## Single Scale Filter Responses to Mathematical Line Models

The line measure generalizes X^23 in Eq. (10.5) and Xg—mean23 in Eq. (10.6). An alternative measure is to use the arithmetic mean of — X2 and — X3, which is given by

To compare these three measures, let us consider a 3-D line image with elliptic (nonisotropic Gaussian) cross sections given by

^elliptic(x; ax, ay) = exp j — ^2- + IL^ J * (10.21)

When ax = ay, lelliptic(x; ax,ay) can be regarded as an ideal line, that is, £line(x; ax). Figure 10.8 shows the plots of the three measures and the eigenvalue variations of the Hessian matrix along the x-axis for the ideal line (ax = ay = 4 in Eq. (10.21), af = 4) and the sheet-like (ax = 20, ay = 3, af = 4) cases. The directions of the three eigenvectors at the points on the x-axis are identical to the x-axis, y-axis, and z-axis in the 3-D images modeled by Eq. (10.21). Let ex, ey, and ez be eigenvectors whose directions are identical to the x-axis, y-axis, and z-axis, respectively, and let Xx, Xy, and Xz be the respective corresponding eigenvalues. Figures 10.8(a) and 10.8(b) show the plots of Xg—mean^ Xa—mean^ and Xmini3 as well as the original profiles, while Figs. 10.8(c) and 10.8(d) show the plots of Xx, Xy, and Xz. In the ideal line case, both X2 and X3 are negative with large absolute values near the line centers. Since X3 tends to have a larger absolute value in the sheet-like case than in the line case (Fig. 10.8(d)), Xa—mean^ still gives a high response in the sheet-like case even if X2 has a small absolute value. Figure 10.9 shows the responses of the three measures at the center of

x coordinate

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x coordinate x coordinate 20 25 x coordinate Figure 10.8: Responses of eigenvalues and 3-D line filters to £elliptic (x; ax, ay) along the x-axis. The eigenvalues and filter responses are normalized so that |k2| and |k3| are one at x = y = 0 when ax = ay = af = 4. (a) kg-mean^ km and the original profile for the ideal line case (ax = ay = 4 in £elliptic (x; ax, ay), af = 4). (b) kg-mean23, ka-mean25, kmin25, and the original profile for the sheet-like case (ax = 20 and ay = 3 in lelliptic(x; ax, ay), af = 4). (c) Eigenvalues for the ideal line case. (d) Eigenvalues for the sheet-like case. 20 25 the line (x = y = 0) when ax and ay in Eq. (10.21) are varied. While kminand Xg-mean23 decrease with deviations from the conditions ax « af and ay « af, K-mean23 gives high responses if ax « ^ or ay « . Thus, Xa-mean23 gives relatively high responses to sheet-like structures while Xmin23 (y23 = 1 in Eq. (10.3)) and Xg-mean23 (y23 = 0.5 in Eq. (10.3)) are able to discriminate line structures from sheet-like structures. Figure 10.9: Responses of 3-D line filters to lelliptic (x; ax, ay) at x = y = 0 with variable ax and ay at the center of the line. The responses are normalized so that |A.2| and |A.3| are one at x = y = 0 when ax = ay = af = 2. (a) Xa—mean23. |

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