Theoretical inquiry into the intrinsic precision of the photogrammetric techniques

121

Theoretical Inquiry into the Intrinsic Precision of the Photogrammetric Techniques by A. M. W A S S E F Survey of Egypt, Giza. "Nothing that is Wrong in principle can be right in practice. Peopleare apt to delude themselves on that point, but the ultimate result will always prove the truth of the maxim .t,, Carl Schurz

Summary. The pressing need for moderately-priced photogrammetric machines led to the development in recent years of two-dimensional instruments in which the errors due to tilt are approximately balanced out by means of adjustable mechanisms t h a t automatically v a r y the separation of the pictures (or, the floating marks) as the model is scanned. T h e underlying mathematical principle is that to the f i r s t degree of approximation the errors of the model are representable by the equations of hyperbolic paraboloids in the photographic coordinates on one picture. The Drobyschew stereometer and the new model of Galileo-Santoni's cartographic stereomicrometer are built upon this principle, whereas the Zeiss Stereotop adopts the easier-to-realize (though basically less powerful) representation by the equation of a hyperboloid. , The present paper is an account of a theoretical inquiry into the goodness of this approach towards the design of inexpensive instruments, and sets to find what sort of results may be expected from instruments based on such representations. The work was stimulated by the remarkably good results which the author had obtained by means of a simple numerical method t h a t called for no more complex instruments than a stereometer (parallax bar) and an "Old Delft" scanning stereoscope [1]. The mathematics of the subject became manageable by using the relative-scale parameter X [2] which is expanded in terms of the orientation elements by means of Taylor's theorem. The f i r s t and the second order terms are fully worked out. Interpreted as a polynomial in the photographic coordinates the expansion can be replaced at little loss of precision by the familiar expression of the hyperbolic paraboloid involving x, y, xy and x 2. It is in fact shown t h a t this expression not only stands for the f i r s t order terms but it absorbs the larger proportion of the influence of the second order terms. An examination of the remainder of Taylor's expansion in its L a g r a n g e ' s form led to similar conclusions. Furthermore, it is shown t h a t the high efficiency of the hyperbolic-paraboloid representation is not seriously challenged in mountainous regions provided t h a t the photographic coordinates are f i r s t divided by the parallax. The investigation, which is supported by numerical analyses, seems to foretell of inexpensive instruments of first-order precision since tilts can now be kept well within two degrees. 1. Preamble. In some types of third-order PlOtting instruments t h e effects of the inclination of the camera axis on height evaluation are approximately neutralized by means of devices t h a t correct the z-parallax as the operator scans the stereoscopic model. The following are popular examples. i. T h e Z e i s s S t e r e o t o p . The correction is introduced.in the form dp = fo + fl x + f2Y + fsxY

(1.1)

which is the equation of a hyperboloid. The correcting mechanism of the instrument is set with the help of four height-control points.

122 The correction at any point within the model is t h e r e a f t e r automatically given as a function of the photographic coordinates x, y of the point on one picture. ii. Drobyschew stereometer [3]. In this instrument the correction is mechanically introduced according to the full formula of the differential of the parallax, t h a t is

dp = fo' + f l 'x + f2'Y + f3 ' x y + f4 'x2

(1.2)

where fo" , . . . . f4" are linear combinations of the elements of orientation. iii. S a n t o n i ' s cartographic stereomicrometer. The correcting mechanism of the latest model of this instrument is based on the property t h a t the hyperbolic paraboloid represented by equation (1.2) is a ruled surface. The surface is physically formed near the instrument. A feeler moves on the surface and communicates the correction corresponding to its position. A. Tewfik already proposed an instrument on the same principle. On the other hand, several numerical and graphical methods have been devised to evaluate heights from simple parallax measurements on similar principles. Of late years, Thompson [4] suggested t h a t the coefficients in equation (1.2) be evaluated by the direct solution of five suitably-distributed control points, with dh instead of dp where dh is the difference between the field value of the height and its value as determined by means of the simple parallax equation. A more rigorous derivation of the functional relationship between the height, the parallax and the photographic coordinates enabled the p r e s e n t writer [1] to develop a simplified analytical method t h a t gave approximately H/6000 for the root mean squ~/re error of height determination under favourable conditions, H being the flying altitude. The method was f u r t h e r developed at the Survey of E g y p t to keep the precision at this level in hilly country and with larger tilts. It was to ascertain the theoretical expectations of these techniques, and to clear the ground for future developments in this field that we set to investigate the general theory of representation of model deformations by such functions of the photographic coordinates. 2. Parametric representation of the stereoscopic model. It was shown in a previous article [2] that the coordinates of a g r o u n d point with reference to a coordinate system attached to the left-hand photograph with origin at the perspective centre are (Z x, XY, Z), where x, y are the photographic coordinates X, Y normalized by dividing each of them by the principal distance of the camera. It follows t h a t the entire subject of model deformations would be covered by an investigation of the influence of the sources of deformation on the p a r a m e t e r ZThe following derivation of 7. is abridged from the article j u s t quoted. Let the system of coordinates O' (x', y', z'), attached to the right-hand photograph with origin at its perspective centre, be rotated to O" ($, 7, ~-) parallel to 0 (x, y, z) of the left-hand picture; and let C~, C~, C z be the components of the base 00". The condition of correct relative orientation is t h a t the plane containing the two perspective centres 0 and O" (Cx, C~, Cz) and the image (x, y, - - 1 ) of a ground point P on the left-hand picture shall also contain the image (~ ÷ C~,~ + C~, ~ + Cz) of P in the right-hand system. Hence,

C~

Cy

x ~:+c~

y ,~+c~

Cz --1 ~-+c,

which gives

z x - - z'~ = C~, z y - - z'~ = C~ , --Z

- - Z ' $ = C~ ,

=

;

O,

123 where, Z=

c~¢ -- c ~ x~'+~

and

Z'=

c~ -- c ~ x~'+~: '

C~ P u t t i n g Cz = b and Cxx = Cz'' we obtain

z

= b ~ Cz" + ( - - ~ )

(2.1)

x (--¢) --~

The transformation of (x', y', z') into (~, J?, ~) is m a d e via the Eulerian angles which are the inclination 8 of the camera axis, and angles V and ~ which define the azimuths of this inclination with reference to the left-hand system and the right-hand system respectively, i.e.

illTI:I 1 w h e r e T is t h e square m a t r i x cos $ sin (p sin v + cos ~ cos v - - c o s S cos ~ sin v + sin (p cos v - - s i n O s i n v 7 - - c o s 8 sin q~ cos V + cos 9~ sin y, cos (~ cos V cos V + sin (p sin V sin 08 cos V J - - s i n 8 sin 9~ cos ~ cos 9v --cos

(2.3)

which m a y be w r i t t e n "-1

F (1---cos 8) sin V + cos A | (1---cos 8) c o s v + s i n A - - s i n 0 sin ~v

( 1 - - c o s 8) sin V - - sin A --(1--cos0) cosv+cosA sin 0 cos ~v

- - s i n ~ sin ~ l/

(2.4) --COS

where V--(P

= A.

(2.5)

3. Polynomial representation of the scale p a r a m e t e r ZL e t us denote by Z(0) t h e value of Z when t h e elements of relative o r i e n t a t i o n are nil; a n d let Z(0, A, Cz') denote its value for non-zero o r i e n t a t i o n elements. Taylor's t h e o r e m for the evaluation of a f u n c t i o n in t h e neighbourhood of a point of r e f e r e n c e gives

1

Z(O, A, Cz') = Z(0) + d z ( 0 ) + ~ T d 2 z ( 0 ) + R

(3.1)

where dz=

0Zo + AZA + Cz'Z v ,

(3.2)

g

and d2z = 02Z80 + d2ZdA ÷ (Cz')2ZCz, Cz ÷ 2 Cz'AZcz, l+ 2 A Ozdo + 2 (~Cz'Ze% .

(3.3)

A s u f f i x e d l e t t e r indicates p a r t i a l d i f f e r e n t i a t i o n with r e s p e c t to the variable r e p r e s e n t e d by the letter. F o r example, 0Z

¢')Z

32Z

ZO= 3 8 ' Zvz'-- 3C~" zAo - 3A bO R is the r e m a i n d e r t e r m .

124 To derive d~ a n d d2z we n e e d t h e f o l l o w i n g f u n c t i o n s : Z=-x(~¢)_~

'

$ = I (1--cos

0) sin (p sin y~ + cos zl] x'-tT[ (1--cos

0) cos (p s i n ~ - - sin A l y " + sin 0 sin y~,

- - $ = ( - - s i n 0 sin~o) x' + (sin 0 cos ~ ) y ' + cos O, f r o m w h i c h we d e r i v e

b('0

2b (--~')

--b z-: = Iz(_$)__Z}

2bx~

z(-,~)(-:)

2,

iz(_¢)_~}

~,

b i - - ~ : - - x (--~') I $

I~(--¢) --~1 s ( sin 0 sin = (--eosOsin = := -

~oo

-

~o s i n y~) ~' ~ sin ~ ) x ' ( - - sin A) x' (--cosA) x'

+ (sin 0 cos ~o sin ~o) y ' + cos 0 s i n ~ , , + ( e o s O c o s qo s i n N) y ' - - s i n O s i n y ~ ,

+ +

( - - cos A) y ' , (sin A) i f ,

~0~ ( - - cos 0 s i n ~ ) x ' ÷ (--~) o = (--$) a a = (sin 0 sin ~ ) x" - (--~) is i n d e p e n d e n t of A .

(cos 0 cos ~o) y ' - - sin 0 , (sin 0 cos ~o) y ' - - cos 0 ,

Furthermore,

= z ~ 0 + z(-~,) ( - - $ ) 0 ,

zo Zz~ Zc , z

ZOO ZAA

z%,c z, ZzIC ,

b$ - x(--¢) --~' = z ~ ($0) 2 + z ( - ~ ) ( - ~ )

(--~-0) 2 + 2 z~(-.;) ( - - ~ o ) ~ e + x ~ o a + z(-,~) (--¢) oo,

~0~ = Ix(__~)__~12'

z

ZAO ZOc •

I~(--~)--~l

z

When O= g =C z'=O, Z

= b/p,

$ t0 ~00

= = = =

~ ~zz ~o~ .~ Z~ Z~(-~)

~

i~(--¢)--~12

we have

x', sin ~ , - - s i n 2 x' -t- (sin ~v cos ~o)y', --y',

=--x', = o, = b/P2 , = 2b/p3, = b (.---¢c,--x')/pS.

(--¢) (--~') 0 (--~') 00 . (--¢)z (--OA~ (--Oo~ Z(-~)

= = = =

1, - - s i n x ' - - cos i f , --1 , o,

=0, =0, = --bx'/P 2,

Z ( - ¢ ) (--¢) = 2 b x x ' / p a ,

125 Hence, xe

= b (sin y + x’s sin y -

XA XO#’

= --by’lP = bx’lp

xee

= (2b/p3) +

,

sina y + (2bxs’/p3)

(2b/p3)

(x +

= 2byQjp3 =o

XAA x 0 ‘C ’ z z

= -bxy’/$

xOzfA

x’ y’ cos v)/ps,

,

(-x’ sin Y + y’cos y)s + x’) (x’ sins y - #sin y cos y) + -I- (h/p? (--1;’ sins y -I- y’ sin y cos y)

,

X?O

= -(2by’/p3)

X00,’

=

(bx/ln2)

Assembling

the

sin y , sin y (bxx’/p2)

terms

of

(3.1)

(--2’

in

the

sin y + y’ cos W) . form

of polynomials

in x and

y we

i.

x(O)

= b/p,

ii.

dX(O)

iii.

l/2 dsX(O)

= (b/p2) ( 0 sin y + px’C,’ - y’A - x’y’0 cos y + ~‘2 0 sin 7~) , = (b/p3) (A2 + xx’02 cos’+)y’2 + + + P/2 blP3) 1 (P -2x-2x’-4xx’2)(Osiny)(Ocosy) -2pxC,‘A-44 (Osiny) -2pxx’C,‘(Ocosy)]~~‘+ + (l/2 b/p3) [ 2 ( 0 sin y)s + px’( 02 - 42) + -x’(p-22x--2x’-2xx’a)(Osiny)s+2p(x

An

inspection

of these

results

shows

+

US

now

consider

the

small

a hyperbolic

The

exact

value

A and C,‘, but when graph) and the function

the

form

R, in (3.1).

x (to, tA, tC;),

where

(3.6)

In 0 <

Lagrange’s t

<

form:

1.

of t is difficult to determine and depends on the actual values of 0, these are small, t is nearly 1/4 (see remark at the end of this paraof the representative polynomial is closely indicated by the trend of a3x

3! i

values

xx’s)C,‘(Osin~.f~)].

of l/s d2x (0) are respectively a paraa parabola of the fourth order; of these parabolas are smaller than

term,

remainder

03 -__ The

(3.4) (3.5)

paraboloid;

1

R = zd3

obtain:

that

1. dx(0) is in the usual form of the equation of 2. the second order terms are parabolic in y’; 3. the coefficients of ~‘2, y’ and the third term bola in x, a parabola of the third order and 4. the contributions of C,’ and A to the order the contribution of 0. Let

bx’/@ ,

+

bdJp,

a 03

!

e=o

of b”x (4>‘03

for the array below:

of points

at which

X =

ezo

0, 5, 10 and

;

;I

g

A

Y = -10, 6 10 0 18 0

0, 10 are

shown

schemetically

126 from which it appears t h a t the simple polynomial 1X4

10--~

(X-- 5)Y

gives the plotted valueswithin seven units equivalentto 7

~- (.017)a(0 o )3 which is below 1/20,000 when each of the tip and the tilt is 2 °. Remark. The function ~ (0, A, Cz' ) may be written in either of the equivalent forms 1 z(O) + dz(O) + 1/2d2z(O ) +~.~ daZ (tO, tA, tCz'), or

1 z(O) + dz(O) + 1/2d2z(O) +d32,(0) + ~ d4x (uO, uA, uCz'), where u is also a fraction between 0 and 1. In other words, there exists a fraction t and another fraction u such t h a t the identity 1

1

1

3.t d3z (tO, tZ], tCz' ) = 3~. daz(0) -}-4! d4z (uO, uA, uCz' ) holds true for any values of 0, A and Cz'. P u t t i n g A - - C z ' = O, this identity reduces to 03 03 04

~(. z(3)(tO) = ~ . Z(8)(0) + 4fz(4)(u0), or

Z~a)(tO) = Z(3) (0)

~-

1/40Z(4) (U),

where Z(r) is w r i t t e n . f o r

~r z 0~ The left-hand member of this equation may similarly be written

z(3)(tO) = z(3)(0)+ tOz(4){v(tO)],

o < v < 1.

Hence,

t=

1/4 Z(4) I v(tO) } Z(4) (uO)

which tends to 1/4 as 0 approaches zero. 4. Goodness of the representation of :~ by the equation of a hyperbolic paraboloid. We shall now proceed to estimate the goodness of fit of the five-term expression

a + fix + 7Y" + ~xy" + ex 2 (4.1) the coefficients of which contain the measured parallaxes, p, as a parameter, i t will be assumed t h a t the principal distance of the camera is 15 cm and that p = b = 10/15. The model is defined by 0 __
127 by X

X ( 0 cos %0)2

does n o t exceed 1 50 4 - " 22~ ( 0 c ° s %0)2

(Ocos%0) 2 18

w h i c h is t h e v a l u e it r e a c h e s at X = 1/2 P, Y = 0, 10 a n d - - 1 0 . To t h i s w e s h o u l d a d d 50 A2 (10)2 A2 = ~ - . F o r a r e l a t i v e tilt 0 cos %0 of 1 ° t h e m a x i m u m e r r o r is .017452

17

18

1000,000

T h e e f f e c t of a n e q u a l A is c o m p a r a t i v e l y large, b e i n g 150/1000,000. It s h o u l d h o w e v e r he n o t e d t h a t w h e n t h e p h o t o g r a p h s a r e c o r r e c t l y b a s e - l i n e d a n d t h e c o o r d i n a t e s a n d p a r a l l a x e s a r e r e f e r r e d to t h e p r i n c i p a l - p o i n t s b a s e , A is v e r y s m a l l a n d t h e t e r m is ~egligible. 2. T h e a b s o l u t e v a l u e of t h e e x p r e s s i o n

--2 bx (x - -

p) 2 y, 02 (sin %0cos %0) pa

is less t h a n

X (X

112(p2~

2 ff \if- --

/ \ - ~ 7 ) ( 0 sin %0) ( 0 cos %0)

which reaches the maximum value

± (L12 ,_oo 2 3 \3

]

1

225 (0e°s%0)(0sin%0) = ~6

(0sin%0)(0c°s%0)

a t X = 1/3 P. T h e e r r o r due to n e g l e c t i n g t h i s t e r m w h e n e a c h of t h e t i p a n d t h e t i l t is one d e g r e e , is less t h a n 40/1000,000. T h i s e r r o r m a y be r e d u c e d to one h a l f of i t s v a l u e by r e p l a c i n g t h e t e r m b y

--Y'(O .....

sin %0) ( 0 cos %0) 15 P

. . . .

3. T h e t e r m

--bxx" y'C z' ( 0 cos %0) 1)2

is n u m e r i c a l l y g r e a t e s t on t h e e d g e s of t h e o v e r l a p a t X = 1/2 P. T h e m a x i m u m v a l u e is 10 ~2 1

1

I

lff) ~ ' ~ C ~ ' ( 0 cos%o) = ~C~'(0 cos%o). T h i s a m o u n t s to 34/1000,000 w h e n 0 cos %0 = C zt__ -- 1 °, b u t it c a n be r e d u c e d to one h a l f of i t s v a l u e b y a d d i n g 1

18 Cz'( 0 cos %0) to t h e coefficient of y" in t h e l i n e a r p a r t of t h e e q u a t i o n . T h e m a x i m u m v a l u e of t h e r e s i d u a l is t h u s r e d u c e d to 17/1000,000.

128 4. The X and X Y terms of the equation of the hyperbolic paraboloid or the hyperboloid absorb the effects of the second-order terms A C z" and A (0 sin y~). 5. The term bx ( x - - p ) 8 (0 sinv2)2

p3-

"

may be written p

1--~-

(~ sin ~v) 2

which reaches the maximum numerical value 1 ( 3 ) 3100 225 (0 sin yJ) 2

3 (0 sin~v) 2 64

at X = x J4 P. The maximum value is 1411000,000 when the tip is 1 °. t t a l f of it can be taken away by adjusting the constant term. 6. The term / p2\ , bx(x--p)2Cz'(Osin~V)p2 = X ( 1 - X ) ( - f 2 ) C z ( O s i n y 3 ) at X = 1/3 P has its maximum value 1 ( 2~2(10t2

•

1

3] \15~] Cz ( ~ sin V) = 1-~Cz" ( 0 sin V) %

at X = I / 3 P . For Cz ' = l and O s i n y J = l ° the error due to neglecting the whole term does not exceed 2011000,000. The addition of Cz" ( 8 sin V) 30 to the constant term reduces the maximum error to one half of its value. Tabte 1 sums up the situation when the equation of a hyperbolic paraboloid is fitted to the scale parameter Z. The first column shows those terms of the polynomial that do TABLE 1. Maximum Deviation of the Scale .Parameter from the Fitted Equation

of a Hyperbolic Paraboloid. (M = 111000,000.) I

Term of the Polynomial bxx

~-

p

y'2 ( 0 COS~/))2

bxx'a ~-

( 0 sin *g) 2

-.4bxx'2 p3

Correcting Term

M( 7 M

20 y' (0 sin yO(O cos ~v)/2

Maximum Residual

cos~v)2~ 1 - -

~ (tilt°) ~ 7 ~ (tip°) 2

(0 ° sinyJ) 2 Y'

(~° sin yJ) (0 ° cosyD ~-

20 ~ (tip °) (tilt °)

--bxx" - -p~- y'C~" ( 0 cos ~)

Y" 17 Cz"° (0° cos~v)~ M

bxx'2 ~ - - C z' (0 sin yJ)

~ C z "° ( 0 ° sin yJ)

~ (Cz"°) (tip °)

b p~ y'2 d 2

1/2 A2

152 M A°

10

17 ~ (C~'°) (tilt °) 10

(See § 5.)

129 not appear in the equation of the surface (that is, the hyperbolic paraboloid). Half the maximum errors due to these terms can be taken up by additive correcting functions to the terms of the equation o f the surface. The correcting functions are given in the second column. The maximum residual discrepancies are shown in the last column. 5. Accuracy of representation of ;~ by the equation of a hyperboloid. The p a r t of the polynomial expression of Z which can not be directly incorporated in the equation of a hyperboloid z = a + ,Sx + 7Y + ~xy

is bx '2

bxx' y "2

p~ ( 0 s i n ~ ) ÷ - - - - p ~

2 bxx'2y"

( 0 c o s y~)2

p3

bx~'y"

(0sinyJ)(0cosyJ) + bxx "2

p2

C.' (0cosy3) + - - pa

bx" (p

2x - - 2x' - - 2xx'2)

p3

C z' ( O s i n y )

+

1/2 (0 sin ~v) 2.

The second term was reduced to a parabola in X (§ 4). As this is not admitted in the equation of the hyperboloid we shall substitute ( X / P ) ( 1 - - X / P ) by li4 which changes the distribution of the residual errors in the model without affecting" their maximum value. The third, fourth and f i f t h terms may be treated as shown in connection with the hyperbolic paraboloid. The sixth t e r m may be written

The coefficient of 1/2 ( 0 sin ~v) 2 ranges from 3 a t X ~ 0 to 0 at X -----P. The expression 3 ( 1 - - X ] P ) would fit at the two terminals giving 1.5 at X = l / 2 P whereas the correct value is .472. The resulting error has the maximum value 304 1 156 (0 ° siny~) 2 1.028~-2(0 ° sin~p)2 = -'M .... The largest error however results from the replacement of the f i r s t term (which is linear in 0) by a linear function of x. The term may be written 2 X2 ~ ( 1 - - ~ - ) ( 0 sin ~o). The largest difference between ( 1 - - . X / P ) 2 and 1 - - X / P the largest discrepancy is 2 (.25 (.01745) (0 ° sinyJ) --

is .25 at X = li2P. Hence,

2908 ( 0 ° sin ~) 1 M -- 344

per degree of tip. For one degree of tip the second-order t e r m contributes 156/M, giving a total of 3064/M, or 1/326. If the range of X is split up in the middle and straight lines are fitted to the ends of each half separately the maximum error due to one degree tip is reduced to one quarter of this value, causing a height error of H/130ff in f l a t country. 6. Case of mountainous regions. The polynomial expressions of Z include p as a p a r a m e t e r ; and it was shown that, f e r constant p, the function a + 13~-I-7y + (~xy'+ sx 2 reproduces the error in Z with g r e a t fidelity. We shall now show t h a t the accuracy of this representation can be maintained in mountainous regions by using X [ P and Y ' ] P in the places of x and y" respectively - - t h a t is to say, by changing the scale of the coordinates according to the measured parallax. F o r instance: 9 *

130 TABLE 2. The Difference between the Cow'feet and the A p p r o x i m a t e

X

Y

Zapp

10

1.049447

~--~app

1.050790 2327

8

1.047120

1.048366 2327

--6

1.044793

4

1.042467

2

1.040140

2424

2326

1.045944

2422

0

1.037813

1.041100 2327 2327

--2

1.035486

--4

1.033159

1.033832

1.030833 2327

--8

1.036254

2327 2326

6

1.038676

t.028506

1.031412 1.028990

2327 --lO

10

1.042931

8

1.041070

6

1.037347

1.042024

1.035485

0

1.033624

--2

1.031762

--4

1.029901

--6

1.028040

--8

1.026178

--10

1.024317

1.038187

1.037347

8

1.035951 1.034555

4

1.033159

2

1.031763

0

1.030367

1861

--4

1.027575

--6

1.026179

--8

1.024783

--10

1.023387

1929

1932

61

590 522 455

763 1423

1427

1431

1.031005

1.023810

24 27

32 35

1437

601 560

1438

1.025255

71

673 638

1.028135

1396

67

31

1433

1.026697

68

705

1428

1396

63 65

736

1.035291

1396

6i

779

787

1.033864

1396

59

840

142o

1.029572

95

384

1.036714

1396

95

55

655

1.038134

1.032436

94

55

718

1929

1396

95

899 1920

1.024701

1396

97 95

954

1926

1.026633

95

1009 1916

1.032417

1.028562

96

389

1.030491

1861

579

1925

1861

1862

673

484

1922

1396

1.028971

2422

1.034342 1862

1396

--2

2420

1.036264

1396 6

768 2422

1923

186i

10

863

1917

1862

2

960

2422

1.040107 1861

4

2424

2422

1862

1.039208

1151

2422

1.043940 1861

95

1055

1.026568

1.026179

97

1246 2422

1.043522

2327

1343

1442 1445

518 472 423

37 41 42 46 49

131 V a l u e s of Z in the Case o f T i p = T i l t = l

X

Y

~app

6

10

1.032693

°.

Z

1.033343 931

8

1.031762

6

1.030832

4

1.029901

1.032412 930

1.031475

1.028970 1.028040

930

---4

1.029598

1.027708 930

1.026179 931

--6

1.025248

--8

1.024318

--lO

1.023387

10

1.028971

8

1.028506

930 931

1.026760 1.025809 1.024855

1.029547 465

1.029096

1.028040 1.027575 1.027109

0

1.026644

465

I0

--2

1.026179

--4

1.025713

466 466

1.026825 1.026366

1.024782

---10

1.024317

10

1.026179

1.026729

8

1.026179

1.026761

1.026179

465 465

1.025903 1.025441

1.026793

451 450

599 581 561 537

576

1.026179

1.026856

0

1.026179

1.026889

1.026179

1.026922

----4

1.026179

1.026952

--6

1.026179

1.026985

~8

1.026179

1.027019 1.027052

13 16 18 20

24 24

--16

619

457 459

463 462 465

--14

590 606

630 638 646 653 656 659

--13 --11

--8 --8 --7 --3 --3 0

659

--32 --32 --32

1.026825

2

1.026179

615

513

1.024976

---2

--10

955

457

1.027282

--8

4

954

628

455

1.025247

1.026179

951

1~27739

--6

6

948

1.028194

465

9

452

466

2

947

1.028646 465

4

943

1.023900

466 6

6

637 940

1.028655

1.027109

7

937

931 --2

0

650 643

1.030538 931

0

650

931 937

931 2

~--~app

--31 --33 --33 --30 --33 --34 --33

550 582 614

--32 --32 --32

646 677 710 743 773 806 840 873

--31 --33 --33 --30 --33 ---34 --33

132 b

dz(O) = ~ -

[0sin W + p(x--p)C~'--y'A--

(x--p)y" (0cosw) + (x--p)~(Osinw) }

may be written:

B,f2

X

•

~) (0 sin ~v) I =

~--p-~ (0 sinyJ) + F 1 \ p ,

p 1,

where F 1 is of the form

It should however be pointed out t h a t the term (B]f)(f/P)LJ (Y'/P) does not fit into this picture in view of the presence of the coefficient f/P, but its influence can be kept within close tolerances by careful base-lining of the photographs to reduce /t. The equation for l/2d2z(0 ) may, with the exception of the t e r m s which include A, be written

B]'2 li2d2•(0 ) = ~ -

B

(XY')

(0sin~)2 +~F 2 ~,p-

.

If we adopt an average value Po of P in the coefficient B/P of /72 the second t e r m will he subject to an error --+ AP/P of its value, where AP is h a l f the actual range of P. It will be seen from the numerical examples which follow in § 7 t h a t the difference between the exact formula of Z and its f i r s t approximation Z ( 0 ) + dz(0) is 1347/M when 0 ° sin y) = 0 ° cos y~ = 1 °. Approximately one half of this difference is accounted for by the term Bf2(Osiny~) 2 (15)-" 684 p3 - 10 ('01745)2 ( 0 ° s i n y j ) 2 = M - " The remainder, 663/M, corresponds to the term (B/P)F2(X/P,Y'/P) and is therefore subject to the error AP/P of its value due to the variation of P from the average value adopted in the coefficient of' F 2. The error under consideration remains less than 1110,000 so long as [ P - - P 0 [ < .15P. 7. Numerical study of special cases. To establish data for numerical analysis, and to check the derivation of the formulas, Z was calculated from the exact formula (2.1).every two centimetres of an overlap defined by 0 _< X ~< 10, i l 0 ~< Y ~< 10 assuming P ~ 10 em an d f = 15 cm for two sets of values of the tip and the tilt. The approximate value of Z was calculated according to the formula Zapp = Z ( O ) +

dz(O)"

Case I: tip = tilt = 1°. The correct and the approximate values of Z and their differences when each of the tip and the tilt is one degree are given in Table 2. i. The parabola which fits the difference Z--Zapp at X = 0 and Y = 10, 0, - - 1 0 may be determined by L a g r a n g e ' s interpolation formula ( Y - - O ) (Y + 10)

(Y'--IO) (Y + lO)

.........

(Z--Zapp)x=o +

( 1 0 - - 0 ) (10 + 10) (Z--Zapp)X=0 Y~10 -~ ( 0 - - 1 0 ) (0 + 10) (Y--IO) (Y--O) -t ( - - 1 0 - - 1 0 ) ( - - 1 0 - - 0 ) (Z--Zapp)X:O Y= --10 which gives (~-

Zapp)X=O = 863.0 + 48.10 Y.

r=0

133 Similarly =

638.0

17.20 Y ~

(Z --

Zapp)X=4

~-

(Z-

Xapp) x = l o = 7 1 0 . 0 - 1 9 . 1 5

.330 Y~

and Y - - . 0 1 5 y2.

S u b s t i t u t i n g in t h e e q u a t i o n

( X - - 4 ) (X--10) Z--Zapp=

(0--4)

(0--10) +

( X - - 0 ) (X--10) (Z--Zapp)X=O

4

(4--0)

(4--10)

( Z - - X a p p ) X = 4 q-

( x - - 0 ) ( x - - 4)

(10--0)

(10--4)

(X--Zapp)x=lo,

we o b t a i n (Z --

Zapp) tip = tilt = 1 o

(6.825 X 2-83.549 X ÷ 863.0) + + ( .1666 X 2 8.392 X + 48.1)Y+ -i- ( .01475 X 2 .1500 X -I.0030)y2.

T h e . l a s t t e r m r a n g e s f r o m a b o u t 0 £o - - 3 8 / M . H a l f of t h i s v a l u e c a n be t a k e n u p by t h e c o n s t a n t t e r m . T h e coefficient of Y m a y be w r i t t e n . 1 6 6 6 X ( X - - 1 0 ) - - 6 . 7 2 6 X + 48.1 a n d c a n be replaced b y - - 6 . 7 2 6 X + 46.0 w i t h a n e r r o r s m a l l e r t h a n .1666 I X (X - - 1 0 ) ] max 2 Hence, t h e m a x i m u m r e s i d u a l is 21/M. T h e e q u a t i o n (863.0 - - 19.0) - - 83.5 X + 46.0 Y - - 6.726 X Y ~- 6.825 40/1000,000. ii. L e t u s n o w c a l c u l a t e t h e r e s i d u a l s w h i c h r e m a i n to t h e v a l u e s of Z a t five p o i n t s , s a y X = 0, Y = ~ 1 0 ; S o l v i n g t h e c o r r e s p o n d i n g five e q u a t i o n s , we o b t a i n Ztip ~ tilt -- 1 o =

--2.1. of t h e h y p e r b o l i c p a r a b o l o i d b e c o m e s X 2, t h e m a x i m u m d i s c r e p a n c y b e i n g a f t e r f i t t i n g a hyperbolic p a r a b o l o i d X = 4, Y = 0 a n d X = 10, Y = ± 1 0 .

1.038679 - - . 0 0 2 4 0 8 4 X + .0012110 Y - - . 0 0 0 1 2 2 7 2 X Y + .00012295 X 2.

T h e d i s c r e p a n c i e s a r e s h o w n in F i g s . l a , . . . , l e in t h e f o r m of h e i g h t c o n t o u r s in m e t r e s a s s u m i n g t h a t t h e p h o t o g r a p h s a r e t a k e n a t H = 10 k m w i t h a c a m e r a of focal l e n g t h 150 ram. T h e e r r o r s a r e less t h a n 0.8 / / / 1 0 0 0 0 . Case I I : tip = tilt = 2% T h e c a s e of two d e g r e e s of tilt a n d two d e g r e e s of tip w a s s i m i l a r l y e x a m i n e d . T a b l e 3 g i v e s t h e d i f f e r e n c e s b e t w e e n t h e e x a c t a n d t h e a p p r o x i m a t e v a l u e s of X. i. T h e p a r a b o l a w h i c h f i t s t h e d i f f e r e n c e s Z - - Z a p p a t Y = 10, 0, - - 1 0 f o r e a c h of t h e X - v a l u e s 0, 4 a n d 10 a r e a s follows ( Z - - Z a l ) p ) x = o = 3583 + 1 9 9 . 7 Y + .16 y2, ( Z - - Z a p p ) x = 4 = 2651 ÷ 76.1 Y - - 1.39 y2, (Z--Zapp)x=10= 2961÷ 68.0Y÷ .15Y2, f r o m w h i c h we f i n d t h a t )~ --,~app ~ +( ÷ (

28.467 X 2 - - 346.870 X + 3583.0 -I.6884 X 2 - - 33.6538 X + 1 9 9 . 7 ) Y ÷ .06442 X 2 - .64520 X ÷ .155) y2.

T h e v a l u e of t h e l a s t t e r m r a n g e s f r o m 0 to 1 0 0 i . 0 6 4 4 2 ( 2 5 ) - - . 6 4 5 2 0 (5) + .1551 = - - 1 4 7 / M , Its r e p l a c e m e n t b y t h e c o n s t a n t - - 1 4 7 / 2 M w o u l d c u t d o w n t h i s m a x i m u m e r r o r to - - 7 4 / M ; b u t a s m o o t h e r d i s t r i b u t i o n o f t h e r e s i d u a l s w o u l d be o b t a i n e d if t h e t e r m is r e p l a c e d by 50(.06442 X 2 - - . 6 4 5 2 0 X + .155) = 3.221 X 2 - - 3 2 . 2 6 0 X ÷ 7.75 w h i c h l e a v e s t h e m a x i m u m e r r o r a t 74/M. T h e c o e f f i c i e n t of Y m a y be w r i t t e n . 6 8 8 4 X ( X - - 1 0 ) 4-

134 TABLE 3.

X

Y

Xapp

O

10

1.098883

The Difference between the Correct and ~he A p p r o x i m a t e ~

5058

1.094230

1.099420 4653

6

1.089577

4

1.084923

2 0

1.080270 1.075617

--2

1.070964

--4

1. 066311

--6

1.061657

--8

1.057004

--10

1.052351

5595

1.104478 4653

8

~--Xapp

!

! 4654

I

4653

[

1.094364

1.074148

4653

1.069097

4654

1.064048

4653

1.059000

4653

--2

4787

4

4383

2 o

3982

5053 5052 5051 5049 5048 5047

405 2 403

--1 404

5054

1.084252 1.079200

4653

5190

5058

1.089306 4653

2 5056

3 401 2 399

3583

1

3184

2

2786

1

2391

.1

1996

O 399

1

398

3

395

O

395

1

394

1602

1.053953 i

10

1:085854

8

1.082131

6

1.078409

4

1.074686

2

1.070964

0

1.067241

"1.090055 3723 3722

I

3723

[

3722 3723

1.082140 1.078173

i

1.074199

]

1.063518

1.070218

1.059796

--6

1.056073

--8

1.052351

!

1.066230

--10

1.048628

10

1.074686

3995

3723

]

3722

I

3723

1 i

1.062235

1.058233 1.054226

t

1.071894

6

1.069102

4

1.066310

2

1.063518

0

1.060726

2792

--4 --6

1.055142 1.052350

--8

1.049558

--10

1.046766

3487

--7

3235

--7

2977

--7

2712

--7

2439

--5

2160

---8.

1875

4015

1583

2917

--3

3172

--11

3058

--11

2933

1.066315 2938

1.063377 2948

1.060429 2962

1.057467

2792

1.054495

2792

1.051515

2792 i

1.048521

240 244 252 258 265 273 279 285

--10 --4 --8 --6 --7 --8 --6 --6 ~7

292

2972 2980 2994

106 114 125

--8 --11 --11

136

2928

2792

230

3278 2898

1.069243

2792

I

4007

2906

2792

1.057934

4002

1.072160 2792

--2

3731

--7

1.077964 1.075066

2792

3971

--5

1.050211

2792

8

3981 3988

3722

--4

3962

3974

~

!

--9 3967

I

3723

--2

4201 3953

1.086102

--lO

2797

--lO

2651

--14

2495

--lO

2325

-- 3

2145

--14

1957

~10 146

--19 156

--10 180 188 202

1755

--14

170

--8 --14

135 Values

X

o f X i n t h e C a s e o f T i p = T i l t = 2 °.

Y

X--Zapp

Xapp

! 10

1.065379

8

1.063518

1861

6

1.061657

4

1.059796

2

1.057934

0

1.056073 L054212

--2 --4

1.052350

--6

1.050489

--8

1.048628

--10

1.046767

10

1.057935

1861 1861 1862 1861 1861 1862 1861 1861 1861

1870 1.066215 1.064334 1.062441 1.060539 1.058627 1.056702 1.054768 1.052823 1.050865

4

1.055143

2

1.054213

10

1.052351

---4

1.051421

--6

1.050490

1934 1945 1958 1967

1.056832 931 930 931

931

1,055938 1.055040 1.054134 1.053220

1.049559

--10

1.048629

1.051374

10

1.052351

1.054648

8

1.052351

1.054777

2645

--lO --13

2605 i

6

1.052351

1.054910

4

1.052351

1.055042

930

2554

--9

2490

--11

2418

--13

2334

--9

2237

L

2131

1.052300

2

1.052351

1.055175

0

1.052351

1.055312 1.055448

--4

1.052351

1.055585

--6

1.052351

1.055725

--8

1.052351

1.055866

--10

1.052351

1.056007

9 20 32 40 51 64 72 84 97

--11 --12 --8 --11 --13 --8 --12 --13 --9

106

2400 --8

2464

--4

2521

--7

2574

--9

2619

885 894

2656

4 898 --8

2689

--8

2713

--6

2730

906

914 920 --6

2741

926

2745

--129

1.052351

--9

878

1.057717

--8

--2

1925

1.058595

931

--2

1912

2677

874

930

1.053282

1902

1.059469 931

0

1898

--12

866

931

1.056074

2697

1881

1.060335

1.057005

6

--11

1.048898

930

8

2706

1.068085

--133 --182

±133 --137

--141

--53 --45

4 --8 --8

--37 --33 --24 --17 --11

i --9

--7 --6

--7

4

--129 2426

--1

2559

4

--133

1

2691

4

l i

2824

--1

i

2961

1

i

3097

--137

--141

--7

---57

2297 4

--136

--140

--64

--1 --132 --133 --137 --136

1 4 --1 1

--137

3

3234

1

3374

0

3515

--140

1 --141

--141 ~I

[

3656

3

o

136 6.884 X - - 3 3 . 6 5 3 8 X ÷ 199.7 w h i c h m a y be replaced by --26.7698 X ÷ 1/2(.6884 ) ( 5 ) ( - - 5 ) + 199.7 = --26.7698 X ÷ 191.095 w i t h an e r r o r not exceeding 8.6. Hence t h e m a x i m u m e r r o r s a t t h e edges a r e ± 86/M. I t follows t h a t the f u n c t i o n 3 5 9 0 - - 3 7 9 . 1 X + 191.i Y - - 2 6 . 7 7 X Y + 31.69 X 2 r e p r e s e n t s ( ~ - - ~ a p p ) t i p = t i l t = 2 o w i t h i n 160/M or 1/6250. F i t t i n g t h e equation of a hyperbolic paraboloid to t h e c o r r e c t values of Z a t t h e nine p o i n t s by t h e m e t h o d of least s q u a r e s left no residuals exceeding 153/M. ii. F o r comparison we have computed (a) t h e hyperboloid t h a t fits at the f o u r c o r n e r points of the overlap, (b, c) the hyperboloic paraboloid which fits X on five points one of which is in the middle in one case and on an edge in the o t h e r case and (d, e) the h y p e r bolic paraboloid t h a t b e s t fits Z a t six points and at nine points respectively. The disc~'epancies a r e shown in F i g u r e s 1 ( a - - e ) . 8. Representation of the errors in the evaluated coordinates by polynomials in the photographic coordinates. In t h i r d - o r d e r p l o t t i n g i n s t r u m e n t s and some numerical m e t h o d s the e f f e c t s o f t h e relative o r i e n t a t i o n and those of the absolute o r i e n t a t i o n are l u m p e d together. The i n s t r u m e n t a l m o v e m e n t s (or the algebraic p a r a m e t e r s in n u m e r i c a l techniques) a r e a d j u s t e d directly on the c e n t r a l points. In this section we shall consider t h e e r r o r s w h i c h arise w h e n the model e r r o r s a r e r e p r e s e n t e d by the equation of a hyperbolic paraboloid in t h e presence of non-zero elements of absolute orientation. The s u r v e y coordinates X , Y, h a r e given in t e r m s of t h e p h o t o g r a p h i c coordinates x, y and the scale p a r a m e t e r X by t h e equation ~

Xm X

= (fs)R o

x = (fs)R o

(8.1)

w h e r e R o is the m a t r i x of o r i e n t a t i o n of t h e l e f t - h a n d p i c t u r e w i t h r e f e r e n c e to t h e s u r v e y axes. R o is o f t h e s a m e f o r m as (2.4) b u t is d i s t i n g u i s h e d f r o m it by s u f f i x i n g zero to t h e elements, t h u s ILo sin Vo + cos zJo R o = ILocosy~o + sinzt o [ - - s i n 9% sin 0 o

M 0 sin ~ ' o - sin A o - - M e cos~, o + cosz] o cos ~oo sin 0 o

- - s i n 0 o sin Y~07 sin ~oCOSy~o I ---cos 0 o ]

w h e r e L o = (1 - - cos 80) sin ~Oo, M o = (1 - - cos ~o) cos ~oo and z] o =JY~o--~°o"

(8.2)

(8.3)

It is clear f.rom (8.1) t h a t if a polynomial o f degree n in z a n d m in y r e p r e s e n t s Z to a specified degree of precision then only a polynomial of h i g h e r degree will e v a l u a t e t h e g r o u n d coordinates w i t h equal precision. N e g l e c t i n g t e r m s c o n t a i n i n g cubes of the o r i e n t a t i o n e l e m e n t s we a t once see f r o m (3.6), (8.1) and (8.2) t h a t t h e t e r m s of t h e h e i g h t equation t h a t do n o t show up in t h e equation of the hyperbolic paraboloid a r e b p~ ( - - y'zJ - - x'y' ~ cos yJ -t- x "2 0 sin ~ ) ( - - 0 o sin q0oX -b ~9oCOS~0Y)-

(8.4)

F o u r t e r m s o f this p r o d u c t have opposite n u m b e r s in polynomial e x p a n s i o n of Z shown in Table 1 and can be similarly t r e a t e d . T h e n e w t e r m s a r e b

(Co cos ~0y) (-- x'~'0 cosy)

and b / ~ (00 cos~oY ) (x'2~ sin ~g).

137

~

_IND"

~) c~ ~o

"3

°

4~

~o

fJ

~ o.~ .~ .-~ ~ o o' °

m

~

, ~ .~ 4.~ -~-~

~.~,

N ~ "3 ...~~ ~ .o ~ , ~

o

o

oool

°

0001[

--

000~,

I:

O00&

¸0009

~:~

~o ~

-5000

~o~

4000-3000 ~0~

0

-

~OJ

138

/ ~, Fig. ld. of y~ per boloid is in the

~ °t

Fig. le. The discrepancies in the values of X per million when a hyperboloid is fitted to nine symmetrically distributed points in the presence of 2°-tip and 2 °tilt.

The discrepancies in the values million when a hyperbolic parafitted to six points as in inset presence of 2°-tip and 2°-tilt.

The f i r s t may be replaced by (21250

-- \3/

X"

i-oo" ~ (

00cos~o) ( 0 cos~)

not exceeding 68/M times the product of the absolute and the relative tilts measured in degrees. The second term may similarly be replaced by 100 " P- ( O° cos ~oo) ( 0 sin V), the maximum discrepancy being relative tip measured in degrees.

68/M times

the product of the absolute tilt and the

Acknowledgements. My thanks are due to Engineer G. F. Yassa for his assistance in the preparation of this paper, and to the s t a f f of the photogrammetry section for carrying out p a r t of the numerical work.

139 References. [1] A . M . Wassef, Analytical P h o t o g r a m m e t r y in Practice; P a p e r read at the Eighth International Congress of P h o t o g r a m m e t r y in Stockholm on 23 July, 1956. [2] A . M . Wassef, Some Recent Developments in Analytical Photogrammetry. The Use of Eulerian Angles and Computational Procedure; Photogrammetria, X, 2, 1953/54, page 76. [3] H.G. Jerie, Stereoger~ite 3. Ordnung in der Sowjetunion; P h o t o g r a m m e t r i a , XI, 4, 1954/55, page 127. [4] E . H . Thompson, Heights from Parallax Measurements; Photogrammetric Record, October 1954.

I.T.C. International Bibliography of Photogrammetry by Ir. F. L. CORTEN International Training Centre for Aerial Survey, Delft, Netherlands. Abstract. Experience has shown that it is impossible to read all publications of importance in aerial survey; in addition, language difficulties prevent t h a t the existing literature is accessible to the potential user. This proofs the necessity of an international bibliography. The I.T.C. is editing a multilingual bibliography consisting of abstract cards in a specified order of classification. It is organized in international cooperation of different national authorities. Introduction. In every field of science and of technique there is, to-day, such a flow of publications t h a t it is virtually impossible for anyone to be completely informed and to stay abreast of all new developments and applications in his profession. Even if there were no limitations of time, the practical limitation of languages often prevents us from taking cognizance of important information. In addition, in many branches of applied science such as photogrammetry there is no reference to literature at all; this means t h a t there is no general access to publications about experience made by others in the past. These facts are felt as serious limitations by most professionals in photogrammetry and aerial survey. A way out of this difficulty would be an international bibliography and this was the subject of discussion in Commission VI of the International Society of Photogrammetry. In 1955 Prof. Dr. Ir. W. Schermerhorn offered to this Society the aid of the I.T.C. and recently he took f u r t h e r action. We now s t a r t the edition of the I.T.C. I N T E R N A T I O N A L BIBLIOGRAPHY OF PHOTOGRAMMETRY with the cooperation of the National Societies and of specialized organizations in different countries. Edition of the International Bibliography,

Form. The Bibliography will be edited in the form of a card possible entrances (such as alphabet, chronology, author, subject, etc.) we chose a double entrance according to author and to subjects. onto more than one card (as an average 3 . . . 4 cards per title) and alphabetical order in the "author set" and in classification order in

index. Of the many journal, geography, Each title is printed will appear once in the "subject set" in