MATHEMATICAL FORMULATION & DIGITAL ANALYSIS 1361
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Fic.6. Close-range photogrammetry using vertical height differences as controls.
TABLE l. ExAMPLES OF FIVE CLOSE-RANGE PROBLEMS.
Measured Arbitrarily Defined Unknown
Cases Parameters ' Parameters Parameters
1 x,y image coordinates; None wi OK X Yi, Z{for i=1m
X,Y,Z object-space X,,Y,,Z; for j-2l,n
coordinates of at least 3 points
2 x,y image coordinates; wy = 90°, $17 «, =0° €, 0, Ki, X1, Y 5,Zi for i2 2,m
S,,S, distances Xi = Yı= Zi = 100.00 X,Y,Z; forj = In
3 x,y image coordinates;
S,,$5,$4 distances As in case 2. As in case 2.
4 x,y image coordinates; X,=Y, =7Z,= 1000.00
angle Bjr Z; = Z, = 1,000.00 As in case 1.
distance S;; o; = 10°
5 x,y image coordinates; XX, =X. Y=Y,
length along three plumb X; 2 Y; 2 Z, — 1000.00 As in case 1.
lines: Sj, Su, and Sn oy = 20°
coordinate system (X,Y, and Z with the origin at the center of the earth), and local-space-
rectangular coordinate system (X,Y, andZ with the origin at or near the center of the area being
mapped). In data input, the object-space coordinates ofthe control points may be described in
any one of the three systems. All the computations, however, are performed in the local-
space-rectangular coordinate system. The computed coordinates of the pass points are listed
in all three coordinate systems in the output.
With some minor modifications to improve the computation efficiency and to use plane
coordinates for all survey measurements, the program can be a powerful tool for data reduction
in close-range applications.
FULLY ANALYTICAL SOLUTION USING DIRECT LINEAR TRANSFORMATION
Tbe method of direct linear transformation (DLT) from comparator coordinates into object-
space coordinates was developed by Abdel-Aziz and Karara!. Itis based on the following pairs
of equations:
L,X+LaY+L30+L
rt 2 «.)2 d ao — Li X+LoY +Lsl +L,
x +(x Xo)r K,+(r +2(x Xe) )P,+2(y Yo) (x Xo)P2 Lo +L4cY+L41Z +1
L;X+LeY+L,Z+L
Vy y) Ka 37x) (y 7s) Path 3( 7) Pa TS Y e
(5)
in which x and y are the measured comparator coordinates of an image point; x, and y, define
the position of the principle point in the comparator coordinate system; r? = (x?+y2); K,, P,,
