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Multiresolution volume data structures allow the extraction of adaptively triangulated isosurfaces in large volume data sets at interactive frame rates, providing the user the ability to visualize the volume data in real-time. Important topological properties such as the genus, or structure and number of connected components of an isosurface are often neglected in existing multiresolution isosurface extraction and rendering algorithms. In this project we develop an efficient technique which provides control over topological changes, in particular preservation of topolgical properties, in an interactive multiresolution isosurface extraction and rendering framework.