The Concept of scales in Gabor filter

Question

I've been reading about Gabor filters, and there is a concept of scale. While I understand that Gabor features help in analyzing both the frequency and orientations of an image, what is the idea of scale? It appears that the original image is resized and then the filter is applied, for each scale. Some elaboration on this would be much helpful.

Answer 1

A Gabor filter is a modulated Gaussian. Or said in another way, it is a (complex) wave function windowed by a Gaussian. The wave function determines the frequency and direction, the Gaussian determines a scale.

With a larger Gaussian, the filter averages over a larger area, and so requires a larger region showing the matching frequency. The output will be smoother, the location where the frequencies match will be less precisely highlighted. But because the wave has to match over a larger area, the frequency has to match more precisely.

With a smaller Gaussian, the output will be less smooth, so it shows the location more precisely. But more similar frequencies will match well.

So, the scale balances spatial localization vs frequency localization. You can be precise in the one or in the other, not both at the same time.

Oh, indeed, the scale doesn’t involve downsampling the image. With a large sigma, where the output is smooth, you can downsample this output if you like. But you cannot downsample the input without potentially affecting the frequencies you are interested in detecting (see “aliasing”).