If your edge/line is the only segmented object (only the line consists of ones) in your image then if M is your binary image (containing ones for white) then: [I,J]=find(M); %equivalent to find(M==1) gives you the I and J indices or image coordinates of the whole line. If your line is straight then the mean of these coordinates is the same as the mean of the first and last coordinate and thus gives you the middle.
If you have multiple objects in the image with ones, not just your line. Then you could filter those out based on size/shape criteria and do the above. Alternatively could create an adjacency matrix whereby you check whether pixels are one and also have neighbours in the line direction that are one. For the end points of the line the latter will be false which allows you to find them. But this depends on the quality and thickness of your line depictions. So not sure if this would work. Also you could develop a type of mask (filter) that will highlight the ends of lines. But I would go for the simple mean of the object approach.