In this case, a5 in equations (1) becomes zero and the rela-

tionship between comparator coordinates and image coordi-

nates can be expressed as:

x– = a

1

+ a

2

x + a

3

y

y– = a

4

+ a

6

y

(10)

Combining equations (10) and (3) one gets

a

11

(X – X

0

) + a

12

(Y – Y

0

) + a

13

(Z – Z

0

)

a

31

(X – X

0

) + a

32

(Y – Y

0

) + a

33

(Z – Z

0

)

a

1

+ a

2

x + a

3

y – c · = 0

(11)

a

21

(X – X

0

) + a

22

(Y – Y

0

) + a

23

(Z – Z

0

)

a

31

(X – X

0

) + a

32

(Y – Y

0

) + a

33

(Z – Z

0

)

a

4

+ a

6

y – c · = 0

Equation (11) has 12 unknowns, but they are not linearly

independent. These unknowns can be reduced to 11 linearly

independent unknowns by eliminating a

2

and a

6

and intro-

ducing two unknowns C

x

and C

y

to replace C;

( C

c

a

C

c

a

x

y

=

=

2

6

;

, C

x

and C

y

reflect possible differential linear

distortions along x and y comparator axes). Equation 11 can

thus be rewritten as:

a–

1

+ a–

2

x – C

x

= 0

a

11

(X – X

0

) + a

12

(Y – Y

0

) + a

13

(Z – Z

0

)

a

31

(X – X

0

) + a

32

(Y – Y

0

) + a

33

(Z – Z

0

)

(12)

a–

3

+ y – C

y

= 0

a

11

(X – X

0

) + a

12

(Y – Y

0

) + a

13

(Z – Z

0

)

a

31

(X – X

0

) + a

32

(Y – Y

0

) + a

33

(Z – Z

0

)

Equations (12) represent the basic equations in the conven-

tional (collinearity) approach. As explained above, these

equations take into consideration the non-prependicularity

between comparator axes, and differential linear distortions

along x and y axes.

4. Observation Equations

Expanding equations (12) by Taylor’s series and neglecting

second and higher order items, one gets:

V

x

+ a

1y

V

y

+ b

1x

Δω

+ b

2x

Δϕ

+ b

3x

Δκ

+ b

4x

Δ

X

0

+ b

5x

Δ

Y

0

+ b

6x

Δ

Z

0

+

b

7x

Δ

C

x

+ b

8x

Δ

C

y

+ b

9x

Δ

a–

1

+ b

10x

Δ

a–

2

+ b

11x

Δ

a–

3

+ F

x

° = 0

and

(13)

V

y

+ b

1y

Δω

+ b

2y

Δϕ

+ b

3y

Δκ

+ b

4y

Δ

Y

0

+ b

5y

Δ

Y

0

+ b

6y

Δ

Z

0

+ b

7y

Δ

C

y

+ b

8y

Δ

C

y

+ b

9y

Δ

a–

1

+ b

10y

Δ

a–

2

+ b

11y

Δ

a–

3

+ F

y

° = 0

Where

V

x

, V

y

are errors in x and y

a

1y

= a

2

, is partial derivative of F

x

w.r.t. y

b

1x

, b

1y

are the partial derivatives of F

x

and F

y

(see footnote

below) w.r.t.

ω

b

2x

, b

2y

are the partial derivatives of F

x

and F

y

w.r.t.

ϕ

b

3x

, b

3y

are the partial derivatives of F

x

and F

y

w.r.t.

κ

b

4x

, b

4y

are the partial derivatives of F

x

and F

y

w.r.t. X

0

b

5x

, b

5y

are the partial derivatives of F

x

and F

y

w.r.t. Y

0

b

6x

, b

6y

are the partial derivatives of F

x

and F

y

w.r.t. Z

0

b

7x

, b

7y

are the partial derivatives of F

x

and F

y

w.r.t. C

x

b

8x

, b

8y

are the partial derivatives of F

x

and F

y

w.r.t. C

y

b

9x

, b

9y

are the partial derivatives of F

x

and F

y

w.r.t. a–

1

b

10x

, b

10y

are the partial derivatives of F

x

and F

y

w.r.t. a–

2

b

11x

, b

11y

are the partial derivatives of F

x

and F

y

w.r.t. a–

3

F

x

o

and F

y

o

are functions of approximate values of the un-

known parameters.

Equations (13) represent the observation equations in the con-

ventional collinearity approach. The observation equations in

the proposed direct approach may be obtained by expanding

equations (9) and including all the zero terms (e.g. 0

l

5

and 0

l

2

)

for ease of reference:

w

1

v

x

+ X

l

1

+ Y

l

2

+ Z

l

3

+

l

4

+ 0

l

5

+ 0

l

6

+ 0

l

7

+ 0

l

8

+ xX

l

9

+ xY

l

10

+ xZ

l

11

+ x = 0,

and

(14)

w

2

v

y

+ 0

l

1

+ 0

l

2

+ 0

l

3

+ 0

l

4

+ X

l

5

+ Y

l

6

+ Z

l

7

+

l

8

+ yX

l

9

+ yY

l

10

+ yZ

l

11

+ y = 0.

In equations (14) the factor w1 and w2 may be considered as

weight factors, and their value can be easily determined in

the solution.

A comparison between equations (13) and equations (14)

indicate the simplicity of the proposed solution.

5. Analysis of Errors

Both the conventional and proposed approaches are influ-

enced by the following errors:

a. Uncertainties in comparator measurements and errors

in object space coordinates of control points.

b. Errors in mathematical modeling of film and lens dis-

tortions (random errors as well as unrepresented – or

residual – systematic errors).

In addition, the conventional iterative approach is subject to

computational errors due to:

a. Iteration criteria

b. Neglecting of second and higher terms in the lineariza-

tion of the observation equations (13).

Obviously, the proposed direct solution is not subject to

these computational errors.

6. Fictitious Data Tests

A number of fictitious data tests were conducted to assess

the capabilities of the proposed solution (equations 9) and

compare them to the capabilities of the conventional ap-

proach (equations 12.) As a datum for comparison of the

two approaches, data from the collinearity approach with 9

unknowns (only parameters of inner and outer orientation are

included, errors due to comparator adjustment, lens distor-

tion, and film deformation are not considered) were used. The

test covered the following aspects: handling of differential

linear distortions along the x and y comparator coordinate

axes, correction for non-prependicularity of the comparator

axes, accuracy of determination of the unknowns (standard

error of unit weight), and computer executing time.

Tables I through V summarize the results of these prelimi-

nary tests.

a

11

(X – X

0

) + a

12

(Y – Y

0

) + a

13

(Z – Z

0

)

a

31

(X – X

0

) + a

32

(Y – Y

0

) + a

33

(Z – Z

0

)

F

x

= a–

1

+ x + a–

2

y – C

x

· = 0

a

11

(X – X

0

) + a

12

(Y – Y

0

) + a

13

(Z – Z

0

)

a

31

(X – X

0

) + a

32

(Y – Y

0

) + a

33

(Z – Z

0

)

F

y

= a–

3

+ y – C

y

· = 0

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105

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