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.

Extracting surfaces, representing boundaries of objects of interest, from volumetric images, has important applications in various scientific domains, from medicine to geology. In this thesis, I introduce novel mathematical, computational, and algorithmic machinery for extraction of sheet-like surfaces (with boundary), whose boundary is unknown a-priori, a particularly important case in applications that has no convenient methods. This case of a surface with boundaries has applications in extracting faults (among other geological structures) from seismic images in geological applications. Another application domain is in the extraction of structures in the lung from computed tomography (CT) images. Although many methods have been developed in computer vision for extraction of surfaces, including level sets, convex optimization approaches, and graph cut methods, none of these methods appear to be applicable to the case of surfaces with boundary. The novel methods for surface extraction, derived in this thesis, are built on the theory of Minimal Paths, which has been used primarily to extract curves in noisy or corrupted images and have had wide applicability in 2D computer vision. This thesis extends such methods to surfaces, and it is based on novel observations that surfaces can be determined by extracting topological structures from the solution of the eikonal partial differential equation (PDE), which is the basis of Minimal Path theory. Although topological structures are known to be difficult to extract from images, which are both noisy and discrete, this thesis builds robust methods based on Morse theory and computational topology to address such issues. The algorithms have run-time complexity O(NlogN), less complex than existing approaches. The thesis details the algorithms, theory, and shows an extensive experimental evaluation on seismic images and medical images. Experiments show out-performance in accuracy, computational speed, and user convenience compared with related state-of-the-art methods. Lastly, the thesis shows the methodology developed for the particular case of surfaces with boundary extends to surfaces without boundary and also surfaces with different topologies, such as cylindrical surfaces, both important cases for many applications in medical image analysis.