Without having time to try this, I could imagine something like this:

Iterate over all the lines and calculate the slopes.

Sort the lines by their slopes

If any two lines have a roughly similar slope, they are either parallel or the same line with a gap in it e.g. the almost vertical lines on the left. To estimate, calculate where they will intercept the x or y-axis. If they intercept at the same point, they are the same line and should be merged into one line. If not, put them in a set of (roughly) parallel lines.

Now compare each set of parallel lines with each other set, and calculate if they intersect (possibly off-screen).

Apply some application-dependent criterium to pick the best.

The running time of this depends a lot on the number of detected lines and the number of sets of parallel lines. You could improve a little by considering only lines with a minimum length, playing with the threshold for which lines are considered parallel, etc.

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