In this chapter, we restrict the scope of our investigation to the sheet model described in section 10.5.1 (Fig. 10.16), that is, a sheet structure with constant thickness t and orientation re^. We define the thickness measured from the MR imaged sheet structure as the distance between both sides of image edges along the sheet normal vector. As long as the sheet model shown in Fig. 10.16 is considered, other definitions of measured thickness, for example, the shortest distance between both sides of the image edges, generally give the same thickness value. We define the image edges as the zero-crossings of the second directional derivatives along the sheet normal vector, which is equivalent to the Canny edge detector [42]. Gaussian blurring is typically combined with the second directional derivatives to adjust scale as well as reduce noise.

The partial second derivative combined with Gaussian blurring for the MR image f (x), for example, is given by fxx(x; a) = gxx(x; a) * f (x), (10.50)

where d2

in which Gauss(x; a) is the isotropic 3-D Gaussian function with SD a. The second directional derivative along re$$ is represented as f"(x; a, n$) = gxx(%; a) * f (x), (10.52)

where H = Re$$x, in which Re$$ denotes a 3 x 3 matrix representing rotation which enables the normal orientation of the sheet s0(x; t), i.e. the x-axis, correspond to re$$ (Fig. 10.16(b)). Similarly, the first directional derivative along re$$ is represented as f'(x; a, re, $) = gx(x'; a) * f (x) , (10.53)

Practically, the first and second directional derivatives can be calculated in a computationally efficient manner using the Hessian matrix V2 f (x; a ) and gradient vector V f (x; a ), respectively. The first and second directional derivatives along the normal direction re,$ of the sheet structure are given by f'(x; a , re,= rl$Vf (x; a) , (10.54)

and f"(x; a, re,$) = rJ^V2f (x; a)n,$ , (10.55)

respectively.

Thickness of sheet structures can be determined by analyzing 1-D profiles of f "(x; a, re, $) and f '(x; a, re, $) along straight line given by x = s -re,$, (10.56)

where s is a parameter representing the position on the straight line. By substituting Eq. (10.56) for x in f"(x; a, re$$) and f'(x; a, re$), f "(s) = f "(s -re,$; a, ?e,$), (10.57)

are derived respectively. Figure 10.17(a) shows a schematic diagram for the 1-D profile processing. Both sides of the boundaries for sheet structures can be

Measured thickness

imaging

Original MR data

imaging

Sine interpolation m

Ilk mm

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