Densely sampled triangular meshes have recently become a commonplace in computer graphics, medical imaging, and numerical analysis. Multiresolution methods offer a practical solution to handle the complexity of the processing algorithms for such meshes. The purpose of irregular subdivision is to provide smooth reconstruction on unstructured mesh hierarchies that can be used as an important component of many multiresolution processing tasks such as editing, compression, and noise removal. In this thesis we address the theoretical questions of regularity estimates for functions produced by irregular subdivision schemes as well as practical applications of such schemes.; We first introduce a simple model of irregular subdivision on the real line and obtain strict regularity estimates for a class of one-dimensional interpolating subdivision schemes. We then use the intuition from the one-dimensional case to introduce a general framework for the regularity analysis of interpolating subdivision in the multivariate setting, and use this framework to construct new subdivision schemes producing smooth functions on unstructured mesh hierarchies. In the last chapter we employ these irregular subdivision schemes to build a multiresolution representation for unstructured meshes and present a number of signal processing applications of our approach.
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