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where (x, y,7) € F ,(x',y',z') € G , R(P,®,K) is 
: ; T. ; 
the orthogonal rotation matrix, [# a" af. is the translation 
vector, and m is the scale factor. 
In order to perform least squares estimation, the true error 
vector Vx. y.z) is introduced to describe the discrepancy 
between the conjugate surfaces: 
V(x,yz)yeG(x, v2) PF v.z) (3) 
As continuous 3D surfaces have to be sampled with discrete 
coordinates, a 3D surface matching is automatically translated 
into a registration of point clouds. So the true error vector of 
equation (3) can be approximately expressed with the 
Euclidean distance of conjugate points, and the aim of least 
squares estimation is defined as follows: 
DI dd || = min (4) 
If the square of the distance is set to D=d 2 the new 
mathematical model of 3D surface matching can be simply 
defined as follows: 
D -(x-xYy *(y-yy *(z-zy (5) 
Where (x, V,Z ) is the point coordinate of the template 
surface, and the X', y ,Z')is the point coordinate of the 
>» p 
searching surface. 
Since equation (5) is nonlinear, it must be linearized by the 
Taylor expansion, ignoring 2nd and higher order terms. 
D+V=D, De +20, +204 
ot, ot, ot, 
oD oD oD oD 
+— do +— do +— dk + — dm 
op ow OK om 
where V can be considered as residual of the Taylor 
expansion, and the Euclidean distance of conjugate points will 
be set to zero at the end of the LS3D routine. Hence, the 
observation error equation can be simplified as: 
oD oD D 
V=D +—dt +—dt 36D y + 
dele LOL 
(7) 
oD D D 
ED laua 
Op ow OK om 
The matrix form of equation (7) is as follows: 
V=A4AX+1, P (8) 
where A is the design matrix, 
X I , 1, 10,0. K, ml is the parameter vector, and 
l= D, is the constant vector that consists of the Euclidean 
distances between the template and searching surface elements, 
P is the weight matrix of the error observation vector. 
The distribution of the random variable is 
V- N(0, 0) , with the statistical expectation 
E(V) «0, E{VV"} = 00, = oll’ . Hence, system 
(8) is a typical Gauss-Markoff estimation model. In order to 
control the estimation quality, an additional error observation 
vector of the unknown parameters could be imported 
(Robert,2004; Gruen and Akca, 2005). 
V=IX+L, P (9) 
Where / is the identity matrix, and L , is the constant vector 
of the error equation, P is a priori weight matrix of unknown 
parameters. If zero weight (2 ); — 0) is set, the i-th 
parameter is assigned as free variable, and if an infinite weight 
value (D )i — OO) is set, the i-th parameter is assigned as 
constant Combining equation (8) and equation (9), the 
maximum likelihood solution of unknown parameters can be 
estimated as follows: 
Nec PACPY'GÉPLCPL) — Qo 
25:2 
a, = PL +V. PV) x (11) 
Vou AX 41 (12) 
V =IX+L, (13) 
where À is the final estimation value of least squares 
— 2 
routine, O, is the mean square error of the weighted units of 
the observations, M is the number of error observation 
equations, and 4 is the number of unknown parameters, 
Ky =N—U are the components of abundant observation. 
2.2 Conjugate Points Rules 
Since it is rather difficult to locate feature points in a local 
window on 3D surfaces, how to establish the conjugate points 
between 3D overlapping regions, is the core strategy in the 3D 
surface matching procedure. In our method, the conjugate 
points rule is unlimited. We could define some new rules for 
specified applications, because the mathematical model of the 
adjustment, defined in 2.1, is generic for available rules. And, 
it is the major advantages of our proposed method compared 
with existing methods. 
In this section, we list three strategies for establishing conjug- 
ate points on 3D surfaces. The first definition called LND rule 
is the same as the Least Normal Distance method (Robert, 2004; 
Gruen and Akca, 2005), using pedal point of triangle in normal 
direction. The second definition called LZD rule is the same as 
the Least Z-Difference method (Rosenholm, 1988), using the 
intersection point of a triangle in vertical direction. The last 
definition called ICP rule is the same as the Iterative Closest 
Point method (Besl and McKay, 1992), using two, the closest 
points in the entire point sets as conjugate points. With the 3D 
surface representation of triangulated irregular network 
    
  
   
   
  
   
   
  
  
  
  
   
     
  
    
  
   
   
  
   
  
   
   
    
   
  
   
   
  
    
   
   
   
   
   
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structure, the conjugate point rules can be listed as follows: 
x E 
T # 7 x 
(a) (b) (c) 
Figure 1. Conjugate points definitions for surface matching 
with TIN structure: (a) LND rule, (b) LZD rule, (c) ICP rule. 
  
  
  
where A, JB and C are the 3 vertexes of the candidate 
conjugate region on the searching surface, and 7? is the normal 
vector of the located triangle, V is the vertical vector of the 
located triangle, $, . .$,. S, are behalf of the Euclidean 
distances from the interpolation point to the 3 vertexes, and the