Simplifying Assumptions

Note that in Eq. 2, we assumed that $D_{i+m,j+n}$ is always valid. However, there are two instances where $D_{i+m,j+n}$ will not be valid.

The location $i+m,j+n$ may not be inside the grid. When the filtering is done close to the grid boundary, i.e., close to the edge of the image, not all the points under the elliptical filter will exist; the average should be computed on only those points that exist. However, taking this boundary effect into account will cause a non-linearity since the divisor, $N_k$, is not a constant after all.

What happened to this assumption on $D_{i+m,j+n}$? To get to the Discrete Fourier Transform (DFT) in Eq. 9, we assumed that the data values are periodic beyond the image edge. This is patently nonsense; storms don't repeat every $250~km$ just for mathematical convenience. We will see, therefore, that at the boundary, our assumptions fail and the modified filter produces answers that are totally different from the original filter.

Secondly, even if $i+m,j+n$ is a valid location, the data value that it contains may not be valid. Wolfson et al. (1999) omit such pixels from the average calculation. However, in that case, the average will have to be computed on different numbers of pixels depending on the location within the image. Consequently, the filter will become shift-variant.

The duality of spatial convolution to Fourier Transform multiplication holds only for linear, shift-invariant filters. So, both these simplifying assumptions have real consequences in the large-scale filtering context.

To retain shift-invariance, a numerical value has to be assigned to every pixel in the image. After this ad-hoc assignment, all pixels are assumed to contain valid information. This will introduce differences in the results that we obtain with the transform method described in this paper and the results that are obtained using the method described by Wolfson et al. (1999).