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Does SIFT work the same on different images sizes?

I was wondering if the result of any keypoint detector & descriptor algorithm is "input-image-size" dependent.

For example, with the SIFT algoritm, I don't have the same kp and descriptor (some are, but lots are not) when working on full size sized images or (down-)resized images (original: 1000px, resized: 500px e.g.).

On bad quality pictures (foggy landscapes e.g.), it may be an issue when working on reduced size images (in order to improve computation time when dealing with thousands of input images).

Is there a way to improve that?