Simultaneous regression of multiple curves in the same dataset or image?
There are many possible methods.  In your picture, there aren't too many dots, so the following algorithm might be good enough:


For each pair of points, find the line through them.  Count the number of points that are approximately on that line.
Choose the line that has the most number of points on it.  Output that line.  Delete all points that are approximately on that line from the figure.
Repeat until no points remain.


For fancier methods (e.g., if you have a huge number of points, or some prior model for what kinds of lines you expect to see), you can look at RANSAC and the expectation maximization (EM) algorithm, but given your sample picture, the above simple method might be adequate in your setting.