In the study of level set methods, several significant problems were neglected all along, such as the existence, uniqueness and singularities of level set methods. In this article we give the proof that in a neighborhood of the initial zero level set, for the level set equations with the restriction of distance function, there exists a unique solution, which must be the signed distance function with respect to the evolving surface. We also present the analysis of singular points' effect on level set evolution and give an adaptive narrow banding algorithm. The detailed numerical analysis and a simplified definition for singular points are presented. We give an adaptive narrow banding algorithm, which avoids the singular points and is proved to be robust and efficient in segmentation of CT data and synthesized images.