Let f (X) be an intensity function of a volume, where x = (x, y, z). Its Hessian matrix V2 f is given by fxx(Xx) fxy(Xx) fxz(Xx) V2 f (X) = fyx(X) fyy(x) fyz(x) , (10.1)

where partial second derivatives of f (X) are represented as fXX(x) = dx f (X), fyz(X) = dydz f (X), and so on. The Hessian matrix V2 f (Xo) at X0 describes the second-order variations around x0 [3-8,11-14,19]. The rotational invariant measures of the second-order local structure can be derived through the eigenvalue analysis of V2 f (Xo).

Let the eigenvalues of V2 f (X) be X^X), X2(X), X3(X) (A.i(X) > X2(X) > X3(X)), and their corresponding eigenvectors be e1(X), e2(X), e3(X), respectively. The eigenvector e1, corresponding to the largest eigenvalue À1, represents the direction along which the second derivative is maximum, and X1 gives the maximum second-derivative value. Similarly, X3 and e3 give the minimum directional second-derivative value and its direction, and X2 and e2 the minimum directional second-derivative value orthogonal to e3 and its direction, respectively. X2 and e2 also give the maximum directional second-derivative value orthogonal to e1 and its direction.

X1, X2, and X3 are invariant under orthonormal transformations. X1, X2, and X3 are combined and associated with the intuitive measures of similarity to

Table 10.1: Basic conditions for each local structure and representative anatomical structures. Each structure is assumed to be brighter than the surrounding region

Structure Eigenvalue condition Decomposed condition Example(s)

Sheet

Line

Blob

AB « 0 Ab « A2 — 0 A3 « Al — 0 A3 « 0 A3 — A2

Cortex Cartilage

Nodule local structures. Three types of second-order local structures—sheet, line, and blob—can be classified using these eigenvalues. The basic conditions of these local structures and examples of anatomical structures that they represent are summarized in Table 10.1, which shows the conditions for the case where structures are bright in contrast with surrounding regions. Conditions can be similarly specified for the case where the contrast is reversed. Based on these conditions, measures of similarity to these local structures can be derived. With respect to the case of a line, we have already proposed a line filter that takes an original volume f into a volume of a line measure [7] given by

|A31 ■ f (A2; A3) ■ «(Ai; A2) A3 < A2 < 0

0, otherwise,

where ^ is a weight function written as f (As; At) — J VAt

in which Yst controls the sharpness of selectivity for the conditions of each local structure (Fig. 10.1(a)), and m is written as

(l + Al)Yst At < As < 0 (l - « TAÏÏ )Yst ¥ > As > 0

otherwise,

// | ||

TO |
/ / | |

CD |
/ / | |

g |
/ / | |

I / |
Y = 0.5, a = 0.25- | |

1/ |
Y = 1.0, a = 0.25----- \ | |

0 |

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