International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
B vint: 
jio mo eg atis y A p def 
  
dx + 
(4) 
de (X, 3. Z) . 08 (X, Y,Z) 
- dy + —— dz 
dy oz 
where 
)x dy 07 
dx = ox dp, , dy= 23 qp, , dz= Za (5) 
i Op, Op; 
where p; € {t, » ty , L , M, @, Q, x} is the i-th transformation 
parameter in Equation (3). Differentiation of Equation (3) gives: 
dx 2 dt, - a, dm + a, d0 + aj; d * a; dk 
dy = dt, + a, dm + a,, d0 + a5, dQ * a», d (6) 
dz = dt, + a ,p dm * a4, do * aj) dp - a3, d 
where aj are the coefficient terms. In the context of adjustment 
of observation equations, each measurement is related with the 
function whose variables are unknown parameters. This 
function constitutes the functional model of the whole 
mathematical model. In the following definition, the terms 
(gx. gy. g,) are 1* derivatives of this function, which is itself of 
the search surface patch g(x,y,z). In other words these terms are 
local surface gradients on the search surface. Using the 
following notation, 
- Og" y.z) 
X 
_ 9g"(x.y,2) 
dy 
_ 0g’ (x,y,2) 
(7) 
0c 
y 3 42 
0x oz 
and substituting Equations (6), Equation (4) gives the following 
equation: 
q 
* 
—e(x, y,z) = g,dt, + g,dt, + g,dt, * (g,ajg * gya»o * g,449)dm 
(84a; * ,4321 * 31 )do + 
(8,912 + £yà»» * 8,255 )dQ 
> 0 
(8,315 + gyda + 87433 )dK — ( Í (X,y.Z) —-2 (x, Y,Z)) 
(8) 
In matrix notation 
-—zAX-/ , P (9) 
where A is the design matrix, x' =[dt, dt, dt, dm do do dx] 
is the parameter vector, and f = f(x.y.z)-g (x.y.z) is the 
observation vector that consists of the Euclidean distances 
between the transformed point using current transformation 
parameters and its coincident surface element on the other 
surface. With the statistical expectation operator E{} and the 
assumptions 
e- N(6,020,) ; 030 =00P; =K1= E fee! | (10) 
the system (9) and (10) is a Gaup-Markov estimation model. 
The unknown 3D similarity transformation parameters are 
treated as stochastic quantities using proper weights. This 
extension gives advantages of control over the estimating 
parameters (Gruen, 1986). In the case of poor initial 
approximations for unknowns or badly distributed 3D points 
along the principal component axes of the surface, some of the 
unknowns, especially the scale factor m, may converge to a 
wrong solution, even if the scale factors between the surface 
patches are same. 
D 
The least squares solution of the joint system Equations (9) and 
(11) gives the unbiased minimum variance estimation for the 
parameters 
4 Y PROS | 
x=(A 'PA+P,) (A Pf+P,/,) solution vector (12) 
aa v! Pv 4 vi Pv, ; 3 
Gc prams variance factor (13) 
{ 
= Ax—/ residual vector for surface observations (14) 
v,=l X—/, residual vector for additional observations (15) 
where stands for the Least Squares (LS) Estimator. The 
function values g(x,y,z) in Equation (2) are actually stochastic 
quantities. This fact is neglected here to allow the use of the 
Gaufi-Markov model and to avoid unnecessary complications, 
as typically done in LSM (Gruen, 1985a). 
Since the functional model is non-linear, the solution iteratively 
approaches to a global minimum. In the first iteration the initial 
approximations for the parameters must be provided: 
0 0 0 0 
Pr Sit t.t 
Kay 29 
( ‘ 
m. o. 0°, x (16) 
The iteration stops if each element of the alteration vector x in 
Equation (12) falls below a certain limit: 
lAu|«e; ; i21(02.—7 (17) 
The theoretical precision of the estimated parameters can be 
evaluated by means of the covariance matrix 
K,, 2620, - 6;N S GQ(A'PA&P,)" (18) 
In a least squares adjustment of indirect observations whose 
functional model is non-linear, the 1* derivatives (25 and 
higher order terms are generally neglected in the Taylor 
expansion) with respect to unknowns are very important terms, 
since they direct the estimation towards a global minimum. The 
terms {g, , g, , g,} are numeric derivatives of the unknown 
surface patch g(x,y,z). Its calculation depends on the analytical 
representation of the surface elements. As a first method, let us 
represent the search surface elements as plane surface patches, 
which are constituted by fitting a plane to 3 neighboring knot 
points, in the implicit form 
°(x,y,z)= Ax+By+Cz+D=0 (19) 
c 
~ 
where A, B, C, and D are parameters of the plane. Using the 
. on Re ~ . . . st 
mathematical definition of the derivation, the numeric I’ 
derivation according to the x-axis is 
0 ; 0 3 D 07 : 
dg (x,y.Z) g (x Ax.y,z)-g (X.y.Z) (20) 
x = lim = 
: ox Ax—0 AX 
ga 
where the numerator term of the equation is simply the distance 
between the plane and the off-plane point (x+Ax,y,z). Then 
using the point-to-plane distance formula, 
  
  
A(x+Ax)+By+Cz+D A 
ER 2 = (21) 
MAAR AB +C JA THB 00 
is obtained. Similarly g, and g, are calculated numerically. 
B C 32 
gy E , gs m. 5 5 5 
y M Br JAZ +B? + C2 
  
  
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