Figure 10.1: Weight functions in measures of similarity to local structures.

(a) ^(Xs; Xt), representing the condition Xt — Xs, where Xt < Xs. ^(Xs; Xt) = 1

when Xt = Xs. ^ (Xs; Xt) = 0 when Xs = 0. (b) w(Xs; Xt), representing the condition

Xt ^ Xs — 0. w(Xs; Xt) = 1 when Xs = 0. w(Xs; Xt) = 0 when Xt = Xs ^ 0 or Xs(>

t in which 0 < a < 1 (Fig. 10.1(b)). a is introduced in order to give m(Xs; Xt) an asymmetrical characteristic in the negative and positive regions of Xs.

Figure 10.2(a) shows the roles of weight functions in representing the basic conditions of the line case. In Eq. (10.2), |X31 represents the condition X3 ^ 0, ^ (X2; X3) represents the condition X3 ~ X2 and decreases with deviation from the condition X3 ~ X2, and a>(X\; X2) represents the condition X2 ^ Xi ~ 0 and decreases with deviation from the condition X1 ~ 0 which is normalized by X2. By multiplying |X3|, ^(X2; X3), and «(X1; X2), we represent the condition for a line shown in Table 10.1. For the line case, the asymmetric characteristic of m is based on the following observations:

• When X1 is negative, the local structure should be regarded as having a blob-like shape when |X11 becomes large (lower right in Fig. 10.2(a)).

• When X1 is positive, the local structure shouldbe regarded as being stenotic in shape (i.e., part of a vessel is narrowed), or it may be indicative of signal loss arising from the partial volume effect (lower left in Fig. 10.2(a)).

Therefore, when X1 is positive, we make the decrease with the deviation from the X1 ~ 0 condition less sharp in order to still give a high response to a stenosislike shape. We typically used a = 0.25 and yst = 0.5 (or 1) in our experiments. Extensive analysis of the line measure, including the effects of parameters yst and a, can be found in [7].

ei e2

O - 0 in negative domain ei

O - 0 in negative domain ei

positive domain-

positive domain-

O - 0 in negative domain

Figure 10.2: Schematic diagrams of measures of similarity to local structures. The roles of weight functions in representing the basic conditions of a local structure are shown. (a) Line measure. The structure becomes sheet-like and the weight function ^ approaches zero with deviation from the condition X3 ~ X2, blob-like and the weight function m approaches zero with transition from the condition X2 ^ Ai ~ 0to X2 ~ Ai ^ 0, and stenosis-like and the weight function m approaches zero with transition from the condition X2 ^ A1 ~ 0 to X1 ^ 0. (b) Blob measure. The structure becomes sheet-like with deviation from the condition X3 ~ X2, and line-like with deviation from the condition X2 ~ A1. (c) Sheet measure. The structure becomes blob-like, groove-like, line-like, or pitlike with transition from X3 ^ X1 ~ 0 to X3 ~ X1 ^ 0, X3 ^ X1 ~ 0 to X1 ^ 0, X3 < X2 ~ 0 to X3 ~ X2 < 0, or X3 < X2 ~ 0 to X2 > 0, respectively. (© 2004 IEEE)

The specific shape given in Eq. (10.3) representing the condition Xt ~ Xs (where t = 3 and s = 2 for the line case) is based on the need to generalize the two line measures Xmin,23 and Xg-mean23 [3]:

min(-X2, —X3) = —X2 X2 < 0 and X3 < 0 "Xmin 23 = { (l0-5)

0 otherwise.

0 otherwise, for the cases X2 < 0 and X3 < 0, Xmin^ can be rewritten as

and Xg-mean23 as

"Xmin23 = —X2 = |X2| = |X3| j — ) , (10.7)

These measures take into account the conditions X3 ^ 0 and X3 ~ X2. | X31 ■ f(X2; X3) in Eq. (10.2) is equal to ^/X3X2 and —X2 when y23 = 0.5 and y23 = 1, respectively. In this formulation [7], the same type of function shape as that in Eq. (10.3) is used for Eq. (10.4) to add the condition X2 ^ X1 ~ 0.

We can extend the line measure to the blob and sheet cases. In the blob case, the condition X3 ~ X2 ~ Xi ^ 0 can be decomposed into X3 ^ 0 and X3 ~ X2 and X2 ~ X1. By representing the condition Xt ~ Xs using f(Xs; Xt), we can derive a blob filter given by c f,, [|X3|-^ CX2; X3) ■ f CX1; X2) X3 < X2 < X1 < 0 Sblob i J ) = < , C10.9)

In the sheet case, the condition X3 ^ X2 ~ X1 ~ 0 can be decomposed into X3 ^ 0 and X3 ^ X2 ~ 0 and X3 ^ X1 ~ 0. By representing the condition Xt ^ Xs ~ 0 using w(Xs; Xt), we can derive a sheet filter given by

Figures 10.2(b) and 10.2(c) show the relationships between the eigenvalue conditions and weight functions in the blob and sheet measures.

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