Detecting line positions on an edged detected photo

I am having an issue trying to learn the pixal location of certain points along a curved line that has been created due to the use of the the canny edge detection method.

I need a way to pick out these curved lines (there are only 2 I need) from a mess of other shorter lines also created by the edge detection. More specifically when I locate these lines I need a way to locate the max and min points along the lines and and find out the location in terms of pixals of these points.

Unforunatly I am unable to show the exact photo I will be using but here I can show a demo photo: http://courses.cs.washington.edu/courses/cse576/09sp/projects/final/artifacts/eharveyj/project4-paper-site_files/image006.gif

Also no code yet extists for this problem as I am unsure how to go about it.

Is this possible to do?

Thank you in advance!