A FAST AND ROBUST FRAMEWORK FOR IMAGE FUSION AND ...

The gradient of the cost in (2.10) is:
Gp = ∂
�
N�
∂Z
=
N�
k=1
k=1
�DF(k)Z − Y (k)� p p
�
F T (k)D T sign(DF(k)Z − Y (k)) ⊙|DF(k)Z − Y (k)| p−1 , (2.11)
where operator ⊙ is the element-by-element product of two vectors.
The vector � Z which minimizes the criterion (2.10) will be the solution to G p =0.
There is a simple interpretation for the solution: The vector � Z is the weighted mean of all
measurements at a given pixel, after proper zero filling 2 and motion compensation.
To appreciate this fact, let us consider two extreme values of p. Ifp =2, then
N�
G2 = F T (k)D T (DF(k) � Zn − Y (k)) = 0, (2.12)
k=1
which is proved in [22] to be the pixelwise average of measurements after image registration. If
p =1then the gradient term will be:
G 1 =
N�
F T (k)D T sign(DF(k) � Z − Y (k)) = 0. (2.13)
k=1
We note that F T (k)DT copies the values from the low-resolution grid to the high-resolution
grid after proper shifting and zero filling, and DF(k) copies a selected set of pixels in high-
resolution grid back on the low-resolution grid (Figure 2.3 illustrates the effect of upsampling
and downsampling matrices D T , and D). Neither of these two operations changes the pixel
values. Therefore, each element of G 1, which corresponds to one element in � Z, is the aggregate
of the effects of all low-resolution frames. The effect of each frame has one of the following
three forms:
1. Addition of zero, which results from zero filling.
2. Addition of +1, which means a pixel in � Z was larger than the corresponding contributing
pixel from frame Y (k).
2 The zero filling effect of the upsampling process is illustrated in Figure 2.3.
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