Can you extract line segment length from image FFT?

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I am trying to determine the average angle and length of a bunch of line segments in an image, like the one below.

image with angled line segments

This is a toy image--the real data I am working with is much nosier. I have gotten good results for the average angle by applying a Radon transform to the FFT of the image--see below.

fft of image above

radon transform showing dominant angle

I intuitively think there must be a way to also extract the average length using the fft of the image, but I don't know how. Any help from "nope you can't do that using FFT" to "here's how to do that using FFT" would be greatly appreciated!

Note: I have tried applying a Hough transform to the Canny output of the image, but my data is noisy to the point that the output is not reliable.

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