Edges from centerline

Dear all,

I'm trying to write an algorithm that computes the edges of a vessel starting from its centerline.
This is the problem:

I have a binary image that represents a complex vessel structure. With a skin operation I have obtained the centerline of the whole structure. I have to consider the centerline of each vessel separately from the other centerlines. This problem is easy, because I can delete the branching points of the tree and with bwconn select each segment of the centerline tree.
But my question is: given one of these segments (which represents the centerline of a single vessel), how can I obrain the edges of this particular vessel?

Any suggestion?

Thank you all in advance,

Alfonso