Full text: Mapping without the sun

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 
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) 
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" 
where V 2 E is the Hessian matrix of the objective 
function E(z, m, O). 
(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 
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 

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