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A method for determining the modulation transfer function from thick microwire profiles measured with x-ray microcomputed tomography
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10.1118/1.4729711
/content/aapm/journal/medphys/39/7/10.1118/1.4729711
http://aip.metastore.ingenta.com/content/aapm/journal/medphys/39/7/10.1118/1.4729711

Figures

Image of FIG. 1.
FIG. 1.

Characteristics of correction factor α(R) in Eq. (1) . The parameter d is the wire diameter.

Image of FIG. 2.
FIG. 2.

Profiles of the tungsten wire with diameter of 0.1 mm. (a) Reconstruction kernel: FC13. (b) Reconstruction kernel: FC51. The center of the horizontal axis was set to the peak position P, which is designated the center of the wire image. The ROI (N × N) in which P is at the center was selected. D 1 and D 2, respectively, denote the edge positions of the two different regions of N st × N st and N end × N end as presented in Appendix A . N st and N end were used to specify the neighborhood of the ROI and N st was set to N. N, N st, and, N end were, respectively, set to 50, 50, and 60 pixels in (a). N, N st, and N end were, respectively, set to 40, 40, and 50 pixels in (b).

Image of FIG. 3.
FIG. 3.

PSFs and MTFs obtained from the wire images shown in Fig. 2 . (a) Reconstruction kernel: FC13. (b) Reconstruction kernel: FC51. Upper row: PSF. Lower row: MTF.

Image of FIG. 4.
FIG. 4.

MTF curves obtained using the two-Lévy model for two different reconstruction kernels (FC13 and FC51). (a) MTF curves. (i): MTF curves of the two-Lévy model in Eq. (9) . (ii): curves of the first term of the two-Lévy model in Eq. (9) . (iii): curves of the second term of the two-Lévy model in Eq. (9) . (b) PSF curves obtained from the inverse Fourier transforms of MTF curves shown in (a). (i): PSF curves obtained from the MTF curves. (ii): curves obtained from the curves (ii) shown in (a). (iii): curves obtained from the curves (iii) shown in (a). Left: reconstruction kernel FC13. Right: reconstruction kernel FC51. Upper row: MTF. Lower row: PSF.

Image of FIG. 5.
FIG. 5.

Relationship between bias values and MTFs. (a) Kernel function: FC13. (b) Kernel function: FC51. (Left) ROI size N = 40. (Right) ROI size N = 50.

Image of FIG. 6.
FIG. 6.

Effect of bias values that are simply determined by measuring the background image value at some distance from the wire. (a) MTFs obtained using four bias values obtained in four square regions S1, S2, S3, and S4 for kernel function FC13. MTFs for S2 and S3 completely overlap. (b) MTFs obtained using four bias values obtained in four square regions S1, S2, S3, and S4 for kernel function FC51.

Image of FIG. 7.
FIG. 7.

Bias selection for two kernel functions, FC13 and FC51. (a) Relationship between bias and L values. (b) MTF curves for three square regions (N r × N r ) within the neighborhood of the ROI (N × N) using the final bias values. (Left) kernel function FC13. (Right) kernel function FC51. In 7(b) , three MTF curves for three different square region sizes completely overlap.

Image of FIG. 8.
FIG. 8.

Effect of ROI size on the bias value and MTF. (a) Bias values for different ROI sizes. (b) MTFs obtained using bias values shown in (a) for different ROI sizes. (Left) reconstruction kernel FC13. (Right) reconstruction kernel FC51. In (b), MTF curves for different ROI sizes almost overlap.

Image of FIG. 9.
FIG. 9.

MTF curves along three directions in frequency space. These directions correspond to the lines u = 0, v = 0, and u = v. (a) MTF curves of clinical CT for the kernel function FC13 using the thin wire phantom (0.1 mm in diameter). (b) MTF curves of SRμCT for the Shepp–Logan filter using the thin wire phantom (3 μm in diameter). The MTF curves along three directions almost overlap.

Image of FIG. 10.
FIG. 10.

MTFs determined using the two-Gaussian and two-Lévy models from profiles of the thin wire phantom of 0.1 mm in diameter. (a) Reconstruction kernel: FC13. (b) Reconstruction kernel: FC51. The MTF curve obtained using the two-Lévy model completely overlaps the gold standard of the MTF curve.

Image of FIG. 11.
FIG. 11.

Determination of the MTF from profiles of the 2-mm-diameter wire phantom. Reconstruction kernel: FC13. (a) |G(u,0)|, |W(u,0)|, and |G(u,0)|/|W(u,0)| curves. The curve of |G(u,0)|/|W(u,0)| was drawn by avoiding the zero value of |W(u,0)|. (b) Result of the determination of MTF(u,0) from the |G(u,0)|/|W(u,0)| curve without outliers.

Image of FIG. 12.
FIG. 12.

| G(u,0)|/|W(u,0)| curves of the thick wire phantoms (1, 2, and 3 mm in diameter). (a) Reconstruction kernel: FC13. (b) Reconstruction kernel: FC51. Gold standard: MTF obtained from profiles of the thin wire phantom of 0.1 mm in diameter.

Image of FIG. 13.
FIG. 13.

MTFs obtained from profiles of three different wire phantoms (1, 2, and 3 mm in diameter) measured by the clinical CT system. (a) Reconstruction kernel: FC13. (b) Reconstruction kernel: FC51. Gold standard: MTF obtained from profiles of the thin wire phantom of 0.1 mm in diameter.

Image of FIG. 14.
FIG. 14.

MTFs from profiles of three different wire phantoms (10 and 30 μm in diameter) measured by the SRμCT system. (a) Reconstruction kernel: Shepp–Logan. (b) Reconstruction kernel: Ramachandran–Lakshminarayanan.

Image of FIG. 15.
FIG. 15.

The third derivative of |G(u,0)|/|W(u,0)| obtained from profiles of the wire phantom (1 mm in diameter) measured by the clinical CT system. The spatial frequency is denoted by u. Reconstruction kernel: FC51. (a) |G(u,0)|, |W(u,0)|, and |G(u,0)|/|W(u,0)| curves. The curve of |G(u,0)|/|W(u,0)| was drawn by avoiding the zero value of |W(u,0)|. (b) The third derivative of the |G(u,0)|/|W(u,0)| curve. (c) Extended graph of the curve in (b). Among the dense frequencies at which the third-derivative curve intersects the frequency axis several times, the lower end of the spatial frequencies is denoted by u st. (I)end denotes the upper end of the range (I) in (c).

Image of FIG. 16.
FIG. 16.

The third derivative of |G(u,0)|/|W(u,0)| obtained from profiles of the wire phantom (2 mm in diameter) measured by the clinical CT system. The spatial frequency is denoted by u. Reconstruction kernel: FC51. (a) |G(u,0)|, |W(u,0)|, and |G(u,0)|/|W(u,0)| curves. The curve of |G(u,0)|/|W(u,0)| was drawn by avoiding the zero value of |W(u,0)|. (b) The third derivative of the |G(u,0)|/|W(u,0)| curve. (c) Extended graph of the curve in (b) with u ranging between 0 and 0.8. Among the dense frequencies around which the third-derivative curve intersects the frequency axis several times, the lower end of the frequencies is labeled u st. (d) Extended graph of the curve in (b) with u ranging between 0.4 and 1.4. Among the dense frequencies around which the third derivative curve intersects the frequency axis several times, the upper end of the frequencies is denoted by u end. (I)end denotes the upper end of range (I) in (c). (II)st and (II)end denote the lower end and upper end of range (II) in (d), respectively.

Tables

Generic image for table

Generic image for table
TABLE I.

Comparison between performance of the MTFs determined using the two-Gaussian and two-Lévy models. The values 10% MTF, RE, and RMSE obtained for the thin wire phantom of 0.1 mm in diameter were measured by the clinical CT system. The gold standard was obtained using a wire phantom of 0.1 mm in diameter.

Generic image for table
TABLE II.

Accuracy of MTFs obtained from profiles of three different wire phantoms (1, 2, and 3 in diameter) measured by the clinical CT system. The gold standard was obtained using a wire phantom of 0.1 mm in diameter.

Generic image for table
TABLE III.

Accuracy of MTFs obtained by the traditional method from profiles of three different wire phantoms (1, 2, and 3 mm in diameter) measured by the clinical CT system. The gold standard was obtained using a wire phantom of 0.1 mm in diameter.

Generic image for table
TABLE IV.

Accuracy of MTFs obtained from profiles of two different wire phantoms (10 and 30 μm in diameter) measured by SRμCT with the reconstruction kernels of Shepp–Logan and Ramachandran–Lakshminarayanan (R–L). The gold standard was obtained using a wire phantom of 3 μm in diameter.

Generic image for table
TABLE V.

Accuracy of MTFs obtained by the traditional method from profiles of two different wire phantoms (10 and 30 μm in diameter) measured by SRμCT with the reconstruction kernels of Shepp–Logan and Ramachandran–Lakshminarayanan (R–L). The gold standard was obtained using a wire phantom of 3 μm in diameter.

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/content/aapm/journal/medphys/39/7/10.1118/1.4729711
2012-06-27
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: A method for determining the modulation transfer function from thick microwire profiles measured with x-ray microcomputed tomography
http://aip.metastore.ingenta.com/content/aapm/journal/medphys/39/7/10.1118/1.4729711
10.1118/1.4729711
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