## 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 [7] and [11] for detailed discussions).

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