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
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succi
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2.
Figure
(c)enh: