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Independent model triangulation using different transformations
l'h,*togrammerr:.. 30( 1974-'75):07 74 ~i Elsevier Scientific Publishing Company, Amsletdam -- P)'inted in 'i he Netherlands
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 USING D I F F E R E N T TRANSFORMATIONS K. C. SAXFNA
Survey o/India. Dehradun (India) (Accepted for publication January 15, 1974) ABSTRACT Saxena, K. C., 1975. Independent Model Triangulation using different transformations. Photogrammetria, 30:67-74. Linkage of models in the semi-analytical method of aerial triangulation is best achieved by the use of three-dimensional projective transformation. Similarity transformation being used at present gives an approximate fit and even affine is better than this. The projective transformation, having twelve parameters requires at least five or more points for least-square adjustment. Use of six points in the model and the common perspective centre is recommended since this gives a more rigid connection to the models than the use of only three model points.
INTRODUCTION
Independent Model Triangulation (I.M,T.) is a semi-analytical method of aerial triangulation in which the models are formed in the machines and the strip formation is carried out analytically by connecting all the models of the strip. Normally the linkage of models is carried out by using a similarity transformation and so far as is known to the author, two methods based on this transformation have been used. One method developed in I.T.C., The Netherlands, is an iterative method and the other one has been evolved by G. H. Schut of N.R.C.. Canada. I.T.C. uses the simple transformation given as: x,
Y,
::
Z1
>
- K
,t, I
K
2
- S? I
- '1)
~_2
2
x.,
Y~,. Z..,
Sx t
4-
Sy Sz
i
where X 1 , Y I , Z~ are the coordinates af model (1); X._,, Y._.,Z., are the coordinates of model (2); and Sx, Sy, Sz are the three shifts. Schut uses the matrix given below which provides better approximations for the elements of the similarity transformation matrix: I
R -- d~+--aU~77b~-+-c'-'
[ d=' ~ a-' - b" - c=' 2 a b - 2 c d
I 2 a b +- 2 c d 2 ac - 2 bd
2 a c -~ 2 b d
d" - a'-' + b'-' - c 2 2 b c - 2 a d 2be+ 2 ad aVZ _ a 2 _ b', 4. c='
68 While studying the results of connection of models by Schut's method, the author found that the residuals left in the Z coordinates were rather high. He has therefore tried to investigate which transformation could provide the best model connection. TRANSFORMATIONS AVAILABLE
1. Orthogonal; 2. Similarity; 3. Affine; 4. Projective. Orthogonal transformation corrects for a small rotation in three-dimensional space by applying a rotation matrix. It does not cater for scale correction and the models do undergo scale change, hence connection of models by orthogonal transformation alone is not feasible. Similarity transformation is the product of a scale factor and an orthogonal matrix. This transformation is being used at present as described above. The basic assumption in the similarity transformation is that the rotation is small and the elements of the rotation matrix, which are complex trigonometrical functions, are replaced by their approximate values. As such this transformation is likely to provide only approximate mathematical model resulting in approximate fit and larger propagation of errors. Affine transJormation corrects for scale changes in different directions and it is a more general solution of similarity transformation. Affine transformation can therefore correct for model deformations and it is expected to give a better solution than the similarity transformation. Projective transformation in three dimensions transforms a three-dimensional space in one system to a three-dimensional space in another system. Since the models are nothing else but three-dimensional spaces in different systems, the 3-D projective transformation should provide the best solution for linkage of models. TESTING THE S U I T A B I L I T Y O F VARIOUS T R A N S F O R M A T I O N S F O R M O D E L CONNECTION
An experiment has been conducted by the author to test the suitability of the three transformations, viz., similarity (using Schut's method), affine and projective in the connection of models as applied to I.M.T. The aim of the experiment was to find out the most satisfactory method for Independent Model Triangulation. MATHEMATICAL FORMULATIONS
Similarity transformation The procedure laid down in Schut (1967) has been followed.
A ffine transformation The affine transformation equations are:
Xl .... a. - a l X 2 i aeY=, aaZ._, YI = b.-~,, bIX=. ~ b:,Y2 } baZ,.. Z1 = ¢',, i c,X=, c~Y., i c:~Z~ This transformation requires at least 4 points, out of which one may be selected as the c o m m o n projection centre as the origin for both the systems (X,. Y:, Z 0 and (X:, Y._,, Z._,) so that a,, -~: b,, = c,, ::~ 1. The equations are then reduced to: X I : a t X 2 + a.2Y._, ~ a:~Z:, Y , : b, X ,2 4 b .2Y ,_, * b ::Z .2 Z1 = (hX._, i c,.Y2 i c:~Z:,
or
Y,
.....
Z,
F (3,1)
-:
0 0
0 0
() 0
0
0
0
0
0
X,2 0
Y ,2 0
Z ,2 0
0 X2
0 Y'2
a 1
Z2j
A X (3,9) (9.1)
a2 a:¢
b:~ c, C. Ca
There are nine parameters so we require at least three points and more than three points for least-square adjustment.
P r o j e c t i v e transJormation
In three-dimensional space, the projective transformation equations are: X, =
Y1 = Zl
--
a,,X,., + a,,eY,., + ataZ2 + a14 a4lX,, + a42Y2 -4- a4aZ 2 ¢. 1 a.,iX.., -f a22Y2 J a,,sZ_, + a2~ a41X.2 + a4.2Y,_, f- a4:~Z._, + 1 a:,,X2 + as,2Y2 + az,sZ2 + az4 a41X:~ + a42Y2 ~ ataZ,, + 1
These equations can be solved and written in the following form: ao -+- a,X,, -+- a=,Y., ~ a:~Z. + a4X2 ~ + ar,XeY.2 + a~¢X.Z., Y , = bo t blX., + b2Ye + b:~Z.2 + a4X2Y2 4- at, Y22 2_ a~Y2Z2 Z1 = co [ c l X 2 -i- ceY2 ¢ caZ'2 + a4X2Z2 -~ asY2Z.2 ! a~Z,., 2 X1 =
Here also if we select the c o m m o n projection centre as the origin for both the systems, then a0 = b, = c,) = 0 and the equations are reduced to:
70 m
Y~
=
Z1
0
0
x2y=, y2 2
Y2Z2 x2 yz Z2 0
0
0
a,,
0
0
x2Z._, Y2Z2
Ze2
y.,
z,,
aa
0
0
0
x2
a4 a5 all bl
b.,
b: F
(3,1)
=
A "X
Cl
(3,12) (12,1)
c., C:~
There are twelve parameters, hence we require at least four points in the model. For least-square adjustment at least five points will be needed. TEST PROCEDURE
Observations for model coordinates were carried out on Wild A-7 and A-8 stereoplotters and computations were done on the Honeywell computer available locally. The coordinates of projection centres were determined by spatial intersection of projected rays on two different projection planes. The measured coordinates of the models were transformed to the coordinates of the previous models by least-square adjustment method using S'mndard Problem 1I for the affine and projective transformations. The method of computation is briefly described below: (1) Select 5 to 6 points common to two models. (2) Reduce the coordinates of the left model to the right-hand projection centre and right model to the left-hand projection centre. (3) Form A and F matrices with the ,'educed coordii~.a~es. (4) Form normal equations by computing A v . A . (5) Compute (A w" A) -1 and A w" F. (6) Compute the parameters J( = (A 'r- A) -1 (A • F); the weight matrix is assumed to be identity. (7) Compute F = A .X. (8) Compute residuals V = /~ - F. These residuals, the maximum value of the residuals and the root mean square values of the residuals of the three coordinates were used to study the comparative effectiveness of the transformations. The results for the connection of three different pairs of models are tabulated in Tables I, II, IlI.
FABLE I Residuals in coordinates Models connected
Schut's m e t h o d Affine transl. Projective transl. .........................................................................................................................................
Points
Vx
l'y
(mm~ i m m t .
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Model 1 (245-246) Model 2 (246-247) P h o t o scale 1:5,600 Model scale 1:10,000 Distribution of points:
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1 2 3 : 4 5 6
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0
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i.03 .05 4 .00 .08 02
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Vz
I/ ):
(ram)
(ram) (ram) (mm)
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Vy
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V2
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Vx
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l/r
I':-
(ram) (mm) n nm~ .
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i.03 -.05 .(16 7 . 2 2 .00 ! .20 .00 .01 . .03 . .26 - .06 .09
-.ll i.06 + .05 .05 ~ .14 ~ .08
-.05 i .10 --.08 -.04 i .04 i .02
! .06 !.01 -.01 -.15 ;.I)2 ! .117
--.05 i.03 + .02 --.01 + .05 • .03
!,01 -.04 i .05 .01 .00 .1t2
,02 Ol .02 - •13 .05 i .(13
4-.08 .21 7.13 +.04 +.02 -.05
-.03 -4-.04 -.02 +.01 - .01 .00
-+ .01 -.112 .00 -,01 •i- ,02 -.01
--.02 .00 F.01 F.01 ~-,01 -,02
-.02 , .02 .00 F.01 -.01 .00
-•[)3 F.05 -.02 F.03 -.115 i .02
,01 .06 .06 .00 --,04 !,05
1020 5O
3O 6O
Model 3 121-22) Model 4 (22-23) P h o t o scale 1:50,000 Model scale 1:25,11011 Distribution of points: 10
1 2 3 4 5 6
-.03 !,06 ~ .04 .-.It - .04 .03
-.1)5 F.02 -.06 .... .18 - ,04 .15
5.20 -.25 -.12 +.27 + .05 .18
.00 f.04 -.04 -.02 - .01 ! .03
-.03 +.05 -.04 --.05 F.08 .... .03
1 2 3 4 5 6
. .05 - .03 -.09 !.t5 !.04 -.07
1.04 -.03 F,02 i .22 !.06 i-.23
-.28 ~-.02 ~ .15 -,16 1.02 F .33
-.05 +.02 +-.03 ~,-.05 - .01 -.04
+.03 .00 --.05 F.05 .00 .06 ~-.06 .00 -.12 -.04 F.08 ~.05
40
3O
6o
Model 5 117-18t Model 6 (18-19) P h o t o scale 1:50,000 Model scale 1:25,000 Distribution of points: 10
4o
3O
6O
T A B L E II M a x i m m n values of residuals Models connected
Schut's m e t h o d
Affine t r a n s f o r m a t i o n
Projective transformation
Remarks
Vxlmax) VY(max) VZirnax) Vx(max) VY(max) VZ(ma~l Vx(max) [/')'(max) Vz(nmx~ (mm) (mm) (ram) (ram) (ram) (ram) (mm) (mm) (mm) 1-2 3-4 5-6
.08 .11 .15
.06 .18 .23
.26 .27 .33
.14 .04 .05
.10 .08 .12
.15 .21 .06
.05 .04 .02
.05 .02 .05
.13 .02 .06
projective gives the best fit projective gives best fit and affine the next best
72 TABLE lIl Root mean square errors Models Schut's method connected
Vx 1-2 3-4 5-6
Vy
Vz
Affine transformation
Vx
Vy
Vz
Projective transformation
Vx
Vy
Vz
(ram) (ram) (ram)
(ram) (ram) (ram)
(ram) (ram) (ram)
.045 .058 .082
.088 .028 .036
.035 .023 .014
,039 .102 ,134
.167 .193 .198
. 0 6 1 .073 .050 .110 .068 .041
.028 .014 .036
Remarks
.059 .014 .044
projective leaves the least r.m.s. errors
CONCLUSION
F r o m the comparison of the residuals, the maximum values of the residuals and the r.m.s, values of the residuals of the three coordinates for the three different methods, it is seen that projective transformation provides the best fit in all the three cases studied and the use of this transformation in model connection is therefore recommended. ACKNOWLEDGEMENTS
The author gratefully acknowledges the help of the Director North Eastern Circle, Survey of India, in this study, by allowing the use of photographs and other data. Grateful acknowledgement is also due to Mr. P. N. Sawhney for his help in the data processing on the electronic computer. REFERENCES Thompson, E. H., 1971. Space resection without interior orientation. Photogramm. Rec., 7 (37): 39-45. Schut, G. H., 1967. Formation of strips from independent models. Rept AP-PR 36, N.R.C. Canada. APPENDIX I
Linearisation of 3-D projective transformation equations The 3-D projective transformation equations are: X1 =
all X2 + a12 Y2 + a13 Z2 + a14 a~l X2 -~- a42 Y2 + a4:~Z., + 1
l (a)
Y, =
a~l X., + a22 Y._, + a.,.~ Z,, + a,_,~ a4l X.2 + a42 Y,2 -~- a4:~Z., + l
1 (b)
Zj =
a:~ X~ ÷ a:~2 Y._, + a:,.~Z., + a:t4 a41 X o + a42 Y._, + a~:~Z2 + 1
1(c)
W h e n the rotations are small tending to zero, the above transformation formulae tend to: X~ : Xo i" 2X,2 Y~ = Yo !- ./ Y~ Z~ : : Z,. ~- 2 Z=,
2(a) 2(b) 2(c)
where 2 is the scale factor and Xo, go, Zo are the three shifts. Now equations l(a), l(b), and l(c) can be written in the form: a14
alj
+
Y1
--
a~. Yz + a13 Z,, a4~ X., -+- a4,, Y., + a43 Z.2 -~ I
3(a)
a24. . . . . . . . . . . . . . . . . . . . . @ a2z y,. a4! X,2 @ a42 Y,2 ,@ a~a Ze + 1 a41 X2 @ a42 Y'2 @ a43 Z2 -~- 1 a ~ X~_ + a,_,:~Z.2 -k3(6) a41 X,. + a4z Y._, + a4:~ Z2 + 1
Z1 . . . . . . . . . . . . . . . . . a:~ ................ a41 X,2 q- a4,., Y'2 -!- a.~:~Z,, + l _+
+.
................. a:c~.................. Z.: a41 X z ÷ a4~ Y'2 4- a4:~ Z,2 4 1
a:n X._, + a32 Y'2 a41 X.z + a4~ Y._, + a4:~Z,_, + 1
3(C)
C o m p a r i n g 2(a), 2(b), 2 ( c ) w i t h 3(a), 3(b) and 3(c), respectively, we get: all ~v( 2; a,22 ~ ,~; a:n ::( )~. i.e., all ~-~ a2.., -~- a:~3 := K 2
(4)
(a12, ala); (a..,1, az:d; (a:~l, a:r,) are small quantities. Further a~l X._, 4- a4~ Yz ~ a4:~Z.2 + 1 ---) I i.e., a41, a42, a4:~ and (a41 X.2 + a4~ Y.2 -~ a4.~ Ze) are also small quantities. E q . l ( a ) can be written in the form: X l = (all X,, + av, Y.~. + al:~Z~., + a14) [1 + (a41Xz + a42 Y2 -}- a4:~Z'0] - t E x p a n d i n g [1 + (a4~ X.., -~- a4~ Y., ~ a4:~ Zz)] -~ and neglecting higher-order terms: Xl = (all X2 -~- alz Y2 ~- a13 Z2 q- aa4) [1 - (a4~ x.e + a4e Y,2 + a4a z~)] : all X.2 -[- a12 Y2 + ai:~ Z2 -~- al4 - aal )(2 (a41 X2 + a42 Y2 + a4z Z2) -- a14 (a41 X2 @ a42 Y2 + a43 Z2) (neglecting the terms containing aa2 a4a, a~2 a4e, av., a4~, aa3 a4~, a~3 a4,.,, a13 a43).
Repiacing a n by K ~: = (an-
a14 a~)X.2 + (a12- a ~ a~,~) Y._, ÷ ( a a 3 - aa4 a4z)Z2 + a ~ - K 2 X2 (all X2 q- a42 Y2 + a~a Z2)
= ao q a ~ X z + a.~ Y., ,~ a3Z,~ + a 4 X z 2 + a ~ X 2 Y z + a n X z Z o
5(a)
74 Eq.l(b) can similarly be solved: Y, = (a21 Xz -t- a22 Y2 4- a23 Z2 -k a24) [1 - (a~ x,2 ÷ a4=, Y.2 q- a43 Z2] : ( a 2 l - a24 a41) X2 ~- (a22 - a24 a42) Y2 -t- (a23 - a24 a43) Z2 - K 2 Y2 (a41 Xe + a42 Y2 -~ a4:~;Z2) (neglecting the terms containing coefficients a2, a~l, a,,1 a42, a.,, a.,:~, a23 a41, a28 a42, a23 a43) Replacing a22 by K 2: Y1 = bo d- bl X2 q- b2 Y2 + b3 Zz + a4 X2 Y2 ÷ as y,.,2 _f_ a(; Y._,Z,_, 5(b) Similarly: Z1 = Co -[- C1 X2 2v C2 Y2 + Ca Z2 + a4 X2 Z2 ~- a5 Y2 Z., + a~; Z._,2 5(c)
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 USING D I F F E R E N T TRANSFORMATIONS K. C. SAXFNA
Survey o/India. Dehradun (India) (Accepted for publication January 15, 1974) ABSTRACT Saxena, K. C., 1975. Independent Model Triangulation using different transformations. Photogrammetria, 30:67-74. Linkage of models in the semi-analytical method of aerial triangulation is best achieved by the use of three-dimensional projective transformation. Similarity transformation being used at present gives an approximate fit and even affine is better than this. The projective transformation, having twelve parameters requires at least five or more points for least-square adjustment. Use of six points in the model and the common perspective centre is recommended since this gives a more rigid connection to the models than the use of only three model points.
INTRODUCTION
Independent Model Triangulation (I.M,T.) is a semi-analytical method of aerial triangulation in which the models are formed in the machines and the strip formation is carried out analytically by connecting all the models of the strip. Normally the linkage of models is carried out by using a similarity transformation and so far as is known to the author, two methods based on this transformation have been used. One method developed in I.T.C., The Netherlands, is an iterative method and the other one has been evolved by G. H. Schut of N.R.C.. Canada. I.T.C. uses the simple transformation given as: x,
Y,
::
Z1
>
- K
,t, I
K
2
- S? I
- '1)
~_2
2
x.,
Y~,. Z..,
Sx t
4-
Sy Sz
i
where X 1 , Y I , Z~ are the coordinates af model (1); X._,, Y._.,Z., are the coordinates of model (2); and Sx, Sy, Sz are the three shifts. Schut uses the matrix given below which provides better approximations for the elements of the similarity transformation matrix: I
R -- d~+--aU~77b~-+-c'-'
[ d=' ~ a-' - b" - c=' 2 a b - 2 c d
I 2 a b +- 2 c d 2 ac - 2 bd
2 a c -~ 2 b d
d" - a'-' + b'-' - c 2 2 b c - 2 a d 2be+ 2 ad aVZ _ a 2 _ b', 4. c='
68 While studying the results of connection of models by Schut's method, the author found that the residuals left in the Z coordinates were rather high. He has therefore tried to investigate which transformation could provide the best model connection. TRANSFORMATIONS AVAILABLE
1. Orthogonal; 2. Similarity; 3. Affine; 4. Projective. Orthogonal transformation corrects for a small rotation in three-dimensional space by applying a rotation matrix. It does not cater for scale correction and the models do undergo scale change, hence connection of models by orthogonal transformation alone is not feasible. Similarity transformation is the product of a scale factor and an orthogonal matrix. This transformation is being used at present as described above. The basic assumption in the similarity transformation is that the rotation is small and the elements of the rotation matrix, which are complex trigonometrical functions, are replaced by their approximate values. As such this transformation is likely to provide only approximate mathematical model resulting in approximate fit and larger propagation of errors. Affine transJormation corrects for scale changes in different directions and it is a more general solution of similarity transformation. Affine transformation can therefore correct for model deformations and it is expected to give a better solution than the similarity transformation. Projective transformation in three dimensions transforms a three-dimensional space in one system to a three-dimensional space in another system. Since the models are nothing else but three-dimensional spaces in different systems, the 3-D projective transformation should provide the best solution for linkage of models. TESTING THE S U I T A B I L I T Y O F VARIOUS T R A N S F O R M A T I O N S F O R M O D E L CONNECTION
An experiment has been conducted by the author to test the suitability of the three transformations, viz., similarity (using Schut's method), affine and projective in the connection of models as applied to I.M.T. The aim of the experiment was to find out the most satisfactory method for Independent Model Triangulation. MATHEMATICAL FORMULATIONS
Similarity transformation The procedure laid down in Schut (1967) has been followed.
A ffine transformation The affine transformation equations are:
Xl .... a. - a l X 2 i aeY=, aaZ._, YI = b.-~,, bIX=. ~ b:,Y2 } baZ,.. Z1 = ¢',, i c,X=, c~Y., i c:~Z~ This transformation requires at least 4 points, out of which one may be selected as the c o m m o n projection centre as the origin for both the systems (X,. Y:, Z 0 and (X:, Y._,, Z._,) so that a,, -~: b,, = c,, ::~ 1. The equations are then reduced to: X I : a t X 2 + a.2Y._, ~ a:~Z:, Y , : b, X ,2 4 b .2Y ,_, * b ::Z .2 Z1 = (hX._, i c,.Y2 i c:~Z:,
or
Y,
.....
Z,
F (3,1)
-:
0 0
0 0
() 0
0
0
0
0
0
X,2 0
Y ,2 0
Z ,2 0
0 X2
0 Y'2
a 1
Z2j
A X (3,9) (9.1)
a2 a:¢
b:~ c, C. Ca
There are nine parameters so we require at least three points and more than three points for least-square adjustment.
P r o j e c t i v e transJormation
In three-dimensional space, the projective transformation equations are: X, =
Y1 = Zl
--
a,,X,., + a,,eY,., + ataZ2 + a14 a4lX,, + a42Y2 -4- a4aZ 2 ¢. 1 a.,iX.., -f a22Y2 J a,,sZ_, + a2~ a41X.2 + a4.2Y,_, f- a4:~Z._, + 1 a:,,X2 + as,2Y2 + az,sZ2 + az4 a41X:~ + a42Y2 ~ ataZ,, + 1
These equations can be solved and written in the following form: ao -+- a,X,, -+- a=,Y., ~ a:~Z. + a4X2 ~ + ar,XeY.2 + a~¢X.Z., Y , = bo t blX., + b2Ye + b:~Z.2 + a4X2Y2 4- at, Y22 2_ a~Y2Z2 Z1 = co [ c l X 2 -i- ceY2 ¢ caZ'2 + a4X2Z2 -~ asY2Z.2 ! a~Z,., 2 X1 =
Here also if we select the c o m m o n projection centre as the origin for both the systems, then a0 = b, = c,) = 0 and the equations are reduced to:
70 m
Y~
=
Z1
0
0
x2y=, y2 2
Y2Z2 x2 yz Z2 0
0
0
a,,
0
0
x2Z._, Y2Z2
Ze2
y.,
z,,
aa
0
0
0
x2
a4 a5 all bl
b.,
b: F
(3,1)
=
A "X
Cl
(3,12) (12,1)
c., C:~
There are twelve parameters, hence we require at least four points in the model. For least-square adjustment at least five points will be needed. TEST PROCEDURE
Observations for model coordinates were carried out on Wild A-7 and A-8 stereoplotters and computations were done on the Honeywell computer available locally. The coordinates of projection centres were determined by spatial intersection of projected rays on two different projection planes. The measured coordinates of the models were transformed to the coordinates of the previous models by least-square adjustment method using S'mndard Problem 1I for the affine and projective transformations. The method of computation is briefly described below: (1) Select 5 to 6 points common to two models. (2) Reduce the coordinates of the left model to the right-hand projection centre and right model to the left-hand projection centre. (3) Form A and F matrices with the ,'educed coordii~.a~es. (4) Form normal equations by computing A v . A . (5) Compute (A w" A) -1 and A w" F. (6) Compute the parameters J( = (A 'r- A) -1 (A • F); the weight matrix is assumed to be identity. (7) Compute F = A .X. (8) Compute residuals V = /~ - F. These residuals, the maximum value of the residuals and the root mean square values of the residuals of the three coordinates were used to study the comparative effectiveness of the transformations. The results for the connection of three different pairs of models are tabulated in Tables I, II, IlI.
FABLE I Residuals in coordinates Models connected
Schut's m e t h o d Affine transl. Projective transl. .........................................................................................................................................
Points
Vx
l'y
(mm~ i m m t .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Model 1 (245-246) Model 2 (246-247) P h o t o scale 1:5,600 Model scale 1:10,000 Distribution of points:
.
.
.
.
.
.
1 2 3 : 4 5 6
.
.
0
.
.
.
.
.
.
.
i.03 .05 4 .00 .08 02
.
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.
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.
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.
Vz
I/ ):
(ram)
(ram) (ram) (mm)
.
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Vy
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V2
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Vx
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l/r
I':-
(ram) (mm) n nm~ .
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.
i.03 -.05 .(16 7 . 2 2 .00 ! .20 .00 .01 . .03 . .26 - .06 .09
-.ll i.06 + .05 .05 ~ .14 ~ .08
-.05 i .10 --.08 -.04 i .04 i .02
! .06 !.01 -.01 -.15 ;.I)2 ! .117
--.05 i.03 + .02 --.01 + .05 • .03
!,01 -.04 i .05 .01 .00 .1t2
,02 Ol .02 - •13 .05 i .(13
4-.08 .21 7.13 +.04 +.02 -.05
-.03 -4-.04 -.02 +.01 - .01 .00
-+ .01 -.112 .00 -,01 •i- ,02 -.01
--.02 .00 F.01 F.01 ~-,01 -,02
-.02 , .02 .00 F.01 -.01 .00
-•[)3 F.05 -.02 F.03 -.115 i .02
,01 .06 .06 .00 --,04 !,05
1020 5O
3O 6O
Model 3 121-22) Model 4 (22-23) P h o t o scale 1:50,000 Model scale 1:25,11011 Distribution of points: 10
1 2 3 4 5 6
-.03 !,06 ~ .04 .-.It - .04 .03
-.1)5 F.02 -.06 .... .18 - ,04 .15
5.20 -.25 -.12 +.27 + .05 .18
.00 f.04 -.04 -.02 - .01 ! .03
-.03 +.05 -.04 --.05 F.08 .... .03
1 2 3 4 5 6
. .05 - .03 -.09 !.t5 !.04 -.07
1.04 -.03 F,02 i .22 !.06 i-.23
-.28 ~-.02 ~ .15 -,16 1.02 F .33
-.05 +.02 +-.03 ~,-.05 - .01 -.04
+.03 .00 --.05 F.05 .00 .06 ~-.06 .00 -.12 -.04 F.08 ~.05
40
3O
6o
Model 5 117-18t Model 6 (18-19) P h o t o scale 1:50,000 Model scale 1:25,000 Distribution of points: 10
4o
3O
6O
T A B L E II M a x i m m n values of residuals Models connected
Schut's m e t h o d
Affine t r a n s f o r m a t i o n
Projective transformation
Remarks
Vxlmax) VY(max) VZirnax) Vx(max) VY(max) VZ(ma~l Vx(max) [/')'(max) Vz(nmx~ (mm) (mm) (ram) (ram) (ram) (ram) (mm) (mm) (mm) 1-2 3-4 5-6
.08 .11 .15
.06 .18 .23
.26 .27 .33
.14 .04 .05
.10 .08 .12
.15 .21 .06
.05 .04 .02
.05 .02 .05
.13 .02 .06
projective gives the best fit projective gives best fit and affine the next best
72 TABLE lIl Root mean square errors Models Schut's method connected
Vx 1-2 3-4 5-6
Vy
Vz
Affine transformation
Vx
Vy
Vz
Projective transformation
Vx
Vy
Vz
(ram) (ram) (ram)
(ram) (ram) (ram)
(ram) (ram) (ram)
.045 .058 .082
.088 .028 .036
.035 .023 .014
,039 .102 ,134
.167 .193 .198
. 0 6 1 .073 .050 .110 .068 .041
.028 .014 .036
Remarks
.059 .014 .044
projective leaves the least r.m.s. errors
CONCLUSION
F r o m the comparison of the residuals, the maximum values of the residuals and the r.m.s, values of the residuals of the three coordinates for the three different methods, it is seen that projective transformation provides the best fit in all the three cases studied and the use of this transformation in model connection is therefore recommended. ACKNOWLEDGEMENTS
The author gratefully acknowledges the help of the Director North Eastern Circle, Survey of India, in this study, by allowing the use of photographs and other data. Grateful acknowledgement is also due to Mr. P. N. Sawhney for his help in the data processing on the electronic computer. REFERENCES Thompson, E. H., 1971. Space resection without interior orientation. Photogramm. Rec., 7 (37): 39-45. Schut, G. H., 1967. Formation of strips from independent models. Rept AP-PR 36, N.R.C. Canada. APPENDIX I
Linearisation of 3-D projective transformation equations The 3-D projective transformation equations are: X1 =
all X2 + a12 Y2 + a13 Z2 + a14 a~l X2 -~- a42 Y2 + a4:~Z., + 1
l (a)
Y, =
a~l X., + a22 Y._, + a.,.~ Z,, + a,_,~ a4l X.2 + a42 Y,2 -~- a4:~Z., + l
1 (b)
Zj =
a:~ X~ ÷ a:~2 Y._, + a:,.~Z., + a:t4 a41 X o + a42 Y._, + a~:~Z2 + 1
1(c)
W h e n the rotations are small tending to zero, the above transformation formulae tend to: X~ : Xo i" 2X,2 Y~ = Yo !- ./ Y~ Z~ : : Z,. ~- 2 Z=,
2(a) 2(b) 2(c)
where 2 is the scale factor and Xo, go, Zo are the three shifts. Now equations l(a), l(b), and l(c) can be written in the form: a14
alj
+
Y1
--
a~. Yz + a13 Z,, a4~ X., -+- a4,, Y., + a43 Z.2 -~ I
3(a)
a24. . . . . . . . . . . . . . . . . . . . . @ a2z y,. a4! X,2 @ a42 Y,2 ,@ a~a Ze + 1 a41 X2 @ a42 Y'2 @ a43 Z2 -~- 1 a ~ X~_ + a,_,:~Z.2 -k3(6) a41 X,. + a4z Y._, + a4:~ Z2 + 1
Z1 . . . . . . . . . . . . . . . . . a:~ ................ a41 X,2 q- a4,., Y'2 -!- a.~:~Z,, + l _+
+.
................. a:c~.................. Z.: a41 X z ÷ a4~ Y'2 4- a4:~ Z,2 4 1
a:n X._, + a32 Y'2 a41 X.z + a4~ Y._, + a4:~Z,_, + 1
3(C)
C o m p a r i n g 2(a), 2(b), 2 ( c ) w i t h 3(a), 3(b) and 3(c), respectively, we get: all ~v( 2; a,22 ~ ,~; a:n ::( )~. i.e., all ~-~ a2.., -~- a:~3 := K 2
(4)
(a12, ala); (a..,1, az:d; (a:~l, a:r,) are small quantities. Further a~l X._, 4- a4~ Yz ~ a4:~Z.2 + 1 ---) I i.e., a41, a42, a4:~ and (a41 X.2 + a4~ Y.2 -~ a4.~ Ze) are also small quantities. E q . l ( a ) can be written in the form: X l = (all X,, + av, Y.~. + al:~Z~., + a14) [1 + (a41Xz + a42 Y2 -}- a4:~Z'0] - t E x p a n d i n g [1 + (a4~ X.., -~- a4~ Y., ~ a4:~ Zz)] -~ and neglecting higher-order terms: Xl = (all X2 -~- alz Y2 ~- a13 Z2 q- aa4) [1 - (a4~ x.e + a4e Y,2 + a4a z~)] : all X.2 -[- a12 Y2 + ai:~ Z2 -~- al4 - aal )(2 (a41 X2 + a42 Y2 + a4z Z2) -- a14 (a41 X2 @ a42 Y2 + a43 Z2) (neglecting the terms containing aa2 a4a, a~2 a4e, av., a4~, aa3 a4~, a~3 a4,.,, a13 a43).
Repiacing a n by K ~: = (an-
a14 a~)X.2 + (a12- a ~ a~,~) Y._, ÷ ( a a 3 - aa4 a4z)Z2 + a ~ - K 2 X2 (all X2 q- a42 Y2 + a~a Z2)
= ao q a ~ X z + a.~ Y., ,~ a3Z,~ + a 4 X z 2 + a ~ X 2 Y z + a n X z Z o
5(a)
74 Eq.l(b) can similarly be solved: Y, = (a21 Xz -t- a22 Y2 4- a23 Z2 -k a24) [1 - (a~ x,2 ÷ a4=, Y.2 q- a43 Z2] : ( a 2 l - a24 a41) X2 ~- (a22 - a24 a42) Y2 -t- (a23 - a24 a43) Z2 - K 2 Y2 (a41 Xe + a42 Y2 -~ a4:~;Z2) (neglecting the terms containing coefficients a2, a~l, a,,1 a42, a.,, a.,:~, a23 a41, a28 a42, a23 a43) Replacing a22 by K 2: Y1 = bo d- bl X2 q- b2 Y2 + b3 Zz + a4 X2 Y2 ÷ as y,.,2 _f_ a(; Y._,Z,_, 5(b) Similarly: Z1 = Co -[- C1 X2 2v C2 Y2 + Ca Z2 + a4 X2 Z2 ~- a5 Y2 Z., + a~; Z._,2 5(c)