Gaussian Derivatives of MR Imaged Sheet Structure

The Fourier transform of the second derivative of Gaussian of x is given by

Gxx(a>; a) = (V2^a)3(2nj«x)2Gauss (r; , (10.68)

and that of the second directional derivative along is represented as

where a>' = r, in which denotes a 3 x 3 matrix representing rotation which enables the &>x-axis correspond to One-dimensional frequency component of G"(a>; a, re^) affecting S(cos) is given by

Similarly, 1-D component of the first directional derivative of Gaussian, G'(cos), is obtained.

Finally the Fourier transforms of f"(s) and f'(s) are derived and given by


The 1-D profiles along the sheet normal direction of the Gaussian derivatives of MR imaged sheet structures (Eqs. (10.57) and (10.58)) are obtained by inverse Fourier transform of Eqs. (10.71) and (10.72), and then thickness is determined according to the procedure shown in Fig. 10.17(a). While simulating the MR imaging and Gaussian derivative computation described in section 10.5.1 essentially requires 3-D convolution in the spatial domain, only 1-D computation is necessary in the frequency domain, which drastically reduces computational cost. In the following sections, we examine the effects of various parameters, which are involved in the sheet model, MR imaging resolution, and thickness determination processes, on measurement accuracy. Efficient computational methods of simulating MR imaging and postprocessing thickness determination processes are essential, and thus simulating the processes by 1-D signal processing in the frequency domain is regarded as the key to comprehensive analysis.

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