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Analytical aerial triangulation with analogue instruments
279
Analytical Aerial Triangulation with Analogue Instruments by N. I. L E V Y I.T.C., Delft 1.1. So f a r analytical aerial triangulation has been applied with a special i n s t r u m e n t only, namely a comparator. However, there is no reason w h y the usually available universal instruments or precise plotters should not also be exploited for this task. Hereby their use is extended and the execution of analytical triangulation is made possible with these instruments over the whole range of photography, from normal to superwide angle. 1.2. This paper gives a description of a general method of analytical aerial triangulation for analogue instruments. The method is actually only a p a r t of a whole system of analytical triangulation developed by the author for application in the I.T.C. The system is general and applicable to all commonly used instruments, comparators as well a s analogue, whether universal or plotters. Here, however, we shall limit ourselves to a short description of that p a r t dealing with the analogue instruments, except for concepts of a general character. F o r more details the reader is referred to the author's ITC publication. 1) 2.1. The analogue i n s t r u m e n t is used as a device for measuring image coordinates. The photographs are set horizontally and coordinates of image points are measured in the projection in the nan-exact model thus formed. Each point is measured stereoseopically with the help of X, Y, Z and by base component. Whenever by is not available, lateral rotation w is used instead. In universal instruments, base inside and base outside a r e used in alternate models, maintaining the position of the common photograph. In plotters, however, this is not possible, so the triangulation is carried out by independent models. The common photograph is t r a n s f e r r e d from one plate to the other r a t h e r than being rotated through 2O0g, because it avoids certain error accumulation. It should however be noted t h a t this t r a n s f e r of the photographs does not entail, as in some other triangulation methods with plotters, the t r a n s f e r of any orientation element. To ensure a better precision of the triangulation, corrections are applied analytically for instrumental errors incorporated in the projection and measuring system, as well as for erroneous horizontal setting of the photographs. A purely analytical orientation method is used for the computation of the relative orientation. (Here the orientation method of C. M. A. v. d. Hour has been applied, although others are obviously applicable). 2.2. F o r the reduction of the observation data a general reduction formula has been derived. This formula makes use of formal parameters and actual to formal representation technique. Let, r, p, q, ~r, ~)x, ~ , ¢)z and ~)c be formal parameters. " r " represents the element used to measure the y parallax. "p and q" are coefficients of this measurement in X and Y respectively. ~r, Qx, ~y and Qz are t r a n s f o r m a t i o n coefficients of r, X, Y and Z respectively. Qc denotes the camera used for the measurements of the y parallax, i.e. = 0 when camera was held fixed, = 1 when it was used. 1) N. I. Levy: Analytical Aerial Triangulation with Analogue Instruments, I.T.C. Publication A/31, summer 1964. 1 8
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Photogrammetria, X I X , No. 7
280
If:
AX = q ( r - - t o )
;
A Y = p ( r - - r o)
(1)
~A= = @=~r ~%
;
@~v =
(2)
~v ~r ~c
and K is a scale factor, then the following formula is general and holds for any instrument, w h e t h e r analogue or comparator.
V
a22 j [ [ ( Y - - Y o + A Y @ ~ y )
a21
In analogue i n s t r u m e n t s K -
K]~ v
C ( Z - - Z o) ~ whereas for c o m p l r a t o r s K = 1.
*v"
÷X
_. 7 - - ÷ u '
I/
-b~ o-
/
Fig. i 2.3. To a d a p t this formula to a specific instrument it is thus only necessary to substitute the actual values of the formal p a r a m e t e r s corresponding to t h a t particular instrument. Formulae (3) has two significant practical properties. Firstly, it is general and thus applicable to any i n s t r u m e n t under consideration, and secondly, it is identical for each camera associated in the measurements, as one is free to allocate different Values to the p a r a m e t e r s of different cameras. These actually lead to the development of a Universal Triangulation P r o g r a m m e (UTRIP-1) which is applicable to all commonly used instruments. As an example, table 1 represents the actual values of the p a r a m e t e r s of each camera of the Wild A-7 and Wild A-8 instruments. Table 1
W//o'A-7
b/.÷I: O
*I -I ¢ l ~ l
kV/ZdA-8
u~" -I @
~" ~I -I rl .o -I o
~ ~z ~ e r s
[22]
/ 0 H rel'heP ~hen i~o Z
0
-I
0
+l
0
+I V x ' x & ( ~ 1
0
~
r~Iz
+I
[(z-z,~, :'~'/~r~fl-
+I
Levy, Analytical Aerial Triangulation with Analogue Insfruments
281
A s s u m i n g t h a t t h e fiducial centre coincides with the principal point, X o and Yo c a n be determined f r o m m e a s u r e m e n t s of the 4 fiduclal m a r k s . In plotters where this is impossible (base zero c a n n o t be set) X o and Yo are determined f r o m a direct m e a s u r e m e n t of the fiducial centre if premarked, the e n g r a v e d cross on the plate holder, or the point a t which the space rod is vertical. 3.1. Hitherto, theoretical coordinate a x i s s y s t e m a n d error-free i n s t r u m e n t were a s s u m e d . This, however, is never realised in practice. Mechanical a d j u s t m e n t is r a t h e r limited. Moreover, in n o r m a l production work i n s t r u m e n t s are r a r e l y a d j u s t e d or even checked. It therefore seems a d v a n t a g e o u s to apply corrections for the i n h e r e n t i n s t r u m e n t a l e r r o r s analytically. The precision is t h u s increased, and to a c e r t a i n e x t e n t becomes independent of the state of a d j u s t m e n t of the i n s t r u m e n t . 3.2. Since t h e errors involved are u s u a l l y small t h e y can be considered a s being linear over s h o r t distances along the X a n d Y axes. Accordingly a second order correction ~urface is applied. Two such correction s u r f a c e s , one for X a n d one for Y, are derived f r o m grid m e a s u r e m e n t s for each of the i n s t r u m e n t cameras, These s u r f a c e s are superimposed analytically on each m e a s u r e d p h o t o g r a p h d u r i n g computation. Correction to image points a r e derived u s i n g f o r m u l a (4) below, (see also Fig. 2).
n
m
u p = Au 1 + ~ ( z l u 2 - - A u 1) + ~ - ( A u 4 - - A u n
mn 0 + ~-(dUl--Z]u2
m
% = Avl + ~ (Ave-- Avl) + s
+ AUa--Au4)
(4)
mn
( A % - - Av 1) + ~ -
(Av 1 - A v e +
Ava " Avi)
In which : zJuo and
are the corrections of point ~ in u an v directions respectively, and A v I...... A v 4 are the corrections of the four related grid points, m = u I - - uo, n = v I - - vo, and s = the distance between two successive grid points. To illustrate, a correction surface of one component is depicted graphically in Figure 2. Av o
Au I...... Au 4
v
×/ I//Z LZI/
I
coO','~½a/~$ Fig. 2
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Photogramme~ria, X1X, No. 7
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A 25 point grid is used, as it seems to be a practical compromise between maximum precision and practical considerations. Having measured the grid at a certain projection distance, two grid points along the Y (or the X) axis are remeasured at a different projection distance for the determination of C c and Z o in the i n s t r u m e n t . 3.3. An i m p o r t a n t feature of this correction method is t h a t it is general and independent of the i n s t r u m e n t type. F u r t h e r m o r e it comprises the total effect of instrumental e r r o r s , including errors incorporated in the projection, as represented at the grid points. Consequently various analogue instruments including those based on the affine principle can be applied, and a simple and easy measuring procedure is achieved. To a certain extend one need not w o r r y about the state of a d j u s t m e n t of the instrument nor about exact zero setting of orientation elements, 3.4. Corrections are also derived for film shrinkage, radial lens distortion, atmospheric refraction and earth curvature. For film shrinkage an affine t r a n s f o r m a t i o n based on the calibrated coordinates of the four fiduciat marks is applied. This is obviously possible only with those instruments in which the four fiducial m a r k s of every photo can be measured, i.e. universal instruments. For the radial lens distortion a set of third order curves are applied to represent the symmetric p a r t while a set of second order curves represent the asymmetry. A combined correction is applied for the effect of earth curvature and atmospheric refraction. 3.5. Let A U and A V represent the combined value of all corrections in U and V expressed on the image plane then the "ideal" image coordinates U V are derived as follows (see formula (3)) :
[1[
=
all
av,
a21
a22
[ [ ( X - - X o + A X e ~ ) K] e.~
J.
[ [ (Y - - Yo + d Y e~v) K ] Qv
I +
AV
(5)
4.1. In both universal i n s t r u m e n t s and precise plotters it is possible to measure independent models. This, however, is an inefficient procedure for the former instruments. It is more favourable (and thus applied here) to follow the conventional aeropolygon procedure of base inside and base outside in alternate models. In plotters this is not possible and therefore independent models a r e measured. To strengthen the connection o5 adjacent models it is advisable to prick two points on V axis of each photograph so t h a t they can be observed in each of the corresponding models. 5.1. A programme r e f e r r e d to as UTRIP-1 (Universal Triangulation Programme) has been w r i t t e n for the medium size Stantec-Zebra computer. The programme accomodates the p a r a m e t e r s of a number of analogue i n s t r u m e n t s most commonly used and can thus be applied to each one of them. These i n s t r u m e n t s are: Wild A-7 Autograph, Zeiss Stereoplanigraph C-8, Santoni Stereometrograph IV, Nistri Photostereograph ill2, Poivilliers Stereotopograph B, Wild A-8 Stereo Plotting Machine, Santoni Simplex III, Thompson-Watts Plotter and Zeiss J e n a Stereometrograph. To apply the programme to other instruments it is only necessary to introduce heir corresponding p a r a m e t e r s prior to computation. 5.2. To cope with practical difficulties and to achieve a reliable and simple procedure the programme has been made as general and as flexible as possible and various checks have been incorporated. Thereby the output of the analogue i n s t r u m e n t can be used directly as input for the computer without the need for any revisions or corrections. E v e r y model is checked for its completeness and stability and every registration for its approval.
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Levy, Analytical Aerial Triangulation with Analogue Instruments
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Most analogue i n s t r u m e n t s are not equipped with automatic registration for by or co. Although the time involved in manual registration of one of these elements is insignificant, the provision of automatic registration is recommended. It would f u r t h e r simplify and economize the application of this method. 6.1. The most significant advantage of the described method is t h a t it provides the possibility to apply the normal analogue instruments to analytical aerial triangulation. This is of a special importance to photogrammetric organizations who do not have sufficient triangulation work to j u s t i f y the purchase of a special instrument. Such organizations are now offered the possibility of applying analytical triangulation with their available analogue instruments, whether universal or plotters. I t is a purely analytical method and as such possesses all properties and advantages of analytical triangulation. The method is simple and general. It is applicable to all commonly used analogue instruments. One need not worry, to a certain extent, about the state of adjustment of the instrument, nor about zero settings of orientation elements. The analytical correction method applied for instrumental errors permits practically a r b i t r a r y settings of the orientation elements and corrects numerically for instrumental errors. No t r a n s f e r of scale or any orientation element is involved in proceeding from one model to the other, even in the case of a plotter where independent models are triangulated.
18.
284
Discussion on the papers of InghiUeri, B r a z i e r and L e v y
Visser: I think it is e a s y to u n d e r s t a n d w h y Prof. T h o m p s o n propounds the method of i n d e p e n d e n t models, because in the M a r k II T h o m p s o n W a t t s plotter it is no longer possible to t r a n s f e r the vertical t h r o u g h the strip as it w a s in the M a r k I. B u t in the case of the A8 and the Stereosimplex III there a r e simple m e a n s of t r a n s f e r r i n g the vertical t h r o u g h the strip, for i n s t a n c e by m e a n s of t h e cross levels, which give v e r y good precision a n d which a r e discarded in t h i s case. P e r h a p s Mr. Brazier or Prof. Inghilleri could explain w h y t h e y do not w a n t to use these cross levels? W h e n t h e y a r e employed the s u b s e q u e n t computation is so simple t h a t it can a l w a y s be done by a desk computer. Brazier: I believe t h a t i f you h a v e a c o m p u t e r you m i g h t as well use it, and I t h i n k the observing procedure in the A8 is considerably simplified if you do not h a v e to m a k e a comparison between one model a n d the next. E a c h model is literally independently observed. The only t h i n g t h a t you keep is the calibration of the Z s e t t i n g on the i n s t r u m e n t a n d the base. Inghilleri: If I understood well, you have asked w h y we do not use the cross levels in the Stereosimplex I I I ? We h a v e had no experience in the use of the cross level. It is not only m y opinion t h a t the cross level device does not have the s a m e a c c u r a c y as analytical f o r m a t i o n of t h e model. The cross level device h a s a n a c c u r a c y t h a t I a s s u m e to be about 2 m i n u t e s . Also, the i n s t r u m e n t work is more complicated. We, for instance, have been able to use a n unskilled operator. Thompson: I h a v e v e r y good reasons to t h i n k t h a t these devices should not be used. F i r s t of all as r e g a r d s accuracy, I do not t h i n k t h i s is really a decisive point because I h a d seriously t h o u g h t a t one time of u s i n g autocollimators w i t h m y i n s t r u m e n t . I c a n n o t u s e levels in m y i n s t r u m e n t because the line in t h e i n s t r u m e n t which r e p r e s e n t s the a i r b a s e is in f a c t held horizontal in the model space, b u t I can use autocollimators a n d t h e y c e r t a i n l y have a n y desired precision one can possibly wish. B u t m y r e a s o n s for not u s i n g t h e m are, f i r s t of all, t h a t the i n s t r u m e n t r e m a i n s in exactly the s a m e condition of a d j u s t m e n t if you do not touch the absolute orientation e l e m e n t s ; and second, t h a t if you add these devices t h e y still do not g e t rid of the computation, because you still h a v e to t r a n s f o r m the coordinates as f a r as t h e v a l u e s X and Y are concerned. You. h a v e t h e r e f o r e still to do some computation a n d I h a v e come to the conclusion you m i g h t as well do the whole lot. Erez: I h a v e a r e m a r k about the distortion s u r f a c e s which Mr. Levy spoke about. E v e r y o n e who h a s a d j u s t e d i n s t r u m e n t s knows v e r y we]l t h a t there is d a n g e r in distortion s u r f a c e s , especially when you touch one of the c a r d a n s of the i n s t r u m e n t , because there could be some s t r a i n in one of the c a r d a n s of the i n s t r u m e n t . T h e n the r e a d i n g s you obtain v a r y w i t h c h a n g i n g h e i g h t or with change of position. So I t h i n k when you use those distortion s u r f a c e s , the i n s t r u m e n t h a s to be calibrated several times, not only one set of 25 points, b u t repeated several times in d i f f e r e n t directions. Levy: I did not m e n t i o n t h a t for the grid m e a s u r e m e n t s we use f o r w a r d and back w a r d m e a s u r e m e n t s and take the m e a n of them, a s I t h o u g h t this w a s quite obvious. It is obvious t h a t if you t a k e a v e r y dense r e s e a u or g r i d a n d m e a s u r e t h e errors a t very
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Discussion on the papers of InghiUerie, B r a z i e r and L e v y
285
s h o r t d i s t a n c e s you will g e t a b e t t e r r e p r e s e n t a t i o n of t h e e r r o r s in t h e i n s t r u m e n t , b u t t h i s is n o t p r a c t i c a l a n d t h e r e f o r e we p r e f e r to u s e a k i n d of c o m p r o m i s e b y t a k i n g 25 points. It is obviously possible t h a t if t h e i n s t r u m e n t is u n s t a b l e a n d y o u r e p e a t r e a d i n g s in one d i r e c t i o n only, y o u will n o t d e t e r m i n e all e r r o r s or e r r o r s o u r c e s , t h e r e f o r e we u s e f o r w a r d a n d b a c k w a r d m e a s u r e m e n t s . I do t h i n k t h a t t h i s a n a l y t i c a l c o r r e c t i o n m e t h o d is f a r b e t t e r t h a n a m e c h a n i c a l a d j u s t m e n t , c e r t a i n l y for a r e a s o n a b l e m a g n i t u d e of errors, as you cannot adjust mechanically as accurately as you can determine and correct the errors analytically. Jerie : R e g a r d i n g Mr. B r a z i e r ' s c o m m e n t s c o n c e r n i n g t h e a d v a n t a g e s of t h e m e t h o d of i n d e p e n d e n t m o d e l s ; I do n o t d e n y t h e s e a d v a n t a g e s , e s p e c i a l l y b e c a u s e o f t h e s i m p l i c i t y of t h e m e t h o d . H o w e v e r , t h e r e w e r e two a r g u m e n t s g i v e n by M r . B r a z i e r w i t h w h i c h ! cannot agree. F i r s t l y , h e s t a t e s t h a t u s i n g t h i s m e t h o d one c a n s t a r t f r o m t h e b e g i n n i n g or t h e e n d o f t h e s t r i p , b o t h o f w h i c h you c a n also do in s t r i p t r i a n g u l a t i o n . H o w e v e r , I do n o t c o m p l e t e l y a g r e e t h a t y o u c a n s t a r t in t h e middle o f t h e s t r i p j u s t a easily, b e c a u s e y o u c e r t a i n l y lose t h e i n n e r o r i e n t a t i o n of t h e one p h o t o g r a p h , w h i c h h a s t h e n to be p u t in t h e p l a t e h o l d e r twice. Secondly, he s t a t e s t h a t a l e s s e r q u a l i f i c a t i o n is r e q u i t e d f o r t h e o p e r a t o r . I c a n n o t a g r e e w i t h t h i s , b e c a u s e one h a s to c a r r y o u t e s s e n t i a l l y t h e s a m e o p e r a t i o n s in t h e m e t h o d of i n d e p e n d e n t p a i r s a s in t h e c a s e of c o n v e n t i o n a l s t r i p t r i a n g u l a t i o n in u n i v e r s a l i n s t r u m e n t s , viz. r e l a t i v e o r i e n t a t i o n a n d m e a s u r e m e n t s of coordinates. T h e o n l y d i f f e r e n c e b e t w e e n t h e two m e t h o d s is t h a t t h e scale t r a n s f e r is n o t c a r r i e d o u t in t h e i n s t r u m e n t a l p h a s e . H o w e v e r , t h e scale t r a n s f e r is a c h i e v e d i n e s s e n t i a l l y t h e s a m e w a y in both m e t h o d s , viz. b y m e a s u r e m e n t s o f c o o r d i n a t e s . I t h i n k t h e real a d v a n t a g e of t h e m e t h o d of i n d e p e n d e n t m o d e l s is t h a t it is s i m p l e r f r o m t h e o r g a n i s a t i o n a l p o i n t of view a n d t h a t i n s t r u m e n t s c a n be u s e d w h i c h a r e also u s e d ~n t h e s u b s e q u e n t p l o t t i n g p h a s e . S c h u t : I w o u l d like to go b a c k once m o r e to t h e a c c u r a c y of i n d e p e n d e n t m o d e l t r i a n g u l a t i o n . P r o f . H a l l e r t p u b l i s h e d a p a p e r on a n i n v e s t i g a t i o n o f t h e a c c u r a c y of d i f f e r e n t m e t h o d s o f t r i a n g u l a t i o n , a n d he f o u n d t h a t t h e m e t h o d w h e r e t h e m o d e l s a r e c o n n e c t e d v i a t h e o r i e n t a t i o n e l e m e n t s of t h e c o m m o n p h o t o g r a p h w a s t h e m o s t a c c u r a t e . I t s e e m s f r o m t h i s t h a t we m u s t conclude t h a t we s h o u l d only u s e t h e i n d e p e n d e n t m o d e l t r i a n g u l a t i o n if we do n o t h a v e a n i n s t r u m e n t a v a i l a b l e in w h i c h t h i s c o n t i n u o u s t r i a n g u l a t i o n c a n be p e r f o r m e d . F u r t h e r , Mr. B r a z i e r s t a t e d t h a t t h e r e s u l t of t h e i n d e p e n d e n t m o d e l t r i a n g u l a t i o n is i n d e p e n d e n t of t h e s e q u e n c e of m e a s u r e m e n t of t h e models, w h i l e t h i s w o u l d n o t be t h e c a s e w i t h c o n t i n u o u s t r i a n g u l a t i o n on a u n i v e r s a l i n s t r u m e n t . I see no r e a s o n f o r t h i s d i f f e r e n c e . A s f a r a s s y s t e m a t i c d e f o r m a t i o n is c o n c e r n e d t h e o n l y b a s i c d i f f e r e n c e b e t w e e n t h e two t r i a n g u l a t i o n m e t h o d s is t h a t w i t h t h e i n d e p e n d e n t m o d e l t r i a n g u l a t i o n t h e u n i t is one model, w h i l e w i t h t h e c o n t i n u o u s t r i a n g u l a t i o n t h e u n i t is two m o d e l s , one w i t h b a s e - i n a n d o n e w i t h b a s e - o u t . I n b o t h c a s e s t h e t r i a n g u l a t i o n i n s t r u m e n t c a n be t h e c a u s e o f a s y s t e m a t i c d e f o r m a t i o n of t h e t r i a n g u l a t i o n s t r i p , a n d in b o t h c a s e s t h i s d e f o r m a t i o n d e p e n d s u p o n t h e d i r e c t i o n of t h e t r i a n g u l a t i o n . van der Weele : I j u s t h a v e one s m a l l q u e s t i o n f o r Mr. B r a z i e r , w h o p o i n t e d o u t t h a t t h e c o m m o n p o i n t s b e t w e e n a d j a c e n t m o d e l s a r e m a r k e d w i t h c r o s s e s of a r a t h e r l a r g e size, 6 m m is m e n t i o n e d . H e said t h a t t h e size of t h e l i n e s is s u c h t h a t y o u c a n p u t t h e m e a s u r i n g m a r k v e r y a c c u r a t e l y on t h e cross. B u t as f a r as I u n d e r s t o o d y o u also h a v e to o b s e r v e s t e r e o c o p i c a l l y on t h o s e points. M y q u e s t i o n is, does n o t t h i s r a t h e r l a r g o cross disturb the stereoscopy?
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Photogrammetria, XIX, No. 7
Schermerhorr~: One o f the a r g u m e n t s for the m e a s u r e m e n t of independent models is t h a t you use the i n s t r u m e n t in the s a m e position, t h u s t h e r e is no mechanical difference, as with base-in and base-out t r i a n g u l a t i o n . I now d r a w y o u r a t t e n t i o n to the new Santoni i n s t r u m e n t which will be shown ix Lisbon, n a m e l y the S t e r e o c a r t o g r a p h V. W i t h this you can m e a s u r e with base-in and base-out, a n d still m a i n t a i n t h e s a m e distribution of w e i g h t s in the i n s t r u m e n t . I believe t h i s is of g r e a t importance, and it eliminates one of the a d v a n t a g e s which h a s a l w a y s been m e n t i o n e d in f a v o u r of m e a s u r e m e n t s of i n d e p e n d e n t pairs. Zarzycki: I f we use t h e A7 or s i m i l a r t y p e of i n s t r u m e n t t h e n the a d v a n t a g e of i n d e p e n d e n t p a i r s can be debated. B u t if we use the A8 or a simpler i n s t r u m e n t , such as the B8 or the Kelsh plotter, where it is v e r y inconvenient to t r a n s f e r the scale w i t h the height, a n d g r a p h i c a l methods are u s u a l l y employed for scale t r a n s f e r , t h e n I t h i n k the independent p a i r s method h a s m e r i t s and can produce good results. The t r a n s f e r of scale would be done a n a l y t i c a l l y a n d this is obviously more a c c u r a t e t h a n a g r a p h i c a l method of t r a n s f e r r i n g t h e scale. Mr. B r a z i e r told u s t h a t he bridged about 150 overlaps and t h a t he had 200 control points. T h i s is more t h a n one point per model. I would like to know how long the strips were t h a t he bridged, and w h a t w a s the distribution of t h e vertical control points. Also, I would like to know w h e t h e r he h a s employed his m e t h o d in b r i d g i n g long strips, let us s a y 15 or 20 models between control. Levy: I h a v e a question for Mr. Brazier. In his paper, p a r a g r a p h 16, in c o m p a r i n g the a n a l y t i c a l to the a n a l o g u e methods, he w r i t e s t h a t " t h e model is determined f r o m relative orientation which is derived f r o m the observation of i n n u m e r a b l e points of detail". Could:he c l a r i f y this declaration, is it b e t t e r t h a n the c o m p a r a t o r relative orientation ? Kure: I h a v e done some tests a n d found less a c c u r a t e r e s u l t s with independent model t r i a n g u l a t i o n t h a n with n o r m a l aerial t r i a n g u l a t i o n , which in m y opinion is m a i n l y due to the less a c c u r a t e t r a n s f e r of scale. In i n d e p e n d e n t model t r i a n g u l a t i o n the t r a n s f e r of scale is done u s i n g the coordinates of the two wing points, and in n o r m a l aerial t r i a n g u l a t i o n it is done on heights. A s you h a v e m o r e u n k n o w n s in independent model t r i a n g u l a t i o n you get less a c c u r a c y a n d the only w a y to improve t h a t would be to use m a y be two sets of w i n g points in order to improve the t r a n s f e r of scale in the independent model t r i a n g u l a t i o n . Thompson: The l a s t speaker m a i n t a i n s t h a t the i n a c c u r a c y arises f r o m the scale t r a n s f e r . M y own look a t the t h e o r y of this showed m e t h a t the actual theoretical acc u r a c y is g r e a t e r when scaling f r o m w i n g points t h a n it is f r o m heights. I h a v e always r e g a r d e d the e r r o r f r o m the w i n g points as being due to more questions of identification t h a n to a n y theoretical reason. W h e n scaling f r o m a c e n t r a l point you do not have to i d e n t i f y the points a s accurately. B u t if one uses a point m a r k e d on the middle picture of three in such a w a y t h a t it is a small circle, a p p r o x i m a t e l y or exactly the s a m e size a s the stereoscopic m a r k , I do not t h i n k t h a t the error of identification is at all serious. I f I m a y , I would like to m a k e one or two other points. A s f a r as I understood Prof. Inghilleri, he did n o t c a r r y out relative orientation by a computional method. I personally t h i n k t h a t all relative orientations should be carried out at least p a r t i a l l y by computation, a n d t h a t c e r t a i n l y the z]eo element should be determined in this way. It is not only more a c c u r a t e , b u t also quicker, especially when the g r o u n d is not v e r y f l a t and w h e n the p h o t o g r a p h s a r e v e r y tilted. It enables one to distribute a n y error properly over the model, even if you simply confine the computation to the zion.
D8]
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C o r r e c t i o n s for d i s t o r t i o n a r e n o t p a r t i c u l a r l y c o n n e c t e d w i t h t h e m e t h o d of indep e n d e n t models, it is a g e n e r a l point. H o w e v e r , a s Mr. L e v y did s h o w c u r v e s of d i s t o r t i o n s , I t h i n k t h e r e is a p o i n t h e r e t h a t is overlooked. W h e n t h e r e is a v a r y i n g r a d i a l d i s t o r t i o n , lLs he h a d in h i s e x a m p l e , y o u m u s t h a v e a s s o c i a t e d t a n g e n t i a l d i s t o r t i o n s , p o s s i b l y n o t of t h e s a m e a m o u n t , b u t c e r t a i n l y s i g n i f i c a n t a n d of t h e s a m e o r d e r . It s e e m s to m e t h a t one h a s e i t h e r to m e a s u r e t h e s e d i s t o r t i o n s or to p r o d u c e s o m e k i n d of t h e o r y to e n a b l e t h e m to be d e d u c e d f r o m t h e v a r y i n g r a d i a l d i s t o r t i o n s , a t h i n g I do n o t t h i n k impossible. I h a v e done s o m e w o r k on t h e q u e s t i o n of v a r y i n g r a d i a l d i s t o r t i o n s a n d h a v e f o u n d some v e r y i n t e r e s t i n g r e s u l t s . I f o u n d f o r e x a m p l e t h a t f o r r a d i a l d i s t o r t i o n s t h a t v a r y a s one g o e s a r o u n d in a z i m u t h , t h e s i m p l e m o d e l of a w e d g e p r i s m is c e r t a i n l y n o t s u f f i c i e n t to a c c o u n t f o r t h e f a c t s . I f y o u do a F o u r i e r a n a l y s i s of t h e d i s t o r t i o n as y o u go r o u n d in a z i m u t h , I h a v e f o u n d n o t o n l y t h e u n e v e n t e r m s b u t also t h e e v e n t e r m s of a F o u r i e r series. I h a v e d o n e t h i s f o r a n A v i o g o n lens.
Ackermann: Mr. L e v y h a s r e f e r r e d to correction o f f i l m ~ h r i n k a g e b a s e d on m e a s u r e m e n t s a n d c a l i b r a t i o n d a t a of t h e f i d u c i a l m a r k s . T h i s is c e r t a i n l y t h e only w a y to f i n d o u t a b o u t f i l m s h r i n k a g e f o r p h o t o g r a p h s of t h e n o n - r e s e a u type. H o w e v e r , one h a s to be r a t h e r c a r e f u l w i t h s u c h a p r o c e d u r e . I t s s u c c e s s d e p e n d s on w h e t h e r t h e s h r i n k a g e a t t h e location of t h e f i d u c i a l m a r k s is r e p r e s e n t a t i v e o f t h e whole p h o t o g r a p h . I c a n show two e x a m p l e s o f s u p p o s e d l y i l l - t r e a t e d f i l m w h o s e s h r i n k a g e p a t t e r n s a r e s u c h t h a t t h e d i s p l a c e m e n t s o f t h e f i d u c i a l m a r k s a r e n o t r e p r e s e n t a t i v e of t h e t o t a l a r e a . I n s u c h c a s e s n u m e r i c a l c o r r e c t i o n s b a s e d on f i d u c i a l m a r k s (especially on t h o s e in t h e c o r n e r s ) w o u l d h a v e a n a d v e r s e e f f e c t a n d w o u l d i n t r o d u c e s e r i o u s e r r o r s .
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Levy: I c o m p l e t e l y a g r e e w i t h Dr. A c k e r m a n n ' s r e m a r k . T h i s is a c t u a l l y one of t h e r e a s o n s w h y I h a v e u s e d a t r a n s f o r m a t i o n h a v i n g 6 p a r a m e t e r s r a t h e r t h a n one h a v i n g 8 p a r a m e t e r s w h i c h l e a v e s no r e d u n d a n c y , t h a t is to s a y no check. Thompson: I w a s p a r t i c u l a r l y c a r e f u l t h i s m o r n i n g w h e n g i v i n g t h e f i g u r e of 14 i~m f o r t h e s t e r e o c o m p a r a t o r r e s u l t s n o t to m e n t i o n w h e t h e r c o r r e c t i o n s h a d been made for the reseau. In fact the photographs carried a reseau and those results of 14 i~m w e r e t h e r e s u l t s we o b t a i n e d u s i n g t h e r e s e a u . H o w e v e r , one o f t h e m a i n l e a s o n s [29]
288 for this experiment the photograph an amount of perhaps second g r i d . T h e r e not think anything t h e r c s e a u . In f a c t
Photogrammetria, XIX, No. 7 w a s to f i n d o u t t h e e f f e c t of t h e r e s e a u . W e h a d s a n d w i c h e d u p w i t h accurate grid, which was displaced from the reseau crosses by an 0.1 m m . M e a s u r e m e n t s w e r e m a d e v i r t u a l l y s i m u l t a n e o u s l y on t h i s w a s o n l y one r e a d i n g to photo p o i n t s a n d two r e a d i n g s to g r i d s . I do could be b e t t e r d e s i g n e d to s h o w t h e i n c r e a s e in a c c u r a y due to u s i n g w i t h o u t t h e r e s e a u we got a b e t t e r a n s w e r !
Schut: I w o u l d like to m a k e a r e m a r k a b o u t t h e s e f o r m u l a e w h i c h t o d a y h a v e b e e n p e r s i s t e n t l y called S c h u t ' s f o r m u l a e . I n t h e f i r s t place, t h e y h a v e been d e s i g n e d f o r t h e a b s o l u t e o r i e n t a t i o n of a model in a o n e - s t e p p r o c e d u r e a n d t h e y c a n do t h a t f o r a n y r o t a t i o n t h a t is r e q u i r e d . H o w e v e r , t h e y do n o t g i v e a l e a s t s q u a r e s a d j u s t m e n t in t h e s e n s e t h a t t h e s u m of t h e s q u a r e s of t h e d i f f e r e n c e s b e t w e e n t h e c o o r d i n a t e s of t i e - p o i n t s becomes a m i n i m u m . It w o u l d be i n t e r e s t i n g to see h o w m u c h t h e r e s u l t d i f f e r s f r o m a least squares adjustment. Thompson: I e n t i r e l y a g r e e w i t h w h a t Mr. S c h u t s a y s , b u t on t h e o t h e r h a n d I do n o t t h i n k t h e m a t t e r is a n i m p o r t a n t one. T h e v a l u e in h i s f o r m u l a e to m y m i n d , a r i s e s e n t i r e l y f r o m t h e f a c t t h a t t h e y a r e e x a c t a n d e a s i l y solved a n d t h a t y o u c a n d e t e r m i n e l a r g e r o t a t i o n s w i t h t h e g r e a t e s t of ease. Once y o u h a v e d e t e r m i n e d t h i s l a r g e r o t a t i o n I t h i n k t h e r e s i d u a l d i s c r e p a n c i e s a r e q u i t e u n i m p o r t a n t . T h i s s u r e l y is a m a t t e r f o r t h e f i n a l block a d j u s t m e n t a n d I do n o t t h i n k t h a t a t t h i s s t a g e you h a v e to w o r r y a b o u t l e a s t s q u a r e s f i t s b e t w e e n models. Inghilleri: I c o n g r a t u l a t e Mr. B r a z i e r on t h e r e s u l t s he got w i t h t h e i n d e p e n d e n t model t r i a n g u l a t i o n , b u t t h e e r r o r s t h a t I h a v e c o m p u t e d f o r m y w o r k a r e t w i c e a s l a r g e a s his. I t is r a t h e r d i f f i c u l t to c o m p a r e r e s u l t s w i t h o u t k n o w i n g t h e e x p e r i m e n t a l conditions. I w o u l d like to s a y t h a t in o u r case t h e t e r r a i n w a s m o u n t a i n o u s , t h a t t h e p a s s p a i n t s w e r e n o t m a r k e d a n d also t h e m a n n e r of a d j u s t m e n t m a y h a v e been d i f f e r e n t . Mr. B r a z i e r did n o t m e n t i o n h i s m e t h o d . I w o u l d also like to k n o w if M r . B r a z i e r ' s A 8 w a s e q u i p p e d w i t h a r e c o r d i n g device. I w o u l d like to p o i n t o u t t h a t t h e r e s u l t s t h a t we a n d o t h e r E u r o p e a n p h o t o g r a m m e t r i c i n s t i t u t e s o b t a i n e d f r o m t h e s a m e s t r i p of p h o t o s a g r e e d to a g r e a t e x t e n t . T h i s w a s o u r f i r s t e x p e r i m e n t in t h i s field a n d its a i m w a s to s h o w t h a t it is possib~le to o b t a i n b y i n d e p e n d e n t model t r i a n g u l a t i o n on t h e S t e r e o s i m p l e x III, r e s u l t s t h a t a r e a s good a s w h e n u s i n g t h e m o r e e x p e n s i v e u n i v e r s a l i n s t r u m e n t s . I n a n s w e r to P r o f . T h o m p s o n t h e r e l a t i v e o r i e n t a t i o n w a s b y e m p i r i c a l methods. Schermerhorn: I d r a w t h e a t t e n t i o n o f t h e a u d i e n c e to t h e f a c t t h a t t h e c o m p a r i s o n of r e s u l t s of d i f f e r e n t t r i a n g u l a t i o n s is a b s o l u t e l y u s e l e s s w h e n it is n o t c e r t a i n t h a t t h e s a m e p h o t o g r a p h i c m a t e r i a l h a s been used. W e h a v e s u f f e r e d s u f f i c i e n t l y f r o m t h i s k i n d of t r o u b l e in p h o t o g r a m m e t r i c d i s c u s s i o n s to convince m e t h a t we m u s t stop t h i s here. It is f o r t h i s r e a s o n t h a t we h a v e u n d e r t a k e n t h e t e r r i b l e job of c o m p u t i n g a n a r t i f i c i a l block, o f w h i c h you c a n all m a k e u s e f o r all k i n d s of m e t h o d s . All k i n d s o f e r r o r s h a v e been i n t r o d u c e d a n d we k n o w w h a t t h e a n s w e r should be a f t e r t h e comp u t a t i o n . I r e c o m m e n d t h a t you m a k e u s e o f it, it is a v a i l a b l e f r o m t h e I.T.C., a n d u s i n g t h i s y o u c a n c o m p a r e m e t h o d s a n d i n s t r u m e n t s , etc. b a s e d on t h e s a m e n e u t r a l m a t e r i a l . Thompson: Could I m a k e one p o i n t ? J u s t in s e l f d e f e n c e m a y I s a y t h a t all t h e f i g u r e s I h a v e q u o t e d t h i s m o r n i n g a n d t h i s a f t e r n o o n , h a v e been o b t a i n e d f r o m t h e s a m e set o f p l a t e s . I i n t e n d t h e s e f i g u r e s s i m p l y a s c o m p a r i s o n s b e t w e e n each o t h e r . T h i s is n o t i n t e n d e d to be a c o m p a r i s o n w i t h r e s u l t s t h a t e i t h e r Mr. B r a z i e r o r a n y o n e else h a s q u o t e d t o d a y , t h e y a r e only c o m p a r i s o n s w i t h i n t h e e x p e r i m e n t s I h a v e done.
[8o3
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289
Brazier: Mr. Schut m a d e a r a t h e r i m p o r t a n t s t a t e m e n t t h a t I do not t h i n k is quite correct, a l t h o u g h I h a v e never m y s e l f tested it. In the p a s t we did a lot of b r i d g i n g by a n a l o g u e methods, and our experience w a s t h a t w h e n the n u m b e r of p h o t o g r a p h s w a s large, t h e bz-curve developed v e r y rapidly. It is quite obvious t h a t t h i s curve is not symmetrical, as one end is v e r y m u c h more curved t h a n the other. I can not agree with Mr. Schut in t h a t i f you p e r f o r m an a n a l o g u e operation f r o m s a y 1--20 in an A7 or C8, you will get the s a m e bridge as o p e r a t i n g f r o m 20--1. In other words these two bridges cannot be superimposed in space, one on the other. T h e point I w a s m a k i n g this m o r n i n g as f a r as u n i q u e n e s s w a s concerned w a s a c o m p u t a t i o n a l one. H e r e we have the independent models observed, a n d you m a y take a n y p a i r of these models, a t the beginning, end or even in the middle and p e r f o r m the computational operations between them, a n d the r e s u l t i n g a r r a y of X Y Z points will be precisely t r a n s f o r m a b l e by Schut's equations f r o m one position to another. These two models are exactly t h e s a m e shape: t h a t is w h a t I m e a n by a unique shape to the bridge. T h i s is not so in a n a l o g u e methods. I will touch on one other point which really s u r p r i s e d me. W h e n we f i r s t t h o u g h t of u s i n g S c h u t ' s e q u a t i o n s in a least s q u a r e s a d j u s t m e n t I did not use t h e m quite as blindly as Mr. Schut imagined. It will be found t h a t Schut's equations are i n v a r i a n t u n d e r d i f f e r e n t i a t i o n with respect to the q u a n t i t i e s a, b, c, a n d t h a t small c h a n g e s of rotation a n d scale c h a n g e a r e m u t u a l l y independent. T h i s m e a n s t h a t , once two s y s t e m s have been b r o u g h t n e a r l y t o g e t h e r by t h e d e t e r m i n a t i o n of t h e g r o s s scale change a n d orientation, it is l e g i m a t e to adopt a least s q u a r e s technique to determine the small rotation a n d small scale c h a n g e to obtain a best f i t in t e r m s of vector displacements. My own feeling is t h a t the least s q u a r e s m e t h o d s is a correct a s s u m p t i o n because you h a v e a s s u m e d t h a t the corrections a r e going to be small a n d t h e r e f o r e t h i s linear equation is correct to the f i r s t order. To a n s w e r Prof. v a n der Weele, provided you p u t only t h r e e crosses on each photog r a p h in the centre of the photograph, I do n o t t h i n k t h a t the f a c t t h a t those crosses exist in one p h o t o g r a p h of the p a i r u p s e t s your stereovision, which depends on a r e a viewing r o u n d the point. In fact, w h e n you p u t the model up these t h r e e crosses plus the three on t h e other p h o t o g r a p h are not s i g n i f i c a n t in relation to the relative orientation of t h a t model. They m a y slightly u p s e t your eyesight a t the six points t h a t you require, b u t w h e n you p u t y o u r f l o a t i n g m a r k down, you p u t it on the centre of t h a t cross, and so f a r we h a v e not experienced a n y difficulty in d e t e r m i n i n g Z. Schermerhorn:
W h a t is " P h o t o p a k e " ?
Brazier: It is the t r a d e n a m e of a fluid used for m a s k i n g detail on negatives. I t h i n k t h i s is really a question for Mr. W i l l i a m s a t Lisbon. He is the p h o t o g r a m m e t r i c expert, I a m the c o m p u t i n g expert. He will h a v e to j u s t i f y a n y p h o t o g r a m m e t r i c techniques t h a t he employs and I think p e r h a p s I h a d better leave the question for h i m to answer. B u t I can say t h a t our observers do not s e e m to experience a n y difficulty in observing on these m a r k s . A n s w e r i n g Dr. Zarzycki's question on the control for the N i g e r i a n job, it would be v e r y nice to be given 150 p h o t o g r a p h s and nine h u n d r e d control points or even two h u n d r e d photo points distributed l l / a per photograph. The t o p o g r a p h y in Nigeria is not like this, however, a n d in f a c t w h a t we n o r m a l l y h a d w a s a r e d u n d a n c y of control on a p a r t i c u l a r overlap where a t r a v e r s e crossed it, g i v i n g p e r h a p s h a l f a dozen points on t h a t photograph, a n d t h e n a h u g e blank space full of swamp, etc., until we hit the n e x t road across the p h o t o g r a p h , five or six p h o t o g r a p h s away. W h e n we s a y we h a v e a large a m o u n t of control, it would be l a r g e if it were distributed evenly, b u t in f a c t
[31]
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Photogrammetria, X1X, No. 7
it w a s o n l y s u f f i c i e n t for t h e job. On a l o n g b r i d g e o f t w e n t y two o v e r l a p s I r e m e m b e r t h a t in f a c t w e h a d f o u r b a n d s of control s p a c e d e v e n l y a l o n g t h e s t r i p . S u r v e y o r s love p u t t i n g p o i n t s in, b u t it is v e r y d i f f i c u l t f o r a p h o t o g r a m m e t r i s t to g e t t h e m in t h e r i g h t places. T h e y a r e inclined to p u t t h e m n e a r t h e u r b a n a r e a s a n d n o t p e r h a p s in t h e desolate s p o t s w h e r e we w o u l d like t h e m to be. I k n o w t h a t c o m p a r i s o n s a r e odious a n d I hope I h a v e n o t o f f e n d e d P r o f . I n g h i l l e r i by s a y i n g t h a t t h e a n s w e r a p p e a r e d to be s l i g h t l y b e t t e r t h a n his. It is t r u e to s a y t h a t we h a v e a r e a d - o u t on t h e A8, w h i c h m e a n s t h a t y o u do n o t m a k e m i s t a k e s in t r a n s c r i p tion. T h e c o o r d i n a t e s come o u t a u t o m a t i c a l l y once t h e p o i n t i n g h a s been m a d e , a n d u n t i l t h e f i n a l a n s w e r come o u t of t h e c o m p u t e r t h e y a r e " u n t o u c h e d b y h u m a n h a n d s " . W e h a d a v e r y good c a m e r a , a n A v i o g o n , on t h i s job a n d so we feel t h a t t h e r e is no g r e a t s o u r c e of e r r o r f r o m t h e p h o t o g r a p h i c side. T h e o t h e r g r e a t point a b o u t it w a s t h a t t h e t e r r a i n w a s r e a s o n a b l y flat, a s I said before. T h e a r e a w a s r o u g h l y t e n m i l e s l o n g a n d t e n m i l e s wide.
de Masson d ' A u t u m e : J ' a i u n e q u e s t i o n ~ p o s e r ~ M. B r a z i e r . L a m 6 t h o d e qu'il u t i l i s e p o u r r a c c o r d e r d e u x m o d u l e s c o n s 6 c u t i f s e s t u n e m 6 t h o d e bas6e s u r les m o i n d r e s carr6s, c'est-~t-dire qu'il y r e n d m i n i m u m la s o m m e des c a r r 6 s de q u e l q u e chose. Quelle e s t la q u a n t i t 6 qu'il c h e r c h e $ r e n d r e m i n i m u m ? Brazier: T h e d i s t a n c e t h a t we m a d e a m i n i m u m w a s t h e vector d i s t a n c e b y w h i c h we c h a n g e d t h e position of e a c h point. de Masson d ' A u t u m e : L a chose q u ' o n m i n i m i s e e s t doric la s o m m e des c a r r 6 s des d i s c o r d a n c e s des c o o r d o n n ~ e s - d ~ p a r t s et des c o o r d o n n ~ e s - a r r i v 6 e s . J e dois f a i r e r e m a r q u e r que, d u p o i n t de v u e th6orique, ce n ' e s t p a s t o u t - S - f a i r c o r r e c t p o u r p l u s i e u r s r a i s o n s . L a p r e m i e r e c ' e s t q u ' o n f a i t j o u e r a u x t r o i s p o i n t s de la f i g u r e de r a c c o r d e m e n t u n rSle identique. A u c e n t r e p e r s p e c t i f , c o m m u n a u x d e u x m o d u l e s c o n s 6 c u t i f s et a u x d e u x p o i n t s a u sol, on f a i t j o u e r e x a c t e m e n t te m ~ m e rSle, ce qui n e me p a r a l t p a s t r ~ s logique. On s u p p o s e en o u t r e q u e les coordonn6es du s o m m e t p e r s p e c t i f et des d e u x p o i n t s a u sot s o n t e n t i ~ r e m e n t i n d 6 p e n d a n t e s , ce qui e s t m a n i f e s t e m e n t f a u x , c a r il y a n u n e c o r r e l a t i o n t r ~ s f o r t e e n t r e les coordonn6es de ces t r o i s points. Darts la m 6 t h o d e que n o u s u t i l i s o n s ~. I ' I G N n o u s n e c h e r c h o n s p a s ~ r e n d r e m i n i m u m la s o m m e des c a r r 6 s des d i s c o r d a n c e s e n t r e les coordonn6es. N o u s c 0 n s i d S r o n s en e f f e t q u ' e n t r e d e u x m o d u l e s c o n s 6 c u t i f s ce qu'il y a de c o m m u n c ' e s t l ' e n s e m b l e d e s r a y o n s i s s u s du p o i n t de r u e c e n t r a l et a b o u t i s s a n t a u x p o i n t s c o m m u n s a u x d e u x mod$1es; c ' e s t doric en s o m m e u n f a i s c e a u p e r s p e c t i f (rfiduit $ q u e l q u e s r a y o n s c o m m u n s ) et la r o t a t i o n doit fitre telle que cette f i g u r e , qui e s t r i g o u r e u s e m e n t ta m ~ m e d a n s les d e u x modules p u i s q u ' o n a vis6 tes m ~ m e s p o i n t s , coincide d a n s les modules a v a n t et a r r i ~ r e . I1 m e semble q u e cette conception e s t p l u s c o r r e c t e que celle qui e s t bas~ s u r l ' e m p l o i de f o r m u l e s de Schut. Inghilleri: I a g r e e w i t h Mr. de M a s s o n d ' A u t u m e , b e c a u s e f r o m t h e t h e o r e t i c a l p o i n t of v i e w t h e c o m p u t a t i o n p r o c e d u r e I u s e d is n o t v e r y r e f i n e d , a n d some c o n s i d e r a t i o n s c a n be i g n o r e d . B u t in m y opinion t h e m e t h o d is good if t h e c o m p u t a t i o n s a r e simple. To solve t h e p r o b l e m c o r r e c t l y f r o m t h e t h e o r e t i c a l p o i n t o f view, r e q u i r e s in m y e x p e r i e n c e , a l a r g e r a m o u n t of c o m p u t a t i o n .
[323
Analytical Aerial Triangulation with Analogue Instruments by N. I. L E V Y I.T.C., Delft 1.1. So f a r analytical aerial triangulation has been applied with a special i n s t r u m e n t only, namely a comparator. However, there is no reason w h y the usually available universal instruments or precise plotters should not also be exploited for this task. Hereby their use is extended and the execution of analytical triangulation is made possible with these instruments over the whole range of photography, from normal to superwide angle. 1.2. This paper gives a description of a general method of analytical aerial triangulation for analogue instruments. The method is actually only a p a r t of a whole system of analytical triangulation developed by the author for application in the I.T.C. The system is general and applicable to all commonly used instruments, comparators as well a s analogue, whether universal or plotters. Here, however, we shall limit ourselves to a short description of that p a r t dealing with the analogue instruments, except for concepts of a general character. F o r more details the reader is referred to the author's ITC publication. 1) 2.1. The analogue i n s t r u m e n t is used as a device for measuring image coordinates. The photographs are set horizontally and coordinates of image points are measured in the projection in the nan-exact model thus formed. Each point is measured stereoseopically with the help of X, Y, Z and by base component. Whenever by is not available, lateral rotation w is used instead. In universal instruments, base inside and base outside a r e used in alternate models, maintaining the position of the common photograph. In plotters, however, this is not possible, so the triangulation is carried out by independent models. The common photograph is t r a n s f e r r e d from one plate to the other r a t h e r than being rotated through 2O0g, because it avoids certain error accumulation. It should however be noted t h a t this t r a n s f e r of the photographs does not entail, as in some other triangulation methods with plotters, the t r a n s f e r of any orientation element. To ensure a better precision of the triangulation, corrections are applied analytically for instrumental errors incorporated in the projection and measuring system, as well as for erroneous horizontal setting of the photographs. A purely analytical orientation method is used for the computation of the relative orientation. (Here the orientation method of C. M. A. v. d. Hour has been applied, although others are obviously applicable). 2.2. F o r the reduction of the observation data a general reduction formula has been derived. This formula makes use of formal parameters and actual to formal representation technique. Let, r, p, q, ~r, ~)x, ~ , ¢)z and ~)c be formal parameters. " r " represents the element used to measure the y parallax. "p and q" are coefficients of this measurement in X and Y respectively. ~r, Qx, ~y and Qz are t r a n s f o r m a t i o n coefficients of r, X, Y and Z respectively. Qc denotes the camera used for the measurements of the y parallax, i.e. = 0 when camera was held fixed, = 1 when it was used. 1) N. I. Levy: Analytical Aerial Triangulation with Analogue Instruments, I.T.C. Publication A/31, summer 1964. 1 8
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280
If:
AX = q ( r - - t o )
;
A Y = p ( r - - r o)
(1)
~A= = @=~r ~%
;
@~v =
(2)
~v ~r ~c
and K is a scale factor, then the following formula is general and holds for any instrument, w h e t h e r analogue or comparator.
V
a22 j [ [ ( Y - - Y o + A Y @ ~ y )
a21
In analogue i n s t r u m e n t s K -
K]~ v
C ( Z - - Z o) ~ whereas for c o m p l r a t o r s K = 1.
*v"
÷X
_. 7 - - ÷ u '
I/
-b~ o-
/
Fig. i 2.3. To a d a p t this formula to a specific instrument it is thus only necessary to substitute the actual values of the formal p a r a m e t e r s corresponding to t h a t particular instrument. Formulae (3) has two significant practical properties. Firstly, it is general and thus applicable to any i n s t r u m e n t under consideration, and secondly, it is identical for each camera associated in the measurements, as one is free to allocate different Values to the p a r a m e t e r s of different cameras. These actually lead to the development of a Universal Triangulation P r o g r a m m e (UTRIP-1) which is applicable to all commonly used instruments. As an example, table 1 represents the actual values of the p a r a m e t e r s of each camera of the Wild A-7 and Wild A-8 instruments. Table 1
W//o'A-7
b/.÷I: O
*I -I ¢ l ~ l
kV/ZdA-8
u~" -I @
~" ~I -I rl .o -I o
~ ~z ~ e r s
[22]
/ 0 H rel'heP ~hen i~o Z
0
-I
0
+l
0
+I V x ' x & ( ~ 1
0
~
r~Iz
+I
[(z-z,~, :'~'/~r~fl-
+I
Levy, Analytical Aerial Triangulation with Analogue Insfruments
281
A s s u m i n g t h a t t h e fiducial centre coincides with the principal point, X o and Yo c a n be determined f r o m m e a s u r e m e n t s of the 4 fiduclal m a r k s . In plotters where this is impossible (base zero c a n n o t be set) X o and Yo are determined f r o m a direct m e a s u r e m e n t of the fiducial centre if premarked, the e n g r a v e d cross on the plate holder, or the point a t which the space rod is vertical. 3.1. Hitherto, theoretical coordinate a x i s s y s t e m a n d error-free i n s t r u m e n t were a s s u m e d . This, however, is never realised in practice. Mechanical a d j u s t m e n t is r a t h e r limited. Moreover, in n o r m a l production work i n s t r u m e n t s are r a r e l y a d j u s t e d or even checked. It therefore seems a d v a n t a g e o u s to apply corrections for the i n h e r e n t i n s t r u m e n t a l e r r o r s analytically. The precision is t h u s increased, and to a c e r t a i n e x t e n t becomes independent of the state of a d j u s t m e n t of the i n s t r u m e n t . 3.2. Since t h e errors involved are u s u a l l y small t h e y can be considered a s being linear over s h o r t distances along the X a n d Y axes. Accordingly a second order correction ~urface is applied. Two such correction s u r f a c e s , one for X a n d one for Y, are derived f r o m grid m e a s u r e m e n t s for each of the i n s t r u m e n t cameras, These s u r f a c e s are superimposed analytically on each m e a s u r e d p h o t o g r a p h d u r i n g computation. Correction to image points a r e derived u s i n g f o r m u l a (4) below, (see also Fig. 2).
n
m
u p = Au 1 + ~ ( z l u 2 - - A u 1) + ~ - ( A u 4 - - A u n
mn 0 + ~-(dUl--Z]u2
m
% = Avl + ~ (Ave-- Avl) + s
+ AUa--Au4)
(4)
mn
( A % - - Av 1) + ~ -
(Av 1 - A v e +
Ava " Avi)
In which : zJuo and
are the corrections of point ~ in u an v directions respectively, and A v I...... A v 4 are the corrections of the four related grid points, m = u I - - uo, n = v I - - vo, and s = the distance between two successive grid points. To illustrate, a correction surface of one component is depicted graphically in Figure 2. Av o
Au I...... Au 4
v
×/ I//Z LZI/
I
coO','~½a/~$ Fig. 2
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282
A 25 point grid is used, as it seems to be a practical compromise between maximum precision and practical considerations. Having measured the grid at a certain projection distance, two grid points along the Y (or the X) axis are remeasured at a different projection distance for the determination of C c and Z o in the i n s t r u m e n t . 3.3. An i m p o r t a n t feature of this correction method is t h a t it is general and independent of the i n s t r u m e n t type. F u r t h e r m o r e it comprises the total effect of instrumental e r r o r s , including errors incorporated in the projection, as represented at the grid points. Consequently various analogue instruments including those based on the affine principle can be applied, and a simple and easy measuring procedure is achieved. To a certain extend one need not w o r r y about the state of a d j u s t m e n t of the instrument nor about exact zero setting of orientation elements, 3.4. Corrections are also derived for film shrinkage, radial lens distortion, atmospheric refraction and earth curvature. For film shrinkage an affine t r a n s f o r m a t i o n based on the calibrated coordinates of the four fiduciat marks is applied. This is obviously possible only with those instruments in which the four fiducial m a r k s of every photo can be measured, i.e. universal instruments. For the radial lens distortion a set of third order curves are applied to represent the symmetric p a r t while a set of second order curves represent the asymmetry. A combined correction is applied for the effect of earth curvature and atmospheric refraction. 3.5. Let A U and A V represent the combined value of all corrections in U and V expressed on the image plane then the "ideal" image coordinates U V are derived as follows (see formula (3)) :
[1[
=
all
av,
a21
a22
[ [ ( X - - X o + A X e ~ ) K] e.~
J.
[ [ (Y - - Yo + d Y e~v) K ] Qv
I +
AV
(5)
4.1. In both universal i n s t r u m e n t s and precise plotters it is possible to measure independent models. This, however, is an inefficient procedure for the former instruments. It is more favourable (and thus applied here) to follow the conventional aeropolygon procedure of base inside and base outside in alternate models. In plotters this is not possible and therefore independent models a r e measured. To strengthen the connection o5 adjacent models it is advisable to prick two points on V axis of each photograph so t h a t they can be observed in each of the corresponding models. 5.1. A programme r e f e r r e d to as UTRIP-1 (Universal Triangulation Programme) has been w r i t t e n for the medium size Stantec-Zebra computer. The programme accomodates the p a r a m e t e r s of a number of analogue i n s t r u m e n t s most commonly used and can thus be applied to each one of them. These i n s t r u m e n t s are: Wild A-7 Autograph, Zeiss Stereoplanigraph C-8, Santoni Stereometrograph IV, Nistri Photostereograph ill2, Poivilliers Stereotopograph B, Wild A-8 Stereo Plotting Machine, Santoni Simplex III, Thompson-Watts Plotter and Zeiss J e n a Stereometrograph. To apply the programme to other instruments it is only necessary to introduce heir corresponding p a r a m e t e r s prior to computation. 5.2. To cope with practical difficulties and to achieve a reliable and simple procedure the programme has been made as general and as flexible as possible and various checks have been incorporated. Thereby the output of the analogue i n s t r u m e n t can be used directly as input for the computer without the need for any revisions or corrections. E v e r y model is checked for its completeness and stability and every registration for its approval.
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283
Most analogue i n s t r u m e n t s are not equipped with automatic registration for by or co. Although the time involved in manual registration of one of these elements is insignificant, the provision of automatic registration is recommended. It would f u r t h e r simplify and economize the application of this method. 6.1. The most significant advantage of the described method is t h a t it provides the possibility to apply the normal analogue instruments to analytical aerial triangulation. This is of a special importance to photogrammetric organizations who do not have sufficient triangulation work to j u s t i f y the purchase of a special instrument. Such organizations are now offered the possibility of applying analytical triangulation with their available analogue instruments, whether universal or plotters. I t is a purely analytical method and as such possesses all properties and advantages of analytical triangulation. The method is simple and general. It is applicable to all commonly used analogue instruments. One need not worry, to a certain extent, about the state of adjustment of the instrument, nor about zero settings of orientation elements. The analytical correction method applied for instrumental errors permits practically a r b i t r a r y settings of the orientation elements and corrects numerically for instrumental errors. No t r a n s f e r of scale or any orientation element is involved in proceeding from one model to the other, even in the case of a plotter where independent models are triangulated.
18.
284
Discussion on the papers of InghiUeri, B r a z i e r and L e v y
Visser: I think it is e a s y to u n d e r s t a n d w h y Prof. T h o m p s o n propounds the method of i n d e p e n d e n t models, because in the M a r k II T h o m p s o n W a t t s plotter it is no longer possible to t r a n s f e r the vertical t h r o u g h the strip as it w a s in the M a r k I. B u t in the case of the A8 and the Stereosimplex III there a r e simple m e a n s of t r a n s f e r r i n g the vertical t h r o u g h the strip, for i n s t a n c e by m e a n s of t h e cross levels, which give v e r y good precision a n d which a r e discarded in t h i s case. P e r h a p s Mr. Brazier or Prof. Inghilleri could explain w h y t h e y do not w a n t to use these cross levels? W h e n t h e y a r e employed the s u b s e q u e n t computation is so simple t h a t it can a l w a y s be done by a desk computer. Brazier: I believe t h a t i f you h a v e a c o m p u t e r you m i g h t as well use it, and I t h i n k the observing procedure in the A8 is considerably simplified if you do not h a v e to m a k e a comparison between one model a n d the next. E a c h model is literally independently observed. The only t h i n g t h a t you keep is the calibration of the Z s e t t i n g on the i n s t r u m e n t a n d the base. Inghilleri: If I understood well, you have asked w h y we do not use the cross levels in the Stereosimplex I I I ? We h a v e had no experience in the use of the cross level. It is not only m y opinion t h a t the cross level device does not have the s a m e a c c u r a c y as analytical f o r m a t i o n of t h e model. The cross level device h a s a n a c c u r a c y t h a t I a s s u m e to be about 2 m i n u t e s . Also, the i n s t r u m e n t work is more complicated. We, for instance, have been able to use a n unskilled operator. Thompson: I h a v e v e r y good reasons to t h i n k t h a t these devices should not be used. F i r s t of all as r e g a r d s accuracy, I do not t h i n k t h i s is really a decisive point because I h a d seriously t h o u g h t a t one time of u s i n g autocollimators w i t h m y i n s t r u m e n t . I c a n n o t u s e levels in m y i n s t r u m e n t because the line in t h e i n s t r u m e n t which r e p r e s e n t s the a i r b a s e is in f a c t held horizontal in the model space, b u t I can use autocollimators a n d t h e y c e r t a i n l y have a n y desired precision one can possibly wish. B u t m y r e a s o n s for not u s i n g t h e m are, f i r s t of all, t h a t the i n s t r u m e n t r e m a i n s in exactly the s a m e condition of a d j u s t m e n t if you do not touch the absolute orientation e l e m e n t s ; and second, t h a t if you add these devices t h e y still do not g e t rid of the computation, because you still h a v e to t r a n s f o r m the coordinates as f a r as t h e v a l u e s X and Y are concerned. You. h a v e t h e r e f o r e still to do some computation a n d I h a v e come to the conclusion you m i g h t as well do the whole lot. Erez: I h a v e a r e m a r k about the distortion s u r f a c e s which Mr. Levy spoke about. E v e r y o n e who h a s a d j u s t e d i n s t r u m e n t s knows v e r y we]l t h a t there is d a n g e r in distortion s u r f a c e s , especially when you touch one of the c a r d a n s of the i n s t r u m e n t , because there could be some s t r a i n in one of the c a r d a n s of the i n s t r u m e n t . T h e n the r e a d i n g s you obtain v a r y w i t h c h a n g i n g h e i g h t or with change of position. So I t h i n k when you use those distortion s u r f a c e s , the i n s t r u m e n t h a s to be calibrated several times, not only one set of 25 points, b u t repeated several times in d i f f e r e n t directions. Levy: I did not m e n t i o n t h a t for the grid m e a s u r e m e n t s we use f o r w a r d and back w a r d m e a s u r e m e n t s and take the m e a n of them, a s I t h o u g h t this w a s quite obvious. It is obvious t h a t if you t a k e a v e r y dense r e s e a u or g r i d a n d m e a s u r e t h e errors a t very
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s h o r t d i s t a n c e s you will g e t a b e t t e r r e p r e s e n t a t i o n of t h e e r r o r s in t h e i n s t r u m e n t , b u t t h i s is n o t p r a c t i c a l a n d t h e r e f o r e we p r e f e r to u s e a k i n d of c o m p r o m i s e b y t a k i n g 25 points. It is obviously possible t h a t if t h e i n s t r u m e n t is u n s t a b l e a n d y o u r e p e a t r e a d i n g s in one d i r e c t i o n only, y o u will n o t d e t e r m i n e all e r r o r s or e r r o r s o u r c e s , t h e r e f o r e we u s e f o r w a r d a n d b a c k w a r d m e a s u r e m e n t s . I do t h i n k t h a t t h i s a n a l y t i c a l c o r r e c t i o n m e t h o d is f a r b e t t e r t h a n a m e c h a n i c a l a d j u s t m e n t , c e r t a i n l y for a r e a s o n a b l e m a g n i t u d e of errors, as you cannot adjust mechanically as accurately as you can determine and correct the errors analytically. Jerie : R e g a r d i n g Mr. B r a z i e r ' s c o m m e n t s c o n c e r n i n g t h e a d v a n t a g e s of t h e m e t h o d of i n d e p e n d e n t m o d e l s ; I do n o t d e n y t h e s e a d v a n t a g e s , e s p e c i a l l y b e c a u s e o f t h e s i m p l i c i t y of t h e m e t h o d . H o w e v e r , t h e r e w e r e two a r g u m e n t s g i v e n by M r . B r a z i e r w i t h w h i c h ! cannot agree. F i r s t l y , h e s t a t e s t h a t u s i n g t h i s m e t h o d one c a n s t a r t f r o m t h e b e g i n n i n g or t h e e n d o f t h e s t r i p , b o t h o f w h i c h you c a n also do in s t r i p t r i a n g u l a t i o n . H o w e v e r , I do n o t c o m p l e t e l y a g r e e t h a t y o u c a n s t a r t in t h e middle o f t h e s t r i p j u s t a easily, b e c a u s e y o u c e r t a i n l y lose t h e i n n e r o r i e n t a t i o n of t h e one p h o t o g r a p h , w h i c h h a s t h e n to be p u t in t h e p l a t e h o l d e r twice. Secondly, he s t a t e s t h a t a l e s s e r q u a l i f i c a t i o n is r e q u i t e d f o r t h e o p e r a t o r . I c a n n o t a g r e e w i t h t h i s , b e c a u s e one h a s to c a r r y o u t e s s e n t i a l l y t h e s a m e o p e r a t i o n s in t h e m e t h o d of i n d e p e n d e n t p a i r s a s in t h e c a s e of c o n v e n t i o n a l s t r i p t r i a n g u l a t i o n in u n i v e r s a l i n s t r u m e n t s , viz. r e l a t i v e o r i e n t a t i o n a n d m e a s u r e m e n t s of coordinates. T h e o n l y d i f f e r e n c e b e t w e e n t h e two m e t h o d s is t h a t t h e scale t r a n s f e r is n o t c a r r i e d o u t in t h e i n s t r u m e n t a l p h a s e . H o w e v e r , t h e scale t r a n s f e r is a c h i e v e d i n e s s e n t i a l l y t h e s a m e w a y in both m e t h o d s , viz. b y m e a s u r e m e n t s o f c o o r d i n a t e s . I t h i n k t h e real a d v a n t a g e of t h e m e t h o d of i n d e p e n d e n t m o d e l s is t h a t it is s i m p l e r f r o m t h e o r g a n i s a t i o n a l p o i n t of view a n d t h a t i n s t r u m e n t s c a n be u s e d w h i c h a r e also u s e d ~n t h e s u b s e q u e n t p l o t t i n g p h a s e . S c h u t : I w o u l d like to go b a c k once m o r e to t h e a c c u r a c y of i n d e p e n d e n t m o d e l t r i a n g u l a t i o n . P r o f . H a l l e r t p u b l i s h e d a p a p e r on a n i n v e s t i g a t i o n o f t h e a c c u r a c y of d i f f e r e n t m e t h o d s o f t r i a n g u l a t i o n , a n d he f o u n d t h a t t h e m e t h o d w h e r e t h e m o d e l s a r e c o n n e c t e d v i a t h e o r i e n t a t i o n e l e m e n t s of t h e c o m m o n p h o t o g r a p h w a s t h e m o s t a c c u r a t e . I t s e e m s f r o m t h i s t h a t we m u s t conclude t h a t we s h o u l d only u s e t h e i n d e p e n d e n t m o d e l t r i a n g u l a t i o n if we do n o t h a v e a n i n s t r u m e n t a v a i l a b l e in w h i c h t h i s c o n t i n u o u s t r i a n g u l a t i o n c a n be p e r f o r m e d . F u r t h e r , Mr. B r a z i e r s t a t e d t h a t t h e r e s u l t of t h e i n d e p e n d e n t m o d e l t r i a n g u l a t i o n is i n d e p e n d e n t of t h e s e q u e n c e of m e a s u r e m e n t of t h e models, w h i l e t h i s w o u l d n o t be t h e c a s e w i t h c o n t i n u o u s t r i a n g u l a t i o n on a u n i v e r s a l i n s t r u m e n t . I see no r e a s o n f o r t h i s d i f f e r e n c e . A s f a r a s s y s t e m a t i c d e f o r m a t i o n is c o n c e r n e d t h e o n l y b a s i c d i f f e r e n c e b e t w e e n t h e two t r i a n g u l a t i o n m e t h o d s is t h a t w i t h t h e i n d e p e n d e n t m o d e l t r i a n g u l a t i o n t h e u n i t is one model, w h i l e w i t h t h e c o n t i n u o u s t r i a n g u l a t i o n t h e u n i t is two m o d e l s , one w i t h b a s e - i n a n d o n e w i t h b a s e - o u t . I n b o t h c a s e s t h e t r i a n g u l a t i o n i n s t r u m e n t c a n be t h e c a u s e o f a s y s t e m a t i c d e f o r m a t i o n of t h e t r i a n g u l a t i o n s t r i p , a n d in b o t h c a s e s t h i s d e f o r m a t i o n d e p e n d s u p o n t h e d i r e c t i o n of t h e t r i a n g u l a t i o n . van der Weele : I j u s t h a v e one s m a l l q u e s t i o n f o r Mr. B r a z i e r , w h o p o i n t e d o u t t h a t t h e c o m m o n p o i n t s b e t w e e n a d j a c e n t m o d e l s a r e m a r k e d w i t h c r o s s e s of a r a t h e r l a r g e size, 6 m m is m e n t i o n e d . H e said t h a t t h e size of t h e l i n e s is s u c h t h a t y o u c a n p u t t h e m e a s u r i n g m a r k v e r y a c c u r a t e l y on t h e cross. B u t as f a r as I u n d e r s t o o d y o u also h a v e to o b s e r v e s t e r e o c o p i c a l l y on t h o s e points. M y q u e s t i o n is, does n o t t h i s r a t h e r l a r g o cross disturb the stereoscopy?
[27]
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Photogrammetria, XIX, No. 7
Schermerhorr~: One o f the a r g u m e n t s for the m e a s u r e m e n t of independent models is t h a t you use the i n s t r u m e n t in the s a m e position, t h u s t h e r e is no mechanical difference, as with base-in and base-out t r i a n g u l a t i o n . I now d r a w y o u r a t t e n t i o n to the new Santoni i n s t r u m e n t which will be shown ix Lisbon, n a m e l y the S t e r e o c a r t o g r a p h V. W i t h this you can m e a s u r e with base-in and base-out, a n d still m a i n t a i n t h e s a m e distribution of w e i g h t s in the i n s t r u m e n t . I believe t h i s is of g r e a t importance, and it eliminates one of the a d v a n t a g e s which h a s a l w a y s been m e n t i o n e d in f a v o u r of m e a s u r e m e n t s of i n d e p e n d e n t pairs. Zarzycki: I f we use t h e A7 or s i m i l a r t y p e of i n s t r u m e n t t h e n the a d v a n t a g e of i n d e p e n d e n t p a i r s can be debated. B u t if we use the A8 or a simpler i n s t r u m e n t , such as the B8 or the Kelsh plotter, where it is v e r y inconvenient to t r a n s f e r the scale w i t h the height, a n d g r a p h i c a l methods are u s u a l l y employed for scale t r a n s f e r , t h e n I t h i n k the independent p a i r s method h a s m e r i t s and can produce good results. The t r a n s f e r of scale would be done a n a l y t i c a l l y a n d this is obviously more a c c u r a t e t h a n a g r a p h i c a l method of t r a n s f e r r i n g t h e scale. Mr. B r a z i e r told u s t h a t he bridged about 150 overlaps and t h a t he had 200 control points. T h i s is more t h a n one point per model. I would like to know how long the strips were t h a t he bridged, and w h a t w a s the distribution of t h e vertical control points. Also, I would like to know w h e t h e r he h a s employed his m e t h o d in b r i d g i n g long strips, let us s a y 15 or 20 models between control. Levy: I h a v e a question for Mr. Brazier. In his paper, p a r a g r a p h 16, in c o m p a r i n g the a n a l y t i c a l to the a n a l o g u e methods, he w r i t e s t h a t " t h e model is determined f r o m relative orientation which is derived f r o m the observation of i n n u m e r a b l e points of detail". Could:he c l a r i f y this declaration, is it b e t t e r t h a n the c o m p a r a t o r relative orientation ? Kure: I h a v e done some tests a n d found less a c c u r a t e r e s u l t s with independent model t r i a n g u l a t i o n t h a n with n o r m a l aerial t r i a n g u l a t i o n , which in m y opinion is m a i n l y due to the less a c c u r a t e t r a n s f e r of scale. In i n d e p e n d e n t model t r i a n g u l a t i o n the t r a n s f e r of scale is done u s i n g the coordinates of the two wing points, and in n o r m a l aerial t r i a n g u l a t i o n it is done on heights. A s you h a v e m o r e u n k n o w n s in independent model t r i a n g u l a t i o n you get less a c c u r a c y a n d the only w a y to improve t h a t would be to use m a y be two sets of w i n g points in order to improve the t r a n s f e r of scale in the independent model t r i a n g u l a t i o n . Thompson: The l a s t speaker m a i n t a i n s t h a t the i n a c c u r a c y arises f r o m the scale t r a n s f e r . M y own look a t the t h e o r y of this showed m e t h a t the actual theoretical acc u r a c y is g r e a t e r when scaling f r o m w i n g points t h a n it is f r o m heights. I h a v e always r e g a r d e d the e r r o r f r o m the w i n g points as being due to more questions of identification t h a n to a n y theoretical reason. W h e n scaling f r o m a c e n t r a l point you do not have to i d e n t i f y the points a s accurately. B u t if one uses a point m a r k e d on the middle picture of three in such a w a y t h a t it is a small circle, a p p r o x i m a t e l y or exactly the s a m e size a s the stereoscopic m a r k , I do not t h i n k t h a t the error of identification is at all serious. I f I m a y , I would like to m a k e one or two other points. A s f a r as I understood Prof. Inghilleri, he did n o t c a r r y out relative orientation by a computional method. I personally t h i n k t h a t all relative orientations should be carried out at least p a r t i a l l y by computation, a n d t h a t c e r t a i n l y the z]eo element should be determined in this way. It is not only more a c c u r a t e , b u t also quicker, especially when the g r o u n d is not v e r y f l a t and w h e n the p h o t o g r a p h s a r e v e r y tilted. It enables one to distribute a n y error properly over the model, even if you simply confine the computation to the zion.
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Discussion on the papers of Inghillerie, Brazier and Levy
C o r r e c t i o n s for d i s t o r t i o n a r e n o t p a r t i c u l a r l y c o n n e c t e d w i t h t h e m e t h o d of indep e n d e n t models, it is a g e n e r a l point. H o w e v e r , a s Mr. L e v y did s h o w c u r v e s of d i s t o r t i o n s , I t h i n k t h e r e is a p o i n t h e r e t h a t is overlooked. W h e n t h e r e is a v a r y i n g r a d i a l d i s t o r t i o n , lLs he h a d in h i s e x a m p l e , y o u m u s t h a v e a s s o c i a t e d t a n g e n t i a l d i s t o r t i o n s , p o s s i b l y n o t of t h e s a m e a m o u n t , b u t c e r t a i n l y s i g n i f i c a n t a n d of t h e s a m e o r d e r . It s e e m s to m e t h a t one h a s e i t h e r to m e a s u r e t h e s e d i s t o r t i o n s or to p r o d u c e s o m e k i n d of t h e o r y to e n a b l e t h e m to be d e d u c e d f r o m t h e v a r y i n g r a d i a l d i s t o r t i o n s , a t h i n g I do n o t t h i n k impossible. I h a v e done s o m e w o r k on t h e q u e s t i o n of v a r y i n g r a d i a l d i s t o r t i o n s a n d h a v e f o u n d some v e r y i n t e r e s t i n g r e s u l t s . I f o u n d f o r e x a m p l e t h a t f o r r a d i a l d i s t o r t i o n s t h a t v a r y a s one g o e s a r o u n d in a z i m u t h , t h e s i m p l e m o d e l of a w e d g e p r i s m is c e r t a i n l y n o t s u f f i c i e n t to a c c o u n t f o r t h e f a c t s . I f y o u do a F o u r i e r a n a l y s i s of t h e d i s t o r t i o n as y o u go r o u n d in a z i m u t h , I h a v e f o u n d n o t o n l y t h e u n e v e n t e r m s b u t also t h e e v e n t e r m s of a F o u r i e r series. I h a v e d o n e t h i s f o r a n A v i o g o n lens.
Ackermann: Mr. L e v y h a s r e f e r r e d to correction o f f i l m ~ h r i n k a g e b a s e d on m e a s u r e m e n t s a n d c a l i b r a t i o n d a t a of t h e f i d u c i a l m a r k s . T h i s is c e r t a i n l y t h e only w a y to f i n d o u t a b o u t f i l m s h r i n k a g e f o r p h o t o g r a p h s of t h e n o n - r e s e a u type. H o w e v e r , one h a s to be r a t h e r c a r e f u l w i t h s u c h a p r o c e d u r e . I t s s u c c e s s d e p e n d s on w h e t h e r t h e s h r i n k a g e a t t h e location of t h e f i d u c i a l m a r k s is r e p r e s e n t a t i v e o f t h e whole p h o t o g r a p h . I c a n show two e x a m p l e s o f s u p p o s e d l y i l l - t r e a t e d f i l m w h o s e s h r i n k a g e p a t t e r n s a r e s u c h t h a t t h e d i s p l a c e m e n t s o f t h e f i d u c i a l m a r k s a r e n o t r e p r e s e n t a t i v e of t h e t o t a l a r e a . I n s u c h c a s e s n u m e r i c a l c o r r e c t i o n s b a s e d on f i d u c i a l m a r k s (especially on t h o s e in t h e c o r n e r s ) w o u l d h a v e a n a d v e r s e e f f e c t a n d w o u l d i n t r o d u c e s e r i o u s e r r o r s .
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Levy: I c o m p l e t e l y a g r e e w i t h Dr. A c k e r m a n n ' s r e m a r k . T h i s is a c t u a l l y one of t h e r e a s o n s w h y I h a v e u s e d a t r a n s f o r m a t i o n h a v i n g 6 p a r a m e t e r s r a t h e r t h a n one h a v i n g 8 p a r a m e t e r s w h i c h l e a v e s no r e d u n d a n c y , t h a t is to s a y no check. Thompson: I w a s p a r t i c u l a r l y c a r e f u l t h i s m o r n i n g w h e n g i v i n g t h e f i g u r e of 14 i~m f o r t h e s t e r e o c o m p a r a t o r r e s u l t s n o t to m e n t i o n w h e t h e r c o r r e c t i o n s h a d been made for the reseau. In fact the photographs carried a reseau and those results of 14 i~m w e r e t h e r e s u l t s we o b t a i n e d u s i n g t h e r e s e a u . H o w e v e r , one o f t h e m a i n l e a s o n s [29]
288 for this experiment the photograph an amount of perhaps second g r i d . T h e r e not think anything t h e r c s e a u . In f a c t
Photogrammetria, XIX, No. 7 w a s to f i n d o u t t h e e f f e c t of t h e r e s e a u . W e h a d s a n d w i c h e d u p w i t h accurate grid, which was displaced from the reseau crosses by an 0.1 m m . M e a s u r e m e n t s w e r e m a d e v i r t u a l l y s i m u l t a n e o u s l y on t h i s w a s o n l y one r e a d i n g to photo p o i n t s a n d two r e a d i n g s to g r i d s . I do could be b e t t e r d e s i g n e d to s h o w t h e i n c r e a s e in a c c u r a y due to u s i n g w i t h o u t t h e r e s e a u we got a b e t t e r a n s w e r !
Schut: I w o u l d like to m a k e a r e m a r k a b o u t t h e s e f o r m u l a e w h i c h t o d a y h a v e b e e n p e r s i s t e n t l y called S c h u t ' s f o r m u l a e . I n t h e f i r s t place, t h e y h a v e been d e s i g n e d f o r t h e a b s o l u t e o r i e n t a t i o n of a model in a o n e - s t e p p r o c e d u r e a n d t h e y c a n do t h a t f o r a n y r o t a t i o n t h a t is r e q u i r e d . H o w e v e r , t h e y do n o t g i v e a l e a s t s q u a r e s a d j u s t m e n t in t h e s e n s e t h a t t h e s u m of t h e s q u a r e s of t h e d i f f e r e n c e s b e t w e e n t h e c o o r d i n a t e s of t i e - p o i n t s becomes a m i n i m u m . It w o u l d be i n t e r e s t i n g to see h o w m u c h t h e r e s u l t d i f f e r s f r o m a least squares adjustment. Thompson: I e n t i r e l y a g r e e w i t h w h a t Mr. S c h u t s a y s , b u t on t h e o t h e r h a n d I do n o t t h i n k t h e m a t t e r is a n i m p o r t a n t one. T h e v a l u e in h i s f o r m u l a e to m y m i n d , a r i s e s e n t i r e l y f r o m t h e f a c t t h a t t h e y a r e e x a c t a n d e a s i l y solved a n d t h a t y o u c a n d e t e r m i n e l a r g e r o t a t i o n s w i t h t h e g r e a t e s t of ease. Once y o u h a v e d e t e r m i n e d t h i s l a r g e r o t a t i o n I t h i n k t h e r e s i d u a l d i s c r e p a n c i e s a r e q u i t e u n i m p o r t a n t . T h i s s u r e l y is a m a t t e r f o r t h e f i n a l block a d j u s t m e n t a n d I do n o t t h i n k t h a t a t t h i s s t a g e you h a v e to w o r r y a b o u t l e a s t s q u a r e s f i t s b e t w e e n models. Inghilleri: I c o n g r a t u l a t e Mr. B r a z i e r on t h e r e s u l t s he got w i t h t h e i n d e p e n d e n t model t r i a n g u l a t i o n , b u t t h e e r r o r s t h a t I h a v e c o m p u t e d f o r m y w o r k a r e t w i c e a s l a r g e a s his. I t is r a t h e r d i f f i c u l t to c o m p a r e r e s u l t s w i t h o u t k n o w i n g t h e e x p e r i m e n t a l conditions. I w o u l d like to s a y t h a t in o u r case t h e t e r r a i n w a s m o u n t a i n o u s , t h a t t h e p a s s p a i n t s w e r e n o t m a r k e d a n d also t h e m a n n e r of a d j u s t m e n t m a y h a v e been d i f f e r e n t . Mr. B r a z i e r did n o t m e n t i o n h i s m e t h o d . I w o u l d also like to k n o w if M r . B r a z i e r ' s A 8 w a s e q u i p p e d w i t h a r e c o r d i n g device. I w o u l d like to p o i n t o u t t h a t t h e r e s u l t s t h a t we a n d o t h e r E u r o p e a n p h o t o g r a m m e t r i c i n s t i t u t e s o b t a i n e d f r o m t h e s a m e s t r i p of p h o t o s a g r e e d to a g r e a t e x t e n t . T h i s w a s o u r f i r s t e x p e r i m e n t in t h i s field a n d its a i m w a s to s h o w t h a t it is possib~le to o b t a i n b y i n d e p e n d e n t model t r i a n g u l a t i o n on t h e S t e r e o s i m p l e x III, r e s u l t s t h a t a r e a s good a s w h e n u s i n g t h e m o r e e x p e n s i v e u n i v e r s a l i n s t r u m e n t s . I n a n s w e r to P r o f . T h o m p s o n t h e r e l a t i v e o r i e n t a t i o n w a s b y e m p i r i c a l methods. Schermerhorn: I d r a w t h e a t t e n t i o n o f t h e a u d i e n c e to t h e f a c t t h a t t h e c o m p a r i s o n of r e s u l t s of d i f f e r e n t t r i a n g u l a t i o n s is a b s o l u t e l y u s e l e s s w h e n it is n o t c e r t a i n t h a t t h e s a m e p h o t o g r a p h i c m a t e r i a l h a s been used. W e h a v e s u f f e r e d s u f f i c i e n t l y f r o m t h i s k i n d of t r o u b l e in p h o t o g r a m m e t r i c d i s c u s s i o n s to convince m e t h a t we m u s t stop t h i s here. It is f o r t h i s r e a s o n t h a t we h a v e u n d e r t a k e n t h e t e r r i b l e job of c o m p u t i n g a n a r t i f i c i a l block, o f w h i c h you c a n all m a k e u s e f o r all k i n d s of m e t h o d s . All k i n d s o f e r r o r s h a v e been i n t r o d u c e d a n d we k n o w w h a t t h e a n s w e r should be a f t e r t h e comp u t a t i o n . I r e c o m m e n d t h a t you m a k e u s e o f it, it is a v a i l a b l e f r o m t h e I.T.C., a n d u s i n g t h i s y o u c a n c o m p a r e m e t h o d s a n d i n s t r u m e n t s , etc. b a s e d on t h e s a m e n e u t r a l m a t e r i a l . Thompson: Could I m a k e one p o i n t ? J u s t in s e l f d e f e n c e m a y I s a y t h a t all t h e f i g u r e s I h a v e q u o t e d t h i s m o r n i n g a n d t h i s a f t e r n o o n , h a v e been o b t a i n e d f r o m t h e s a m e set o f p l a t e s . I i n t e n d t h e s e f i g u r e s s i m p l y a s c o m p a r i s o n s b e t w e e n each o t h e r . T h i s is n o t i n t e n d e d to be a c o m p a r i s o n w i t h r e s u l t s t h a t e i t h e r Mr. B r a z i e r o r a n y o n e else h a s q u o t e d t o d a y , t h e y a r e only c o m p a r i s o n s w i t h i n t h e e x p e r i m e n t s I h a v e done.
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Discussion on the papers of Inghillerie, Brazier and Levy
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Brazier: Mr. Schut m a d e a r a t h e r i m p o r t a n t s t a t e m e n t t h a t I do not t h i n k is quite correct, a l t h o u g h I h a v e never m y s e l f tested it. In the p a s t we did a lot of b r i d g i n g by a n a l o g u e methods, and our experience w a s t h a t w h e n the n u m b e r of p h o t o g r a p h s w a s large, t h e bz-curve developed v e r y rapidly. It is quite obvious t h a t t h i s curve is not symmetrical, as one end is v e r y m u c h more curved t h a n the other. I can not agree with Mr. Schut in t h a t i f you p e r f o r m an a n a l o g u e operation f r o m s a y 1--20 in an A7 or C8, you will get the s a m e bridge as o p e r a t i n g f r o m 20--1. In other words these two bridges cannot be superimposed in space, one on the other. T h e point I w a s m a k i n g this m o r n i n g as f a r as u n i q u e n e s s w a s concerned w a s a c o m p u t a t i o n a l one. H e r e we have the independent models observed, a n d you m a y take a n y p a i r of these models, a t the beginning, end or even in the middle and p e r f o r m the computational operations between them, a n d the r e s u l t i n g a r r a y of X Y Z points will be precisely t r a n s f o r m a b l e by Schut's equations f r o m one position to another. These two models are exactly t h e s a m e shape: t h a t is w h a t I m e a n by a unique shape to the bridge. T h i s is not so in a n a l o g u e methods. I will touch on one other point which really s u r p r i s e d me. W h e n we f i r s t t h o u g h t of u s i n g S c h u t ' s e q u a t i o n s in a least s q u a r e s a d j u s t m e n t I did not use t h e m quite as blindly as Mr. Schut imagined. It will be found t h a t Schut's equations are i n v a r i a n t u n d e r d i f f e r e n t i a t i o n with respect to the q u a n t i t i e s a, b, c, a n d t h a t small c h a n g e s of rotation a n d scale c h a n g e a r e m u t u a l l y independent. T h i s m e a n s t h a t , once two s y s t e m s have been b r o u g h t n e a r l y t o g e t h e r by t h e d e t e r m i n a t i o n of t h e g r o s s scale change a n d orientation, it is l e g i m a t e to adopt a least s q u a r e s technique to determine the small rotation a n d small scale c h a n g e to obtain a best f i t in t e r m s of vector displacements. My own feeling is t h a t the least s q u a r e s m e t h o d s is a correct a s s u m p t i o n because you h a v e a s s u m e d t h a t the corrections a r e going to be small a n d t h e r e f o r e t h i s linear equation is correct to the f i r s t order. To a n s w e r Prof. v a n der Weele, provided you p u t only t h r e e crosses on each photog r a p h in the centre of the photograph, I do n o t t h i n k t h a t the f a c t t h a t those crosses exist in one p h o t o g r a p h of the p a i r u p s e t s your stereovision, which depends on a r e a viewing r o u n d the point. In fact, w h e n you p u t the model up these t h r e e crosses plus the three on t h e other p h o t o g r a p h are not s i g n i f i c a n t in relation to the relative orientation of t h a t model. They m a y slightly u p s e t your eyesight a t the six points t h a t you require, b u t w h e n you p u t y o u r f l o a t i n g m a r k down, you p u t it on the centre of t h a t cross, and so f a r we h a v e not experienced a n y difficulty in d e t e r m i n i n g Z. Schermerhorn:
W h a t is " P h o t o p a k e " ?
Brazier: It is the t r a d e n a m e of a fluid used for m a s k i n g detail on negatives. I t h i n k t h i s is really a question for Mr. W i l l i a m s a t Lisbon. He is the p h o t o g r a m m e t r i c expert, I a m the c o m p u t i n g expert. He will h a v e to j u s t i f y a n y p h o t o g r a m m e t r i c techniques t h a t he employs and I think p e r h a p s I h a d better leave the question for h i m to answer. B u t I can say t h a t our observers do not s e e m to experience a n y difficulty in observing on these m a r k s . A n s w e r i n g Dr. Zarzycki's question on the control for the N i g e r i a n job, it would be v e r y nice to be given 150 p h o t o g r a p h s and nine h u n d r e d control points or even two h u n d r e d photo points distributed l l / a per photograph. The t o p o g r a p h y in Nigeria is not like this, however, a n d in f a c t w h a t we n o r m a l l y h a d w a s a r e d u n d a n c y of control on a p a r t i c u l a r overlap where a t r a v e r s e crossed it, g i v i n g p e r h a p s h a l f a dozen points on t h a t photograph, a n d t h e n a h u g e blank space full of swamp, etc., until we hit the n e x t road across the p h o t o g r a p h , five or six p h o t o g r a p h s away. W h e n we s a y we h a v e a large a m o u n t of control, it would be l a r g e if it were distributed evenly, b u t in f a c t
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Photogrammetria, X1X, No. 7
it w a s o n l y s u f f i c i e n t for t h e job. On a l o n g b r i d g e o f t w e n t y two o v e r l a p s I r e m e m b e r t h a t in f a c t w e h a d f o u r b a n d s of control s p a c e d e v e n l y a l o n g t h e s t r i p . S u r v e y o r s love p u t t i n g p o i n t s in, b u t it is v e r y d i f f i c u l t f o r a p h o t o g r a m m e t r i s t to g e t t h e m in t h e r i g h t places. T h e y a r e inclined to p u t t h e m n e a r t h e u r b a n a r e a s a n d n o t p e r h a p s in t h e desolate s p o t s w h e r e we w o u l d like t h e m to be. I k n o w t h a t c o m p a r i s o n s a r e odious a n d I hope I h a v e n o t o f f e n d e d P r o f . I n g h i l l e r i by s a y i n g t h a t t h e a n s w e r a p p e a r e d to be s l i g h t l y b e t t e r t h a n his. It is t r u e to s a y t h a t we h a v e a r e a d - o u t on t h e A8, w h i c h m e a n s t h a t y o u do n o t m a k e m i s t a k e s in t r a n s c r i p tion. T h e c o o r d i n a t e s come o u t a u t o m a t i c a l l y once t h e p o i n t i n g h a s been m a d e , a n d u n t i l t h e f i n a l a n s w e r come o u t of t h e c o m p u t e r t h e y a r e " u n t o u c h e d b y h u m a n h a n d s " . W e h a d a v e r y good c a m e r a , a n A v i o g o n , on t h i s job a n d so we feel t h a t t h e r e is no g r e a t s o u r c e of e r r o r f r o m t h e p h o t o g r a p h i c side. T h e o t h e r g r e a t point a b o u t it w a s t h a t t h e t e r r a i n w a s r e a s o n a b l y flat, a s I said before. T h e a r e a w a s r o u g h l y t e n m i l e s l o n g a n d t e n m i l e s wide.
de Masson d ' A u t u m e : J ' a i u n e q u e s t i o n ~ p o s e r ~ M. B r a z i e r . L a m 6 t h o d e qu'il u t i l i s e p o u r r a c c o r d e r d e u x m o d u l e s c o n s 6 c u t i f s e s t u n e m 6 t h o d e bas6e s u r les m o i n d r e s carr6s, c'est-~t-dire qu'il y r e n d m i n i m u m la s o m m e des c a r r 6 s de q u e l q u e chose. Quelle e s t la q u a n t i t 6 qu'il c h e r c h e $ r e n d r e m i n i m u m ? Brazier: T h e d i s t a n c e t h a t we m a d e a m i n i m u m w a s t h e vector d i s t a n c e b y w h i c h we c h a n g e d t h e position of e a c h point. de Masson d ' A u t u m e : L a chose q u ' o n m i n i m i s e e s t doric la s o m m e des c a r r 6 s des d i s c o r d a n c e s des c o o r d o n n ~ e s - d ~ p a r t s et des c o o r d o n n ~ e s - a r r i v 6 e s . J e dois f a i r e r e m a r q u e r que, d u p o i n t de v u e th6orique, ce n ' e s t p a s t o u t - S - f a i r c o r r e c t p o u r p l u s i e u r s r a i s o n s . L a p r e m i e r e c ' e s t q u ' o n f a i t j o u e r a u x t r o i s p o i n t s de la f i g u r e de r a c c o r d e m e n t u n rSle identique. A u c e n t r e p e r s p e c t i f , c o m m u n a u x d e u x m o d u l e s c o n s 6 c u t i f s et a u x d e u x p o i n t s a u sol, on f a i t j o u e r e x a c t e m e n t te m ~ m e rSle, ce qui n e me p a r a l t p a s t r ~ s logique. On s u p p o s e en o u t r e q u e les coordonn6es du s o m m e t p e r s p e c t i f et des d e u x p o i n t s a u sot s o n t e n t i ~ r e m e n t i n d 6 p e n d a n t e s , ce qui e s t m a n i f e s t e m e n t f a u x , c a r il y a n u n e c o r r e l a t i o n t r ~ s f o r t e e n t r e les coordonn6es de ces t r o i s points. Darts la m 6 t h o d e que n o u s u t i l i s o n s ~. I ' I G N n o u s n e c h e r c h o n s p a s ~ r e n d r e m i n i m u m la s o m m e des c a r r 6 s des d i s c o r d a n c e s e n t r e les coordonn6es. N o u s c 0 n s i d S r o n s en e f f e t q u ' e n t r e d e u x m o d u l e s c o n s 6 c u t i f s ce qu'il y a de c o m m u n c ' e s t l ' e n s e m b l e d e s r a y o n s i s s u s du p o i n t de r u e c e n t r a l et a b o u t i s s a n t a u x p o i n t s c o m m u n s a u x d e u x mod$1es; c ' e s t doric en s o m m e u n f a i s c e a u p e r s p e c t i f (rfiduit $ q u e l q u e s r a y o n s c o m m u n s ) et la r o t a t i o n doit fitre telle que cette f i g u r e , qui e s t r i g o u r e u s e m e n t ta m ~ m e d a n s les d e u x modules p u i s q u ' o n a vis6 tes m ~ m e s p o i n t s , coincide d a n s les modules a v a n t et a r r i ~ r e . I1 m e semble q u e cette conception e s t p l u s c o r r e c t e que celle qui e s t bas~ s u r l ' e m p l o i de f o r m u l e s de Schut. Inghilleri: I a g r e e w i t h Mr. de M a s s o n d ' A u t u m e , b e c a u s e f r o m t h e t h e o r e t i c a l p o i n t of v i e w t h e c o m p u t a t i o n p r o c e d u r e I u s e d is n o t v e r y r e f i n e d , a n d some c o n s i d e r a t i o n s c a n be i g n o r e d . B u t in m y opinion t h e m e t h o d is good if t h e c o m p u t a t i o n s a r e simple. To solve t h e p r o b l e m c o r r e c t l y f r o m t h e t h e o r e t i c a l p o i n t o f view, r e q u i r e s in m y e x p e r i e n c e , a l a r g e r a m o u n t of c o m p u t a t i o n .
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