Since p(y,z) is independent of x , it can be considered a
constant and removed from the maximum function:
X z arg max p(x)p(y,z| x)
Xx
- argmax p(x)p(y | x)p(z | x, y) (9)
X
= argmax p(x) p(y | x) p(z | x)
X
Since z and y are both known quantities, so it is tenable
for p(z] x y) 7 p(z| x) in (9).
The function p(y|x) provides a measure of the
conformance of the estimated image x to the observed image
y according to the observation model (2). Assuming that the
noise is zero-mean Gaussian noise, and each image is
independent
K, B,,
201 = 1] [1120 1x) (10)
k=l b=l
where K, is the image number of y, B, , is the band number
of y, ,, and
1 2
POs | x)= WE ex|-]».. -A,,,x)| Lm (11)
Qza, ,,) rt
where a, , is the variance of the noise ny; ;, N,;, and
N,,k,vy are the image dimensions of y, .
The function p(z | x) is determined by the probability density
of the noise vector n. , , in (6), and is expressed as:
K, 5}
pc =] 111260. (12)
k=1 b=1
1
Dux) NN 77 XP
Az pp) UU ^
3 (13)
Heo 74 kpX— 7468-40] / 20 4)
where a.,, is the variance of the noise n, , , B, , is the
band number of z,,, and N,; , and N,, are the image
dimensions of z, .
An edge-preserving Huber-Markov image model (Schultz and
Stevenson, 1996; Shen and Zhang, 2009) is employed for
density function p(x)
B,
$2 Pd.) 20. 5t (14)
p=] [ NUUS exp
pi (2705 5) ij éay
where &, is the model parameter of the bth band, & is a
local group of pixels called a clique, and y. is the set of all the
cliques, N, , and N, are the image dimensions of x . The
quantity de(x,(1,j)) is a spatial activity measure to
pixel x; (i, j) , and the following finite second-order differences
are computed in two adjacent cliques for every location
(i,j) in the image.
di (xpi, J) = xp (i= 1, J) = 2x (i, /) + (i +1, /) (15)
dj) = (i,j =D = 2%, (i, )) + xp (0, j +1) (16)
In (13), p(-) is the Huber function defined as:
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia
n? h| €
a Ae (17)
2u|h|-4^ Mu
where 4 is a threshold parameter separating the quadratic and
linear regions. When 4 approaches 4-00 , the prior becomes the
Gauss-Markov, which has similar spatial constraints to the
Laplacian prior.
Substituting (10)-(14) in (9) and implementing the monotonic
logarithm function, after some manipulation, N,, , N,, »
Ny» Ny, Nz 4, and N.,,can be safely dropped, and
the maximization of this posterior probability distribution is
equivalent to the following regularized minimum problem:
X=arg min[ E(x) | (18)
where
K, 8, ;
E(x) MY ris - 4,0% +
k=1 bal ^ eb
K Bj
Ci thd] XE 3 peu» (9)
HH ij Say
In this paper, we assume a, 4 5, @; x; and a, are invariable.
Thus, minimum function can be se as
55,
Elo) = AY ]»;- Aral Ed Z kB: "i
pp pp
B,
+2) 2 pd (xi, )) (20)
b=1 i,j Sey
where 4; and A; are two regularization parameters. At last, the
steepest gradient descent method(Shen and Zhang, 2009; Shen
et al., 2010) is employed to solve the fusion images.
4. EXPERIMENTAL RESULTS
The proposed method was tested using simulated images, and
the experimental images and results are illustrated in Fig.1. We
used one HS image to simulate one PAN image, one MS image
(four bands) and four degraded HS images. Fig.1(a)-(c) show
the PAN image, the cubic interpolated versions of the MS
image and HS image respectively. The fusion method were
implemented in four cases that the input images are respectively
four HS images, HS image + MS image, one HS image + PAN
image, and all the simulated images. The fused results are
shown in Fig.1(d)-(g). By visual inspection, each of the fusion
results enhances the spatial resolution. Moreover, the image of
the integrated fusion method has the best visual quality.
The fused images are evaluated using five quality indices.
These are the root mean square error (RMSE), correlation
coefficient (CC), universal image quality index (UIQI), means
relative dimensionless global error (ERGAS) and spectral angle
(SA). They are defined by equations (21)-(25), respectively.
