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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
64 
Poros and Peterson, 1985), but we permit the existence of 
linear-assumption error as 
P(g IZ) = -77-exp j—-Az- B) T K~\g -Az- Æ) j (6) 
M 
Sx,y - A x,y z x,y + B x,y + n x,) 
(1) 
where A xy . and B x y are the relative gain and offset parameters 
respectively, n x y is the sum of linear-assumption error and 
sensor noise. In matrix-vector form, the relation between the 
observed image and the desired image can be expressed as 
g = Az + B + n 
(2) 
In the model, g is the lexicographically ordered vector of the 
observed image, z represents the desired image, A is a 
diagonal matrix with diagonal elements being the gains of all 
pixels, B is the offset vector, and n represents the noise vector. 
2.2 MAP Reconstruction Model 
In recent years, the Maximum A Posteriori (MAP) estimation 
method has been central to the solution of ill-posed inverse 
problems in a wide range of applications (Borman and 
Stevenson, 1998 ), such as image denoising (Hamza and Krim, 
2004), deblurring (Ferrari et al., 1995), super resolution 
reconstruction (Shen et al., 2007), and others. Our purpose is to 
realize the MAP estimate of a destriped or inpainted image z , 
given the degraded image g . It can be computed by 
z = argmax/?(z | g) 
Applying Bayes’ rule, equation (3) becomes 
(3) 
where A/j is a constant, and K is the covariance matrix that 
describes the noise. Since the noise is assumed 
independent, K is a diagonal matrix containing the noise 
variances. Thus, we can further rewrite equation (6) as 
P(g I z) =—exp] --||Q(g -Az-B) 
(7) 
where Q is also a diagonal matrix. 
The second density function in (5) is the image prior which 
imposes the spatial constrains on the image. This may include 
such constraints such as positivity, smoothness and so on. Here, 
we employ an edge-preserving Huber-Markov image prior 
model. This prior model can effectively preserve the edge and 
detailed information in the image (Schultz and Stevenson, 1996; 
Shen et al., 2007, doi:10.1093/comjnl/bxm028). It is denoted as 
/>(z) = —exp 
m 2 
x,y ceC j 
(8) 
In this expression, M 2 is a constant, c is a clique within the set 
of all image cliques C, the quantity d c (z xv ) is a spatial activity 
measure to pixel z x y which is often formed by first-order or 
second-order differences, and /?(•) is a Huber function defined 
as 
z = arg max 
P(g I z)p(z) 
* Pig) 
(4) 
/0(0 = • 
2/00-/' 
(9) 
Since p(z | g) is independent of g , p(g) can be considered a 
constant and hence equation (4) can be rewritten as 
z = arg max p(g \ z)p(z) ■ 
(5) 
where p is a threshold parameter separating the quadratic and 
linear regions. 
As for the d c (z x v ), we compute the following finite second- 
order differences in four adjacent cliques for every 
location (x,y) in the image 
The first probability density functions (PDF) in (5) is the 
likelihood density function. It is determined by the probability 
density of the noise vector in (2), i.e., p(g \ z) = p(n) . Since 
different pixels may be degraded to different degrees in the 
destriping and inpainting problems, we assume the noise is not 
identical, but still independent. Under these assumptions, the 
probability density is given by 
z x-\,y 2z x y + z x+ i y 
(10) 
Z X,y-1 — 2 Z x,y + z x,y+1 
(11) 
z x-\,y-\ — 2 Z x,y + Z X+l,>’+l] 
(12) 
Z X-l,y+l — 2 Z x,y + z x+l,y-l J 
(13)