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.