We have assumed that Gaussian blurring combined with derivative computation is isotropic as shown in Eq. (10.51). Another choice is to use anisotropic Gaussian blurring corresponding to voxel anisotropy, which is given by d 2
9xx(x; axy, az) = Gauss(x, y; aXy)Gauss(z; az), (10.75)
where az and axy are determined so as to satisfy — = A, and thus az = Azaxy because we assumed Axy = 1. Figure 10.20(c) shows plots of measured thickness obtained using anisotropic Gaussian blurring when Az = 2 and axy = The plots using anisotropic Gaussian blurring were closer to the ideal for t > 4 and any 0, while those using isotropic one were closer for t > 2 and 0 < 30°.
0 15 30 45 60 75 90 Sheet normal orientation 0 (deg)
T = 6 
1 1 _' 
5 

T = 4__  
CO  
T = 2  
T = 1 
Az = 2  
0 15 30 45 60 75 90 Sheet normal orientation 0 (deg)
0 15 30 45 60 75 90 Sheet normal orientation 0 (deg)
0 15 30 45 60 75 90 Sheet normal orientation 0 (deg)
T = 
6 
1 1 1 
T = 
5  
T = 
4  
T = 
3  
T = 
2  
1 
Az = 2  
0 15 30 45 60 75 90 Sheet normal orientation 0 (deg)
0 15 30 45 60 75 90 Sheet normal orientation 0 (deg)
0 15 30 45 60 75 90 Sheet normal orientation 0 (deg)
Figure 10.20: Effects of voxel anisotropy Az in MR imaging and anisotropic
Gaussian blurring on measured thickness T. The unit is Axy. a = 22 and $ = 0°. (a) Relations between measured thickness T and sheet normal orientation 0 with different t values. (b) Plots of maximum 0 at which error magnitude  E is guaranteed to satisfy E < 0.1, E < 0.2, and E < 0.4 for t = 2 while voxel anisotropy Az is varied (where E = T — t). (c) Relations between true thickness t and measured thickness T with the use of anisotropic Gaussian blurring based i i on voxel anisotropy. axy = and az = Az. (© 2004 IEEE)
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