Abstract: The existing axis transformation algorithms assume that all the points on the boundary have equal importance, and thus a small change of the boundary topology is likely to produce a totally different medial axis skeleton. Therefore, an importance driven axis theory, as well as an effective generation algorithm, is proposed to extend the traditional axis theory. First, we densely sample the boundary of the given shape and set a weight for each sampling point according to the problem requirements. Then we compute the power diagram with regard to the weighted sampling points, and obtain an initial axis by cutting off the portion outside of the boundary. Finally, an extra pruning operation is required to remove those unimportant branches. Applications in medical image processing and shape clustering exhibit usefulness of the importance driven medial axis.