Non-Uniform Grids

The process of convolution is symmetrical. It combines two functions. We usually refer to one as the signal and the other as the filter but it does not matter which is which. For filtered noise we usually think of the function as a set of points and the filter as a circular, spherical or hyperspherical footprint with higher values in the centre. We get the same result if the filter is a single point with value 1.0 and the function is a set of overlapping circular or spherical regions:

These regions could be made different sizes to achieve the effect of better packing. We can visualize this as packed circles with different diameters. But remember each circle is now smaller than the footprint it represents. We can achieve better packing with non-uniform circles or spheres but how should we place them to make filtered noise that is not artificially shaped by our choice of non-uniform pattern? One answer is to use a slice from a space of higher dimension.