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August  2021, 15(4): 683-721. doi: 10.3934/ipi.2021010

Bragg scattering tomography

James W. Webber , and  Eric L. Miller

161 College Avenue, Halligan Hall, Medford, MA 02155, USA

* Corresponding author: James W. Webber

Received  April 2020 Revised  October 2020 Published  August 2021 Early access  January 2021

Fund Project: This material is supported by the U.S. Department of Homeland Security, Science and Technology Directorate, Office of University Programs, under Grant Award 2013-ST-061-ED0001

Here we introduce a new forward model and imaging modality for Bragg Scattering Tomography (BST). The model we propose is based on an X-ray portal scanner with linear detector collimation, currently being developed for use in airport baggage screening. The geometry under consideration leads us to a novel two-dimensional inverse problem, where we aim to reconstruct the Bragg scattering differential cross section function from its integrals over a set of symmetric $ C^2 $ curves in the plane. The integral transform which describes the forward problem in BST is a new type of Radon transform, which we introduce and denote as the Bragg transform. We provide new injectivity results for the Bragg transform here, and describe how the conditions of our theorems can be applied to assist in the machine design of the portal scanner. Further we provide an extension of our results to $ n $-dimensions, where a generalization of the Bragg transform is introduced. Here we aim to reconstruct a real valued function on $ \mathbb{R}^{n+1} $ from its integrals over $ n $-dimensional surfaces of revolution of $ C^2 $ curves embedded in $ \mathbb{R}^{n+1} $. Injectivity proofs are provided also for the generalized Bragg transform.

Keywords: Bragg scattering, tomography, Volterra integral equations, generalized Radon transforms.
Mathematics Subject Classification: Primary: 45Axx, 44A12.
Citation: James W. Webber, Eric L. Miller. Bragg scattering tomography. Inverse Problems and Imaging, 2021, 15 (4) : 683-721. doi: 10.3934/ipi.2021010
References:
[1]

G. Ambartsoumian, R. Gouia-Zarrad and M. A. Lewis, Inversion of the circular Radon transform on an annulus, Inverse Problems, 26 (2010), 105015, 11pp. doi: 10.1088/0266-5611/26/10/105015.

[2]

H. Andrade-Loarca, G. Kutyniok and O. Öktem, Shearlets as feature extractor for semantic edge detection: The model-based and data-driven realm, Proc. A., 476 (2020), 841–866. arXiv: 1911.12159. doi: 10.1098/rspa.2019.0841.

[3]

D. Ballantine Jr, R. M. White, S. J. Martin, A. J. Ricco, E. Zellers, G. Frye and H. Wohltjen, Acoustic Wave Sensors: Theory, Design and Physico-chemical Applications, Elsevier, 1996.

[4]

G. Beylkin, The inversion problem and applications of the generalized Radon transform, Communications on Pure and Applied Mathematics, 37 (1984), 579-599.  doi: 10.1002/cpa.3160370503.

[5]

R. Bryan, International Tables for Crystallography, Vol. C. Mathematical, physical and chemical tables edited by A. J. C. Wilson, 1993.

[6]

A. M. Cormack, The Radon transform on a family of curves in the plane, Proc. Amer. Math. Soc., 83 (1981), 325-330.  doi: 10.1090/S0002-9939-1981-0624923-1.

[7]

A. M. Cormack, Radon's problem for some surfaces in $\mathbb{R}^n$, Proc. Amer. Math. Soc., 99 (1987), 305-312.  doi: 10.2307/2046630.

[8]

J. J. DeMarco and P. Suortti, Effect of scattering on the attenuation of X-rays, Physical Review B, 4 (1971), 1028.

[9]

A. Denisyuk, Inversion of the generalized Radon transform, Translations of the American Mathematical Society-Series 2, 162 (1994), 19-32.  doi: 10.1090/trans2/162/02.

[10]

M. J. Ehrhardt, K. Thielemans, L. Pizarro, D. Atkinson, S. Ourselin, B. F. Hutton and S. R. Arridge, Joint reconstruction of PET-MRI by exploiting structural similarity, Inverse Problems, 31 (2014), 015001, 23pp. doi: 10.1088/0266-5611/31/1/015001.

[11]

J. A. Greenberg, K. Krishnamurthy and D. Brady, Snapshot molecular imaging using coded energy-sensitive detection, Optics express, 21 (2013), 25480-25491.  doi: 10.1364/OE.21.025480.

[12]

A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240.  doi: 10.1215/S0012-7094-89-05811-0.

[13]

M. Hassan, J. A. Greenberg, I. Odinaka and D. J. Brady, Snapshot fan beam coded aperture coherent scatter tomography, Optics Express, 24 (2016), 18277-18289. 

[14]

J. Hubbell, W. J. Veigele, E. Briggs, R. Brown, D. Cromer and d. R. Howerton, Atomic form factors, incoherent scattering functions, and photon scattering cross sections, Journal of Physical and Chemical Reference Data, 4 (1975), 471-538.  doi: 10.1063/1.555523.

[15]

J. H. Hubbell and I. Overbo, Relativistic atomic form factors and photon coherent scattering cross sections, Journal of Physical and Chemical Reference Data, 8 (1979), 69-106.  doi: 10.1063/1.555593.

[16]

C.-Y. Jung and S. Moon, Inversion formulas for cone transforms arising in application of Compton cameras, Inverse Problems, 31 (2015), 015006, 20pp. doi: 10.1088/0266-5611/31/1/015006.

[17]

H. P. Klug and L. E. Alexander, X-ray Diffraction Procedures: For Polycrystalline and Amorphous Materials, X-Ray Diffraction Procedures: For Polycrystalline and Amorphous Materials, 2nd Edition, by Harold P. Klug, Leroy E. Alexander, pp. 992. Wiley-VCH, May 1974., 992.

[18]

K. MacCabe, K. Krishnamurthy, A. Chawla, D. Marks, E. Samei and D. Brady, Pencil beam coded aperture X-ray scatter imaging, Optics Express, 20 (2012), 16310-16320.  doi: 10.1364/OE.20.016310.

[19]

F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001. doi: 10.1137/1.9780898719284.

[20]

M. Nguyen and T. Truong, Inversion of a new circular-arc Radon transform for Compton scattering tomography, Inverse Problems, 26 (2010), 099802, 1 p. doi: 10.1088/0266-5611/26/9/099802.

[21]

S. J. Norton, Compton scattering tomography, Journal of Applied Physics, 76 (1994), 2007-2015.  doi: 10.1063/1.357668.

[22]

V. P. Palamodov, A uniform reconstruction formula in integral geometry, Inverse Problems, 28 (2012), 065014, 15pp. doi: 10.1088/0266-5611/28/6/065014.

[23]

V. Palamodov, An analytic reconstruction for the Compton scattering tomography in a plane, Inverse Problems, 27 (2011), 125004, 8pp. doi: 10.1088/0266-5611/27/12/125004.

[24]

M. Plancherel and M. Leffler, Contribution à l'étude de la représentation d'une fonction arbitraire par des intégrales définies, Rendiconti del Circolo Matematico di Palermo (1884-1940), 30 (1910), 289-335. 

[25]

E. T. Quinto, The dependence of the generalized Radon transform on defining measures, Transactions of the American Mathematical Society, 257 (1980), 331-346.  doi: 10.1090/S0002-9947-1980-0552261-8.

[26]

G. Rigaud, 3D Compton scattering imaging: Study of the spectrum and contour reconstruction, arXiv preprint, arXiv: 1908.03066.

[27]

G. Rigaud and B. N. Hahn, 3D Compton scattering imaging and contour reconstruction for a class of Radon transforms, Inverse Problems, 34 (2018), 075004, 22pp. doi: 10.1088/1361-6420/aabf0b.

[28]

G. Rigaud, M. K. Nguyen and A. K. Louis, Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography, Inverse Problems in Science and Engineering, 20 (2012), 809-839.  doi: 10.1080/17415977.2011.653008.

[29]

A. Smakula and J. Kalnajs, Precision determination of lattice constants with a Geiger-counter X-ray diffractometer, Physical Review, 99 (1955), 1737. doi: 10.1103/PhysRev.99.1737.

[30]

A. A. Taha and A. Hanbury, Metrics for evaluating 3D medical image segmentation: Analysis, selection, and tool, BMC Medical Imaging, 15 (2015), 29. doi: 10.1186/s12880-015-0068-x.

[31]

A. Taylor and H. Sinclair, On the determination of lattice parameters by the Debye-Scherrer method, Proceedings of the Physical Society, 57 (1945), 126. doi: 10.1088/0959-5309/57/2/306.

[32]

F. G. Tricomi, Integral Equations, Dover Publications, Inc., New York, 1985.

[33]

T. Truong and M. Nguyen, Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality, Inverse Problems, 27 (2011), 125001, 23pp. doi: 10.1088/0266-5611/27/12/125001.

[34]

T. T. Truong and M. K. Nguyen, New properties of the V-line Radon transform and their imaging applications, Journal of Physics A: Mathematical and Theoretical, 48 (2015), 405204, 28pp. doi: 10.1088/1751-8113/48/40/405204.

[35]

T. T. Truong, M. K. Nguyen and H. Zaidi, The mathematical foundations of 3D Compton scatter emission imaging, International Journal of Biomedical Imaging, 2007 (2007), Article ID 092780. doi: 10.1155/2007/92780.

[36]

N. Wadeson, Modelling and Correction of Scatter in a Switched Source Multi-Ring Detector X-ray CT Machine, PhD thesis, The University of Manchester (United Kingdom), 2011.

[37]

B. E. Warren, X-ray Diffraction, Courier Corporation, 1990.

[38]

J. Webber and E. L. Miller, Compton scattering tomography in translational geometries, Inverse Problems, 36 (2020), 025007, 20pp. doi: 10.1088/1361-6420/ab4a32.

[39]

J. Webber and E. T. Quinto, Microlocal analysis of a Compton tomography problem, SIAM J. Imaging Sci., 13 (2020), 746–774, arXiv: 1902.09623. doi: 10.1137/19M1251035.

[40]

J. Webber, E. T. Quinto and E. L. Miller, A joint reconstruction and lambda tomography regularization technique for energy-resolved X-ray imaging, Inverse Problems, 36 (2020), 074002, 32 pp. doi: 10.1088/1361-6420/ab8f82.

[41]

J. W. Webber and E. T. Quinto, Microlocal analysis of generalized Radon transforms from scattering tomography, arXiv preprint, arXiv: 2007.00208.

show all references

References:
[1]

G. Ambartsoumian, R. Gouia-Zarrad and M. A. Lewis, Inversion of the circular Radon transform on an annulus, Inverse Problems, 26 (2010), 105015, 11pp. doi: 10.1088/0266-5611/26/10/105015.

[2]

H. Andrade-Loarca, G. Kutyniok and O. Öktem, Shearlets as feature extractor for semantic edge detection: The model-based and data-driven realm, Proc. A., 476 (2020), 841–866. arXiv: 1911.12159. doi: 10.1098/rspa.2019.0841.

[3]

D. Ballantine Jr, R. M. White, S. J. Martin, A. J. Ricco, E. Zellers, G. Frye and H. Wohltjen, Acoustic Wave Sensors: Theory, Design and Physico-chemical Applications, Elsevier, 1996.

[4]

G. Beylkin, The inversion problem and applications of the generalized Radon transform, Communications on Pure and Applied Mathematics, 37 (1984), 579-599.  doi: 10.1002/cpa.3160370503.

[5]

R. Bryan, International Tables for Crystallography, Vol. C. Mathematical, physical and chemical tables edited by A. J. C. Wilson, 1993.

[6]

A. M. Cormack, The Radon transform on a family of curves in the plane, Proc. Amer. Math. Soc., 83 (1981), 325-330.  doi: 10.1090/S0002-9939-1981-0624923-1.

[7]

A. M. Cormack, Radon's problem for some surfaces in $\mathbb{R}^n$, Proc. Amer. Math. Soc., 99 (1987), 305-312.  doi: 10.2307/2046630.

[8]

J. J. DeMarco and P. Suortti, Effect of scattering on the attenuation of X-rays, Physical Review B, 4 (1971), 1028.

[9]

A. Denisyuk, Inversion of the generalized Radon transform, Translations of the American Mathematical Society-Series 2, 162 (1994), 19-32.  doi: 10.1090/trans2/162/02.

[10]

M. J. Ehrhardt, K. Thielemans, L. Pizarro, D. Atkinson, S. Ourselin, B. F. Hutton and S. R. Arridge, Joint reconstruction of PET-MRI by exploiting structural similarity, Inverse Problems, 31 (2014), 015001, 23pp. doi: 10.1088/0266-5611/31/1/015001.

[11]

J. A. Greenberg, K. Krishnamurthy and D. Brady, Snapshot molecular imaging using coded energy-sensitive detection, Optics express, 21 (2013), 25480-25491.  doi: 10.1364/OE.21.025480.

[12]

A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240.  doi: 10.1215/S0012-7094-89-05811-0.

[13]

M. Hassan, J. A. Greenberg, I. Odinaka and D. J. Brady, Snapshot fan beam coded aperture coherent scatter tomography, Optics Express, 24 (2016), 18277-18289. 

[14]

J. Hubbell, W. J. Veigele, E. Briggs, R. Brown, D. Cromer and d. R. Howerton, Atomic form factors, incoherent scattering functions, and photon scattering cross sections, Journal of Physical and Chemical Reference Data, 4 (1975), 471-538.  doi: 10.1063/1.555523.

[15]

J. H. Hubbell and I. Overbo, Relativistic atomic form factors and photon coherent scattering cross sections, Journal of Physical and Chemical Reference Data, 8 (1979), 69-106.  doi: 10.1063/1.555593.

[16]

C.-Y. Jung and S. Moon, Inversion formulas for cone transforms arising in application of Compton cameras, Inverse Problems, 31 (2015), 015006, 20pp. doi: 10.1088/0266-5611/31/1/015006.

[17]

H. P. Klug and L. E. Alexander, X-ray Diffraction Procedures: For Polycrystalline and Amorphous Materials, X-Ray Diffraction Procedures: For Polycrystalline and Amorphous Materials, 2nd Edition, by Harold P. Klug, Leroy E. Alexander, pp. 992. Wiley-VCH, May 1974., 992.

[18]

K. MacCabe, K. Krishnamurthy, A. Chawla, D. Marks, E. Samei and D. Brady, Pencil beam coded aperture X-ray scatter imaging, Optics Express, 20 (2012), 16310-16320.  doi: 10.1364/OE.20.016310.

[19]

F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001. doi: 10.1137/1.9780898719284.

[20]

M. Nguyen and T. Truong, Inversion of a new circular-arc Radon transform for Compton scattering tomography, Inverse Problems, 26 (2010), 099802, 1 p. doi: 10.1088/0266-5611/26/9/099802.

[21]

S. J. Norton, Compton scattering tomography, Journal of Applied Physics, 76 (1994), 2007-2015.  doi: 10.1063/1.357668.

[22]

V. P. Palamodov, A uniform reconstruction formula in integral geometry, Inverse Problems, 28 (2012), 065014, 15pp. doi: 10.1088/0266-5611/28/6/065014.

[23]

V. Palamodov, An analytic reconstruction for the Compton scattering tomography in a plane, Inverse Problems, 27 (2011), 125004, 8pp. doi: 10.1088/0266-5611/27/12/125004.

[24]

M. Plancherel and M. Leffler, Contribution à l'étude de la représentation d'une fonction arbitraire par des intégrales définies, Rendiconti del Circolo Matematico di Palermo (1884-1940), 30 (1910), 289-335. 

[25]

E. T. Quinto, The dependence of the generalized Radon transform on defining measures, Transactions of the American Mathematical Society, 257 (1980), 331-346.  doi: 10.1090/S0002-9947-1980-0552261-8.

[26]

G. Rigaud, 3D Compton scattering imaging: Study of the spectrum and contour reconstruction, arXiv preprint, arXiv: 1908.03066.

[27]

G. Rigaud and B. N. Hahn, 3D Compton scattering imaging and contour reconstruction for a class of Radon transforms, Inverse Problems, 34 (2018), 075004, 22pp. doi: 10.1088/1361-6420/aabf0b.

[28]

G. Rigaud, M. K. Nguyen and A. K. Louis, Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography, Inverse Problems in Science and Engineering, 20 (2012), 809-839.  doi: 10.1080/17415977.2011.653008.

[29]

A. Smakula and J. Kalnajs, Precision determination of lattice constants with a Geiger-counter X-ray diffractometer, Physical Review, 99 (1955), 1737. doi: 10.1103/PhysRev.99.1737.

[30]

A. A. Taha and A. Hanbury, Metrics for evaluating 3D medical image segmentation: Analysis, selection, and tool, BMC Medical Imaging, 15 (2015), 29. doi: 10.1186/s12880-015-0068-x.

[31]

A. Taylor and H. Sinclair, On the determination of lattice parameters by the Debye-Scherrer method, Proceedings of the Physical Society, 57 (1945), 126. doi: 10.1088/0959-5309/57/2/306.

[32]

F. G. Tricomi, Integral Equations, Dover Publications, Inc., New York, 1985.

[33]

T. Truong and M. Nguyen, Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality, Inverse Problems, 27 (2011), 125001, 23pp. doi: 10.1088/0266-5611/27/12/125001.

[34]

T. T. Truong and M. K. Nguyen, New properties of the V-line Radon transform and their imaging applications, Journal of Physics A: Mathematical and Theoretical, 48 (2015), 405204, 28pp. doi: 10.1088/1751-8113/48/40/405204.

[35]

T. T. Truong, M. K. Nguyen and H. Zaidi, The mathematical foundations of 3D Compton scatter emission imaging, International Journal of Biomedical Imaging, 2007 (2007), Article ID 092780. doi: 10.1155/2007/92780.

[36]

N. Wadeson, Modelling and Correction of Scatter in a Switched Source Multi-Ring Detector X-ray CT Machine, PhD thesis, The University of Manchester (United Kingdom), 2011.

[37]

B. E. Warren, X-ray Diffraction, Courier Corporation, 1990.

[38]

J. Webber and E. L. Miller, Compton scattering tomography in translational geometries, Inverse Problems, 36 (2020), 025007, 20pp. doi: 10.1088/1361-6420/ab4a32.

[39]

J. Webber and E. T. Quinto, Microlocal analysis of a Compton tomography problem, SIAM J. Imaging Sci., 13 (2020), 746–774, arXiv: 1902.09623. doi: 10.1137/19M1251035.

[40]

J. Webber, E. T. Quinto and E. L. Miller, A joint reconstruction and lambda tomography regularization technique for energy-resolved X-ray imaging, Inverse Problems, 36 (2020), 074002, 32 pp. doi: 10.1088/1361-6420/ab8f82.

[41]

J. W. Webber and E. T. Quinto, Microlocal analysis of generalized Radon transforms from scattering tomography, arXiv preprint, arXiv: 2007.00208.

Figure 1.  The portal scanner geometry. The scanned object is labelled as $ f $. The detectors are collimated to planes, and the scattering events occur along lines $ L = \{x_2 = a, x_3 = 0\} $, for some $ -1<a<1 $. The scatter from $ L $ is measured by detectors $ \mathbf{d}\in\{x_2 = 1, x_3 = \epsilon\} $, for some $ \epsilon>0 $
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Figure 2.  A scattering event occurs at a scattering site $ \mathbf{x} $, for photons emitted from a source $ \mathbf{s} $ and recorded at a detector $ \mathbf{d} $. The initial photon energy is $ E $ and the scattered energy is $ E_s $. Here $ \mathbf{v} $ is the direction normal to the detector surface, displayed as a square in the $ x_1x_3 $ plane. The scattering angle is $ \omega = 2\theta $. where $ \theta $ is the Bragg angle
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Figure 3.  Plot of the curves of integration for the Bragg transform for varying $ E $ and $ x_2 $. $ s_1 = 0 $ is fixed
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Figure 4.  A circle with center $ \mathbf{c} = (-\sqrt{r^2-1}, 0) $, radius $ r $ is pictured. The circle intersects the dashed line at two points, whose $ x_1 $ coordinates are the solutions to (4.6)
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Figure 5.  Bragg scanning modality in the $ n = 2 $ case. A square detector $ \hat{ \mathbf{d}} = ( \mathbf{s}, 1) $ is shown opposite a source $ \hat{ \mathbf{s}} = ( \mathbf{s}, -1) $, and collects photons (shown as wavy lines) scattered from points $ \hat{ \mathbf{x}} = ( \mathbf{x}, 1-|\textbf{b}-\hat{ \mathbf{s}}|) $ on the crystal plane. The crystal sample (the red plane) is placed between, and is parallel to the $ \hat{ \mathbf{s}} $-plane and $ \hat{ \mathbf{d}} $-plane. The center of the base of the cone is $ \textbf{b} $, the source opening angle is $ \beta $ (as in figure 1a), and the source width is $ w = |\textbf{b}-\hat{ \mathbf{s}}|\tan\beta $. The momentum transfer is $ q = Eq_1(| \mathbf{x}- \mathbf{s}|) = E\sin\frac{\omega}{2} $
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Figure 6.  $ q_1 $ curve examples. For the decreasing curves displayed we would have to choose $ w>1 $ for the injectivity of $ \mathfrak{B}_1 $ to hold
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Figure 7.  Plot of the curves of integration for the offset Bragg transform for varying $ E $ and $ \frac{E_M}{q_1(w)} $. The line $ \{q = E_m\} $ is displayed in black
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Figure 8.  Invertible design regions $ S $ (the blue regions on the top row) and possible linear $ \Phi $ (the red lines on the bottom row) for varying $ \beta $. Note that the $ \epsilon $ scales of the figures are different for each $ \beta $
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Figure 9.  Venetian blind detector configurations for varying source opening angles $ \beta $. We show 21 detector arrays at $ \epsilon = \Phi(x_2) $ for $ x_2\in\left\{-1+\frac{j-1}{10} : 1\leq j\leq 21\right\} $, where for each $ \beta $, $ \Phi $ is the corresponding straight line relationship of figure 8. The blue lines represent the collimation planes which intersect the tunnel at position $ T = 420(-x_2+1) $, where $ x_2 = \Phi^{-1}(\epsilon) $
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Figure 10.  $ F(\cdot, Z) $ plots for varying $ Z $. The plots are normalized in $ L^{\infty} $ (by max value)
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Figure 11.  Top row – 2-spherical phantom. Bottom row – 4-spherical phantom. The image values in the left column are included for visualization (e.g. to distinguish between different materials) and have no physical meaning
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Figure 12.  2-spherical phantom reconstructions for all $ (x_2, c_{\text{avg}})\in\{205,410,615\}\times\{1, 10\} $, and the Ground Truth (GT) on the top row. To clarify, the GT does not vary with $ c_{\text{avg}} $ and is included for comparison
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Figure 13.  4-spherical phantom reconstructions for all $ (x_2, c_{\text{avg}})\in\{205,410,615\}\times\{1, 10\} $, and the Ground Truth (GT) on the top row. To clarify, the GT does not vary with $ c_{\text{avg}} $ and is included for comparison
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Table 1.  Relative least square error values $ \epsilon_{\text{ls}} $ for all experiments conducted. The values in the left-hand columns give the $ x_2 $ coordinates (in mm) of the sphere centers and the equations of the scanning line profiles (as illustrated in figure 11)
$ \epsilon_{\text{ls}} $ $ c_{\text{avg}}=10 $ $ c_{\text{avg}}=1 $ $ \epsilon_{\text{ls}} $ $ c_{\text{avg}} = 10 $ $ c_{\text{avg}} = 1 $
$ x_2=0 $ $ 0.18 $ $ 0.58 $ $ x_2 = 0 $ $ 0.23 $ $ 0.73 $
$ x_2=205 $ $ 0.18 $ $ 0.56 $ $ x_2 = 205 $ $ 0.22 $ $ 0.71 $
$ x_2=615 $ $ 0.18 $ $ 0.57 $ $ x_2 = 615 $ $ 0.22 $ $ 0.71 $
(A) 2-spherical phantom (B)4-spherical phantom
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$ \epsilon_{\text{ls}} $ $ c_{\text{avg}}=10 $ $ c_{\text{avg}}=1 $ $ \epsilon_{\text{ls}} $ $ c_{\text{avg}} = 10 $ $ c_{\text{avg}} = 1 $
$ x_2=0 $ $ 0.18 $ $ 0.58 $ $ x_2 = 0 $ $ 0.23 $ $ 0.73 $
$ x_2=205 $ $ 0.18 $ $ 0.56 $ $ x_2 = 205 $ $ 0.22 $ $ 0.71 $
$ x_2=615 $ $ 0.18 $ $ 0.57 $ $ x_2 = 615 $ $ 0.22 $ $ 0.71 $
(A) 2-spherical phantom (B)4-spherical phantom
Table 2.  Mean $ F_1 $ score results. The values in the left-hand columns give the $ x_2 $ coordinates (in mm) of the sphere centers and the equations of the scanning line profiles (as illustrated in figure 11)
Mean $ F_1 $ score $ c_{\text{avg}}=10 $ $ c_{\text{avg}}=1 $ Mean $ F_1 $ score $ c_{\text{avg}} = 10 $ $ c_{\text{avg}} = 1 $
$ x_2=0 $ $ 0.90 $ $ 0.84 $ $ x_2 = 0 $ $ 0.86 $ $ 0.67 $
$ x_2=205 $ $ 0.90 $ $ 0.90 $ $ x_2 = 205 $ $ 0.89 $ $ 0.70 $
$ x_2=615 $ $ 0.91 $ $ 0.90 $ $ x_2 = 615 $ $ 0.87 $ $ 0.73 $
(A) 2-spherical phantom (B) 4-spherical phantom
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Mean $ F_1 $ score $ c_{\text{avg}}=10 $ $ c_{\text{avg}}=1 $ Mean $ F_1 $ score $ c_{\text{avg}} = 10 $ $ c_{\text{avg}} = 1 $
$ x_2=0 $ $ 0.90 $ $ 0.84 $ $ x_2 = 0 $ $ 0.86 $ $ 0.67 $
$ x_2=205 $ $ 0.90 $ $ 0.90 $ $ x_2 = 205 $ $ 0.89 $ $ 0.70 $
$ x_2=615 $ $ 0.91 $ $ 0.90 $ $ x_2 = 615 $ $ 0.87 $ $ 0.73 $
(A) 2-spherical phantom (B) 4-spherical phantom
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