Embedding Moving Frames

The moving frame is defined by the voxel position Xo, the offset vectors p, and the eigenvectors e1, e2, e3 at each detected point of a medial axis or surface. In order to perform the subsequent processes based on moving frames, each voxel within the candidate regions obtained in Step 1 needs to be related to the moving frame. First, we find the correspondences between each voxel and one of the detected points of a medial axis or surface. Once these correspondences are found, each voxel is directly related to its corresponding moving frame. To find the correspondences, we use the Voronoi tessellation of the detected points.

The territory of the detected point in the Voronoi tessellation can be regarded as the set of voxels to which each discrete moving frame is applied. This process is identical in both the line and sheet cases.

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