## Effects of Gaussian Standard Deviation in Postprocessing

Figure 10.19 shows the effects of the standardg deviation (SD), a, in Gaussian blurring. In Fig. 10.19(a), the relations between true thickness t and measured thickness T are shown for three a values (1, 1) when Az is equal to 1, i.e. in the case of isotropic voxel. The relation is regarded as ideal when T = t , which is the diagonal in the plots of Fig. 10.19(a). For each a value, the relations were plotted using two values of sheet normal orientation 0 (0°, 45°), while $ was fixed to 0°. Strictly speaking, voxel shape is not perfectly isotropic even when Az is equal to 1 because the shape is not spherical. Thus, slight dependence on 0 was observed.

In order to observe the deviation from T = t more clearly, we defined the error as E = T — t. Figure 10.19(b) shows the plots of error E instead of T. With a = 2, considerable ringing was observed for error E. With a = 1, error magnitude | E| was significantly large for small t (around t = 2). With a =

12 3 4 True thickness t

12 3 4 True thickness t

V 1 |
1 1 1 |

9=0° _ | |

- |
v 9=45° _ |

- |
c = 1 - |

1 |
i i i |

12 3 4 True thickness t

12 3 4 True thickness t

Figure 10.19: Effects of Gaussian SD, a in postprocessing for thickness determination with isotropic voxel. The unit is Axy. 0 = 0°. (a) Relations between true thickness t and measured thickness T. (b) Relations between true thickness t and error T - t . (© 2004 IEEE)

however, ringing became small and error magnitude \E\ was sufficiently small around t = 2. a = ^ gave a good compromise optimizing the trade-off between reducing the ringing and improving the accuracy for small t . Actually, error magnitude \E\ is guaranteed to satisfy \E\ < 0.1 for t > 2.0 with a = while |E\ < 0.1 for t > 3.2 with a = 1 and, \E\ < 0.1 for t > 2.9 with a = 1. Based on this result, we used a = ^ in the following experiments if not specified.

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