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defined as the points with the maximum and minimum values of f (s) among those satisfying the condition given by f (s) = 0. Let f (s) have its maximum and minimum values at s = p and s = q, respectively. The measured thickness,

T is defined as the distance between the two detected boundary points, which is given by

The procedures for thickness determination from a volume dataset in the previous section.

10.5.2Frequency Domain Analysis of MR Imaging and Width Quantification

In order to elucidate the effects of MR imaging and postprocessing parameters, observations in the frequency domain are helpful. All the processes to obtain f (x" ; σ, r"θ ,φ ) and f (x" ; σ, r"θ ,φ ) from the original sheet structure s(x" ; τ, r"θ ,φ ) are modeled as linear filtering processes excepting the magnitude operator applied in Eq. (10.49).

10.5.2.1 Modeling a Sheet Structure

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

where F represents the Fourier transform, δ(ω) denotes the unit impulse, and

ω" = (ωx, ωy, ωz). Note that F{Bar(x ; τ )} = τ · Sinc(ωx; τ ) when L+ = L− = 0 and L0 = 1 in Bar(x ; τ ). The Fourier transform of 3-D sheet structure whose normal is r"θ,φ , s(x" ; τ, r"θ ,φ ), is given by

which enables the ωx-axis correspond to r"θ ,φ (Fig. 10.18(a)).

(a)

(b)

(c)

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 xy = x = y. 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. ( c 2004 IEEE)

574 Sato

In the 3-D space of the frequency domain, S(ω"; τ, r"θ ,φ ) has energy only in the

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

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

10.5.2.2 Modeling MR Image Acquisition

where Rect(x" ; ax, ay, az) = Rect(x ; ax) · Rect(y ; ay) · Rect(z ; az) (Fig. 10.18(b)), and

1 − 1 a ≤ x ≤ 1 a

= 2 2

Rect(x ; a) (10.65)

0otherwise.

By substituting Eq. (10.62) for ω" in Eq. (10.64) to obtain 1-D frequency component affecting S(ωs), the following is derived

Thus, the Fourier transform of MR image of the sheet structure, F(ωs) is given by

where F−1 represents the inverse Fourier transform. If F−1{S(ωs)M(ωs)} is a nonnegative function, F(ωs) is given by F(ωs) = S(ωs)M(ωs) and all the processes can be described as linear filtering processes. Deformation of the original signal due to truncation is clearly understandable in the frequency domain (Fig. 10.18(c)).

10.5.2.3 Gaussian Derivatives of MR Imaged Sheet Structure

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

where ω" = Rθ ,φ ω", in which Rθ ,φ denotes a 3 × 3 matrix representing rotation which enables the ωx-axis correspond to r"θ ,φ . One-dimensional frequency component of G (ω"; σ, r"θ ,φ ) affecting S(ωs) is given by

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

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

respectively.

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.

10.5.3 Accuracy Evaluation by Numerical Simulation

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: τ , r"θ ,φ , L−, L0, and L+ for defining sheet structures; x, y, and z for determining MR imaging resolution; and Gaussian SD, σ , 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 r"θ ,φ 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 τ , r"θ ,φ , L−, L0, and L+, MR imaging parameters x, y, and z, and postprocessing parameters

σ , F (ωs) and F (ωs) 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 (ωs) and F (ωs), 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 τ 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 xy(= x = y) 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. xy ≤ z, and

a measure of voxel anisotropy can be defined as z . In the simulations, we

xy

assumed that

We performed the above described numerical simulation with different combinations of τ , r"θ,φ , L−, L0, L+, z, and σ .

The unit of dimension for the following simulation results was xy, i.e., xy =

1 was assumed as described in the previous section. Thus, other parameters (τ , T , z, σ ) were normalized by xy, and voxel anisotropy was represented

as z = ( z = z ). In the simulation, we used L0 = 200 and L− = L+ = 100

xy 1

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.

10.5.3.1Effects of Gaussian Standard Deviation in Postprocessing

Figure 10.19 shows the effects of the standardg deviation (SD), σ , in Gaussian

blurring. In Fig. 10.19(a), the relations between true thickness τ and measured

√

thickness T are shown for three σ values ( 12 , 22 , 1) when z is equal to 1, i.e. in the case of isotropic voxel. The relation is regarded as ideal when T = τ , which is the diagonal in the plots of Fig. 10.19(a). For each σ value, the relations were plotted using two values of sheet normal orientation θ (0◦, 45◦), while φ was fixed to 0◦. Strictly speaking, voxel shape is not perfectly isotropic even when

z is equal to 1 because the shape is not spherical. Thus, slight dependence on

θ was observed.

In order to observe the deviation from T = τ more clearly, we defined the error as E = T − τ . Figure 10.19(b) shows the plots of error E instead of T .

With σ = 1 , considerable ringing was observed for error E. With σ = 1, error

2 √

magnitude |E| was significantly large for small τ (around τ = 2). With σ = 22 ,

however, ringing became small and error magnitude |E| was sufficiently small

√

around τ = 2. σ = 22 gave a good compromise optimizing the trade-off between

reducing the ringing and improving the accuracy for small τ . Actually, error

√

magnitude |E| is guaranteed to satisfy |E| < 0.1 for τ > 2.0 with σ = 22 , while

|E| < 0.1 for τ > 3.2 with σ = 1 and, |E| < 0.1 for τ > 2.9 with σ = 1. Based on

√ 2

this result, we used σ = 22 in the following experiments if not specified.

10.5.3.2 Effects of Voxel Anisotropy in MR Imaging

Figure 10.20(a) shows the effects of sheet normal orientation θ and voxel anisotropy z on measured thickness T . The relations between measured thickness T and sheet normal orientation θ for six values of true thickness τ (1, 2, 3, 4, 5, 6) were plotted when three different values of voxel anisotropy z (1, 2, 4) were used. The relations were regarded as ideal when T = τ for any θ , which is the horizontal in the plots of Fig. 10.20(a). When z = 1, the relations were highly close to the ideal for τ > 2. When z = 2 and z = 4, significant deviations from the ideal were observed for θ > 15◦ and θ > 30◦, respectively.

Figure 10.20(b) shows the plots of the maximum θ at which error magnitude

|E| is guaranteed to satisfy |E| < 0.1, |E| < 0.2, and |E| < 0.4 for τ = 2 with varied voxel anisotropy z. These plots clarify the range of θ where the deviation from the ideal is sufficiently small. There was no significant difference between the plots for τ = 2 and different values of τ (for τ > 2).

10.5.3.3Using Anisotropic Gaussian Blurring Based on Voxel Anisotropy

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

because we assumed xy = 1. Figure 10.20(c) shows plots of measured thick-

√

ness obtained using anisotropic Gaussian blurring when z = 2 and σxy = 22 . The plots using anisotropic Gaussian blurring were closer to the ideal for τ ≥ 4 and any θ , while those using isotropic one were closer for τ ≥ 2 and θ < 30◦.

Figure 10.20: Effects of voxel anisotropy z in MR imaging and anisotropic Gaussian blurring on measured thickness T . The unit is xy. σ = and φ = 0◦.

(a) Relations between measured thickness T and sheet normal orientation θ with different τ values. (b) Plots of maximum θ at which error magnitude |E| is guaranteed to satisfy |E| < 0.1, |E| < 0.2, and |E| < 0.4 for τ = 2 while voxel anisotropy z is varied (where E = T − τ ). (c) Relations between true thickness

τ and measured thickness T with the use of anisotropic Gaussian blurring based

10.5.4Validating the Numerical Simulation by in Vitro Experiments

To validate the numerical simulation, the postprocessing method for thickness determination was used to measure real MR images of two different objects, an acrylic plate phantom.

A phantom of sheet-like objects with known thickness was used. It consisted of four acrylic plates of 80 × 80 (mm2) with thickness τ = 1.0, 1.5, 2.0, and 3.0 (mm), placed parallel to each other with an interval of 30 mm (Fig. 10.21(a)).

(a)

θ = 0° and φ = 0°

θ = 45° and φ = 0°

(b)

Figure 10.21: Acrylic plate phantom and its MR images. (a) Physical appearance. (b) MR images. The horizontal and vertical axes of the images correspond to the x-axis and z-axis, respectively. The voxel size was xy = 0.625 mm and

z = 1.5 mm. As can be easily observed by naked eye, the acrylic plate with

τ = 1 mm appears to be imaged slightly thicker in θ = 45◦ and φ = 0◦ than θ = 0◦ and φ = 0◦. ( c 2004 IEEE)