outliers O k . We use a cyclic coordinate optimization procedure 
solve all the unknowns. The HR image, registration parameters 
and outliers are found in an alternate manner given the two 
others, respectively. A block diagram of the whole optimization 
procedure is shown in Figure 1. 
4.1 Registration 
To solve the registration parameters m k , we first expand the 
cost function (16) by Taylor formula ( For expression 
convenience, we neglect the subscript k) : 
£(m)*£(m”)-K^^) r (Am) + -(Am) r /7"(Am) (17) 
dm" 2 
dE(m)_ an( j Yjn ¿ enote t h e 
where Yl is the iteration number, 
dm n 
gradient matrix and Hessian matrix of E(m) atm” respectively: 
®W=2(/-)'Oy 08) 
dm" * 
H" = 2(7” ) r O n k J n + 2£ a;"0;( 1 9) 
In (18) and (19), r” is the residual vector that is equal 
n dr n 
t0 Sk ~Kk DB k M kZ-h 0 J ,J = denotes the gradient 
dm" 
matrix of r" , and H" is the Hessian matrix of r". For small 
r" , we have the following approximations: 
H" ~ 2(J") T O k J n . Thus, differentiating (6) with respect to 
Am and setting the result equal to zero, after some 
manipulation, we have 
(,J") r O n k J n (Am) = -(J n ) T O k r" (20) 
and 
Am = [(J n fOlJ" r 11 H J" f O n k r n ] (21) 
Then, the parameter vector can be updated by 
m'" ! = m n + Am (22) 
r = -\ k MlB T k D T 0 T k {g k -\ k DB k M k z-\ k I) + Xr (26) 
where /* is the derivative of the regularization term that can be 
solved on a pixel-by-pixel basis. Thus, the HR image is solved 
by employing the successive approximations iteration 
2" +1 =z"-fir" (27) 
where Yl is the iteration number, and J3 is the step size which 
can be solved by 
/? = 
(■r n fr" 
(r") T (V 2 E)r" 
(28) 
where V 2 E is the Hessian matrix of the objective 
function E(z, m, O). 
5. EXPERIMENTAL RESULTS 
(a) (b) 
(c) (d) 
4.2 Outlier Detection 
To detect the outliers, we employ two criteria. The first is a 
geometrical criterion that requires the predicted location of a 
pixel using the current motion parameters is still in the image 
field. The horizontal and vertical predicted locations are judged 
by equation (23) and (24) respectively. 
0 < a 0 + a x x x + a 2 y { < (23) 
0 < b 0 + b x x x + b 2 y x < N 2 (24) 
Here, N x and N 2 are respectively the horizontal and vertical 
size of the observed image. The second criterion is a 
photometrical one. Define f k = h x k DB k M k z~ h 0 k I , the 
photometrical criterion is 
I gk ( x > y) ~ fk (*> y)\<d* cr k (x, y) (25) 
where cr k (x, y) is the standard deviation at site (x, y) of the 
kth image g k , and d in the right hand side acts as a scale 
threshold. 
4.3 SR Reconstruction 
The steepest descent optimization is used to solve the HR 
image z ■ Differentiating (16) with respect to z , we have 
(g) (h) 
Figure, (a)-(f) Images respectively captured on 28/12/2003. 
30/12/2003, 01/01/2004, 04/01/2004, 06/01/2004 and 
08/01/2004, (g) 
reconstmcted ima 
The proposed alg 
images which wer 
6, 8 January 2004 
the Satellite Ren 
University. We cl 
algorithm indeper 
the two regions a: 
Figures 3(a)-(f)- 
Figure 3. (a)-(f) Irr 
30/12/2003, 01/( 
08/01/2004, (g) 
reconstructed imag 
In the experiment 
as: ^ = 1 , X = 0.1 
kernel with unit \ 
employed. We assi 
60