Speed of 2-D Gaussian smoothing

This approximate Gaussian smoothing in two dimensions is very fast. Only eight add-subtract pairs are required per output point, and no multiplies at all are required except for final scaling. The compute time is independent of the widths of the Gaussian(!). (You should understand this if you understood that one-dimensional convolution with a rectangle requires just one add-subtract pair per output point.) Thus this technique should be useful in two-dimensional slant stack.

EXERCISES:

1. Deduce that a 2-D filter based on the subroutine triangle() which produces the 2-D quasi-Gaussian mound in Figure 12 has a gain of unity at zero (two-dimensional) frequency (also known as (k_x,k_y)=0).
2. Let the 2-D quasi-Gaussian filter be known as F. Sketch the spectral response of F.
3. Sketch the spectral response of 1-F and suggest a use for it.
4. The tent filter can be implemented by smoothing first on the 1-axis and then on the 2-axis. The conjugate operator smooths first on the 2-axis and then on the 1-axis. The tent-filter operator should be self-adjoint (equal to its conjugate), unless some complication arises at the sides or corners. How can a dot-product test be used to see if a tent-filter program is self-adjoint?