Scale-invariant image processing by means of scaled transforms or form-invariant, linear shift-variant filters

Scaled transforms have recently been introduced as a powerful mathematical tool for processing (e.g., detecting) images that are arbitrarily scaled or rotated. It will be shown that the scaled transforms are equivalent to a subclass of the linear shift-variant, "form-invariant" filters. The equivalence holds in the sense that, given any scaled transform and any input signal, a linear shift-variant filter can be found such that the outputs of the filter and of the scaled transform are Fourier-transform pairs. The relevance of this result is briefly discussed.