It is, however, certainly true t h a t a complete investigation of these influences requires an immense quantity of observational material t h a t can, most probably, only be obtained by a close international cooperation. The following example may illustrate t h i s - f a c t : a relative orientation of a pair of photographs is based on the measurement of vertical-parallaxes in a n u m b e r of (generally six) points of the image. If we assume t h a t the influences of the instrument and the observer (groups 2 and 3) are completely known, there remains only to be determined the influence of the properties of the photograph on the value of the cofactors. The properties of the photographic image are primarily dependent on the type of camera. It is therefore necessary to investigate the influence of differences in the type of the camera on the photographic image. Even if the camera and its objective have been completely analysed there are still a g r e a t many variables, like: 1) atmospheric conditions ( c o n t r a s t ) ; 2) flying height (movement Of i m a g e ) ; 3) character of the t e r r a i n (definition of details); 4) airplane (vibrations) ; 5) emulsion-base (differential shrinkage) ; 6 ) etc. It will be clear that, notwithstanding the rigorous exclusion of unknown factors, there still remains a large uncertainty. F o r this reason there is little basis for comparison of the results published by various authors unless a very careful and complete description of all circumstances is added. Such a description has still little value if the influence of variations of the circumstances is not known. The foregoing may have made clear why until now the application of the theory of errors to p h o t o g r a m m e t r y has so little practical value. Considering the quantity of factors t h a t influence the accuracy, one might doubt if it will ever come to a theoretically correct application. There seem, however, to be several reasons for which one need not be too pessimistic. It may be expected t h a t in future the existing number of variants in methods and instruments will be limited. On the other hand there exists a tendency in the optical and mechanical performance of cameras and instruments t h a t reduces a number of sources of errors to a level where they may be neglected. Some examples are: 1. the new wide-angle lens of Wild, the Aviogon, has the p r o p e r t y that the diminishing of light at the margin of the photograph is much smaller than with all foregoing types. In addition the sharpness of the image is much more uniform over the whole field; 1) See Appendix.
51 2. the mechanical precision of the universal instruments is gradually increasing so t h a t the number of sources of errors t h a t have an appreciable influence on the observations is getting smaller. That nevertheless an enormous work has to be undertaken to obtain a well-based tensor of cofactors for d i f f e r e n t methods and instruments will be demonstrated in the following chapters. They deal with an investigation of some detailed influences, executed during the years of German occupation with the welcome and v a l u e d financial support of the Delft University F~undation. II. The preliminary purpose Of the investigation was a comparison of the observational-system Of the Wild-A 5 and the Zeiss-Stereoplanigraph. The basic principle of these systems is illustrated in the figures 1 and 2. In the A 5 observation takes place perpendi-
Fig. 1
Fig. 2
cular to the plane of the photograph, with a telescope of a complicated construction. This telescope is sketched in a much simplified-form in fig. 1 since only the p a r t closest to the image is characteristic for its functioning. The stereoplanigraph is based on the Porro-Koppe principle (fig. 2). F u r t h e r characteristics are: a) the auxiliary optical system consists of a positive and a negative lens of equal focal length which are responsible for the formation of a sharp image in a plane through the measuring m a r k ; b) the rotating m i r r o r is automatically directed so t h a t the reflected rays have a fixed horizontal direction to the telescope. If we take the simple case of a true vertical photograph of an object consisting of a rectangular grid, the photograph will contain a reduced but also rectangular image of this grid. I t will be obvious t h a t the image which is observed through the telescope of the A 5 will also be a rectangular and non-distorted grid and t h a t this image will be the same over the whole photograph (The lens-distortion in the observational system is supposed to be negligible). I t may be expected that, in this respect and excluding all other influences, the observational accuracy will be equal for each position of the observed point on the plate.
52 The observational system of the stereoplanigraph has other properties, as will be shown u s i n g fig. 3. The grid on the plate will be projected undistorted on the horizontal plane V. The optical rayS, s t a r t i n g from a n y point (e.g. the point p) of the plate will be t r a n s f o r m e d to a parallel bundle by the lens O. The auxilia r y system (V.S.) produees an optical image in the plane V" which is perpendicular to the line OP (which line coincides with the optical axis of the aux. system). Hence, the optical image of q will fall in Q'. The distance p q will therefore be observed as PQJ but will be measu~'ed equal to PQ. The relation between the observed and the measured distances is of g r e a t importance since the measured values (coordinates of parallaxes) z(.2 are used in practice. This relation can be developed in a formula in which the elements denoting the position of the observed point e n t e r as parameters. Fig. 4 gives the elements t h a t must be considered in developing this formula. The plane V p is the plane of the optical image when
//
Fig. 3 the point P is observed. The position of P is determined bij the angles i i I Q and o. In the plane V' a coordinate-system S-~] I is chosen, so t h a t the Saxis is parallel to the eye-base. The direction of this ~-axis can be determined by considerFig. 4 ing t h a t the optical system of the stereoplanigraph includes a dove-prism which is automatically guided so t h a t the epipolar plane through P is seen as a line parallel to the eye-base. The S-axis is therefore chosen along the intersection-line of the plane V" with the plane t h r o u g h S, T and P, the base-line in fig. 4 being supposed to be parallel to the X-axis. The ~/-axis is perpendicular to the ~-axis. The above-mentioned formula is: h ( X c o s ~ - - h s i n o c o s Q - - Y sinQ s i n v ) cos~ cos~ (h cosQ cos ~ ÷ Y cos~ sinQ + X s i n o ) h (Y c o s Q - - h s i n 6 ) ~/ = cos~ cos~ (h cos~ cos ~ ÷ Y cos~ s i n ~ % X s i n ~ )
i I
(1)
53 f r o m w h i c h we derive b y d i f f e r e n t i a t i o n : dX=tg~ dY -
tga
1 d * / + cos~o d~
(2}
1 cos Q d~/
or d $ : c o s ~ d X ~ d,/ = cos~ dY
sin~ sine dY
(3)
I f we s u p p o s e t h a t t h e o b s e r v a t i o n s of t h e c o o r d i n a t e s ~ a n d ~7 a r e i n d e p e n d e n t a n d e q u a l l y a c c u r a t e so t h a t we m a y p u t : m~ :m~!q :/~ a n d m~TI : 0 we c a n derive f r o m t h e above f o r m u l a " (2) t h a t t h e a c c u r a c y of t h e m e a s u r e d v a l u e s f o r X a n d Y will be e x p r e s s e d b y :
(
? me x
=
• "
1)
tg2o t g 2 Q + c o s 2 ~
#2
1
tg~tg # T h e s e f o r m u l a e a r e valid f o r t h e s t e r e o p l a n i g r a p h in t h e case w h e r e m o n o c u l a r m e a s u r e m e n t s a r e execut e d f o r t h e d e t e r m i n a t i o n of coordin a t e s o f p o i n t s in a h o r i z o n t a l plane• For the A5 zrv2 x ~
m'~ v ~
we m a y e x p e c t to f i n d re2; ~r~xy = O.
In the case that stereoscopic m e a s u r e m e n t s a r e e x e c u t e d we h a v e to deal w i t h two i m a g e s both of w h i c h will i n f l u e n c e t h e a c c u r a c y of t h e m e a s u r e m e n t s . When using the A 5 the c o o p e r a t i o n of t h e t w o i m a g e s will be a s i m p l e one s i n c e t h e y are equal for flat terrain. With the stereoplanigraph m a t t e r s a r e m u c h m o r e complicated. T h e p l a n e s V ' (fig. 4) have different positions for left a n d r i g h t c a m e r a so t h a t / ,)e t h e p r o j e c t i o n s of t h e h o r i z o n t a l p l a n e on t h e inclined p l a n e s will h a v e d i f f e r e n t f o r m s • I t is obvious t h a t t h e mon o c u l a r o b s e r v e d p r o j e c t i o n of a g r i d m u s t h a v e t h e s h a p e of Fig. 6 two i n t e r s e c t i n g cones o f r a y s . Indeed, t h e lines in t h e X - Y p l a n e p a r a l l e l to th~ Y - a x i s will be p r o j e c t e d on t h e V '
54 plane (fig. 4) as lines through the intersection point of V r and the line through S parallel to the Y-axis. The same is true for lines parallel to the X-axis. The image for an arbit r a r i l y chosen point P will therefore have the form given in fig. 5 where the circle represents the limit of the field of view. Fig. 6 g i v e s some examples of the difference between the two images when using stereoscopic ./ 3 vision. It will be interesting to check these results in practice. The following chapters will give an examination in the above-mentioned line for monocular measurements of coordinates in a horizontal plane and for stereoscopic measurements of p a r a l l a x e s .
¢
2
6
8
.9
III. Monocula/r measurement of X and Y coordinates. The measurements reported here were originally ~-----~---'--'-meant for another purpose but they can be used here Fig. 7 too. On a photograph nine points were marked by a small hole according to the scheme of fig. 7. In the stereoplanigraph as well as in the A 5 t h e X and Y coordinates of these points were measured with a constant value of Z, using one projector at a time and assuming t h a t the photograph has a t r u l y vertical axis. Fo~ each point 120 series of 5 observations were made, thus giving 480 superfluous observations. The mean value for X and Y for each series was subtracted from the observed values giving 600 differences from which the mean errors as well as the amount of correlation for X and Y could be computed. Table I gives the result for the same photograph measured in the A 5 and the stereoplanigraph (values in microns). In judging this result, it should be taken into account t h a t the given numbers show the sum of the influence of setting- and reading errors. The counting dials of the A 5 are divided in 0,01 ram, those of the stereoplanigraph in 0,1 mm. F o r the A 5 the two influences cannot be separated since they will have the same influence for each position on the photograph. For the stereoplanigraph a separation is possible according to the following formulae. Assumptions: 1) the mean error of the reading of a coordinate is, for each point, equal to /~1; 2) the readings of X and Y coordinates are not correlated; 3) ~ = m 2 ~ = /~2 and m ~ I = 0. Formula 4 gives in this case: m~=
1 (tg~tg~+~-)~2+/~j2 1
~
= -co~- ~
+ #~
(5)
tgo tg~
~y=
cos~~
The two unknowns /~ and /~ can be found with these formulae from the values of table I. The result is (approximately) ~u~ = 50 micr.• jul~ = 6 0 micr. e
/~ = 0,007 nmL tt~ = 0,008 ram.
55
Table II gives a comparison of the real values and those t h a t are computed with the given values of 14 and /~1" The tables I and II give rise to the following considerations: 1. for the A 5, the correlation between X and Y is indeed very small considering t h a t the given numbers are the 8 q u i r e s of the mean errors. The mean errors themselves have no g r e a t differences (between 7,5 and 9 micron). There are three exceptions namely the points 5, 6 a n d 9. The points 6 and 9 are situated at the same side of the plate and since the values for ~ 2 and ~%~ are too g r e a t for both it m a y be concluded t h a t a mechanical e r r o r of the i n s t r u m e n t causes these discrepancies. In point 5 both m r and mu have g r e a t values. An explanation of this fact can possibly be found in the consideration t h a t point 5 is the only point where the space-rod
•
/ -
\~o
w
"o/
Fig. 8 has a vertical position, so t h a t no pressure is exercised on the base-carriage. Any amount of back-lash can only find expression in the measuring of this point; 2. the results for the stereoplanigraph give the impression that, in addition to the correlation t h a t could be expected on theoretical grounds, there is also another small but nearly constant a m o u n t of correlation. This amount could perhaps be explained by assuming t h a t it is caused by mechanical or optical features of the i n s t r u m e n t (that have not been taken into account) or by the method of observation; 3. the separation of the mean errors by formulae 5 is based on a plausible assumption but a confirmation seems not undesirable. The same computations have therefore been made with a s e t of similar measurements executed (two years earlier) on another photograph. The result is given in table III. It is note worthy t h a t in this case ~ gives the same value as table II but t h a t /~1 is much smaller. An explanation for this fact might be t h a t the observations given in table II were made in J a n u a r y 1944. A t t h a t time the heating of the rooms was very bad so t h a t great differences in t e m p e r a t u r e and humidity occurred. It is not impossible t h a t this fact has had an u n f a v o u r a b l e influence o~ the constancy of the mechanical t r a n s p o r t a t i o n of the movements from measuring-mark to counting dials.
56 Table III shows also a small but approximately constant correlation between X and Y for all points; 4. it is perhaps advisable to stress the fact t h a t the above explanations have no pretension to be true nor to give a sufficient confirmation of the theoretical considerations. A closer investigation and more material will be necessary. The best indication t h a t the theory has at least a sound base can be given in the following way. I f the correlation-amounts resulting from two independent measurements with an i n t e r v a l of nearly two years are represented in a two-dimensional g r a p h (after correction for the constant part) we get the result of fig. 8. According to the theory the points 2, 4, 5, 6 and 8 should coincide with the origine and the points 1, 3, 7 and 9 with the two crosses. Fig. 8 shows at least t h a t it is indeed necessary for the application of tl~e theory of errors to photogrammetrical measurements to take into account the type of observational system of the instrument. IV. Measure~vent of paredlc~xes. Monocular measurements like those treated in the foregoing chapter are not usual. When using double-image instruments the observations consist generally in measurements of parallaxes and coordinates. In this chapter we will confine ourselves to parallax-measurements, and based on the above considerations we can distinguish between coordinate-paxallaxes dX and dY as used in the well-known parallax-formulae and the observational-paq"allaxes d$ and dy as these are observed in the eye-pieces. As we have seen, there is no difference between the two kinds of parallaxes (in the A 5); in the stereoplanigraph this difference exists indeed. If we assume t h a t the observational-parallaxes d~ and d~ are of the same accuracy for every point of the image it will be clear t h a t the precision of the coordinate-parallaxes in the stereoplanigraph will be dependent on the position of the observed point in the spatial image. As a consequence of the increasing inequality of the two images towards the m a r g i n of the photographs and of the increasing difficulty for the eyes to fuse the two images, the question arises of whether a supplementary decrease in the precision must not be expected. This question can only be answered by an experiment. How.ever, direct observations at the stereoplanigraph cannot be used for this purpose for the following reasons: 1. with a normal base-altitude ratio, an observed point lies in the centre of one photograph, but the corresponding point on the second photograph will be situated at the border. This may cause a different pl~otographic image-quality which has no relation to the observational system of the instrument; 2. the same f a c t may cause a difference of illumination; 3. differences in the state of the adjustment of the two complicated observation-telescopes may confuse the results; 4. etc. To eliminate these sources of confusion from the results of the measurements so t h a t only the influence of the observation system remains, an effort was made to construct a new a p p a r a t u s for this purpose. The requirements for this a p p a r a t u s are: 1. the observed projections must have mathematical properties corresponding with formulae 4; 2. the observed point P must be seen b y both eyes under exactly equal circumstances. The photographic image-quality, the optical p a r t s used for the observation, the illumination etc. etc. must be exactly the same; 3. the enlargement m u s t be the same for both eyes;
57 4. t h e above-mentioned features respect to the eye-base.
must
remain
constant for each position of P with
The f i r s t requirement determines the construction principle of the instrument.
..... l .......... ] ...... P
.... s//
p'
J
P
.
D Fig. 10
Fig. 9
Fig. 11 d
The second requirement is satisfied if both eyes observe the same point using only plane surface-mirrors which are very constant in quality and cannot cause any optical aberration. The third requirement involves an equal distance between eye and point for both eyes.
58 T h i s d i s t a n c e m u s t be k e p t c o n s t a n t f o r each position of P to s a t i s f y t h e f o u r t h requirement. If enlarging lenses are not used this distance must correspond approximately with t h e d i s t a n c e o f d i s t i n c t vision. Fig. 9 g i v e s a s c h e m e of t h e - f i n a l c o n s t r u c t i o n , a, b, c a n d d a r e f o u r p a i r s of m i r r o r s of w h i c h b a n d c a r e f i x e d to a slide a n d a r e m o v a b l e in h o r i z o n t a l direction w i t h r e s p e c t to a a n d d. T h e two m i r r o r s d c a n r o t a t e a b o u t a n a x i s p e r p e n d i c u l a r to t h e p l a n e o f d r a w i n g . T h e complete set of m i r r o r s c a n also be tilted a b o u t t h e a x i s L - - R to be able to o b s e r v e in / 2 J tilted epipolar planes. • • It is obvious t h a t t h e 5 shape of the perspective i m a g e o f P is d e t e r m i n e d by t h e position o f L a n d R w i t h r e s p e c t to P. So t h e f i r s t r e q u i r e m e n t is s a t i s fied. Since both eyes observe /0 t h e s a m e p o i n t P, t h e seR 2 cond condition also does n o t • b give d i f f i c u l t i e s . Fig. 12 To s h o w h o w t h e t h i r d condition is m e t , i m a g i n e ~ ,n/, t h a t a f t e r P a p o i n t P" will ~6be observed. B o t h m i r r o r s d m u s t t h e n be r o t a t e d a J ¢ c e r t a i n a m o u n t . T h e diffe- 52r e n c e of l e n g t h LJ I t h a t is c a u s e d in t h i s w a y c a n be 5 0 c o m p e n s a t e d b y a horizon~zStal movement of the mirr o r s b a n d c o v e r a d i s t a n c e ~/6. ~6 O 7 equal to 1/~ LJ l. T h e total d i s t a n c e eye ~'~" - - P is k e p t c o n s t a n t b y a ¢2~ IT vertical m o v e m e n t of t h e 12 e y e - r i n g s O~ a n d O R. r/O~ / To be able to m e a s u r e l i l l l l /~ ' C05/0 parallaxes, a measuringg/ ~2 m a r k is p u t in each of t h e Fig. 1 3 l i g h t - p a t h s to P ( M s a n d M 2 in fig. 10). T h e s e m e a s u r i n g - m a r k s c a n be m o v e d in two directions p e r p e n d i c u l a r to e a c h o t h e r . T h e s e m o v e m e n t s c a n be r e a d on divided s c r e w s u p to 0,001 m m . A d i s a d v a n t a g e o f t h i s m e t h o d is t h a t M a n d P a r e n o t e q u a l l y s h a r p b u t e x p e r i e n c e s h o w s t h a t , if t h e d i s t a n c e M P is s u f f i c i e n t l y s m a l l (2 to 3 m m ) , t h i s d i f f e r e n c e c a n be neglected. T h e m e a s u r i n g - m a r k s c o n s i s t of s m a l l m e t a l s p h e r e s w i t h a d i a m e t e r of 0,1 ram, each f i x e d to a g l a s s p l a t e w i t h v e r y t r a n s p a r a n t shellac. Fig. 10 g i v e s a s c h e m e of t h e cons t r u c t i o n a n d fig. 11 a n i l l u s t r a t i o n o f t h e f i n i s h e d i n s t r u m e n t . T h e m e a s u r e m e n t s w e r e e x e c u t e d in t h e f o l l o w i n g m a n n e r . A s u i t a b l e p o i n t w a s chosen on a p h o t o g r a p h a n d t h i s . p o i n t w a s s h i f t e d to t h e d e s i r e d position u n d e r t h e i n s t r u m e n t . T h e m e a s u r i n g device w a s p u t over t h i s p o i n t so t h a t t h e m o v i n g d i r e c t i o n s w e r e p a r a l l e l to t h e X- a n d Y - a x e s ( X - a x i s p a r a l l e l to e y e - b a s e ) . T h e e l e v a t i o n of t h e p o i n t ( X - p a r a l l a x ) w a s m e a s u r e d 50 t i m e s a n d a f t e r w a r d s 50 o b s e r v a t i o n s w e r e m a d e
Y
59 f o r t h e Y - p a r a l l a x . T h e s a m e p o i n t w a s m e a s u r e d s u c c e s s i v e l y in 12 p o s i t i o n s w i t h r e s p e c t to t h e eye-base, a s s h o w n in fig. 12. T h e s e m e a s u r e m e n t s w e r e a f t e r w a r d s ~'J/C]?
sT~D~yl'n 2$
4" 2
3 2" I 0 5
6
II
~
9
12
\
2s 24' \ 23 \ 22
\
2/
\ /8
\
22
2 . . 3 24`
26
27
28
2.9
30
3/
•.~2
JJ
0"£-~q
In o
Fig. 14 r e p e a t e d in t h e s a m e w a y b u t in r e v e r s e d o r d e r to e l i m i n a t e t h e i n f l u e n c e of i n c r e a s i n g r o u t i n e on t h e o p e r a t o r . T h e t o t a l n u m b e r of o b s e r v a t i o n s w a s 2400 f r o m w h i c h t h e m e a n e r r o r s in t h e Xand Y-parallaxes were computed. To e l i m i n a t e t h e i n d i v i d u a l i n f l u e n c e of t h e o p e r a t o r it w a s p l a n n e d to h a v e 10 p e r s o n s e x e c u t e t h e above p r o g r a m . L a c k of t i m e m a d e it impossible to f i n i s h t h i s p r o g r a m , so t h a t only t h e r e s u l t s of f o u r o b s e r v e r s a r e available. T h e s e r e s u l t s a r e g i v e n in table IV. Formula 3 gave the result: d~ = COS~ dY so t h a t m~tl = cos2~ ~n~v. Since in p r i n c i p l e t h e m e a s u r i n g of a Y - c o o r d i n a t e a n d of a Y - p a r a l l a x is t h e s a m e , a n d a s s u m i n g t h a t m t l is t h e s a m e f o r all positions of t h e p o i n t on t h e i m a g e , we m a y e x p e c t t h a t t h e m e a n e r r o r of a Y - p a r a l l a x o b s e r v a t i o n (~n~v) w o u l d be p r o p o r t i o n a l to 1 cos ~) Fig. 13 i l l u s t r a t e s t h e r e s u l t . T h e expected relation is c o n f i r m e d v e r y s a t i s f a c t o r i l y . It is also r e m a r k a b l e t h a t of t h e p o i n t s w i t h t h e s a m e v a l u e of
~
, those w h i c h
lie closest to t h e b o r d e r h a v e t h e g r e a t e s t m e a n error. H o w e v e r , t h i s s e c o n d a r y e f f e c t m a y be a n a c c i d e n t a l one a n d s h o u l d be c o n f i r m e d b y m o r e m a t e r i a l . W i t h r e s p e c t to t h e X - p a r a l l a x e s , m a t t e r s a r e n o t so simple. Since in one e p i p o l a r p l a n e ~L a n d aR a r e n o t equal, it is n o t a priori clear w h i c h r e l a t i o n c a n be expected b e t w e e n t h e X- a n d t h e S - p a r a l l a x e s . Besides, in f o r m u l a 3, f o r m o n o c u l a r m e a s u r e m e n t s , t h e Y - p a r a l l a x h a s also s o m e influence. W e will c o n f i n e o u r s e l v e s t h e r e f o r e to t h e e x p e r i m e n t a n d t r y to f i n d a n e x p e r i m e n t a l relation. One can, f o r i n s t a n c e , t a k e as a p a r a m e t e r t h e a n g l e a t w h i c h t h e two projectin~ °
60 rays intersect. Indeed, fig. 14 shows t h a t there is a certain relation between this angle and mw. A similar relati/Y~x/r/rnicr /2 P, 8 10 $ zl 3 2 I on, however, seems to I 26 f exist between mp~ and the distance be2Y / tween P and the 2¢ / middle of the base (fig. 15). 2J / The number of observations and the 22. / dispersion of the re2/. /, / sults are too g r e a t to allow a definite de20. 2 termination of the /.q law t h a t is in force here. But this is not the purpose of this /7 article. The important thing is t h a t it will ' I.~6 I, /4 ~ Y /5.8 143 " '.'Z21ZSl~S 186 ~ / h c m now be clear t h a t for Fig. 15 the X- as well as for the Y-parallaxes the accuracy depends on the position of the point in the spatial model. The X - a n d Y-parallaxes were not measured simultaneously so t h a t from the present material it is not possible to conclude a n y t h i n g about the eventual correlation between those quantities.
/
V. Conclusians.
The foregoing chapters give no more than an example of the various problems that must be solved before enough data are gathered for a foundation of a theory of errors of aerial triangulation. The purpose of this article is simply to show: 1. t h a t it is useful to enlarge and deepen our knowledge about the elementary observations t h a t are used in photogrammetry; 2. t h a t this knowledge is of great importance since it is the only mcans for an objective and scientific comparison of the accuracy of methods and instruments; 3. t h a t a prolonged investigation of a g r e a t m a n y influences is necessary, which will require the execution of a large program of observations; 4. t h a t it will still take many years before the exact influence of the m a n y varying circumstances can be known; 5. t h a t it deserves recommendation to t r y to organise an international cooperation for this purpose. I take it for granted t h a t the editorial s t a f f of Photogrammetria will be glad to publish the results of those investigations so t h a t they will be available to all photogrammetrists. Lit. :
(1) W . K . B a c h m a n n . Th4orie des erreurs et compensation des triangulations a6riennes. Lausanne, 1946. (2) J . M . T i e n s t ~ a . A n extension of the technique of the method of least squares to correlated observations. Bull. G~odfisique, nouvelle s~rie, 1947, n r 6, p. 301--335.
61 APPENDIX For the readers who are not familiar with the work of Tienstra, the following short explanation of the principle may be of use. Cofactors are quantities t h a t are named weight-numbers in the classical l i t e r a t m e and are generally denoted by the letter Q (Q~I' Qle' Q2~ etc.). The importance of the ,,tensor of cofactors" is described in the articles of Prof. Tienstra in: Bulletin G~od~sique, nouvelle s~rie, 1947, p. 301--335 and 1948, p. 289-306. In short, the following is said: The classical technic of the method of least squares is based on the assumption t h a t the observations are not correlated, but in reality correlation is nearly always present although the explanation is not always known. This correlation is generally of physical character and the g r e a t e s t problem for an adjustment is the determination of numbers to indicate the amount of correlation and to adjust these correlated observations in the r i g h t way. Prof. Tienstra has given a universal solution for this second problem. He explains t h a t we can t r e a t the physical correlation in the same way as a linear algebraic correlation. A linear algebraic correlation arises from a linear t r a n s f o r m a t i o n applied to noncorrelated observations. Suppose we have three noncorrelated observations xl, x~ and x a. We t r a n s f o r m them w i t h the formulae: X = alx 1 q- a.ax 2 A- aa~ca X a = blx 1 + b..~x2 + b:~xa (1) X -- cloc1 "4- C2x.a + cax a The quantities X , X z and X a are now correlated. Suppose the weights of x~, x z and ( 1 1 ) x a t ° b e g11' gu2 .and gaa' and their cofactors resp. gll, g,'a~ and ga3 g l , = ~77, g.,.,=g.~getc. then the tensor of cofactors of the quantities xl, x2 and x a is gl~ 0 0 0 g22 0 0 0 gaa
(2a) and the weight-tensor
g~l 0 0 I 0 gw., 0 1 0 0 ga'~
(25)
If we name the cofactors of X1, X, etc. resp. G ,t, G 12 etc. then their tensor is: ] G~ G l'a
a13
G2~ Cr31 G 2'a G a2
e~a
(3 a) and the weight-tensor
G33
Gll Grz G~a G, 1 G=,., G2a
G31 632
(:P')
633
The values of (3 a) are found from (1) by means of the well-known special law of propagation of errors, namely:
(~11 = 6611741gll~- ~20,2g,2:2-~ a.~%gaa Gra = a l b l g l l + a.,b..,g ='a ÷ a:~bagaa etc. and the values of G~,, G~ etc. in (3b) are found from (3 a) by the weight-equations: G l l G l l + G.,1G21 + .. :~ 1 l G ll Grz + G ,1G2= + .. = 0 In total 3 sets of 3 equations. I f now q , e~ and e a are the corrections to x , x, and x a the method of least square.s learns t h a t gns~s~ + .q.z2a2e., -k gaJaSa m u s t be a m i n i m u m .
62 P r o f . T i e n s t r a p r o v e s t h a t , w h e n we a d j u s t t h e q u a n t i t i e s X~, X.~ a n d X 3 w i t h t h e c o r r e c t i o n s E l , Eu a n d Ea t h e m i n i m u m - c o n d i t i o n t a k e s t h e f o r m : G l l E 1 E 1 + GI~E~E ~ + G21E2E1 + G~E2E. 2 + . . . .
----> rain.
f r o m w h i c h h e d.evelops t h e f u r t h e r t e c h n i c s of t h e a d j u s t m e n t . T h e k n o w l e d g e - o f t h e t e n s o r (3a) is t h e r e f o r e n e c e s s a r y . I n t h e c a s e o f p h y s i c a l c o r r e l a t i o n t h i s t e n s o r c a n also be d e t e r m i n e d by m e a n s o f a s t a t i s t i c a l e x a m i n a t i o n of t h e o b s e r v a t i o n s . T h e g r e a t i m p o r t a n c e o f t h i s conception c o n s i s t s t h e r e f o r e o f t h e f a c t t h a t , once t h e t e n s o r of c o f a e t o r s o f o u r o b s e r v a t i o n s h a s been d e t e r m i n e d , it is n o t i n t e r e s t i n g if t h e e x i s t i n g c o r r e l a t i o n is of p h y s i c a l or a l g e b r a i c a l n a t u r e . T h e a d j u s t m e n t - r u l e s a r e u n i v e r s a l a n d c a n be u s e d f o r both cases.
Table I. Stereoplanigraph
Autograph A 5 Point • mx2
•
mY2
f
mxy
~t x 2
•
TtlY2
r i
~xy
I
55
73
+ 4
132
180
--40
77
69
--4
126
106
--14
3
93
75
+ 4
155
121
+ 8
4
63
65
+4
95
110
-- 8
5
128
114
÷ 8
103
76
--14
6
83
123
+24
129
137
--16
7
73
81
--12
119
117
+ 8
8
77
70
-- 1
122
166
--26
9
112
121
+45
122
135
--30
85
88
÷ 8
123
128
--15
Average
63
II ~ II ~°
I
÷
÷
÷
÷
I
÷
I
I iI °°
~ o ~ o o÷o ~ o ~ o c~
r~
I
11
64
I
+
÷
+
+
+
+
+
L'-.
+
I]
~,1 ~
L" ~
-~
~
l
~
@'~
L""
L'-'
+
+
+
I
+
~
~
I
÷
÷
+
,I
I
I
÷
li
+
I
I
I
I
i
I. Introduction. The application of the theory of errors to photogrammetry has not yet reached the same intensity as in the other branches of geodesy n o t w i t h s t a n d i n g the f a c t t h a t there is g r e a t need for the results of such an application. The f i r s t purpose of such a study is to find a numerical expression for the accuracy of a certain quantity. This expression may then be used in d i f f e r e n t ways e.g.: a) for a comparison of d i f f e r e n t methods (radial-triangulation versus spatial-triangu: lation, etc.) ; b) for an examination of the influence of d i f f e r e n t circumstances (scale of~ the photographs, type of camera, etc.) ; c) for an examination of the lines along which an improvement of instruments or methods can be obtained in the most efficient way; d) for a prediction of the accuracy t h a t might be expected in later applications; e) etc. etc. These considerations are naturally of g r e a t importance for the development of photogrammetry, and t h e i r practical value is increased if they are combined with economical studies. A t present, what has been done in this field in the Netherlands and abroad is not much more t h a n a modest beginning. There are several reasons for this fact: 1. the p r o b l e m s are very complicated and therefore require an elaborate quantity of work; 2. there are many d i f f e r e n t methods and instruments, s o - t h a t each theoretical investigation is only applicable in a limited number of cases. An example of this fact is the excellent work of Prof. W. K. Bachmann [(1)]. In this book the author applies the theory of errors to a method of relative orientation t h a t is based on theoretical considerations but which will probably be rarely used in practice. The resulting tables and formulae have therefore a v e r y limited practical value; 3. as f a r as I know, none of the authors of the present time have taken into account the real properties of the observations. The consequence is t h a t the results of the theoretical considerations do not correspond with those of practical experience. The weakest point of all theoretical treatise~ in this field lies in t h e f a c t t h a t until now no one has taken the trouble to prove t h a t his hypotheses and idealisations about the character of the observations sufficiently approach reality. There are three kinds of problems to be solved for the application of the theory of errors : A. in the f i r s t place there is the need for computation-rules of general validity. This need is completely filled by the publications of Prof. J. M. Tienstra [(2)]. Based on his theory it is 0nly a question of computation-technics to be able to execute the required work in the most efficient way. In this respect it is possible t h a t in future the electronic computation machines and such-like may render useful service; B. in the second place it is necessary to obtain a sufficiently accurate mathematical d e s c r i p t i o n of the problem a t hand. This may lead to considerable work but there is no principal difficulty in a precise and careful analysis of all observations and manipulations executed a t a given instrument or for a given method;
4,
50 C. at the base of the problem is finally the tensor of cofactors t h a t determines the values of the mean errors and correlations of t h e observations. 1) To be able to set up this tensor it is necessary to go back to experiments. Series of observations must be produced, varying the circumstances as f a r as these may have influence in practice, to conclude with a statistical examination for the determination of the cofactors. This phase of the problem offers a lot of practical difficulties. A f i r s t examination of the photogrammetric problems shows t h a t we may separate the influences o n ' t h e observational accuracy into three groups: (1) the influence of the properties of the photograph; (2) ,, ,, ,, . . . . of the i n s t r u m e n t ; (3) ,. . . . . . . . . of the observer. In which order of succession these groups are i m p o r t a n t cannot be said a priori. It is, however, certainly true t h a t a complete investigation of these influences requires an immense quantity of observational material t h a t can, most probably, only be obtained by a close international cooperation. The following example may illustrate t h i s - f a c t : a relative orientation of a pair of photographs is based on the measurement of vertical-parallaxes in a n u m b e r of (generally six) points of the image. If we assume t h a t the influences of the instrument and the observer (groups 2 and 3) are completely known, there remains only to be determined the influence of the properties of the photograph on the value of the cofactors. The properties of the photographic image are primarily dependent on the type of camera. It is therefore necessary to investigate the influence of differences in the type of the camera on the photographic image. Even if the camera and its objective have been completely analysed there are still a g r e a t many variables, like: 1) atmospheric conditions ( c o n t r a s t ) ; 2) flying height (movement Of i m a g e ) ; 3) character of the t e r r a i n (definition of details); 4) airplane (vibrations) ; 5) emulsion-base (differential shrinkage) ; 6 ) etc. It will be clear that, notwithstanding the rigorous exclusion of unknown factors, there still remains a large uncertainty. F o r this reason there is little basis for comparison of the results published by various authors unless a very careful and complete description of all circumstances is added. Such a description has still little value if the influence of variations of the circumstances is not known. The foregoing may have made clear why until now the application of the theory of errors to p h o t o g r a m m e t r y has so little practical value. Considering the quantity of factors t h a t influence the accuracy, one might doubt if it will ever come to a theoretically correct application. There seem, however, to be several reasons for which one need not be too pessimistic. It may be expected t h a t in future the existing number of variants in methods and instruments will be limited. On the other hand there exists a tendency in the optical and mechanical performance of cameras and instruments t h a t reduces a number of sources of errors to a level where they may be neglected. Some examples are: 1. the new wide-angle lens of Wild, the Aviogon, has the p r o p e r t y that the diminishing of light at the margin of the photograph is much smaller than with all foregoing types. In addition the sharpness of the image is much more uniform over the whole field; 1) See Appendix.
51 2. the mechanical precision of the universal instruments is gradually increasing so t h a t the number of sources of errors t h a t have an appreciable influence on the observations is getting smaller. That nevertheless an enormous work has to be undertaken to obtain a well-based tensor of cofactors for d i f f e r e n t methods and instruments will be demonstrated in the following chapters. They deal with an investigation of some detailed influences, executed during the years of German occupation with the welcome and v a l u e d financial support of the Delft University F~undation. II. The preliminary purpose Of the investigation was a comparison of the observational-system Of the Wild-A 5 and the Zeiss-Stereoplanigraph. The basic principle of these systems is illustrated in the figures 1 and 2. In the A 5 observation takes place perpendi-
Fig. 1
Fig. 2
cular to the plane of the photograph, with a telescope of a complicated construction. This telescope is sketched in a much simplified-form in fig. 1 since only the p a r t closest to the image is characteristic for its functioning. The stereoplanigraph is based on the Porro-Koppe principle (fig. 2). F u r t h e r characteristics are: a) the auxiliary optical system consists of a positive and a negative lens of equal focal length which are responsible for the formation of a sharp image in a plane through the measuring m a r k ; b) the rotating m i r r o r is automatically directed so t h a t the reflected rays have a fixed horizontal direction to the telescope. If we take the simple case of a true vertical photograph of an object consisting of a rectangular grid, the photograph will contain a reduced but also rectangular image of this grid. I t will be obvious t h a t the image which is observed through the telescope of the A 5 will also be a rectangular and non-distorted grid and t h a t this image will be the same over the whole photograph (The lens-distortion in the observational system is supposed to be negligible). I t may be expected that, in this respect and excluding all other influences, the observational accuracy will be equal for each position of the observed point on the plate.
52 The observational system of the stereoplanigraph has other properties, as will be shown u s i n g fig. 3. The grid on the plate will be projected undistorted on the horizontal plane V. The optical rayS, s t a r t i n g from a n y point (e.g. the point p) of the plate will be t r a n s f o r m e d to a parallel bundle by the lens O. The auxilia r y system (V.S.) produees an optical image in the plane V" which is perpendicular to the line OP (which line coincides with the optical axis of the aux. system). Hence, the optical image of q will fall in Q'. The distance p q will therefore be observed as PQJ but will be measu~'ed equal to PQ. The relation between the observed and the measured distances is of g r e a t importance since the measured values (coordinates of parallaxes) z(.2 are used in practice. This relation can be developed in a formula in which the elements denoting the position of the observed point e n t e r as parameters. Fig. 4 gives the elements t h a t must be considered in developing this formula. The plane V p is the plane of the optical image when
//
Fig. 3 the point P is observed. The position of P is determined bij the angles i i I Q and o. In the plane V' a coordinate-system S-~] I is chosen, so t h a t the Saxis is parallel to the eye-base. The direction of this ~-axis can be determined by considerFig. 4 ing t h a t the optical system of the stereoplanigraph includes a dove-prism which is automatically guided so t h a t the epipolar plane through P is seen as a line parallel to the eye-base. The S-axis is therefore chosen along the intersection-line of the plane V" with the plane t h r o u g h S, T and P, the base-line in fig. 4 being supposed to be parallel to the X-axis. The ~/-axis is perpendicular to the ~-axis. The above-mentioned formula is: h ( X c o s ~ - - h s i n o c o s Q - - Y sinQ s i n v ) cos~ cos~ (h cosQ cos ~ ÷ Y cos~ sinQ + X s i n o ) h (Y c o s Q - - h s i n 6 ) ~/ = cos~ cos~ (h cos~ cos ~ ÷ Y cos~ s i n ~ % X s i n ~ )
i I
(1)
53 f r o m w h i c h we derive b y d i f f e r e n t i a t i o n : dX=tg~ dY -
tga
1 d * / + cos~o d~
(2}
1 cos Q d~/
or d $ : c o s ~ d X ~ d,/ = cos~ dY
sin~ sine dY
(3)
I f we s u p p o s e t h a t t h e o b s e r v a t i o n s of t h e c o o r d i n a t e s ~ a n d ~7 a r e i n d e p e n d e n t a n d e q u a l l y a c c u r a t e so t h a t we m a y p u t : m~ :m~!q :/~ a n d m~TI : 0 we c a n derive f r o m t h e above f o r m u l a " (2) t h a t t h e a c c u r a c y of t h e m e a s u r e d v a l u e s f o r X a n d Y will be e x p r e s s e d b y :
(
? me x
=
• "
1)
tg2o t g 2 Q + c o s 2 ~
#2
1
tg~tg # T h e s e f o r m u l a e a r e valid f o r t h e s t e r e o p l a n i g r a p h in t h e case w h e r e m o n o c u l a r m e a s u r e m e n t s a r e execut e d f o r t h e d e t e r m i n a t i o n of coordin a t e s o f p o i n t s in a h o r i z o n t a l plane• For the A5 zrv2 x ~
m'~ v ~
we m a y e x p e c t to f i n d re2; ~r~xy = O.
In the case that stereoscopic m e a s u r e m e n t s a r e e x e c u t e d we h a v e to deal w i t h two i m a g e s both of w h i c h will i n f l u e n c e t h e a c c u r a c y of t h e m e a s u r e m e n t s . When using the A 5 the c o o p e r a t i o n of t h e t w o i m a g e s will be a s i m p l e one s i n c e t h e y are equal for flat terrain. With the stereoplanigraph m a t t e r s a r e m u c h m o r e complicated. T h e p l a n e s V ' (fig. 4) have different positions for left a n d r i g h t c a m e r a so t h a t / ,)e t h e p r o j e c t i o n s of t h e h o r i z o n t a l p l a n e on t h e inclined p l a n e s will h a v e d i f f e r e n t f o r m s • I t is obvious t h a t t h e mon o c u l a r o b s e r v e d p r o j e c t i o n of a g r i d m u s t h a v e t h e s h a p e of Fig. 6 two i n t e r s e c t i n g cones o f r a y s . Indeed, t h e lines in t h e X - Y p l a n e p a r a l l e l to th~ Y - a x i s will be p r o j e c t e d on t h e V '
54 plane (fig. 4) as lines through the intersection point of V r and the line through S parallel to the Y-axis. The same is true for lines parallel to the X-axis. The image for an arbit r a r i l y chosen point P will therefore have the form given in fig. 5 where the circle represents the limit of the field of view. Fig. 6 g i v e s some examples of the difference between the two images when using stereoscopic ./ 3 vision. It will be interesting to check these results in practice. The following chapters will give an examination in the above-mentioned line for monocular measurements of coordinates in a horizontal plane and for stereoscopic measurements of p a r a l l a x e s .
¢
2
6
8
.9
III. Monocula/r measurement of X and Y coordinates. The measurements reported here were originally ~-----~---'--'-meant for another purpose but they can be used here Fig. 7 too. On a photograph nine points were marked by a small hole according to the scheme of fig. 7. In the stereoplanigraph as well as in the A 5 t h e X and Y coordinates of these points were measured with a constant value of Z, using one projector at a time and assuming t h a t the photograph has a t r u l y vertical axis. Fo~ each point 120 series of 5 observations were made, thus giving 480 superfluous observations. The mean value for X and Y for each series was subtracted from the observed values giving 600 differences from which the mean errors as well as the amount of correlation for X and Y could be computed. Table I gives the result for the same photograph measured in the A 5 and the stereoplanigraph (values in microns). In judging this result, it should be taken into account t h a t the given numbers show the sum of the influence of setting- and reading errors. The counting dials of the A 5 are divided in 0,01 ram, those of the stereoplanigraph in 0,1 mm. F o r the A 5 the two influences cannot be separated since they will have the same influence for each position on the photograph. For the stereoplanigraph a separation is possible according to the following formulae. Assumptions: 1) the mean error of the reading of a coordinate is, for each point, equal to /~1; 2) the readings of X and Y coordinates are not correlated; 3) ~ = m 2 ~ = /~2 and m ~ I = 0. Formula 4 gives in this case: m~=
1 (tg~tg~+~-)~2+/~j2 1
~
= -co~- ~
+ #~
(5)
tgo tg~
~y=
cos~~
The two unknowns /~ and /~ can be found with these formulae from the values of table I. The result is (approximately) ~u~ = 50 micr.• jul~ = 6 0 micr. e
/~ = 0,007 nmL tt~ = 0,008 ram.
55
Table II gives a comparison of the real values and those t h a t are computed with the given values of 14 and /~1" The tables I and II give rise to the following considerations: 1. for the A 5, the correlation between X and Y is indeed very small considering t h a t the given numbers are the 8 q u i r e s of the mean errors. The mean errors themselves have no g r e a t differences (between 7,5 and 9 micron). There are three exceptions namely the points 5, 6 a n d 9. The points 6 and 9 are situated at the same side of the plate and since the values for ~ 2 and ~%~ are too g r e a t for both it m a y be concluded t h a t a mechanical e r r o r of the i n s t r u m e n t causes these discrepancies. In point 5 both m r and mu have g r e a t values. An explanation of this fact can possibly be found in the consideration t h a t point 5 is the only point where the space-rod
•
/ -
\~o
w
"o/
Fig. 8 has a vertical position, so t h a t no pressure is exercised on the base-carriage. Any amount of back-lash can only find expression in the measuring of this point; 2. the results for the stereoplanigraph give the impression that, in addition to the correlation t h a t could be expected on theoretical grounds, there is also another small but nearly constant a m o u n t of correlation. This amount could perhaps be explained by assuming t h a t it is caused by mechanical or optical features of the i n s t r u m e n t (that have not been taken into account) or by the method of observation; 3. the separation of the mean errors by formulae 5 is based on a plausible assumption but a confirmation seems not undesirable. The same computations have therefore been made with a s e t of similar measurements executed (two years earlier) on another photograph. The result is given in table III. It is note worthy t h a t in this case ~ gives the same value as table II but t h a t /~1 is much smaller. An explanation for this fact might be t h a t the observations given in table II were made in J a n u a r y 1944. A t t h a t time the heating of the rooms was very bad so t h a t great differences in t e m p e r a t u r e and humidity occurred. It is not impossible t h a t this fact has had an u n f a v o u r a b l e influence o~ the constancy of the mechanical t r a n s p o r t a t i o n of the movements from measuring-mark to counting dials.
56 Table III shows also a small but approximately constant correlation between X and Y for all points; 4. it is perhaps advisable to stress the fact t h a t the above explanations have no pretension to be true nor to give a sufficient confirmation of the theoretical considerations. A closer investigation and more material will be necessary. The best indication t h a t the theory has at least a sound base can be given in the following way. I f the correlation-amounts resulting from two independent measurements with an i n t e r v a l of nearly two years are represented in a two-dimensional g r a p h (after correction for the constant part) we get the result of fig. 8. According to the theory the points 2, 4, 5, 6 and 8 should coincide with the origine and the points 1, 3, 7 and 9 with the two crosses. Fig. 8 shows at least t h a t it is indeed necessary for the application of tl~e theory of errors to photogrammetrical measurements to take into account the type of observational system of the instrument. IV. Measure~vent of paredlc~xes. Monocular measurements like those treated in the foregoing chapter are not usual. When using double-image instruments the observations consist generally in measurements of parallaxes and coordinates. In this chapter we will confine ourselves to parallax-measurements, and based on the above considerations we can distinguish between coordinate-paxallaxes dX and dY as used in the well-known parallax-formulae and the observational-paq"allaxes d$ and dy as these are observed in the eye-pieces. As we have seen, there is no difference between the two kinds of parallaxes (in the A 5); in the stereoplanigraph this difference exists indeed. If we assume t h a t the observational-parallaxes d~ and d~ are of the same accuracy for every point of the image it will be clear t h a t the precision of the coordinate-parallaxes in the stereoplanigraph will be dependent on the position of the observed point in the spatial image. As a consequence of the increasing inequality of the two images towards the m a r g i n of the photographs and of the increasing difficulty for the eyes to fuse the two images, the question arises of whether a supplementary decrease in the precision must not be expected. This question can only be answered by an experiment. How.ever, direct observations at the stereoplanigraph cannot be used for this purpose for the following reasons: 1. with a normal base-altitude ratio, an observed point lies in the centre of one photograph, but the corresponding point on the second photograph will be situated at the border. This may cause a different pl~otographic image-quality which has no relation to the observational system of the instrument; 2. the same f a c t may cause a difference of illumination; 3. differences in the state of the adjustment of the two complicated observation-telescopes may confuse the results; 4. etc. To eliminate these sources of confusion from the results of the measurements so t h a t only the influence of the observation system remains, an effort was made to construct a new a p p a r a t u s for this purpose. The requirements for this a p p a r a t u s are: 1. the observed projections must have mathematical properties corresponding with formulae 4; 2. the observed point P must be seen b y both eyes under exactly equal circumstances. The photographic image-quality, the optical p a r t s used for the observation, the illumination etc. etc. must be exactly the same; 3. the enlargement m u s t be the same for both eyes;
57 4. t h e above-mentioned features respect to the eye-base.
must
remain
constant for each position of P with
The f i r s t requirement determines the construction principle of the instrument.
..... l .......... ] ...... P
.... s//
p'
J
P
.
D Fig. 10
Fig. 9
Fig. 11 d
The second requirement is satisfied if both eyes observe the same point using only plane surface-mirrors which are very constant in quality and cannot cause any optical aberration. The third requirement involves an equal distance between eye and point for both eyes.
58 T h i s d i s t a n c e m u s t be k e p t c o n s t a n t f o r each position of P to s a t i s f y t h e f o u r t h requirement. If enlarging lenses are not used this distance must correspond approximately with t h e d i s t a n c e o f d i s t i n c t vision. Fig. 9 g i v e s a s c h e m e of t h e - f i n a l c o n s t r u c t i o n , a, b, c a n d d a r e f o u r p a i r s of m i r r o r s of w h i c h b a n d c a r e f i x e d to a slide a n d a r e m o v a b l e in h o r i z o n t a l direction w i t h r e s p e c t to a a n d d. T h e two m i r r o r s d c a n r o t a t e a b o u t a n a x i s p e r p e n d i c u l a r to t h e p l a n e o f d r a w i n g . T h e complete set of m i r r o r s c a n also be tilted a b o u t t h e a x i s L - - R to be able to o b s e r v e in / 2 J tilted epipolar planes. • • It is obvious t h a t t h e 5 shape of the perspective i m a g e o f P is d e t e r m i n e d by t h e position o f L a n d R w i t h r e s p e c t to P. So t h e f i r s t r e q u i r e m e n t is s a t i s fied. Since both eyes observe /0 t h e s a m e p o i n t P, t h e seR 2 cond condition also does n o t • b give d i f f i c u l t i e s . Fig. 12 To s h o w h o w t h e t h i r d condition is m e t , i m a g i n e ~ ,n/, t h a t a f t e r P a p o i n t P" will ~6be observed. B o t h m i r r o r s d m u s t t h e n be r o t a t e d a J ¢ c e r t a i n a m o u n t . T h e diffe- 52r e n c e of l e n g t h LJ I t h a t is c a u s e d in t h i s w a y c a n be 5 0 c o m p e n s a t e d b y a horizon~zStal movement of the mirr o r s b a n d c o v e r a d i s t a n c e ~/6. ~6 O 7 equal to 1/~ LJ l. T h e total d i s t a n c e eye ~'~" - - P is k e p t c o n s t a n t b y a ¢2~ IT vertical m o v e m e n t of t h e 12 e y e - r i n g s O~ a n d O R. r/O~ / To be able to m e a s u r e l i l l l l /~ ' C05/0 parallaxes, a measuringg/ ~2 m a r k is p u t in each of t h e Fig. 1 3 l i g h t - p a t h s to P ( M s a n d M 2 in fig. 10). T h e s e m e a s u r i n g - m a r k s c a n be m o v e d in two directions p e r p e n d i c u l a r to e a c h o t h e r . T h e s e m o v e m e n t s c a n be r e a d on divided s c r e w s u p to 0,001 m m . A d i s a d v a n t a g e o f t h i s m e t h o d is t h a t M a n d P a r e n o t e q u a l l y s h a r p b u t e x p e r i e n c e s h o w s t h a t , if t h e d i s t a n c e M P is s u f f i c i e n t l y s m a l l (2 to 3 m m ) , t h i s d i f f e r e n c e c a n be neglected. T h e m e a s u r i n g - m a r k s c o n s i s t of s m a l l m e t a l s p h e r e s w i t h a d i a m e t e r of 0,1 ram, each f i x e d to a g l a s s p l a t e w i t h v e r y t r a n s p a r a n t shellac. Fig. 10 g i v e s a s c h e m e of t h e cons t r u c t i o n a n d fig. 11 a n i l l u s t r a t i o n o f t h e f i n i s h e d i n s t r u m e n t . T h e m e a s u r e m e n t s w e r e e x e c u t e d in t h e f o l l o w i n g m a n n e r . A s u i t a b l e p o i n t w a s chosen on a p h o t o g r a p h a n d t h i s . p o i n t w a s s h i f t e d to t h e d e s i r e d position u n d e r t h e i n s t r u m e n t . T h e m e a s u r i n g device w a s p u t over t h i s p o i n t so t h a t t h e m o v i n g d i r e c t i o n s w e r e p a r a l l e l to t h e X- a n d Y - a x e s ( X - a x i s p a r a l l e l to e y e - b a s e ) . T h e e l e v a t i o n of t h e p o i n t ( X - p a r a l l a x ) w a s m e a s u r e d 50 t i m e s a n d a f t e r w a r d s 50 o b s e r v a t i o n s w e r e m a d e
Y
59 f o r t h e Y - p a r a l l a x . T h e s a m e p o i n t w a s m e a s u r e d s u c c e s s i v e l y in 12 p o s i t i o n s w i t h r e s p e c t to t h e eye-base, a s s h o w n in fig. 12. T h e s e m e a s u r e m e n t s w e r e a f t e r w a r d s ~'J/C]?
sT~D~yl'n 2$
4" 2
3 2" I 0 5
6
II
~
9
12
\
2s 24' \ 23 \ 22
\
2/
\ /8
\
22
2 . . 3 24`
26
27
28
2.9
30
3/
•.~2
JJ
0"£-~q
In o
Fig. 14 r e p e a t e d in t h e s a m e w a y b u t in r e v e r s e d o r d e r to e l i m i n a t e t h e i n f l u e n c e of i n c r e a s i n g r o u t i n e on t h e o p e r a t o r . T h e t o t a l n u m b e r of o b s e r v a t i o n s w a s 2400 f r o m w h i c h t h e m e a n e r r o r s in t h e Xand Y-parallaxes were computed. To e l i m i n a t e t h e i n d i v i d u a l i n f l u e n c e of t h e o p e r a t o r it w a s p l a n n e d to h a v e 10 p e r s o n s e x e c u t e t h e above p r o g r a m . L a c k of t i m e m a d e it impossible to f i n i s h t h i s p r o g r a m , so t h a t only t h e r e s u l t s of f o u r o b s e r v e r s a r e available. T h e s e r e s u l t s a r e g i v e n in table IV. Formula 3 gave the result: d~ = COS~ dY so t h a t m~tl = cos2~ ~n~v. Since in p r i n c i p l e t h e m e a s u r i n g of a Y - c o o r d i n a t e a n d of a Y - p a r a l l a x is t h e s a m e , a n d a s s u m i n g t h a t m t l is t h e s a m e f o r all positions of t h e p o i n t on t h e i m a g e , we m a y e x p e c t t h a t t h e m e a n e r r o r of a Y - p a r a l l a x o b s e r v a t i o n (~n~v) w o u l d be p r o p o r t i o n a l to 1 cos ~) Fig. 13 i l l u s t r a t e s t h e r e s u l t . T h e expected relation is c o n f i r m e d v e r y s a t i s f a c t o r i l y . It is also r e m a r k a b l e t h a t of t h e p o i n t s w i t h t h e s a m e v a l u e of
~
, those w h i c h
lie closest to t h e b o r d e r h a v e t h e g r e a t e s t m e a n error. H o w e v e r , t h i s s e c o n d a r y e f f e c t m a y be a n a c c i d e n t a l one a n d s h o u l d be c o n f i r m e d b y m o r e m a t e r i a l . W i t h r e s p e c t to t h e X - p a r a l l a x e s , m a t t e r s a r e n o t so simple. Since in one e p i p o l a r p l a n e ~L a n d aR a r e n o t equal, it is n o t a priori clear w h i c h r e l a t i o n c a n be expected b e t w e e n t h e X- a n d t h e S - p a r a l l a x e s . Besides, in f o r m u l a 3, f o r m o n o c u l a r m e a s u r e m e n t s , t h e Y - p a r a l l a x h a s also s o m e influence. W e will c o n f i n e o u r s e l v e s t h e r e f o r e to t h e e x p e r i m e n t a n d t r y to f i n d a n e x p e r i m e n t a l relation. One can, f o r i n s t a n c e , t a k e as a p a r a m e t e r t h e a n g l e a t w h i c h t h e two projectin~ °
60 rays intersect. Indeed, fig. 14 shows t h a t there is a certain relation between this angle and mw. A similar relati/Y~x/r/rnicr /2 P, 8 10 $ zl 3 2 I on, however, seems to I 26 f exist between mp~ and the distance be2Y / tween P and the 2¢ / middle of the base (fig. 15). 2J / The number of observations and the 22. / dispersion of the re2/. /, / sults are too g r e a t to allow a definite de20. 2 termination of the /.q law t h a t is in force here. But this is not the purpose of this /7 article. The important thing is t h a t it will ' I.~6 I, /4 ~ Y /5.8 143 " '.'Z21ZSl~S 186 ~ / h c m now be clear t h a t for Fig. 15 the X- as well as for the Y-parallaxes the accuracy depends on the position of the point in the spatial model. The X - a n d Y-parallaxes were not measured simultaneously so t h a t from the present material it is not possible to conclude a n y t h i n g about the eventual correlation between those quantities.
/
V. Conclusians.
The foregoing chapters give no more than an example of the various problems that must be solved before enough data are gathered for a foundation of a theory of errors of aerial triangulation. The purpose of this article is simply to show: 1. t h a t it is useful to enlarge and deepen our knowledge about the elementary observations t h a t are used in photogrammetry; 2. t h a t this knowledge is of great importance since it is the only mcans for an objective and scientific comparison of the accuracy of methods and instruments; 3. t h a t a prolonged investigation of a g r e a t m a n y influences is necessary, which will require the execution of a large program of observations; 4. t h a t it will still take many years before the exact influence of the m a n y varying circumstances can be known; 5. t h a t it deserves recommendation to t r y to organise an international cooperation for this purpose. I take it for granted t h a t the editorial s t a f f of Photogrammetria will be glad to publish the results of those investigations so t h a t they will be available to all photogrammetrists. Lit. :
(1) W . K . B a c h m a n n . Th4orie des erreurs et compensation des triangulations a6riennes. Lausanne, 1946. (2) J . M . T i e n s t ~ a . A n extension of the technique of the method of least squares to correlated observations. Bull. G~odfisique, nouvelle s~rie, 1947, n r 6, p. 301--335.
61 APPENDIX For the readers who are not familiar with the work of Tienstra, the following short explanation of the principle may be of use. Cofactors are quantities t h a t are named weight-numbers in the classical l i t e r a t m e and are generally denoted by the letter Q (Q~I' Qle' Q2~ etc.). The importance of the ,,tensor of cofactors" is described in the articles of Prof. Tienstra in: Bulletin G~od~sique, nouvelle s~rie, 1947, p. 301--335 and 1948, p. 289-306. In short, the following is said: The classical technic of the method of least squares is based on the assumption t h a t the observations are not correlated, but in reality correlation is nearly always present although the explanation is not always known. This correlation is generally of physical character and the g r e a t e s t problem for an adjustment is the determination of numbers to indicate the amount of correlation and to adjust these correlated observations in the r i g h t way. Prof. Tienstra has given a universal solution for this second problem. He explains t h a t we can t r e a t the physical correlation in the same way as a linear algebraic correlation. A linear algebraic correlation arises from a linear t r a n s f o r m a t i o n applied to noncorrelated observations. Suppose we have three noncorrelated observations xl, x~ and x a. We t r a n s f o r m them w i t h the formulae: X = alx 1 q- a.ax 2 A- aa~ca X a = blx 1 + b..~x2 + b:~xa (1) X -- cloc1 "4- C2x.a + cax a The quantities X , X z and X a are now correlated. Suppose the weights of x~, x z and ( 1 1 ) x a t ° b e g11' gu2 .and gaa' and their cofactors resp. gll, g,'a~ and ga3 g l , = ~77, g.,.,=g.~getc. then the tensor of cofactors of the quantities xl, x2 and x a is gl~ 0 0 0 g22 0 0 0 gaa
(2a) and the weight-tensor
g~l 0 0 I 0 gw., 0 1 0 0 ga'~
(25)
If we name the cofactors of X1, X, etc. resp. G ,t, G 12 etc. then their tensor is: ] G~ G l'a
a13
G2~ Cr31 G 2'a G a2
e~a
(3 a) and the weight-tensor
G33
Gll Grz G~a G, 1 G=,., G2a
G31 632
(:P')
633
The values of (3 a) are found from (1) by means of the well-known special law of propagation of errors, namely:
(~11 = 6611741gll~- ~20,2g,2:2-~ a.~%gaa Gra = a l b l g l l + a.,b..,g ='a ÷ a:~bagaa etc. and the values of G~,, G~ etc. in (3b) are found from (3 a) by the weight-equations: G l l G l l + G.,1G21 + .. :~ 1 l G ll Grz + G ,1G2= + .. = 0 In total 3 sets of 3 equations. I f now q , e~ and e a are the corrections to x , x, and x a the method of least square.s learns t h a t gns~s~ + .q.z2a2e., -k gaJaSa m u s t be a m i n i m u m .
62 P r o f . T i e n s t r a p r o v e s t h a t , w h e n we a d j u s t t h e q u a n t i t i e s X~, X.~ a n d X 3 w i t h t h e c o r r e c t i o n s E l , Eu a n d Ea t h e m i n i m u m - c o n d i t i o n t a k e s t h e f o r m : G l l E 1 E 1 + GI~E~E ~ + G21E2E1 + G~E2E. 2 + . . . .
----> rain.
f r o m w h i c h h e d.evelops t h e f u r t h e r t e c h n i c s of t h e a d j u s t m e n t . T h e k n o w l e d g e - o f t h e t e n s o r (3a) is t h e r e f o r e n e c e s s a r y . I n t h e c a s e o f p h y s i c a l c o r r e l a t i o n t h i s t e n s o r c a n also be d e t e r m i n e d by m e a n s o f a s t a t i s t i c a l e x a m i n a t i o n of t h e o b s e r v a t i o n s . T h e g r e a t i m p o r t a n c e o f t h i s conception c o n s i s t s t h e r e f o r e o f t h e f a c t t h a t , once t h e t e n s o r of c o f a e t o r s o f o u r o b s e r v a t i o n s h a s been d e t e r m i n e d , it is n o t i n t e r e s t i n g if t h e e x i s t i n g c o r r e l a t i o n is of p h y s i c a l or a l g e b r a i c a l n a t u r e . T h e a d j u s t m e n t - r u l e s a r e u n i v e r s a l a n d c a n be u s e d f o r both cases.
Table I. Stereoplanigraph
Autograph A 5 Point • mx2
•
mY2
f
mxy
~t x 2
•
TtlY2
r i
~xy
I
55
73
+ 4
132
180
--40
77
69
--4
126
106
--14
3
93
75
+ 4
155
121
+ 8
4
63
65
+4
95
110
-- 8
5
128
114
÷ 8
103
76
--14
6
83
123
+24
129
137
--16
7
73
81
--12
119
117
+ 8
8
77
70
-- 1
122
166
--26
9
112
121
+45
122
135
--30
85
88
÷ 8
123
128
--15
Average
63
II ~ II ~°
I
÷
÷
÷
÷
I
÷
I
I iI °°
~ o ~ o o÷o ~ o ~ o c~
r~
I
11
64
I
+
÷
+
+
+
+
+
L'-.
+
I]
~,1 ~
L" ~
-~
~
l
~
@'~
L""
L'-'
+
+
+
I
+
~
~
I
÷
÷
+
,I
I
I
÷
li
+
I
I
I
I
i