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Existing state-of-the-art minimal path techniques work well to extract simple open curves in images when both endpoints of the curve are given as user input or when one input is given and the total length of the curve is known in advance. Curves which branch require even further prior input from the user, namely, each branch endpoint. In this work, we present a novel minimal path-based algorithm which works on much more general curve topologies with far fewer demands on the user for initial input compared to prior minimal path-based algorithms. The two key novelties and benefits of this new approach are that 1) it may be used to detect both open and closed curves, including more complex topologies containing both multiple branch points and multiple closed cycles without requiring a priori knowledge about which of these types is to be extracted, and 2) it requires only a single input point which, in contrast to existing methods, is no longer constrained to be an endpoint of the desired curve but may in fact be ANY point along the desired curve (even an internal point). We perform quantitative evaluation of the algorithm on 48 images (44 pavement crack images, 1 catheter tube image, and 3 retinal images) against human supplied ground truth. The results demonstrate that the algorithm is indeed able to extract curve-like objects accurately from images with far less prior knowledge and less user interaction compared to existing state-of-the-art minimal path-based image processing algorithms. In the future, the algorithm can be applied to other 2D curve-like objects and it can be extended to detect 3D curves.