When I say template matching, I'm referring to finding occurrences of a small image (the template) in a larger image.
The OpenCV library provides the trivial solution, that slides the template over every possible location of the image. While this implementation provides translation invariance, it provides no stretching/scaling invariance. To get stretch and translation invariance, one would need to stretch the template (or shrink the main image) until iteratively, running the original template check over the image. This increases complexity to $O(S * n^2)$ where S is the number of different resolutions one checks - if one want's to check every possible resolution, the overall complexity is $O(n^4)$. Effectively, you generate $O(n^4)$ subsections and check if they're equal to the template.
From what I was taught, Image Segmentation Networks do just this - however, instead of using the basic template matching (i.e. checking the pixels match) the generated subsection is put through a classifier network - so this would be more expensive than standard templating.
My question is, are my calculations correct - for complete subsection generation, is the complexity $O(n^4)$ and is there really no better algorithm for generating these subsections - used by both image detection algorithms?