Image processing techniques can be applied to solve problems in many fields, including biomedicine, robotics, security, computer vision, etc. In order for the solutions to achieve good performance in terms of effectiveness and efficiency, the problems need to be modeled properly and solved in an efficient manner. In this dissertation, we present new algorithms for several problems in biomedical image processing and computer vision. The problems we study are either not studied by other people before, or the existing solutions are not satisfactory enough, due to improper models chosen, or non-optimal solutions adopted. For the biomedical image processing problems, unlike traditional image processing approaches that are based on signal processing techniques, we design algorithms based on the geometric features of target objects (e.g., blood clots, bones, vessels, etc.), and apply graph algorithms, as well as other algorithms in computational geometry to solve the problems. We also study the image completion problem, which is a common problem in computer vision. We design algorithms based on existing algorithm frameworks, but adopt new models and seek optimal solutions to solve many key sub-problems, while most existing algorithms rely heavily on heuristics, and cannot handle many complicated cases sufficiently well. In our new approaches, we extensively apply graph algorithms, optimization techniques and other algorithms in computational geometry, and achieve better performance than traditional methods. The algorithms we apply, extend or design to solve the problems include clustering, plane sweeping, graph search, maximum-weight independent set on circle graphs, etc. We also perform quantitative analyses on the image data based on the image processing results. The analysis results we produce help physicians and biologists explore unknown mechanisms of the human body, and develop therapeutic strategies to better treat patients.