The research in the thesis investigates the use of minimal path techniques to track and detect cracks, modeled as curves, in critical infrastructure like pavements and bridges. We developed a novel minimal path algorithm to detect curves with complex topology that may have both closed cycles and open sections using an arbitrary point on the curve as the sole input. Specically, we applied the novel algorithm to three problems: semi-automatic crack detection, detection of continuous cracks for crack sealing applications and detection of crack growth in structures like bridges. The current state of the art minimal path techniques only work with prior knowledge of either both terminal points or one terminal point plus total length of the curve. For curves with multiple branches, all terminal points need to be known. Therefore, we developed a new algorithm that detects curves and relaxes the necessary user input to one arbitrary point on the curve. The document presents the systematic development of this algorithm in three stages. First, an algorithm that can detect open curves with branches was formulated. Then this algorithm was modied to detect curves that also have closed cycles. Finally, a robust curve detection algorithm was devised that can increase the accuracy of curve detection. The algorithm was applied to crack images and the results of crack detection were validated against the ground truth. In addition, the algorithm was also used to detect features like catheter tube and optical nerves in medical images. The results demonstrate that the algorithm is able to accurately detect objects that can be modeled as open curves.