Time scale and Fourier transform
Most of the classical linear transformations (even filtering) may have three main types of scalings:

natural or none: sum or integral does not have an explicit scaling factor (but the inverse may need one)
in amplitude: because of linearity, at least one reference &quot;unit&quot; signal should have a unit amplitude in the transformed domain, for an easy read of graphs. Typically a unit amplitude sine (which is not $L_1$-integrable) should have a one amplitude at its frequency in the Fourier domain
in energy: to have constant energy in the original or the transformed domain.

Each of these options (sometimes, two of them are equal) depends on the purpose. Fourier  has a specific relation to scale (Where in “Discrete Time Signal Processing” (Oppenheim et al.) can I find the scale-change theorem?):
$$  s(\alpha t) \mapsto  \frac{1}{|\alpha|} S \left(  \frac{f}{|\alpha|}\right) $$
Obtaining scale-, shift- or orientation-invariant features is a long-lasting issue in DSP, and is still a concern in machine intelligence or deep learing.
Along the Fourier line, you can look at the Fourier-Mellin transform or scale represensation (by L. Cohen), and the many feature representations like SIFT, SURF, ORB, BRIEF: Image Matching Using SIFT, SURF, BRIEF and
ORB: Performance Comparison for Distorted