Figure 1: Examples of meshes created with our algorithm: The surfaces have very complex structure, high genus, high curvature, and multiple disconnected components. The meshes are the coarsest level generated by our algorithm and have been produced without user intervention. Left: smooth shaded view. Right: hidden line view. Inset: zoomed view highlighting the level of adaptivity generated by the coarse mesh extracton algorithm. ×ØÖ ØWe present a new algorithm for extracting adaptive multiresolution triangle meshes from volume datasets. The algorithm guarantees that the topological genus of the generated mesh is the same as the genus of the surface embedded in the volume dataset at all levels of detail. In addition to this "hard constraint" on the genus of the mesh, the user can choose to specify some number of soft geometric constraints, such as triangle aspect ratio, minimum or maximum total number of vertices, minimum and/or maximum triangle edge lengths, maximum magnitude of various error metrics per triangle or vertex, including maximum curvature (area) error, maximum distance to the surface, and others. The mesh extraction process is fully automatic and does not require manual adjusting of parameters to produce the desired results as long as the user does not specify incompatible constraints. The algorithm robustly handles special topological cases, such as trimmed surfaces (intersections of the surface with the volume boundary), and manifolds with multiple disconnected components (several closed surfaces embedded in the same volume dataset). The meshes may self-intersect at coarse resolutions. However, the self-intersections are corrected automatically as the resolution of the meshes increase. We show several