## Medial Axis and Surface Detection

Let f (x) be an intensity function of a volume, where x = (x, y, z), and f (X; a) be its Gaussian smoothed volume with standard deviation a. The second-order approximation of f (xx; a) around x0 is given by fn(x; a) = f + (X — xc)TV f + 1(x — x0)TV2 Mx — X0), (10.29)

where f0 = f (x0), Vf0 = Vf (xx0), and V2f0 = V2 f (x0). Thus, the second-order structures of local intensity variations around each point of a volume can be described by the original intensity, the gradient vector, and the Hessian matrix. The gradient vector of Gaussian smoothed volume f (x; a ) is defined as

Vf (x; a) = (fx(x; a), fy(x; a), f(x; a))T, (10.30)

where partial derivatives of f (x; a) are represented as fx(oc; a) = d^f (x; a), fy(x; a ) = dyf (x; a ), and fz(x; a) = -fzf (^; a ).

The Hessian matrix of Gaussian smoothed volume f (x; a ) is given by

fxxQ%; a) fxyQ%; a) fxz(X; a) fyx (x ; a ) fyy(x ; a ) fyz(x ; a ) , fzx(x ; a) fzy(x ; a) fzz(x ; a) .

where partial second derivatives of f (x; a) are represented as fxx(x; a) =

Let the eigenvalues of V2 f (x ; a) be X1, X2, X3 (X1 > X2 > X3) and their corresponding eigenvectors be e1, e2, e3 (|e1| = |(32| = \e3\ = 1), respectively. For the ideal line, e1 is expected to give its tangential direction and both |X2| and |X3|, directional second derivatives orthogonal to e1, should be large on its medial axis, while e3 is expected to give the orthogonal direction of a sheet and only |X31 should be large on its medial surface (Fig. 10.11). Here, structures of interest are assumed to be brighter than surrounding regions.

The initial regions obtained in Step 1 are searched for medial axes and surfaces, which are detected based on the second-order approximation of f (X ; a ) The medial axis and surface extraction is based on a formal analysis of the second-order 3D local intensity structure. Here, af is the filter scale used in e, e.

Figure 10.11: Line and sheet models with the eigenvectors of the Hessian matrix. (a) Line. (b) Sheet.

medial axis/surface detection, and we assume that the width range of structures of interest is around the width at which the filter with af gives the peak response (see  and  for detailed discussions).

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