Question

Is there a filtering algorithm that converges at some shape? My problem is that I am filtering a two dimensional graph, and applying filtering repeatedly. I’m down-sampling data and re-sampling it using gaussian filters (footprints), but the graph changes its shape with every subsequent filtering. What I need is to achieve some final shape, so that after enough filtering, the graph will no longer change shape.

EDIT: by filtering I mean smoothing out, not dropping some information.

Answer 1

The simple answer is no. Mathematically filtering with a Gaussian means that you’re convolving your data with a Gaussian. But convolving is the same as multiplying in the Fourier domain, so repeated application of the filter is like repeated multiplications in the Fourier domain, and here you can see that things will either blow up or go to zero. There might be something else that’s also validly called filtering that doesn’t do this, and you might be able make or dig something up that will do what you want, but repeated convolutions with the same kernel, Gaussian or otherwise, will not converge.