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This paper presents a comprehensive extension of the Scale Invariant Feature Transform (SIFT), originally introduced in 2D, to volumetric images. While tackling the significant computational efforts required by such multiscale processing of large data volumes, our implementation addresses two important mathematical issues related to the 2D-to-3D extension. It includes efficient steps to filter out extracted point candidates that have low contrast or are poorly localized along edges or ridges. In addition, it achieves, for the first time, full 3D orientation invariance of the descriptors, which is essential for 3D feature matching. An application of this technique is demonstrated to the feature-based automated registration and segmentation of clinical datasets in the context of radiation therapy.