Modeling a Sheet Structure

The Fourier transform of 3-D sheet structure orthogonal to the x-axis, s0(x ; t), is given by

S0(&>; t) = F{Bar(x; t)} ■ 8(a>y) ■ 8(uz), (10.60)

where F represents the Fourier transform, 8(w) denotes the unit impulse, and x = (wx, rny, mz). Note that F{Bar(x; t)} = t ■ Sinc(«x; t) when L + = L_ = 0 and L0 = 1 in Bar(x ; t). The Fourier transform of 3-D sheet structure whose normal is s(x ; t, rg^), is given by

where a>' = Re^cx, in which denotes a 3 x 3 matrix representing rotation which enables the xx-axis correspond to ^ (Fig. 10.18(a)).

Figure 10.18: Frequency domain analysis of sheet structure modeling, MR imaging, and thickness determination. (a) Modeling a sheet structure. In the frequency domain, a sheet structure is basically modeled as the sinc function whose width is inversely proportional to the thickness in the spatial domain. (b) Modeling MR imaging. It is assumed here that Axy = Ax = Ay. The voxel size determines the frequency bandwidth of each axis, which is also inversely proportional to the size in the spatial domain. (c) Modeling MR image acquisition of a sheet structure. In the frequency domain, imaged sheet structure is essentially the band-limited sinc function. (© 2004 IEEE)

In the 3-D space of the frequency domain, S(a>; t, re^) has energy only in the 1-D subspace represented as a straight line given by

where ms is a parameter representing the position on the straight line. By substituting Eq. (10.62) for T in Eq. (10.61), the following is derived

S(Ms) = S(Ms • ?e4; t, ?e^) = f{Bar(x; t)}, (10.63)

where S(a>s) represents energy distribution along Eq. (10.62). Analysis of the degradation of 1-D distribution, S(ms), is sufficient to examine the effects of MR imaging and postprocessing parameters in the subsequent processes. It should be noted that S(a>s) is the 1D sinc function when L _ = L+.

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