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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)