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Triplets: A basic unit for analytical aerotriangulation
367
Triplets: A Basic Unit for Analytical Aerotriangulation by Professor A r t h u r J. MeNAIR Cornel1 University, Ithaca, New York, U.S.A. In the beginning God created man with two eyes. F o r thousands of years, the full capability of this gift of two eyes was not completely understood nor used. The second eye was eonsidered to be a spare in case of loss or damage to one eye. For cons man was ignorant of the principles of stereoscopy and depth perception. It might be fairly said t h a t only in the l a s t one hundred years, approximately corresponding to the period of h i s knowledge of photogrammetry, has man understandingly used his two eyes to measure distances, r a t h e r t h a n simply to e ~ i m a t e them, in three dimensions. In 1899, Sebastian F i n s t e r w a l d e r [1] of Germany established the foundation of analytical photogrammetry. In 1909 the Zeiss-Orel stereoautograph and later the ZeissH u g e r s h o f f aerocartograph established the foundation of instrumental photogrammetry. Thus, since approximately 1900 man has restricted his thinking about photogrammetry to the use of two photographs a t a time viewed with two eyes. The pair of photographs, or stereomodel, has been the limit and the unit for consideration both analytically and instrumentally. With the advent of electronic computers in the late 1950% man could begin to seriously consider the application of computational methods to the solution of his photog r a m m e t r y problems. He soon began to expand his vision and to talk about strips and then about blocks of photography. In 1958, the f i r s t general analytic solution for multiple station triangulation w a s presented by Brown [2]. Also in 1958 the general simultaneous solution for strips and blocks of photographs utilizing either ground control or exposure station data was presented by McNair, Dodge & Rutledge [3]. Other general solutions followed; but as of 1964, the massive amount of compuation of simultaneous equations and the limitations of the size of electronic computers has rendered the completely general solution still i m p r a c t i c a l Now we find m a n able to overcome another of his seeming physical limitations. In 1962 Mikhail [4] suggested "'viewing" three photographs a t a time so as to provide g r e a t e r s t r e n g t h in the tripIet model with a comparatively small increase in the computations required. In 1963 Mikhail [5] went a step f u r t h e r and presented the necessary theory for solution of two-directional triplets with 60 per cent overlap in both directions, t h a t is, the use of 9 photographs as a sub-block. In 1964 Anderson [6] developed the "three-eyed man", produced a working computer programme and obtained quantitative results demonstrating the g r e a t e r strength of triplet models over stereomodels. Let us examine briefly these methods which free man from his former limitations.
Triplets. In the f i r s t presentation of the triplet solution Mikhail [4] uses the colinearity method of Hellmut Schmid. The middle photograph (No. 2) of a t r i p l e t (Nos. 1, 2, 3) is held fixed (co~ = ~2 = ~2 = Yo~ = Zo2 = 0) and the airbase between the f i r s t two photographs is also held fixed (Xol = 0 and Xo2 = 1). Thus seven of the eighteen elements of the triplet are fixed and eleven elements are unknown. In this system the X- and Ycoordinates of points which a p p e a r on two photographs are determined and only in a triplet are all three X-, Y- and Z-coordinates determined. When the second triplet is tied to the first, Z-values are taken from the triplet overlaps and X- and Y-values are checked together from both triplets. The presentation of triplets by Anderson [6] is based upon the coplanarity principle [109]
Photog~ammetria, X I X , No. 7
368
developed b y S c h u t . ( F i g u r e 1). A g a i n t h e six e l e m e n t s of o r i e n t a t i o n a n d r e s e c t i o n o f t h e m i d d l e p h o t o g r a p h a r e held fixed. V a r i o u s s c h e m e s h a v e been t r i e d w i t h r e g a r d to fixing the airbase of the first two photographs: 1. i t m a y be a s s u m e d to be u n i t y a s in t h e p r e v i o u s m e t h o d ; 2. it m a y be a s s u m e d a s t h e e s t i m a t e d e x p o s u r e s t a t i o n d i s t a n c e ; 3. it m a y be a s s u m e d e q u a l to t h e a i r b a s e k n o w n f r o m a u x i l i a r y d a t a . T h e e l e m e n t s o f c a m e r a i n n e r o r i e n t a t i o n a r e a s s u m e d k n o w n . T h e r e f o r e , eleven u n k n o w n p a r a m e t e r s r e m a i n to be d e t e r m i n e d . P/-/O7-0 , "
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McNair, T~iplets: A Basic Unit for Analytical Aerotq'iangulation
369
Intersection condition equations are w r i t t e n for each point in any stereo overlap, (Figure 2) and also scale t r a n s f e r condition equations are w r i t t e n for those points which fall in the triplet overlap. An example is given in Figure 3. These equations are linearized in terms of corrections to the eleven unknown p a r a m e t e r s and the unknown residuals (x and y) for measured plate coordinates. The complete a r r a y of observation equations for triplet i, ], k is shown in F i g u r e 4.
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The matrix representing the syste m of observation equations can be written as: QV+NA+W=O the V matrices are the plate coordinate measurement residuals, the A matrices are the corrections to the eleven unknown parameters., Q and N are coefficient matrices of measured plate coordinate residuals and of the eleven unknown p a r a m e t e r s of the triplet, respectively, and the W matrices are the constants obtained by substitution of approximations for the unknowns. The least squares a d j u s t m e n t consists of minimizing the squares of the weighted plate coordinate residuals, V, and simultaneous,ly solving for the most probable values for the eleven unknowns in the triplet. The maximum size matrices which must be inverted are 3 X 3's f o r normal equations for residuals and 11 X l l ' s for corrections. Output from relative orientation consists of: 1. Exposure station coordinates and orientation matrices for the three photos in the a r b i t r a r y X-, Y-, Z-coordinates system. 2. Plate coordinates measuring residuals for images of all points used in the triangulation. 3. Standard e r r o r of unit weight of the adjustment• A t this point, having the x and y plate coordinate residuals, the photogrammetrist can detect and discard any poorly measured points• He generally would do this by setting up a criterion for rejection based on the size of the indicated m e a s u r i n g errors and where
[1113
Photogrammet~ia, X I X , No. 7
370
programme this for his computer. The computer rejects the points containing large errors and recomputes the unknown orientation parameters. Triplet model coordinates in the a r b i t r a r y system are then computed immediately for all ray intersections. Three methods have been tried for this: 1. approximate coordinates obtained as the averages of observed sectors; 2. most probable coordinates using corrected plate coordinates; 3. most probable coordinates by least squares a d j u s t m e n t using colinearity condition equations. Relative orientation of the next triplet is performed as follows: 1. set the position and orientation of the middle photograph of this triplet equal to the position and orientation just computed f o r it as the third photograph of the preceding triplet; 2. set the X-component of the airbase between the f i r s t two photos equal to the value of the X-component j u s t calculated for the airbase between the second and t h i r d photos of the preceding triplet. Thus full advantage is taken of the 100 per cent overlap of successive common photos without sacrificing the rigour of independent relative orientation for each triplet. Note t h a t assembly of the triplets (strip assembly) takes place at the same time as relative orientation and model coordinate computation proceed along the strip. Therefore, all computed results are automatically in a common, a r b i t r a r y strip coordinate system. Lastly the a r b i t r a r y strip coordinate system is t r a n s f o r m e d to ground control using a linear three-dimensional conformal coordinate t r a n s f o r m a t i o n . In the ordinary case of redundant ground control a linear t r a n s f o r m a t i o n may not provide sufficient accuracy. If this h a p p e n s a secondary adjustment is performed using second degree equations in a three-dimensional simultaneous coordinate t r a n s f o r m a t i o n developed by Mikhail [5]. Two-directional triplets in sub-blocks. If a homogeneous block of photographs over an area (not a strip) is desired, more sidelap than the traditional 20 to 30 per cent will be required. If the ground control occurs in a random pattern, t h a t is, not concentrated along a strip, a g r e a t e r s t r e n g t h of computed or photogrammetric control between observed ground control is desirable. Mikhail [5] has suggested a 60% sidelap of adjacent strips of photography so as to provide the photogrammetrist freedom to extend in any direction between control with equal accuracy. More important, however, is the concept of using a central photograph and tying it and every photograph which touches it into a unified block and calling this a unit or sub-block. This sub-block normally would be composed of nine photographs, one at the center and eight around it, or three strips of three each. A total of 54 elements exist for absolute orientation of the nine photographs. 47 parameters of relative orientation exist so a computer with sufficient capacity to handle a 47 X 47 m a t r i x is required. The general procedure would be like t h a t described above for relative orientation, triplet assembly, coordinate computation, and coordinate t r a n s formation of a single triplet. Space does not permit a thorough explanation and derivation. The theory has been worked out. Cornell University, Ithaca, New York, is currently p r e p a r i n g a computer programme in F O R T R A N 62 on its Control Data Corporation 1604 computer. When this is completed later in 1964 tests will be made on both fictitions and real photography to determine the advantage and limitations of the sub-block approach. No tests have been run as yet. Difficulty is being encountered in finding suitable photography with 60 per cent side lap.
[112]
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0.150 0.150 0.080 0.150 0.080 0.150
Rejection Limit Pass Points mm
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Plwtogrammetria, X I X , No. 7
372
T e s t results for strip a e r o t r i a n g u l a t i o n by triplets. Dr. A n d e r s o n wrote the computer p r o g r a m m e a n d p e r f o r m e d the t e s t s which follow. F i r s t , tests were r u n with the well-known C h u r c h fictitious s t r i p of p h o t o g r a p h s 1 to 12 inclusive: scale 1 : 80,000, f = 150 ram, relief a p p r o x i m a t e l y 10 per cent of f l i g h t height, flown E a s t , u s i n g geocentric coordinates. Strip 1 used 9 points per photo, 6 points per model, in the conventional positions and w a s t r u l y a cantilever extension w i t h 4 complete (horizontal a n d vertical) control points in the f i r s t model. Strip 2 also used 9 points per photo b u t h a d 2 complete g r o u n d control points in each end model. Strip 3 used 15 points per photo, 10 points per model, and 2 complete g r o u n d control points a t each end. A f o u r t h t e s t w a s r u n u s i n g the US Geological S u r v e y fictitious N o r t h - S o u t h strip of photos 0 t h r o u g h 5: scale 1 : 16,000, f = 150 mm, a p p r o x i m a t e l y 10 per cent relief, 25 points per p h o t o g r a p h , 15 points per model, 4 complete g r o u n d control points 2 a t each end of the strip. The r e s u l t s are shown in Table No. 1. The m a x i m u m discrepancies are reported in geocentric X-, Y- a n d Z-directions. The n e x t three columns are the s t a n d a r d errors of u n i t weight in feet c~=
6
where V is the component of the discrepancy between the fictitious and the computed position of the g r o u n d point, P is a weight m a t r i x (unit m a t r i x was used t h r o u g h o u t ) , and 6 is the overdetermination or the n u m b e r of eondition equations m i n u s the n u m b e r of unknowns. A s a result of these t e s t s it w a s concluded: 1. :2. 3.
t h a t the equations were correct; t h a t the computer p r o g r a m m e w a s correct; t h a t a sufficient n u m b e r of s i g n i f i c a n t f i g u r e s were being carried in t h e computer.
Secondly, r a n d o m l y distributed displacements in both X a n d Y were applied to the fictitious plate coordinates. Two sets of t e s t s were p e r f o r m e d with these d a t a to v e r i f y the least s q u a r e s a d j u s t m e n t where r a n d o m errors occur. Set 1 h a d a n o r m a l distribution of errors w i t h a m e a n square value of 0.050 m m a n d a m a x i m u m of 0.150 m m applied to both the x and y coordinates or p h o t o g r a p h s 2 t h r o u g h 10 of the Church strip. F i g u r e 5 shows the distribution of the p a s s points and control. Test cases 5, 6 and 8 used all 15 points per photograph. Test cases 7 a n d 9 h a d a rejection critelion of -+ 0.080 m m applied. T h i s eliminated those p a s s points h a v i n g the l a r g e s t plate coordinate discrepancies and sensibly reduced the m e a n s q u a r e e r r o r of the computed g r o u n d positions. Table No. 2 shows t h e results. The d a t a in Set 1, strip 5 h a d been u s e d previously for cantilever computation by a modified H e r g e t method of stereopairs. A comparison of stereopair and stereotriplet assembles is shown in F i g u r e 6 for strips 5, 6 a n d Herget. A v e r a g e discrepancies of X, Y a n d Z geocentric coordinates a n d of the r e s u l t a n t h a v e been plotted. The discrepancies of 150 microns m a x i m u m , used in Set 1, were a p p r o p r i a t e six to ten y e a r s ago when the f i r s t analytical a e r o t r i a n g u l a t i o n s were performed, b u t t h e y were felt to be unrealistically large with 1964 c o m p a r a t o r i n s t r u m e n t s . Therefore, Set l ( a ) w a s r u n on t e s t cases 10 t h r o u g h 14 u s i n g m a x i m u m x- or y-displacements to t h e fictitious d a t a of 0.015 m m w i t h a m e a n square d i s p l a c e m e n t of 0.005 ram. The distribution of p a s s points and control points is shown in F i g u r e 7 and the r e s u l t s compiled in Table No. 2. In t e s t cases 11, 13 and 14 rejection criteria of 0.008, 0.008 and 0.005 m m respectively were used to eliminate points with the l a r g e s t r a n d o m errors. It will be noted t h a t rejection of these points did not g r e a t l y improve the results. [114]
McNair, Triplets : A Basic Unit for Analytical Aerotriangulation
373
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Photogrammetr~a, X1X, No. 7
376
Thirdly, a single test on real photography was performed. This strip is 17 miles in length at a scale of 1 : 20.000. Photo coordinates had been measured on the Nistri TA-3 stereocomparator and corrected for lens distortion and film shrinkage. Results are listed in Table No. 3 for 15 photographs of the strip. Note t h a t the largest residual is 8.1 microns.
T A B L E No. 3. Summary of results Arizona t e s t strip 7. Std. E r r o r of Unit Wt.
Triplet No.
3
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5 6 7 8 9 10 11 12 13
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0.0040 0.0076 0.0075 0.0061 0.0055 0.0044 0.0098 0.0082 0.0047 0.0060 0.0044 0.0069 0.0055
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0.0035 0.0029 0.0051 0.0028 0.0041 0.0042 0.0041 0.0053 0.0023 0.0021 0.0025 0.0058 0.0025
0.0030 0.0072 0.0103 0.0054 0.0054 0.0031 0.0056 0.0049 0.0053 0.0054 0.0048 0.0081 0.0069
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Conclusions. The completely general, rigorous solution to aerial triangulation of strips and of blocks results in a massive amount of calculation and data processing. A t the other extreme computation of stereopairs, especially in cantilever extension, can be performed on reasonable-sized computers but leaves much to be desired in rigour, accuracy and builtin checks. The stereotriplet, "looking at" three photographs a t a time, overcomes the limitations of stereopairs imposed by physical and i n s t r u m e n t a l characteristics. The central photograph of each triplet is overlapped 100 per cent. Mathematical observation of the three photographs provides detection of large plate coordinate errors since residuals are available for both x- and y-image coordinates. Assembly of the triplets in the order 1-2-3, 2-3-4, 3 - 4 - 5 . . . makes maximum use of the 100 per cent overlap of successive photographs eliminating the "hinge effect" of stereopairs, and simultaneously, with no additional work whatsoever, provides unified model coordinates for the entire strip. Development of a computer programme to test the feasibility of two-directional subblocks is in progress at the Surveying D e p a r t m e n t of Cornell University. A CDC 1604 computer p r o g r a m m e d in F O R T R A N 62 is being used. Both fictitious and real photog r a p h y are to be tested using the programme in 1964. I f the sub-block approach proves to be as much improvement as the triplets have been it should then be widely adopted wherever there is a desire to minimize the amount of ground control required for a block. However, photographic procedures would then have to be changed to provide 60 per cent side lap on all flights r a t h e r t h a n the conventional 20 to 30 per cent.
[118]
McNair, Triplets': A Basic Unit f~r Analytical Aerotriangulation
377
Tests of both fictitious and real photography using the stereotriplet as the basic unit indicate t h a t : 1. "Wild" points can be detected; 2. a criterion can be set up for the rejection of poorly measured image coordinates and the computations can proceed un~nterruptedly using acceptably measured points; 3. an economy of computer time as compared to a general solution is achieved; 4. computed ground control pesitions have only a half to a third of the discrepancies produced using the stereopair as the basic unit. BIBLIOGRAPHY. [1] F i n s t e r w a l d e r, S., Die Geometrischen Grundlagen der Photogrammetrie. J a h r e s bericht der Deutsehen Mathematischen ¥ereinigung, VI, 2, Leipzig 1899. [2] B r o w n , Duane, A solution to the General Problem of Multiple Station Analytical Stereotriangulation. RCA Data Reduction Technical Report No. 43, F e b r u a r y 1958. [3] M c N a i r , A. J., D o d g e , H. F., R u t l e d g e , J. D., A solution of the General Analytical Aerotriangulation Problem. Final Technical Report, E R D L Contact No. DA-44-009 E N G 2986, Cornell University, May 1958. [4] M i k h a i l , E d w a r d M., Use of Triplets for Analytical Aerotriangu!ation. Photogrammetric Engineering, September 1962. [5] M i k h a i l , E d w a r d M., A New Approach to Analytical Aerotriangulation: TwoDirectional Triplets in Sub-Blocks. Ph. D. Thesis, Cornell Univ. June 1963. [6] A n d e r so n, James M., Analytical Aerotriangulation Using Triplets, P h . D . Thesis, Cornell University, Ithaca, N.Y. June 1964.
Discussion Ackermann: I suppose I have to make some remarks because I have committed myself in the invited paper for Comm. III by stating t h a t Prof. Lehmann has proved t h a t the triplet method is not more accurate than the conventional method. I modify this is so f a r as I should have said t h a t it is not substantially better. Prof. Lehmann has shown t h a t with the triplet procedure most of the orientation elements have the same accuracy as with the classical procedure. For instance, the scale t r a n s f e r is exactly the same. There are, however, a few orientation elements, ~ and (o in particular, which have smaller standard errors, in the order of 20%. But this alone is not yet really conclusive, as the correlation p a t t e r n of these orientation elements is also different. It still remains to be shown by how much the model coordinates will be improved. I do not expect striking differences. The important f e a t u r e of the triplet procedure seems to the use of x-parallaxes in the common overlap. However, this is also possible with orientation methods which use stereo-pairs only. It is well known t h a t such a procedure improves the ~ and ~ precision and introduces mutual correlation. The triplet method should therefore be compared with the bridging method which uses also x-parallaxes for relative orientation. As f a r as I have understood the method, the relative orientation of each stereo-pair is computed twice in the triplet method. Both orientations use the same y-parallax information but differ with regard to different x-parallax information. I would like te know how the two d i f f e r e n t orientations of each stero-pair are used for the strip formation. [119]
Photogrammetria, X I X , No. 7
378
W e i g h t m a n : The m a i n value seems to is on three p h o t o g r a p h s . The drawback of point t h a t is on t h r e e p h o t o g r a p h s , you use l a t e r one, a n d the f a c t t h a t it is on the t h r e e t h a t problem a n d t h e r e f o r e I like it.
be t h a t you a r e u s i n g the f a c t t h a t a point the n o r m a l m e t h o d is t h a t , if you h a v e a it once in t h e f i r s t stereogram, a g a i n in the together is ignored. This method g e t s a r o u n d
FSrstne~: I f I understood Dr. A c k e r m a n n correctly he said t h a t the new method is m o r e a c c u r a t e t h a n the old one because o f the relative orientation. T h e r e f o r e ycu h a v e a proof t h a t t h e relative orientation w a s not a t a n o p t i m u m . Otherwise the a c c u r a c y c a n n o t be g r e a t e r t h a n before. Avke~mann: W i t h the use of x - p a r a l l a x e s the relative orientation should be s o m e w h a t m o r e accurate, because one is u s i n g additional i n f o r m a t i o n which is not used in t h e n o r m a l y - p a r a l l a x procedure. There is, hoewever, a certain d a n g e r in u s i n g xp a r a l l a x e s f o r relative orientation, which m a y have p r e v e n t e d its application up to now, a l t h o u g h t h e m e t h o d as such h a s been known for a long time. Where s y s t e m a t i c model d e f o r m a t i o n s a r e p r e s e n t the corresponding e r r o r s in x - p a r a l l a x e s a : e imposed on t h e s u b s e q u e n t relative orientation. Thompson: A few y e a r s ago a method of Bartorelli w a s popular t h a t did t h i s r i g h t t h r o u g h the strip. P e r h a p s someone could tell u s about it? de Masson d ' A u t u m e : D a n s la m~thode IGN qui est expos~e au congr~s de Londres et en suite '& la r~union de Milan on op~re p a r modules m a i s ensuite on utilise la connexion de deux modules cons~qutifs, et en particulier la discordance en Z qu~on constate, p o u r am~liorer l'orientation relative des deux modules. C'est u n peu a n a l o g u e ~ ce que r~alise la m~thode des triplets. I1 reste me semble u n petite difficultY: vous calculez done le t r i p l e t 1, 2, 3. I n c o n t e s t a b l e m e n t l'orientation relative que vous trouvez pour c h a c u n des d e u x modules consgcutives est u n peu meilIeure que si vous n'aviez p a s utilis~ la liaison que vous a v e z i n t r o d u i t comme u n e condition s u p p l e m e n t a i r e avec le poids qui lui revient. Vous obtenez donc deux t r i p l e t s dont les o r i e n t a t i o n s relatives sont meilleures que si vous aviez utilis~ des modules simples, m a i s il n'emp~che que m a t r i c e rotation 2 - - 3 que vous trouvez d a n s le p r e m i e r triplet I, 2, 3 a a u c u n e raison d'etre la m~me que celle qui vous obtenez d a n s le triplet 2--3---4. Vous avez donc le choix entre d e u x m a t r i c e s rotations, prenez-vous la m o y e n n e ou si non c o m m e n t operez-vous? E n s u i t e q u a n d vous raccordez, les triplets, en a d m i t t a n t que vous ayez r~solu cette premiere difficultY, alors vous disposez d ' u n c e r t a i n hombre de t r i p l e t s qui se r e c o u v r e n t m a i s vous constatez forcement des discordances d a n s la partie commune entre deux triplets jointifs. Thompson: M. de Masson d ' A u t u m e , si j'ai bien compri Ia m~thode, ce n'est p a s u n r e c o u v r e m e n t d ' u n module, c'est s i m p l e m e n t r e c o u v r e m e n t d'une seule photo, c ' e s t ~ dire on a 1 - - 2 - - 3 et p u i s 3---4--5. P e r h a p s Prof. M c N a i r could tell us. M v N a i r : The o v e r l a p s are 11 2, 3; 2~ 3, 4; 3, 4, 5; the middle photo of the triplet a s s u m e s the s a m e orientation t h a t w a s j u s t computed f o r it a s 3rd photo in t h e previous triplet. A n d t h e X-coordinate for the base a s s u m e s the s a m e length as j u s t previously computed a s t h e b a s e 2, 3 in the previous triplet. de Masson d ' A u t u m e : Je suis d'accord m a i s ~a n'emp~che que vous devez c o n s t a t e r f o r c e m e n t des discordances d a n s la partie qui se recouvre. McNair: [120]
We do get two relative orientations. We also get the X a n d Y d i s c r e -
Discussion on the paper of McNair
379
pancies a t a n y points which occurred in the f i r s t a n d second triplet. Now we can calculate t r i p l e t model coordinates by u s i n g the individual X, Y coordinates as modified by the discrepancies, a n d t h e n take the a v e r a g e of the r e s u l t s of t h e two d i f f e r e n t models, t h i s is the simplest method. Alternatively, we m a y calculate the m o s t probable coordinates u s i n g corrected plate coordinates, or thirdly we can calculate the most probable coordinates by least s q u a r e s a d j u s t m e n t u s i n g the co-linear condition equations. T h i s is considerable work, b u t does not take long in an electronic calculator. I should h a v e indicated t h a t on this l a s t t e s t we calculated 13 triplets in ten and a fraction m i n u t e s , costing about $ 3 per model.
Roelofs: I would like to m a k e a r a t h e r f u n d a m e n t a l r e m a r k . I would s a y t h a t if you are u s i n g the s a m e observations, if you are a d j u s t i n g these r i g o r o u s l y according to the method of least s q u a r e s you a l w a y s g e t the s a m e a n s w e r w h e t h e r you use stereo pairs, stereo templets or stereo q u a d r u p l e t s or w h a t ever it m a y be. You m u s t take into account in each step t h e correlations you used a n d t h e p r o p a g a t i o n of these correlations and weights. So if you are c o m p a r i n g methods I should s a y you should find out where t h e observations a r e d i f f e r e n t a n d w h a t a p p r o x i m a t i o n s are made. Doyle: I agree w i t h Prof. Roelofs, however, I believe t h a t in the method proposed by Prof. M c N a i r additional observations h a v e been introduced, in t h a t they f i r s t computed p h o t o g r a p h s 1, 2 a n d 3 with atl possible condition e q u a t i o n s a n d t h e n photog r a p h s 2, 3 a n d 4 w i t h all possible condition equations. T h e r e f o r e t h e y h a v e introduced additional observation e q u a t i o n s and should not expect to get t h e s a m e orientation m a t r i x nor t h e same b a s e length. I feel t h a t Prof. M c N a i r h a s not y e t a n s w e r e d the question as to w h a t he does w i t h the two d i f f e r e n t orientation m a t r i c e s a n d with the two d i f f e r e n t base lengths. I would like to add f u r t h e r comment. In m y experience the rejection of a point s i m p l y because the least square residual on t h a t point h a p p e n s to be large is not a good rejection criterion. Quite often you find t h a t for s a m e u n k n o w n r e a s o n t h e residuals a r e crowded into one p a r t i c u l a r point, and t h a t point itself is not poorly m e a s u r e d nor is it erroneously identified. I confess t h a t I don't know a n y better criterion for r e j e c t i n g points, b u t I know t h a t t h i s is not a good one. Jerie: I t h i n k it is quite obvious t h a t t h e only difference between the triplet method a n d conventional a e r o - t r i a n g u l a t i o n lies in the f a c t t h a t x - p a r a l l a x e s a r e also used to determine the relative orientation. In the triplet method one will obtain two d i f f e r e n t relative orientations f o r each model. Is it now correct simply to t a k e the m e a n of these two relative o r i e n t a t i o n s ? Does not this disturb the homogeneity of the strip f o r m a t i o n ? Inghilleri: In our method of computation we use the X - p a r a l l a x e s or h e i g h t s of the orientation points, s o m e w h a t similarly to the t r i p l e t method. The only difference is t h a t the condition equations are not solved s i m u l t a n e o u s l y for two a d j a c e n t models, b u t it n e v e r t h e l e s s gives a n improved connection between t h e three photos t h a t are involved in two a d j a c e n t models. Mikhail: The model by model a p p r o a c h t h a t is used in the U.S. Co~st and Geodetic S u r v e y is to p e r f o r m t h e f i r s t model completely independently f r o m the second model, i.e. w i t h o u t c a r r y i n g p a r a m e t e r s . W h e n I t h o u g h t of the triplets I w a s merely t h i n k i n g of m a k i n g use of the triple overlap, t h e n c a r r y i n g on w i t h the second triplet entirely independently f r o m the first. So one ends up with three i n d e p e n d e n t d e t e r m i n a t i o n s of every p h o t o g r a p h e x c l u d i n g the f i r s t a n d the l a s t p a i r s within t h e strip. Dr. Anderson, however, p e r f o r m s the relative orientation and strip a s s e m b l y in one step; t h u s o b t a i n i n g only two d e t e r m i n a t i o n s for each photograph. I a m not quite sure 24* [121]
Photogrammetria, XIX, No. 7
380 Triplet number
Photo number
1
2
3
4
5
6
7
8
whether he use the average of the two independent determinations or not, and I don't t h i n k the two determinations of each photograph will be the same. To me the g r e a t advantage of t h e triplet method lies in the s t i f f connections between triplets when building up the total strip a f t e r the determination of each triplet. A n o t h e r advantage of the triplet method is t h a t because of the better connection of the models one is not restricted to having the ground control in specified positions as required in the normal aero-triangulation procedure.
[122]
Triplets: A Basic Unit for Analytical Aerotriangulation by Professor A r t h u r J. MeNAIR Cornel1 University, Ithaca, New York, U.S.A. In the beginning God created man with two eyes. F o r thousands of years, the full capability of this gift of two eyes was not completely understood nor used. The second eye was eonsidered to be a spare in case of loss or damage to one eye. For cons man was ignorant of the principles of stereoscopy and depth perception. It might be fairly said t h a t only in the l a s t one hundred years, approximately corresponding to the period of h i s knowledge of photogrammetry, has man understandingly used his two eyes to measure distances, r a t h e r t h a n simply to e ~ i m a t e them, in three dimensions. In 1899, Sebastian F i n s t e r w a l d e r [1] of Germany established the foundation of analytical photogrammetry. In 1909 the Zeiss-Orel stereoautograph and later the ZeissH u g e r s h o f f aerocartograph established the foundation of instrumental photogrammetry. Thus, since approximately 1900 man has restricted his thinking about photogrammetry to the use of two photographs a t a time viewed with two eyes. The pair of photographs, or stereomodel, has been the limit and the unit for consideration both analytically and instrumentally. With the advent of electronic computers in the late 1950% man could begin to seriously consider the application of computational methods to the solution of his photog r a m m e t r y problems. He soon began to expand his vision and to talk about strips and then about blocks of photography. In 1958, the f i r s t general analytic solution for multiple station triangulation w a s presented by Brown [2]. Also in 1958 the general simultaneous solution for strips and blocks of photographs utilizing either ground control or exposure station data was presented by McNair, Dodge & Rutledge [3]. Other general solutions followed; but as of 1964, the massive amount of compuation of simultaneous equations and the limitations of the size of electronic computers has rendered the completely general solution still i m p r a c t i c a l Now we find m a n able to overcome another of his seeming physical limitations. In 1962 Mikhail [4] suggested "'viewing" three photographs a t a time so as to provide g r e a t e r s t r e n g t h in the tripIet model with a comparatively small increase in the computations required. In 1963 Mikhail [5] went a step f u r t h e r and presented the necessary theory for solution of two-directional triplets with 60 per cent overlap in both directions, t h a t is, the use of 9 photographs as a sub-block. In 1964 Anderson [6] developed the "three-eyed man", produced a working computer programme and obtained quantitative results demonstrating the g r e a t e r strength of triplet models over stereomodels. Let us examine briefly these methods which free man from his former limitations.
Triplets. In the f i r s t presentation of the triplet solution Mikhail [4] uses the colinearity method of Hellmut Schmid. The middle photograph (No. 2) of a t r i p l e t (Nos. 1, 2, 3) is held fixed (co~ = ~2 = ~2 = Yo~ = Zo2 = 0) and the airbase between the f i r s t two photographs is also held fixed (Xol = 0 and Xo2 = 1). Thus seven of the eighteen elements of the triplet are fixed and eleven elements are unknown. In this system the X- and Ycoordinates of points which a p p e a r on two photographs are determined and only in a triplet are all three X-, Y- and Z-coordinates determined. When the second triplet is tied to the first, Z-values are taken from the triplet overlaps and X- and Y-values are checked together from both triplets. The presentation of triplets by Anderson [6] is based upon the coplanarity principle [109]
Photog~ammetria, X I X , No. 7
368
developed b y S c h u t . ( F i g u r e 1). A g a i n t h e six e l e m e n t s of o r i e n t a t i o n a n d r e s e c t i o n o f t h e m i d d l e p h o t o g r a p h a r e held fixed. V a r i o u s s c h e m e s h a v e been t r i e d w i t h r e g a r d to fixing the airbase of the first two photographs: 1. i t m a y be a s s u m e d to be u n i t y a s in t h e p r e v i o u s m e t h o d ; 2. it m a y be a s s u m e d a s t h e e s t i m a t e d e x p o s u r e s t a t i o n d i s t a n c e ; 3. it m a y be a s s u m e d e q u a l to t h e a i r b a s e k n o w n f r o m a u x i l i a r y d a t a . T h e e l e m e n t s o f c a m e r a i n n e r o r i e n t a t i o n a r e a s s u m e d k n o w n . T h e r e f o r e , eleven u n k n o w n p a r a m e t e r s r e m a i n to be d e t e r m i n e d . P/-/O7-0 , "
Pl'lO.rOj
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Fig. 2. Scale T r a n s f e r P o i n t
[110]
Area q Area p Area r
la ~
t
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.
Ib
Ia points = (1,2,..., m) II p o i n t s = (1,2 . . . . , t ) Ib p o i n t s = ( 1 , 2 , . . . , l)
Fig. 3.
m = 10 t =
5
l = 10
E x a m p l e T r i p l e t - 25 p o i n t s
McNair, T~iplets: A Basic Unit for Analytical Aerotq'iangulation
369
Intersection condition equations are w r i t t e n for each point in any stereo overlap, (Figure 2) and also scale t r a n s f e r condition equations are w r i t t e n for those points which fall in the triplet overlap. An example is given in Figure 3. These equations are linearized in terms of corrections to the eleven unknown p a r a m e t e r s and the unknown residuals (x and y) for measured plate coordinates. The complete a r r a y of observation equations for triplet i, ], k is shown in F i g u r e 4.
~ 0. "1 •
0
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Fig. 4. System of observation equations for a triplet.
The matrix representing the syste m of observation equations can be written as: QV+NA+W=O the V matrices are the plate coordinate measurement residuals, the A matrices are the corrections to the eleven unknown parameters., Q and N are coefficient matrices of measured plate coordinate residuals and of the eleven unknown p a r a m e t e r s of the triplet, respectively, and the W matrices are the constants obtained by substitution of approximations for the unknowns. The least squares a d j u s t m e n t consists of minimizing the squares of the weighted plate coordinate residuals, V, and simultaneous,ly solving for the most probable values for the eleven unknowns in the triplet. The maximum size matrices which must be inverted are 3 X 3's f o r normal equations for residuals and 11 X l l ' s for corrections. Output from relative orientation consists of: 1. Exposure station coordinates and orientation matrices for the three photos in the a r b i t r a r y X-, Y-, Z-coordinates system. 2. Plate coordinates measuring residuals for images of all points used in the triangulation. 3. Standard e r r o r of unit weight of the adjustment• A t this point, having the x and y plate coordinate residuals, the photogrammetrist can detect and discard any poorly measured points• He generally would do this by setting up a criterion for rejection based on the size of the indicated m e a s u r i n g errors and where
[1113
Photogrammet~ia, X I X , No. 7
370
programme this for his computer. The computer rejects the points containing large errors and recomputes the unknown orientation parameters. Triplet model coordinates in the a r b i t r a r y system are then computed immediately for all ray intersections. Three methods have been tried for this: 1. approximate coordinates obtained as the averages of observed sectors; 2. most probable coordinates using corrected plate coordinates; 3. most probable coordinates by least squares a d j u s t m e n t using colinearity condition equations. Relative orientation of the next triplet is performed as follows: 1. set the position and orientation of the middle photograph of this triplet equal to the position and orientation just computed f o r it as the third photograph of the preceding triplet; 2. set the X-component of the airbase between the f i r s t two photos equal to the value of the X-component j u s t calculated for the airbase between the second and t h i r d photos of the preceding triplet. Thus full advantage is taken of the 100 per cent overlap of successive common photos without sacrificing the rigour of independent relative orientation for each triplet. Note t h a t assembly of the triplets (strip assembly) takes place at the same time as relative orientation and model coordinate computation proceed along the strip. Therefore, all computed results are automatically in a common, a r b i t r a r y strip coordinate system. Lastly the a r b i t r a r y strip coordinate system is t r a n s f o r m e d to ground control using a linear three-dimensional conformal coordinate t r a n s f o r m a t i o n . In the ordinary case of redundant ground control a linear t r a n s f o r m a t i o n may not provide sufficient accuracy. If this h a p p e n s a secondary adjustment is performed using second degree equations in a three-dimensional simultaneous coordinate t r a n s f o r m a t i o n developed by Mikhail [5]. Two-directional triplets in sub-blocks. If a homogeneous block of photographs over an area (not a strip) is desired, more sidelap than the traditional 20 to 30 per cent will be required. If the ground control occurs in a random pattern, t h a t is, not concentrated along a strip, a g r e a t e r s t r e n g t h of computed or photogrammetric control between observed ground control is desirable. Mikhail [5] has suggested a 60% sidelap of adjacent strips of photography so as to provide the photogrammetrist freedom to extend in any direction between control with equal accuracy. More important, however, is the concept of using a central photograph and tying it and every photograph which touches it into a unified block and calling this a unit or sub-block. This sub-block normally would be composed of nine photographs, one at the center and eight around it, or three strips of three each. A total of 54 elements exist for absolute orientation of the nine photographs. 47 parameters of relative orientation exist so a computer with sufficient capacity to handle a 47 X 47 m a t r i x is required. The general procedure would be like t h a t described above for relative orientation, triplet assembly, coordinate computation, and coordinate t r a n s formation of a single triplet. Space does not permit a thorough explanation and derivation. The theory has been worked out. Cornell University, Ithaca, New York, is currently p r e p a r i n g a computer programme in F O R T R A N 62 on its Control Data Corporation 1604 computer. When this is completed later in 1964 tests will be made on both fictitions and real photography to determine the advantage and limitations of the sub-block approach. No tests have been run as yet. Difficulty is being encountered in finding suitable photography with 60 per cent side lap.
[112]
12 13 14
10
6 7 8 9 Herget
5
Number
i 1 I
L
3.70 1.33 1.38 3.20
AX'
3.77 1.77 1.79 0.07
6.03 1.67 1.82 0.06 I
AZ"
AY'
Geocentric M a x i m u m Discrepancies, Ft. I
I Oy,
2.45 1.17 1.04 0.03
O X,
1.14 0.90 0.76 2.19
S t a n d a r d E r r o r s of Unit Weight, Ft.
1.23 0.96 1.15 0.05
9 9 15 25
Pass Points per Photo
Number of Tie Points
No. 1. S u m m a r y of d i s c r e p a n c i e s t e s t c a s e s 1 - - 4 ; t h e o r e t i c a l l y c o r r e c t f i c t i t i o u s d a t a .
Cantilever Bridge Bridge Bridge
Type Extension
+z
,~
'~
r-~
57.15 94.02 122.54 --60.59 --53.72 256.37 11.87 11.85 12.08 12.00 --6.65
201.50 --170.74 78.95 58.92 --55.71 292.81
--13.78 --7.80 6.98 --5.27 --7.20
130.08 109.54 96.32 --41.26 --37.43 77.19
--11.69 --11.85 --12.05 --13.00 --11.00
A Z'
AY"
AX ~
Geocentric M a x i m u m Errors Ft.
77.92 61.00 35.26 39.04 23.33 193.99 4.78 3.29 2.89 2.41 2.93
2.93 3.52 2.37 3.31 2.50
Gy,
57.15 31.01 26.77 29.25 26.77 48.34
~X'
S t a n d a r d E r r o r s of Unit Weight, Ft.
6.26 5.80 4.66 4.94 4.52
2 80 3.23 2.78 2.36
2.76
99.41 75.76 68.38 52.38 45.23 251.26 23.34 32.52 52.12 30.23 28.01 152.20
GZ ,
Resultant M.S.E. in Position Ft.
Cantilever Cantilever Cantilever Bridge Bridge Cantilever Cantilever Cantilever Bridge Bridge Bridge
4 4 4 6 6 4 4 4 6 6 6 0.015 0.008 0.015 0.008 0.005
No. of Tie P o i n t s
0.150 0.150 0.080 0.150 0.080 0.150
Rejection Limit Pass Points mm
No. 2. S u m m a r y of d i s c r e p a n c i e s b e t w e e n f i c t i t i o u s a n d c a l c u l a t e d g e o c e n t r i c c o o r d i n a t e s t e s t c a s e s 5 - - 1 4 . H e r g e t m e t h o d , C h u r c h D a t a R a n d o m Sets 1, l ( a ) .
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Plwtogrammetria, X I X , No. 7
372
T e s t results for strip a e r o t r i a n g u l a t i o n by triplets. Dr. A n d e r s o n wrote the computer p r o g r a m m e a n d p e r f o r m e d the t e s t s which follow. F i r s t , tests were r u n with the well-known C h u r c h fictitious s t r i p of p h o t o g r a p h s 1 to 12 inclusive: scale 1 : 80,000, f = 150 ram, relief a p p r o x i m a t e l y 10 per cent of f l i g h t height, flown E a s t , u s i n g geocentric coordinates. Strip 1 used 9 points per photo, 6 points per model, in the conventional positions and w a s t r u l y a cantilever extension w i t h 4 complete (horizontal a n d vertical) control points in the f i r s t model. Strip 2 also used 9 points per photo b u t h a d 2 complete g r o u n d control points in each end model. Strip 3 used 15 points per photo, 10 points per model, and 2 complete g r o u n d control points a t each end. A f o u r t h t e s t w a s r u n u s i n g the US Geological S u r v e y fictitious N o r t h - S o u t h strip of photos 0 t h r o u g h 5: scale 1 : 16,000, f = 150 mm, a p p r o x i m a t e l y 10 per cent relief, 25 points per p h o t o g r a p h , 15 points per model, 4 complete g r o u n d control points 2 a t each end of the strip. The r e s u l t s are shown in Table No. 1. The m a x i m u m discrepancies are reported in geocentric X-, Y- a n d Z-directions. The n e x t three columns are the s t a n d a r d errors of u n i t weight in feet c~=
6
where V is the component of the discrepancy between the fictitious and the computed position of the g r o u n d point, P is a weight m a t r i x (unit m a t r i x was used t h r o u g h o u t ) , and 6 is the overdetermination or the n u m b e r of eondition equations m i n u s the n u m b e r of unknowns. A s a result of these t e s t s it w a s concluded: 1. :2. 3.
t h a t the equations were correct; t h a t the computer p r o g r a m m e w a s correct; t h a t a sufficient n u m b e r of s i g n i f i c a n t f i g u r e s were being carried in t h e computer.
Secondly, r a n d o m l y distributed displacements in both X a n d Y were applied to the fictitious plate coordinates. Two sets of t e s t s were p e r f o r m e d with these d a t a to v e r i f y the least s q u a r e s a d j u s t m e n t where r a n d o m errors occur. Set 1 h a d a n o r m a l distribution of errors w i t h a m e a n square value of 0.050 m m a n d a m a x i m u m of 0.150 m m applied to both the x and y coordinates or p h o t o g r a p h s 2 t h r o u g h 10 of the Church strip. F i g u r e 5 shows the distribution of the p a s s points and control. Test cases 5, 6 and 8 used all 15 points per photograph. Test cases 7 a n d 9 h a d a rejection critelion of -+ 0.080 m m applied. T h i s eliminated those p a s s points h a v i n g the l a r g e s t plate coordinate discrepancies and sensibly reduced the m e a n s q u a r e e r r o r of the computed g r o u n d positions. Table No. 2 shows t h e results. The d a t a in Set 1, strip 5 h a d been u s e d previously for cantilever computation by a modified H e r g e t method of stereopairs. A comparison of stereopair and stereotriplet assembles is shown in F i g u r e 6 for strips 5, 6 a n d Herget. A v e r a g e discrepancies of X, Y a n d Z geocentric coordinates a n d of the r e s u l t a n t h a v e been plotted. The discrepancies of 150 microns m a x i m u m , used in Set 1, were a p p r o p r i a t e six to ten y e a r s ago when the f i r s t analytical a e r o t r i a n g u l a t i o n s were performed, b u t t h e y were felt to be unrealistically large with 1964 c o m p a r a t o r i n s t r u m e n t s . Therefore, Set l ( a ) w a s r u n on t e s t cases 10 t h r o u g h 14 u s i n g m a x i m u m x- or y-displacements to t h e fictitious d a t a of 0.015 m m w i t h a m e a n square d i s p l a c e m e n t of 0.005 ram. The distribution of p a s s points and control points is shown in F i g u r e 7 and the r e s u l t s compiled in Table No. 2. In t e s t cases 11, 13 and 14 rejection criteria of 0.008, 0.008 and 0.005 m m respectively were used to eliminate points with the l a r g e s t r a n d o m errors. It will be noted t h a t rejection of these points did not g r e a t l y improve the results. [114]
McNair, Triplets : A Basic Unit for Analytical Aerotriangulation
373
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Photogrammetr~a, X1X, No. 7
376
Thirdly, a single test on real photography was performed. This strip is 17 miles in length at a scale of 1 : 20.000. Photo coordinates had been measured on the Nistri TA-3 stereocomparator and corrected for lens distortion and film shrinkage. Results are listed in Table No. 3 for 15 photographs of the strip. Note t h a t the largest residual is 8.1 microns.
T A B L E No. 3. Summary of results Arizona t e s t strip 7. Std. E r r o r of Unit Wt.
Triplet No.
3
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5 6 7 8 9 10 11 12 13
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0.0040 0.0076 0.0075 0.0061 0.0055 0.0044 0.0098 0.0082 0.0047 0.0060 0.0044 0.0069 0.0055
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0.0035 0.0029 0.0051 0.0028 0.0041 0.0042 0.0041 0.0053 0.0023 0.0021 0.0025 0.0058 0.0025
0.0030 0.0072 0.0103 0.0054 0.0054 0.0031 0.0056 0.0049 0.0053 0.0054 0.0048 0.0081 0.0069
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Conclusions. The completely general, rigorous solution to aerial triangulation of strips and of blocks results in a massive amount of calculation and data processing. A t the other extreme computation of stereopairs, especially in cantilever extension, can be performed on reasonable-sized computers but leaves much to be desired in rigour, accuracy and builtin checks. The stereotriplet, "looking at" three photographs a t a time, overcomes the limitations of stereopairs imposed by physical and i n s t r u m e n t a l characteristics. The central photograph of each triplet is overlapped 100 per cent. Mathematical observation of the three photographs provides detection of large plate coordinate errors since residuals are available for both x- and y-image coordinates. Assembly of the triplets in the order 1-2-3, 2-3-4, 3 - 4 - 5 . . . makes maximum use of the 100 per cent overlap of successive photographs eliminating the "hinge effect" of stereopairs, and simultaneously, with no additional work whatsoever, provides unified model coordinates for the entire strip. Development of a computer programme to test the feasibility of two-directional subblocks is in progress at the Surveying D e p a r t m e n t of Cornell University. A CDC 1604 computer p r o g r a m m e d in F O R T R A N 62 is being used. Both fictitious and real photog r a p h y are to be tested using the programme in 1964. I f the sub-block approach proves to be as much improvement as the triplets have been it should then be widely adopted wherever there is a desire to minimize the amount of ground control required for a block. However, photographic procedures would then have to be changed to provide 60 per cent side lap on all flights r a t h e r t h a n the conventional 20 to 30 per cent.
[118]
McNair, Triplets': A Basic Unit f~r Analytical Aerotriangulation
377
Tests of both fictitious and real photography using the stereotriplet as the basic unit indicate t h a t : 1. "Wild" points can be detected; 2. a criterion can be set up for the rejection of poorly measured image coordinates and the computations can proceed un~nterruptedly using acceptably measured points; 3. an economy of computer time as compared to a general solution is achieved; 4. computed ground control pesitions have only a half to a third of the discrepancies produced using the stereopair as the basic unit. BIBLIOGRAPHY. [1] F i n s t e r w a l d e r, S., Die Geometrischen Grundlagen der Photogrammetrie. J a h r e s bericht der Deutsehen Mathematischen ¥ereinigung, VI, 2, Leipzig 1899. [2] B r o w n , Duane, A solution to the General Problem of Multiple Station Analytical Stereotriangulation. RCA Data Reduction Technical Report No. 43, F e b r u a r y 1958. [3] M c N a i r , A. J., D o d g e , H. F., R u t l e d g e , J. D., A solution of the General Analytical Aerotriangulation Problem. Final Technical Report, E R D L Contact No. DA-44-009 E N G 2986, Cornell University, May 1958. [4] M i k h a i l , E d w a r d M., Use of Triplets for Analytical Aerotriangu!ation. Photogrammetric Engineering, September 1962. [5] M i k h a i l , E d w a r d M., A New Approach to Analytical Aerotriangulation: TwoDirectional Triplets in Sub-Blocks. Ph. D. Thesis, Cornell Univ. June 1963. [6] A n d e r so n, James M., Analytical Aerotriangulation Using Triplets, P h . D . Thesis, Cornell University, Ithaca, N.Y. June 1964.
Discussion Ackermann: I suppose I have to make some remarks because I have committed myself in the invited paper for Comm. III by stating t h a t Prof. Lehmann has proved t h a t the triplet method is not more accurate than the conventional method. I modify this is so f a r as I should have said t h a t it is not substantially better. Prof. Lehmann has shown t h a t with the triplet procedure most of the orientation elements have the same accuracy as with the classical procedure. For instance, the scale t r a n s f e r is exactly the same. There are, however, a few orientation elements, ~ and (o in particular, which have smaller standard errors, in the order of 20%. But this alone is not yet really conclusive, as the correlation p a t t e r n of these orientation elements is also different. It still remains to be shown by how much the model coordinates will be improved. I do not expect striking differences. The important f e a t u r e of the triplet procedure seems to the use of x-parallaxes in the common overlap. However, this is also possible with orientation methods which use stereo-pairs only. It is well known t h a t such a procedure improves the ~ and ~ precision and introduces mutual correlation. The triplet method should therefore be compared with the bridging method which uses also x-parallaxes for relative orientation. As f a r as I have understood the method, the relative orientation of each stereo-pair is computed twice in the triplet method. Both orientations use the same y-parallax information but differ with regard to different x-parallax information. I would like te know how the two d i f f e r e n t orientations of each stero-pair are used for the strip formation. [119]
Photogrammetria, X I X , No. 7
378
W e i g h t m a n : The m a i n value seems to is on three p h o t o g r a p h s . The drawback of point t h a t is on t h r e e p h o t o g r a p h s , you use l a t e r one, a n d the f a c t t h a t it is on the t h r e e t h a t problem a n d t h e r e f o r e I like it.
be t h a t you a r e u s i n g the f a c t t h a t a point the n o r m a l m e t h o d is t h a t , if you h a v e a it once in t h e f i r s t stereogram, a g a i n in the together is ignored. This method g e t s a r o u n d
FSrstne~: I f I understood Dr. A c k e r m a n n correctly he said t h a t the new method is m o r e a c c u r a t e t h a n the old one because o f the relative orientation. T h e r e f o r e ycu h a v e a proof t h a t t h e relative orientation w a s not a t a n o p t i m u m . Otherwise the a c c u r a c y c a n n o t be g r e a t e r t h a n before. Avke~mann: W i t h the use of x - p a r a l l a x e s the relative orientation should be s o m e w h a t m o r e accurate, because one is u s i n g additional i n f o r m a t i o n which is not used in t h e n o r m a l y - p a r a l l a x procedure. There is, hoewever, a certain d a n g e r in u s i n g xp a r a l l a x e s f o r relative orientation, which m a y have p r e v e n t e d its application up to now, a l t h o u g h t h e m e t h o d as such h a s been known for a long time. Where s y s t e m a t i c model d e f o r m a t i o n s a r e p r e s e n t the corresponding e r r o r s in x - p a r a l l a x e s a : e imposed on t h e s u b s e q u e n t relative orientation. Thompson: A few y e a r s ago a method of Bartorelli w a s popular t h a t did t h i s r i g h t t h r o u g h the strip. P e r h a p s someone could tell u s about it? de Masson d ' A u t u m e : D a n s la m~thode IGN qui est expos~e au congr~s de Londres et en suite '& la r~union de Milan on op~re p a r modules m a i s ensuite on utilise la connexion de deux modules cons~qutifs, et en particulier la discordance en Z qu~on constate, p o u r am~liorer l'orientation relative des deux modules. C'est u n peu a n a l o g u e ~ ce que r~alise la m~thode des triplets. I1 reste me semble u n petite difficultY: vous calculez done le t r i p l e t 1, 2, 3. I n c o n t e s t a b l e m e n t l'orientation relative que vous trouvez pour c h a c u n des d e u x modules consgcutives est u n peu meilIeure que si vous n'aviez p a s utilis~ la liaison que vous a v e z i n t r o d u i t comme u n e condition s u p p l e m e n t a i r e avec le poids qui lui revient. Vous obtenez donc deux t r i p l e t s dont les o r i e n t a t i o n s relatives sont meilleures que si vous aviez utilis~ des modules simples, m a i s il n'emp~che que m a t r i c e rotation 2 - - 3 que vous trouvez d a n s le p r e m i e r triplet I, 2, 3 a a u c u n e raison d'etre la m~me que celle qui vous obtenez d a n s le triplet 2--3---4. Vous avez donc le choix entre d e u x m a t r i c e s rotations, prenez-vous la m o y e n n e ou si non c o m m e n t operez-vous? E n s u i t e q u a n d vous raccordez, les triplets, en a d m i t t a n t que vous ayez r~solu cette premiere difficultY, alors vous disposez d ' u n c e r t a i n hombre de t r i p l e t s qui se r e c o u v r e n t m a i s vous constatez forcement des discordances d a n s la partie commune entre deux triplets jointifs. Thompson: M. de Masson d ' A u t u m e , si j'ai bien compri Ia m~thode, ce n'est p a s u n r e c o u v r e m e n t d ' u n module, c'est s i m p l e m e n t r e c o u v r e m e n t d'une seule photo, c ' e s t ~ dire on a 1 - - 2 - - 3 et p u i s 3---4--5. P e r h a p s Prof. M c N a i r could tell us. M v N a i r : The o v e r l a p s are 11 2, 3; 2~ 3, 4; 3, 4, 5; the middle photo of the triplet a s s u m e s the s a m e orientation t h a t w a s j u s t computed f o r it a s 3rd photo in t h e previous triplet. A n d t h e X-coordinate for the base a s s u m e s the s a m e length as j u s t previously computed a s t h e b a s e 2, 3 in the previous triplet. de Masson d ' A u t u m e : Je suis d'accord m a i s ~a n'emp~che que vous devez c o n s t a t e r f o r c e m e n t des discordances d a n s la partie qui se recouvre. McNair: [120]
We do get two relative orientations. We also get the X a n d Y d i s c r e -
Discussion on the paper of McNair
379
pancies a t a n y points which occurred in the f i r s t a n d second triplet. Now we can calculate t r i p l e t model coordinates by u s i n g the individual X, Y coordinates as modified by the discrepancies, a n d t h e n take the a v e r a g e of the r e s u l t s of t h e two d i f f e r e n t models, t h i s is the simplest method. Alternatively, we m a y calculate the m o s t probable coordinates u s i n g corrected plate coordinates, or thirdly we can calculate the most probable coordinates by least s q u a r e s a d j u s t m e n t u s i n g the co-linear condition equations. T h i s is considerable work, b u t does not take long in an electronic calculator. I should h a v e indicated t h a t on this l a s t t e s t we calculated 13 triplets in ten and a fraction m i n u t e s , costing about $ 3 per model.
Roelofs: I would like to m a k e a r a t h e r f u n d a m e n t a l r e m a r k . I would s a y t h a t if you are u s i n g the s a m e observations, if you are a d j u s t i n g these r i g o r o u s l y according to the method of least s q u a r e s you a l w a y s g e t the s a m e a n s w e r w h e t h e r you use stereo pairs, stereo templets or stereo q u a d r u p l e t s or w h a t ever it m a y be. You m u s t take into account in each step t h e correlations you used a n d t h e p r o p a g a t i o n of these correlations and weights. So if you are c o m p a r i n g methods I should s a y you should find out where t h e observations a r e d i f f e r e n t a n d w h a t a p p r o x i m a t i o n s are made. Doyle: I agree w i t h Prof. Roelofs, however, I believe t h a t in the method proposed by Prof. M c N a i r additional observations h a v e been introduced, in t h a t they f i r s t computed p h o t o g r a p h s 1, 2 a n d 3 with atl possible condition e q u a t i o n s a n d t h e n photog r a p h s 2, 3 a n d 4 w i t h all possible condition equations. T h e r e f o r e t h e y h a v e introduced additional observation e q u a t i o n s and should not expect to get t h e s a m e orientation m a t r i x nor t h e same b a s e length. I feel t h a t Prof. M c N a i r h a s not y e t a n s w e r e d the question as to w h a t he does w i t h the two d i f f e r e n t orientation m a t r i c e s a n d with the two d i f f e r e n t base lengths. I would like to add f u r t h e r comment. In m y experience the rejection of a point s i m p l y because the least square residual on t h a t point h a p p e n s to be large is not a good rejection criterion. Quite often you find t h a t for s a m e u n k n o w n r e a s o n t h e residuals a r e crowded into one p a r t i c u l a r point, and t h a t point itself is not poorly m e a s u r e d nor is it erroneously identified. I confess t h a t I don't know a n y better criterion for r e j e c t i n g points, b u t I know t h a t t h i s is not a good one. Jerie: I t h i n k it is quite obvious t h a t t h e only difference between the triplet method a n d conventional a e r o - t r i a n g u l a t i o n lies in the f a c t t h a t x - p a r a l l a x e s a r e also used to determine the relative orientation. In the triplet method one will obtain two d i f f e r e n t relative orientations f o r each model. Is it now correct simply to t a k e the m e a n of these two relative o r i e n t a t i o n s ? Does not this disturb the homogeneity of the strip f o r m a t i o n ? Inghilleri: In our method of computation we use the X - p a r a l l a x e s or h e i g h t s of the orientation points, s o m e w h a t similarly to the t r i p l e t method. The only difference is t h a t the condition equations are not solved s i m u l t a n e o u s l y for two a d j a c e n t models, b u t it n e v e r t h e l e s s gives a n improved connection between t h e three photos t h a t are involved in two a d j a c e n t models. Mikhail: The model by model a p p r o a c h t h a t is used in the U.S. Co~st and Geodetic S u r v e y is to p e r f o r m t h e f i r s t model completely independently f r o m the second model, i.e. w i t h o u t c a r r y i n g p a r a m e t e r s . W h e n I t h o u g h t of the triplets I w a s merely t h i n k i n g of m a k i n g use of the triple overlap, t h e n c a r r y i n g on w i t h the second triplet entirely independently f r o m the first. So one ends up with three i n d e p e n d e n t d e t e r m i n a t i o n s of every p h o t o g r a p h e x c l u d i n g the f i r s t a n d the l a s t p a i r s within t h e strip. Dr. Anderson, however, p e r f o r m s the relative orientation and strip a s s e m b l y in one step; t h u s o b t a i n i n g only two d e t e r m i n a t i o n s for each photograph. I a m not quite sure 24* [121]
Photogrammetria, XIX, No. 7
380 Triplet number
Photo number
1
2
3
4
5
6
7
8
whether he use the average of the two independent determinations or not, and I don't t h i n k the two determinations of each photograph will be the same. To me the g r e a t advantage of t h e triplet method lies in the s t i f f connections between triplets when building up the total strip a f t e r the determination of each triplet. A n o t h e r advantage of the triplet method is t h a t because of the better connection of the models one is not restricted to having the ground control in specified positions as required in the normal aero-triangulation procedure.
[122]