1362 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1975 
and P, are lens distortion parameters; X, Y, andZ are the object-space coordinates of the point 
imaged; and L, to L,, are the eleven unknown transformation coefficients. 
Assuming that x, — y, — 0, equation 5 can be reduced to the following form: 
L4,X *L;Y t L3Z *L 
2 20,2 = PE RE Tey Rd 
(6) 
LX+LY+L-Z+<L 
y+yr?K, +2xyP,+(r2+2y2)P2 = E TUS 
The formulation used by Abdel-Aziz and Karara in the development of a least-squares 
solution by means of equation 6 was not fully explained in their article!. Two different 
approaches can be used, and these are discussed in the following paragraphs. 
Direct solution approach. The following expressions can be derived directly from equation 
6: 
IX +LaY +Lgd tL4-xXLg—-xYLa,c-xZLj, 
—xr2AK,—(r2+2x?)AP,—2yxAP2—x=r, 
N Loy 
—yr2AK,+2xyAP+(r2+2y?)AP,—y=r, 
in which A = LX +L,Y +L, Z+] 
and r,,r, are the residual errors on the condition equations. 
Letting AK,=K;, AP,=P; and AP,=P3, equation 7 can be reduced to the following matrix 
equation: 
Ty -X -Y:-Z2-1 0 0 0 0 xX xY xZ x02. (2392x439) 99x m x -0 (8) 
nA 0: 0.9-X -Y -2.—1 uX uXouZ ur? oxy (r2+Qy?) yl 
in which L— [L, Li L5 In KP Pi] 
Each object point which has known object-space coordinates (X,Y,Z) gives rise to two such 
equations. A minimum of 7 control points are needed to solve for all 14 unknown parameters 
included in the L-matrix. It would be a direct solution, and no iteration is needed. 
Iterative solution approach. Let V, and V, represent the small random errors in the 
measured comparator coordinates. The following expressions can be derived from equation 6 
by omitting the effects of V, and V, in the coefficients ofthe lens distortion parameters K;,, P,, 
and P,: 
Vo AL IL 2a A Lat Lost 1 + 21, 
A A A A A A 
+xr2K, + (r2+2x2)P,+2yxP, + =0 
; X y Z 
e ou Lom LO A hack den Alan BEL © 
+ yrK,+2xyP, + (r2+2y2)P2+ À = 0 
in which A = L,X t Lj,oY + La Z + l 
Since equation 9 is non-linear, it must first be linearized and, subsequently, the least-squares 
solution must follow an iterative procedure. 
The DLT solutions, as formulated in the preceding two paragraphs, are intended primarily 
for use with non-metric camera photography. The traditional parameters of interior(x,, y,, and 
focal length) and exterior(w, q, x, X^, Y^, Z^) orientation parameters are embedded in the DLT 
coefficients L, to L,,. Ifthe DLT solution is to be used for data reduction in a photogrammetric 
mapping project, there must be sufficient control points imaged in each photo for the determi- 
nation of the DLT coefficients. Having computed the DLT coefficients for all the photos, the 
object-space coordinates of any object point which is imaged in two or more photos are then 
computed. The formulas for computing object-space coordinates from DLT coefficients of the 
photos were also derived by Abdel-Aziz and Karara?.