The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
65 
Substituting (7) and (8) in (5), using the logarithm function and 
performing some manipulations, M x , M 2 can be safely 
dropped. The maximization of this posterior probability 
distribution is equivalent to the following regularized minimum 
problem 
£ = arg min /11|Q(g - Az - B)f + ^ p(d c (z xy )) (14) 
x,y csC 
where A is called as the regularization parameter. 
2.3 Solution Method 
A gradient descent optimization method is used for the 
minimum problem in (14). Differentiating the cost function 
with respect to z , we have 
r = -A T Q T Q(g-A Z -B) + Ar (15) 
where r is the derivative of the regularization term. Then, the 
desired image is solved by employing the successive 
approximations iteration 
z n+ \=z„-ß n r n (16) 
where n is the iteration number, /3 n is the step size. 
2.4 Parameter Determination 
In order to use the observation model (2), A (gains) 
and B (biases) should be first determined. It is easily 
understood that the gain and bias should be respectively 1 and 0 
for healthy pixels. For dead pixel in image inpainting, the gain 
can be regarded as 0 and the bias the pixel value. For the 
destriping problem, the parameters of pixels in a row or a 
column are often assumed to be the same. We use the moment 
matching method (Gadallah et al., 2000) to obtain the gains and 
biases of the stripe pixels Therefore, the moment matching 
method is a special case of the proposed algorithm 
with yT —>■ oo and Q being a unit matrix in equation (14). 
The matrix Q is diagonal and its elements represent the 
reciprocal of the noise standard deviation in different pixel 
locations. For convenience, we scale the element values to the 
range of 0~1. The difference caused by the scaling can be 
balanced by A ( A is determined heuristically). For all the 
healthy pixels, the corresponding elements are set as the 
maximum value 1. On the contrary, the elements should be 0 
for dead pixels because they do not have any correlation with 
the true scene. The elements of other bad pixels are between 0 
and 1, and they correlate with the local activity level, the 
validity of moment matching and so on. Generally, we can 
select small element values to recovery the information from 
the neighbors using the prior constraint. On the other hand, 
larger element values should be chosen for sharp regions in 
order to retain the high-frequency information. We use the 
standard deviation as the activity measure, and a simple linear 
function is employed to determine element values. 
3. EXPERIMENTAL RESULTS 
3.1 Destriping Experiments 
The proposed algorithm was tested for destriping on images of 
the Moderate Resolution Imaging Spectrometer (MODIS) 
aboard the Terra and Aqua platforms. The Terra MODIS data 
used in this paper was acquired on December 31, 2007, and the 
Aqua MODIS data was acquired on December 28, 2003. 
Sections of size 400x400 were extracted from the original 
images as experimental data. For calculation and display 
convenience, the original data are coded to an 8-byte scale. The 
original images and destriped results of Terra and Aqua are, 
respectively, shown in Figure 1 and 
Figure 2. It can be seen that the moment matching method can 
greatly improve the image quality, but there are still 
considerable radiance fluctuations within the resulting image. 
The proposed algorithm, however, provides a much more robust 
destriping from the visual perspective. 
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(a) original image 
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(b) moment matching 
Figure 1. Destriped results of the Terra MODIS image.