what is meant by support of a filter?

Question

what is meant by support of a filter in context of image processing? In the attached link of a short tutorial for detecting bar in an image by gabor filter,it is said that γ=1 helps in finding the support of the filter. link1

Answer 1

Many filters, particularly linear time-invariant ones, are represented by a kernel function. The output of a filter at a point is calculated by a weighted average of all the points of the input where the weights are specified by the kernel function. The kernel function takes displacements from the current output point we're calculating to other points, so it is always relative to the output point.

The support of a filter is then, intuitively, the "size" of the kernel. Any displacement for which the kernel function is $0$ means that we don't have to consider the corresponding point at all when calculating the value of an output point. Therefore, the most strict definition of support is that it is the set of displacements where the kernel function is non-zero. This could be a quite complicated set in general, and so isn't particularly helpful for understanding or calculation. A looser definition would be the support is any set (of displacements) which contains all the displacements where the kernel function is non-zero. This allows us to conservatively approximate the support with some simple-to-describe shape. This will usually be something like an ellipse or rectangle or square (or their lower-/higher-dimensional analogues). Given such a class of approximate shapes, the support will usually be taken to be the smallest approximate shape which contains all the displacements for which the kernel function is non-zero.

For that page, the support is the smallest ellipse which contains all the non-zero inputs of the kernel function. For discrete kernels that are typically used to implement these filters on a computer, usually a rectangular support is used. This allows calculating the weighted average by iterating only over the points in a (typically much smaller) sub-rectangle of the input, rather than iterating over the entire input while multiplying most values with zero. It also means the kernel function can be described by only specifying its values for that rectangle, and this is usually represented by a matrix called the mask in the 2D case.