## Line Case Medial Axis Detection

We assume that the tangential direction is given by e1 at the voxel around the medial axis. The 2-D intensity function, c(u) (u = (u, v)T), on the cross-sectional plane of f (x ; af ) orthogonal to e1, should have its peak on the medial axis. The second-order approximation of c(u) is given by c(U) = f (Xa; a f ) + UJV c0 + 1 uJV 2c0u,

where ue2 + ve3 = x -x0, Vc0 = (V f ■ e2, V f ■ e3)T (V f is the gradient vector, that is, V f (Xo; af )), and

c(u) should have its peak on the medial axis of the line. The peak is located at the position satisfying d „ d ^ —c(u) = 0 and —c(u) = 0. du dv

By solving Eq. (10.34), we have the offset vector, p = (px, py, pz)J, of the peak position from xo given by p = se2 + te3,

where s = — vf 'e2 and t = — vf 'e. For the medial axis to exist at the voxel

### A2 A3

the peak of c(u) needs to be located in the territory of voxel X> Thus, the medial axis is detected only if |px\ < |py\ < 5, and |pz\ < By combining the voxel position X and offset vector p, the medial axis is localized at subvoxel resolution.

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