I= [fax dv fin, -xja,-5}-1(< 2" }+E, 
The Ei; are uncorrelated random numbers with zero expectation 
value: : 
(yen (e & Yaar 4 in 
Image restoration means the estimation of the scene brightness 
l'(x;y;) on the position x';-iói, y» 7 j& with 8; € Ai, 6 € As. 
For 6< A we have subpixel accuracy. 
To simplify for the integral we use discrete, scalar notation. 
With approximation of the scene with a step function it follows 
[=X h(i=k j=0)d +E =X h 4, +& 
kJ kd 
For pixel resolution (u=v=1) is 
L, = N i dag te, 
and for the 1D case 
=) A LA Hé 
It is meaningful to fit most of the PSF with a Gaussian shape 
function 
h (xx). (=) 
ECAP mel 
Because of different pixel size for the focal plane arrays, a 
reference of the PSF to pixel distance is meaningful. Typical 
values for sigma of the system-PSF are in the range of 0.5 to 
1.0 in units of pixel size in the image plane. An approximate 
determination of the PSF includes optics and the pixel size. An 
additional part in moving direction must be taken into account. 
The following calculation is done only in one dimension. The 
pixel PSF for the most simple case (constant sensitivity, no 
diffusion effects) is i 
[1 : ó Ó 
Nema a 
Hp (x)=13 > 2 
(0 elsewhere 
The optics PSF can be model with a Gaussian shape function 
with 0-0/3. Figure 2 shows the resulting MTF and its 
components for the ADS40. The motion MTF depends on 
flight height and should be in the range of half of a pixel. 
  
  
—E T rrp po : 
L = Motion MTF 
O08 x > Li 
un N. = 
Da <—— Pixel MTF = 
=— Optics MTF 
+ 
| 
D 1 I me 
x | 
0.2 + Total MTF > zl 
- SS ™ 
i ES [i i Em 
  
  
  
    
Figure 2. MTF-estimation for ADS40 
2.3 Image Restoration 
The removal of blur has been investigated with great effort in 
the past (Andrews and Hunt, 1977; Katsaggelos, 1991). Image 
restoration is an ill-posed inverse problem. For this reason, a 
unique solution may not exist, and a small amount of noise can 
lead to large reconstruction errors. To overcome this difficulty, 
     
  
   
   
   
    
    
    
   
     
     
  
   
      
     
    
  
    
   
    
     
   
  
  
  
  
  
  
  
  
  
  
    
    
    
     
    
    
   
    
   
   
   
  
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004 
a priori knowledge on the original scene and PSF as well as 
noise information on the image forming system are necessary. 
Image reconstruction approaches may be distinguished by the 
derivation criteria, by algorithms working in the object and 
Fourier space, by local, recursive and global algorithms, by 
linear and non linear algorithms, and so on. The applied 
algorithm for blur removal depends on the special application, 
the prior information and the used criterion for the restoration 
quality as well as on system PSF and noise. Some examples 
are: Inverse Filter, Least-Squares Filter, SVD-Based Pseudo- 
Inverse Filter, Maximum Entropy Pseudo-Inverse Filter, 
Laplacian-Constrained  Least-Squares Filter and Linear 
Minimum Mean-Square Error Wiener Filter (see Katsaggelos, 
1991). 
The algorithm to be used here is the linear restoring finite 
impulse response (FIR) filter, described e.g. in (Reulke, 1995). 
The FIR or local filter is a fast algorithm to estimate the 
original grey value from the surrounding image values. Since 
the algorithm is a local one, local changes of the camera 
properties (PSF and noise) or the scene statistics may be 
considered. For filter coefficient determination the least mean 
square solution (LMS) should be used. 
The scene is described as a stationary random process which 
contains the prior information for stabilization of the 
reconstruction problem. For the estimation of the scene value 
at position i,j we use the linear FIR filter approach 
/ 
JL, = > à, + K. jl 
Lb! 
KM 
The unknown coefficients are determined by minimizing the 
least mean square error 
Sty- (4-1. T) 
The derivation to the coefficients leads to a set of linear 
equations for the optimal filter coefficients ay. An additional 
parameter &=0/K is essential for stabilizing or regularizing the 
solution. 
2.4 Resolution of staggered CCD-line 
To validate the resolution enhancement of staggered arrays 
theoretical investigations of the ultimate spatial resolution 
taking into account the total point spread function (PSF) and 
Shannon's sampling theorem have been carried through. (Jahn, 
2000). 
Theoretically, the resolution is as good as for a non-staggered 
line with half pixel size A/2 if the pixels of the staggered line 
array have size A/2 too (with pixel distance A). The slightly 
different viewing angle of both lines of a staggered array can 
lead to deteriorations because of aircraft motion and attitude 
fluctuations and a non-flat terrain. 
The sampling process with respect to a special pixel size shall 
be described. We are looking for the sampling of the optical 
signal I(x,y) behind the optics and in front of the focal plane. 
The sampling is realised by rectangular or square detectors 
with pixel size 6x8 arranged in a regular line grid with 
sampling distance A. For staggered arrays this distance is given 
by A = 6/2 (see figure 1). 
The signal values I; = Kxiyj)) (Xi = 1-A, y =j-À, i,j = 0, +1, +2,.) 
can be obtained by a convolution of I(x,y) with the geometrical 
pixel PSF 
Ps. uz ff (x, zXSY -y'y Hx, y')dx'dy' 
The pixel PSF only in one dimension is 
  
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