g k =Kk DB k M k Z + h 0Jc I + n k (2)
where I is the N^xl unit vector, h l k and \ k are
respectively the gain and offset of the photometric parameters
which balance the relative effects of sun zenith angle and
atmosphere condition between different observed images.
3. JOINT MAP RECONSTRUCTION MODEL
It has been proved that the image registration (or motion
estimation) and SR reconstruction are interdependent (Hardie,
Barnard et al. 1997; Segall, Molina et al. 2003; Shen, Zhang et
al. 2007), and a desirable solution is to simultaneously
implement the image registration and SR reconstruction in a
joint framework. Let the full set of P LR images, registration
parameter vectors be denoted by g = {^,£2>••••£/>} »
m = {m v m^,....m P } respectively. Employing the maximum a
posteriori (MAP) framework, the desired HR image and
registration parameters can be computed as
z,m = arg max {p(z, m | g)} (3)
Z,m
Applying Bayes’ rule yields, (3) becomes:
p(g\z,m)p(z,m)}
P(S) J (4 )
P(g\z,m)p(m)p(z)
P(g)
Since p(z,m\g) is independent of g , equation (4) can be
rewritten as
z,« = aigmaxjj~J[/>(& \z,m k )p{m k )]p{z)^ (5)
Assuming the model noise in (2) to be Gaussian with mean zero
and variancecr 2 , the total probability function p(g k \z,m k ) can
be given by
P(g k \z,m k ) = ~e\p<
\ k DB k M k Z h 0k I
2(7 2
- arg max
z,nt
z,m - arg max
z,m
(6)
where O k is a diagonal matrix that denotes which pixels are
outliers that strongly deviate from the observation model. In
order to preserve the edge and detailed information in the
reconstructed image, a Huber-Markov prior is employed
for p(z). The Huber-Markov prior has the form as (Schultz and
Stevenson 1996)
P(z) = — exp
P x,y ceC
(7)
where C is a constant value, ¡5 is a control parameter, C is a
clique within the set of all image cliques C , the quantity
^c( z x ,y) ls a spatial activity measure to pixel Z which is
x,y
often formed by first-order or second-order differences,
a nd /?(■) is the potential Huber function which is defined as
P( 0 =
\ï 1*1 ^ P
\2p\i\-/S \i\>ju
(8)
where p is a threshold parameter separating the quadratic and
linear regions (Schultz and Stevenson 1996). As for
the d c (z x y ) , we compute the following finite second-order
differences in four adjacent cliques for every location (x,>')in
the SR image
d c ( Z x,y) Z x-l,y ^- Z x,y +Z x+l,y
(9)
d c ( Z X,y) = Z x,y-1 ~ ^ Z x,y + z x,y+l
(10)
d c ( Z x,y ) ~ ~Jj\_ Z x-\,y-\ ~ ^ Z x,y + Z x+1,>-+1 J
(11)
d c ( Z x,y ) = ^J^\_ Z x-l,y+l - ^ Z x,y + Z x+l,y-l]
(12)
Generally, the choice of the prior statistical model
registration parameters p(m k ) is highly application
for the
specific
(Hardie, Barnard et al. 1997), and it depends on the assumed
motion model. In this paper, we assume the following global
affine motion model between the observed images
x 2 =a 0 + a x x x + a 2 y x
(13)
y 2 =K + b \X\ +b 2 yi
(14)
where (Xj, ) and (x 2 , y 2 ) are respectively the pixel locations
of the referenced and un-referenced images,
anda 0 ,a [ ,a 2 ,bQ,b ] ,b 2 are the affine parameters. In this case, the
registration parameters are significantly overdetermined by the
data and a prior is not necessary to yield a useful solution
(Hardie, Barnard et al. 1997). Therefore, we select ppt^) as a
constant value in this paper. Thus, substituting (6) and (7) in (5),
after some manipulations, the following minimization cost
function is obtained.
¿map = ar g m in[£(z,/«,0)] (15)
with the objective function E(z,m,0) is
E(z,m,0) = £|| O t ( gi -l ht DB l ,M k z-KJ)t
* (16)
+aXIM(a.,))
x,y ceC
It should be noted that the motion matrix M k embodies the
registration parameter vector m k .
4. OPTIMIZATION METHOD
Figure 1. The optimization procedure
In the cost function (16), there are three large sets of unknowns:
the HR image z , the registration parameters m k and the
