The easiest way might be to submit the mass of data as if they were scattered points, and use one of the scattered data interpolants; see http://www.mathworks.com/help/matlab/scattered-data-interpolation.html
If the lines happen to have been created over a rectangular grid aligned with the X Y axes, or if they can be rotated to be in that alignment, then you could also use one of the gridded interpolants, but it might not be worth doing.
