In order to examine the effects of various parameters on the accuracy of thickness determination, numerical simulation based on the theory described in the previous section was performed. The parameters used in the simulation were classified into the following three categories: t, L_, L0, and L + for defining sheet structures; Ax, Ay, and Az for determining MR imaging resolution; and Gaussian SD, a, used in computer postprocessing for thickness determination.
We assumed that the estimated sheet thickness T was obtained under the condition that the sheet normal orientation was known. The numerical simulation was performed in the frequency domain exactly in the same manner as described in section 10.5.2. Based on sheet structure parameters t, L_, L0, andL+, MR imaging parameters Ax, Ay, and Az, and postprocessing parameters a, F'(a>s) and F"(a>s) were obtained by 1-D computation in the frequency domain according to Eqs. (10.71) and (10.72), respectively. And then, f'(s) and f"(s) were obtained by inverse Fourier transform of F'(a>s) and F"(a>s), respectively. Using f (s) and f "(s), thickness T were estimated using Eq. (10.59). Finally, estimated thickness T was compared with the actual thickness t to reveal the limits on accuracy. It should be noted that only 1-D computation was necessary for 3D thickness determination in our numerical simulation.
In the simulation, the effect of anisotropic resolution of MR volume data was the focus. Let Axy(= Ax = Ay) be the pixel size within the slices. Resolution of MR volume data is typically anisotropic because they usually have lower resolution along the third direction (orthogonal to the slice plane) than within slices. Hence, it can be assumed that the resolution along the z-axis is lower than that in the xy-plane and that pixels in the xy-plane are square, i.e. Axy < Az, and a measure of voxel anisotropy can be defined as A. In the simulations, we
Axxy assumed that
without loss of generality, and thus,
Az Az voxel anisotropy =-= — = Az > 1. (10.74)
We performed the above described numerical simulation with different combinations of t, re, L_, L0, L+, Az, and a.
Hessian-Based Multiscale Enhancement and Segmentation Table 10.3: Parameter values used in numerical simulations
Voxel size e
Postprocessing Gaussian SD a
The unit of dimension for the following simulation results was Axy, i.e., Axy = 1 was assumed as described in the previous section. Thus, other parameters (t, T, Az, a) were normalized by Axy, and voxel anisotropy was represented as Az = (A = A). In the simulation, we used L0 = 200 and L_ = L+ = 100
for the bar profile. These parameter values were determined so that the bar profile was symmetric and the magnitude operator in Eq. (10.49) did not affect the results. Table 10.3 summarizes the parameter values used in the numerical simulations described below.
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