How does multiscale feature matching work? ORB, SIFT, etc

When reading about classic computer vision I am confused on how multiscale feature matching works.

Suppose we use an image pyramid,

1. How do you deal with the same feature being detected at multiple scales? How do you decide which to make a deacriptor for?

2. How do you connected features between scales? For example let's say you have a feature detected and matched to a descriptor at scale .5. Is this location then translated to its location in the initial scale?

I can share something about SIFT that might answer question (1) for you.
I'm not really sure what you mean in your question (2) though, so please clarify?

SIFT (Scale-Invariant Feature Transform) was designed specifically to find features that remains identifiable across different image scales, rotations, and transformations.

When you run SIFT on an image of some object (e.g. a car), SIFT will try to create the same descriptor for the same feature (e.g. the license plate), no matter what image transformation you apply.

Ideally, SIFT will only produce a single descriptor for each feature in an image.
However, this obviously doesn't always happen in practice, as you can see in an OpenCV example here:

OpenCV illustrates each SIFT descriptor as a circle of different size. You can see many cases where the circles overlap. I assume this is what you meant in question (1) by "the same feature being detected at multiple scales".

And to my knowledge, SIFT doesn't really care about this issue. If by scaling the image enough you end up creating multiple descriptors from "the same feature", then those are distinct descriptors to SIFT.

During descriptor matching, you simply brute-force compare your list of descriptors, regardless of what scale it was generated from, and try to find the closest match.
The whole point of SIFT as a function, is to take in some image feature under different transformations, and produce a similar numerical output at the end.

So if you do end up with multiple descriptors of the same feature, you'll just end up having to do more computational work, but you will still essentially match the same pair of feature across two images regardless.

Edit:

If you are asking about how to convert coordinates from the scaled images in the image pyramid back into original image coordinates, then David Lowe's SIFT paper dedicates section 4 on that topic.

The naive approach would be to simply calculate the ratios of the scaled coordinates vs the scaled image dimensions, then extrapolate back to the original image coordinates and dimensions. However, this is inaccurate, and becomes increasingly so as you scale down an image.

> Example: You start with a 1000x1000 pixel image, where a feature is located at coordinates (123,456). If you had scaled down the image to 100x100 pixel, then the scaled keypoint coordinate would be something like (12,46). Extrapolating back to the original coordinates naively would give the coordinates (120,460).

So SIFT fits a Taylor expansion of the Difference of Gaussian function, to try and locate the original interesting keypoint down to sub-pixel levels of accuracy; which you can then use to extrapolate back to the original image coordinates.

Unfortunately, the math for this part is quite beyond me. But if you are fluent in math, C programming, and want to know specifically how SIFT is implemented; I suggest you dive into Rob Hess' SIFT implementation, lines 467 through 648 is probably the most detailed you can get.