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
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