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Topological volume skeletons represent level-set graphs of 3D scalar fields, and have recently become crucial to visualizing the global isosurface transitions in the volume. However, it is still a time-consuming task to extract them, especially when input volumes are large-scale data and/or prone to small-amplitude noise. The paper presents an efficient method for accelerating the computation of such skeletons using adaptive tetrahedralization. The tetrahedralization is a top-down approach to linear interpolation of the scalar fields in that it selects tetrahedra to be subdivided adaptively using several criteria. As the criteria, the method employs a topological criterion as well as a geometric one in order to pursue all the topological isosurface transitions that may contribute to the global skeleton of the volume. The tetrahedralization also allows us to avoid unnecessary tracking of minor degenerate features that hide the global skeleton. Experimental results are included to demonstrate that the present method smoothes out the original scalar fields effectively without missing any significant topological features.