Detect U Shaped Edges in an Image

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I'm trying to detect the location of a fingertip from an image. I've been able to crop out a region in the image where it must have a fingertip, and extract the edges using Canny Edge Detector. However I'm stuck. Since my project description says I can't use the skin color for detection, I cannot find the exact contour of the finger, and will have to try to separate the fingertip with edges alone. Right now I'm thinking since the finger has a curved arch shape/letter U shape, maybe that could be used for detection. But since it has to be rotation/scale invariant, most algorithms I found so far are not up to it. Does anyone have an idea of how to do this? Thanks for anyone that responds! enter image description here

This is the result I have now. I want to put a bounding box around the index fingertip, or the highest fingertip, whichever is the easiest.

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