Full text: Proceedings, XXth congress (Part 3)

    
International Archives ‘ ‘he Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
coarse pixel which affects each fine pixel can be found, such 
that in terms of grey-scale values: 
(0.0) (8.1) (0.2) (0.3) (0,4) (0.5) 
  
  
  
  
  
(L0) 
  
(2.0) 
  
  
  
  
  
  
  
(3.0) 
  
1 
Fine grid system CL CZ. C3 … Coarse Pixels 
Figure 1. Coarse data mapped on the enhancement grid 
C2 =[F(2,3)+0.5*F(1,3)+0.5* F(2,4)+ 0.25*F(1,9)]*p~ — (D) 
Where p is the enhancement ratio , which in this case is 1.5 as 
deduced form Figure 1 . 
The observation model may be represented as 
y = Ax +n (2) 
where y contains the grey values of the low resolution image 
pixels, x are the required high resolution image pixels, À is the 
matrix of the coefficients, n represents the additive noise 
matrix. 
Based on the least squares theory, the solution of (2) results 
from the minimization of 
F(x) «|n| 2l y - Ax I (3) 
So the solution is 
x =[A'AJ'*A"y (4) 
3. IMAGE MATCHING ERROR ANALYSIS 
Image matching is a very important step to the success of the 
resolution enhancement. Therefore, accurate matching methods, 
based on robust motion models should be needed (Park, 2003). 
However, subpixel matching is not accurate enough due to 
many reasons in practice. So the matching error should be 
considered in the resolution enhancement. 
In (2), A is the matrix of the coefficients containing the sub- 
pixel motion information of the low-resolution images, 1t can 
be written as 
A = A+ AA (5) 
where A is the accurate contribution of the high-resolution 
image to the low-resolution image. À is the matrix of the 
weights containing the estimated matching parameters. AA is 
the uncertainty caused by inaccurate matching. As the 
matching error increases, the difference between A and AA 
The difference distorts the reconstructed high- 
increases. 
. Equations (2) and (5) can be written as 
resolution image[11] 
  
y =(A + AA)x +n =A +(AAx+n)=Ax+n (6) 
where n 2 AAx 4 n, includes the intrinsic additive noise and 
the matching error noise. In Lee's paper (Lee, 2003), it is 
empirically proved that the matching error noise has a 
Gaussian type, and that its standard deviation is proportional to 
the degree of the matching error. Therefore, N may be 
regarded as Gaussian type noise in the enhancement process. 
4. PROPOSED ALGORITHM 
The problem of estimating a high-resolution image from some 
low-resolution images is ill-posed, since many solutions satisfy 
the constrains of the observation. In RG Algorithm, the least 
squares solution is advisable if there is not any noise or the 
noise is small enough to neglect. In many cases, however, 
perfect matching is practically impossible to realize. That is to 
say, there may be considerable matching error noise in the 
enhancement process. In this case, the solution of (4) cannot 
satisfy the enhancement demand. 
To solve this problem, solution for high-resolution image is 
constructed by applying regularization technique that involves 
a functional || C x | and a regularization parameter & to the 
minimization problem. The solution results from the 
minimization of? 
F(a,x)=|ly-Ax |’ +a|lCx |’ (7) 
where is the regularization parameter controlling the terms 
2 
lly - Ax II^ and || Cx||', C is a high-pass operator. We select 
2-D Laplacian for C. 
The necessary condition for the minimum is that the derivative 
of F(Œ,X) with respect to x is equal to zero, which is 
a -2A'Ax- 2A'y 120g "Cx 20 (8) 
x 
The solution is 
Xx z[A'A * aC'C]! * A! y (9) 
Regularization parameter & controls the balance between 
fidelity to the data and smoothness of the solution (Lee, 2003). 
If it is too large, the resolution will be too smooth and loss 
fidelity; if it is too small, the noise problem will not be solved 
effectively. To solve this problem, we employ an iterative 
algorithm to estimate the regularization parameter at the same 
time with the enhanced image. 
[n the iteration steps, the choice of & utilizes the information 
available at each iteration step in the enhancement process of 
the high-resolution image. It satisfies the following properties: 
- a(x) is proportional to || y - AX I? 
- a(x) is inversely proportional to IC x|f 
- Q'(x) is larger than zero 
      
  
  
   
  
   
  
   
   
    
   
  
  
   
   
   
  
  
   
   
   
  
  
  
  
  
   
    
   
   
    
  
   
  
   
   
  
   
   
  
  
  
    
   
     
   
     
Intei 
  
To 
func 
whei 
beco 
from 
The 
succi 
The « 
The i 
2. 
  
Figure 
(c)enh: 
   
	        
Waiting...

Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.