matching model. For the target g(x,y) and search q(/,s) 
images, the line and sample coordinates (pixel) are expressed as 
l=a +ax+ay and s=by+bx+b,y , with the ag,a, 
ay, by, b, and b, symbols expressing affinity. 
Differencing pixel values may lead to a gray-level function as 
vj t hg * hq;(ag * aix a5 y, bg * bx b5y) - g;(x, y) 20. Inde 
x i varies for n pixels in a window. Symbol v; denotes a zero- 
mean residual error having the Gaussian distribution, or 
N (0,02) with the c; symbol meaning the standard deviation; 
hy and A, linearly modify pixel values. 
Linear expansion at approximate unknowns results in a system 
of error equations, defined as v-- Ax-1 with 08Q > BY 
referring to Mikhail (1976), one obtains the least-squares 
solution of unknown parameters as 
x-Q,AlQ Q) 
where apart from scaling, the covariance matrix, 
Q, =(ATQ Ay! , results from error propagation. The 
solution for measurement residuals is given by the 1— Ax term, 
resulting in v-Q,Q lI , where Q, (=Q- AQ AT) is the 
corresponding covariance matrix. 
A variance-component estimator is eventually formed by Wu eft 
al. (2009) as 
dis vIQ!c;Q ^v 
iT = — 
tr(Q,QC;Q") 
The simplified BIQUE (Best Invariant Quadratic Unbiased 
Estimator) by Crocetto et al. (2000) is an alternative method. 
Intuitively, adaptive weights in terms of the inverse of 
for ie (L2,...,m) (3) 
Sec, , should be better than weights in Q^! that remain 
unchanged during iterative processing. 
2.3 Reliability of the Matched Points 
As implied, RANSAC (Random Sample Consensus) 
distinguishes inliers from outliers by requiring that any n- 
member subgroup of inlying data points leads to only one set of 
model parameters. When the w symbol stands for the 
probability of a data point consistent with an affine 
transformation model and the z symbol for the probability of 
selecting an error-free n-member set, Fischler and Bolles (1981) 
have shown that the number k of attempts for the removal of 
outliers can be computed by equating the (1— w" yf term with 
the 1—z term. The solution for Æ results in 
k « log(1— z)/log(1 — w") (4) 
According to Schwarz and Kok (1993), the v; residual 
normalized by the standard deviation of v; has a tau 
distribution, which is related to Student's distribution by 
f = vn—u I, -u-l 
-u 7 7p —————— (5) 
WU AL m 
where n—u means the degree of freedom. The normalized 
residuals are treated in hypothesis testing as a test statistic, 
gauged by a threshold at a 5 percent significance level. The 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B3, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
    
blunder is removed one at a time so that the algorithm is termed 
IDS (Iterated Data Snooping). 
As calculation of the covariance matrix for a large number of 
data residuals can grow burdensome, the RANSAC algorithm is 
usually conducted first. For the remaining data points, IDS 
could be invoked to ensure that indeed they are regular samples 
of an experiment at hand. Because of the difference in theory, 
RANSAC and IDS are expected to be complementary. 
2.4 Thin-plate Spline Interpolation 
TPS (Thin-Plate Splines) stands for a flexible function in that it 
emulates the minimized bending energy of a metal plate on 
multiple tie-point constraints. A trend surface stems from a 
global, affine transformation between two overlapping images. 
If the x and y symbols denote the transformed line and 
sample coordinates (pixel), coordinate discrepancies at m 
conjugate points can be expressed as A =(à-5, 
n=X Xm Xa) and 
à - (4-Y,uy2- X», ---> Ym 7 Ym) » Tespectively. For an 
interpolation point, TPS actually involves both trend and 
discrepancy values, conforming to the concept of a remove-and- 
restore operation (Darbeheshti and Featherstone, 2009). 
In determining weights, a special matrix is defined as 
0 Kz): i: KGy,) 
K- e o oa Le (6) 
Bln) = Llp) ss on 0 
where with r;, meaning a Euclidean distance between points j 
and k, both €{1,2,...,m}, K(r) is the so-called fundamental 
solution of the biharmonic equation and takes on the form 
K(r)=r" log? . An interim vector consisting of weight 
coefficients is computed to be w, =K 6 and Wy -K, 
in the respective line and sample directions. 
At a place i other than the locations of the m tie points, a vector 
exists, k/ =(K(r;), K(F;2),…, K(Fim )) - In association with the 
w. and w, weights, the k; 
y i 
interpolation in a field of coordinate discrepancies, as follows 
(Du et al., 2008) 
vector is employed for an 
T T. 1 m 
óy; -W,k; -óy Kk, 
The corresponding values from the trend function have to be 
added to those of Equation 7, or x; ^ óx; and y;  óy;, in order 
to achieve the TPS-based warping and resampling for the 
purpose of image registration. Later, the difference in 
performance between an affine transform and the thin-plate 
spline warping methods will be assessed. 
3. MULTISOURCE IMAGE TESTS 
3.1 SPOT and Formosat images 
A combination of SPOT green (500—590 nm) and Formosat 
infrared (760—900 nm) bands and a plain rural area (Figure 1; 
  
     
  
    
      
  
     
  
   
    
  
   
    
       
   
     
   
      
     
   
    
   
      
     
   
      
   
     
      
   
   
    
    
    
     
   
   
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