Abstract

The refractive-index gradient vector field approach establishes a connection between a tomographic data set of differential phase contrast images and the distribution of the partial spatial derivatives of the refractive index in an object. The reconstruction of the refractive index in a plane requires the integration of its gradient field. This work shows how this integration can be efficiently performed by converting the problem to the Poisson equation, which can be accurately solved even in the case of noisy and large datasets. The performance of the suggested method is discussed and demonstrated experimentally by computing the refractive index distribution in both a simple plastic phantom and a complex biological sample. The quality of the reconstruction is evaluated through the direct comparison with other commonly used methods. To this end, the refractive index is retrieved from the same data set using also (1) the filtered backprojection algorithm for gradient projections, and (2) the regularized phase-retrieval procedure. Results show that the gradient vector field approach combined with the developed integration technique provides a very accurate depiction of the sample internal structure. Contrary to the two other techniques, the considered method does not require a preliminary phase-retrieval and can be implemented with any advanced computer tomography algorithm. In this work, analyzer-based phase contrast images are used for demonstration. Results, however, are generally valid and can be applied for processing differential phase-contrast tomographic data sets obtained with other phase-contrast imaging techniques.

© 2014 Optical Society of America

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  4. F. Pfeiffer, C. Kottler, O. Bunk, and C. Davis, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007).
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2013 (1)

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol 58, R1–R35 (2013).
[Crossref]

2012 (1)

2011 (1)

T. Thuering, P. Modregger, B. R. Pinzer, Z. Wang, and M. Stampanoni, “Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiography,” Opt. Express 19, 25542–25558 (2011).

2009 (1)

A. Beck and M. Teboulle, “A fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems,” SIAM J. Imaging Sciences 2, 183–202 (2009).
[Crossref]

2007 (2)

F. Pfeiffer, C. Kottler, O. Bunk, and C. Davis, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

A. Maksimenko, “Nonlinear extension of the X-ray diffraction enhanced imaging,” Appl. Phys. Lett. 90, 154106 (2007).
[Crossref]

2006 (2)

2005 (2)

A. Maksimenko, M. Ando, S. Hiroshi, and T. Yausa, “Computed tomographic reconstruction based on x-ray refraction contrast,” Appl. Phys. Lett. 86, 124105 (2005).
[Crossref]

T. Weitkamp, A. Diaz, Ch. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express. 13, 6296–6304 (2005).
[Crossref] [PubMed]

2000 (1)

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

1999 (1)

V.A. Bushuev and A.A. Sergeev, “Inverse problem in the X-ray phase contrast method,” Technical Phys. Lett. 25, 83–85 (1999).
[Crossref]

1997 (1)

J. M. Hyman and M. Shashkov, ”Natural discretizations for the divergence, gradient, and curl on logically rectangular grids”, Computers Math. Applic. 33, 81–104 (1997).
[Crossref]

1995 (2)

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2318 (1995).
[Crossref]

A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instr. Meth. Phys. Res. A 352, 622–628 (1995).
[Crossref]

1988 (1)

1983 (1)

A. J. Devaney, “A Computer Simulation study of Diffraction Tomography,” IEEE Trans. Biomed. Eng. 30377–386 (1983).
[Crossref] [PubMed]

1979 (1)

G. R. Hammerstein, D. W. Miller, D. R. White, M. E. Masterson, H. Q. Woodard, and J. S. Laughlin, “Absorbed radiation dose in mammography,” Radiology 130, 485–491 (1979).
[PubMed]

1978 (1)

F. J. Harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform,” Proc. IEEE,  66, 51–83 (1978).
[Crossref]

1961 (1)

R. P. Fedorenko, “A relaxation method for solving elliptic difference equations,” USSR Comput. Math. Math. Phys. 1, 1092 (1961).
[Crossref]

Akatsuka, T.

Ando, M.

Beck, A.

A. Beck and M. Teboulle, “A fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems,” SIAM J. Imaging Sciences 2, 183–202 (2009).
[Crossref]

Beliaevskaya, E. A.

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2318 (1995).
[Crossref]

Bravin, A.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol 58, R1–R35 (2013).
[Crossref]

Bunk, O.

F. Pfeiffer, C. Kottler, O. Bunk, and C. Davis, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

Bushuev, V.A.

V.A. Bushuev and A.A. Sergeev, “Inverse problem in the X-ray phase contrast method,” Technical Phys. Lett. 25, 83–85 (1999).
[Crossref]

Byer, R. L.

Chapman, D.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, Ch. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

Chapman, L. D.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Cloetens, P.

T. Weitkamp, A. Diaz, Ch. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express. 13, 6296–6304 (2005).
[Crossref] [PubMed]

Coan, P.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol 58, R1–R35 (2013).
[Crossref]

David, Ch.

T. Weitkamp, A. Diaz, Ch. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express. 13, 6296–6304 (2005).
[Crossref] [PubMed]

Davis, C.

F. Pfeiffer, C. Kottler, O. Bunk, and C. Davis, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

Devaney, A. J.

A. J. Devaney, “A Computer Simulation study of Diffraction Tomography,” IEEE Trans. Biomed. Eng. 30377–386 (1983).
[Crossref] [PubMed]

Diaz, A.

T. Weitkamp, A. Diaz, Ch. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express. 13, 6296–6304 (2005).
[Crossref] [PubMed]

Dilmanian, F. A.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Dreissi, D.

Faris, G. W.

Fedorenko, R. P.

R. P. Fedorenko, “A relaxation method for solving elliptic difference equations,” USSR Comput. Math. Math. Phys. 1, 1092 (1961).
[Crossref]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C, 2 (Cambridge University, 1992, pp. 871–872).

Hammerstein, G. R.

G. R. Hammerstein, D. W. Miller, D. R. White, M. E. Masterson, H. Q. Woodard, and J. S. Laughlin, “Absorbed radiation dose in mammography,” Radiology 130, 485–491 (1979).
[PubMed]

Harris, F. J.

F. J. Harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform,” Proc. IEEE,  66, 51–83 (1978).
[Crossref]

Hashimoto, E.

Hasnah, M.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, Ch. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

Hiroshi, S.

A. Maksimenko, M. Ando, S. Hiroshi, and T. Yausa, “Computed tomographic reconstruction based on x-ray refraction contrast,” Appl. Phys. Lett. 86, 124105 (2005).
[Crossref]

Hyman, J. M.

J. M. Hyman and M. Shashkov, ”Natural discretizations for the divergence, gradient, and curl on logically rectangular grids”, Computers Math. Applic. 33, 81–104 (1997).
[Crossref]

Hyodo, K.

Ignatyev, K.

Ingal, V. N.

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2318 (1995).
[Crossref]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging(IEEE Press, 1988, Chap. 5).

Kottler, C.

F. Pfeiffer, C. Kottler, O. Bunk, and C. Davis, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

Laughlin, J. S.

G. R. Hammerstein, D. W. Miller, D. R. White, M. E. Masterson, H. Q. Woodard, and J. S. Laughlin, “Absorbed radiation dose in mammography,” Radiology 130, 485–491 (1979).
[PubMed]

Lopez, F. C. M.

Maksimenko, A.

A. Maksimenko, “Nonlinear extension of the X-ray diffraction enhanced imaging,” Appl. Phys. Lett. 90, 154106 (2007).
[Crossref]

T. Yuasa, A. Maksimenko, E. Hashimoto, H. Sugiyama, K. Hyodo, T. Akatsuka, and M. Ando, “Hard-x-ray region tomographic reconstruction of the refractive-index gradient vector field: imaging principles and comparisons with diffraction-enhanced-imaging-based computed tomography,” Opt. Lett. 31, 1818–1820 (2006).
[Crossref] [PubMed]

A. Maksimenko, M. Ando, S. Hiroshi, and T. Yausa, “Computed tomographic reconstruction based on x-ray refraction contrast,” Appl. Phys. Lett. 86, 124105 (2005).
[Crossref]

Masterson, M. E.

G. R. Hammerstein, D. W. Miller, D. R. White, M. E. Masterson, H. Q. Woodard, and J. S. Laughlin, “Absorbed radiation dose in mammography,” Radiology 130, 485–491 (1979).
[PubMed]

Miller, D. W.

G. R. Hammerstein, D. W. Miller, D. R. White, M. E. Masterson, H. Q. Woodard, and J. S. Laughlin, “Absorbed radiation dose in mammography,” Radiology 130, 485–491 (1979).
[PubMed]

Modregger, P.

T. Thuering, P. Modregger, B. R. Pinzer, Z. Wang, and M. Stampanoni, “Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiography,” Opt. Express 19, 25542–25558 (2011).

Momose, A.

A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instr. Meth. Phys. Res. A 352, 622–628 (1995).
[Crossref]

Mondal, I.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, Ch. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

Morgan, K.

O. C. Zienkiewicz and K. Morgan, Finite Elements and Approximation (Dover Pubn. Inc., 2006, Chap. 3).

Munro, P. R. T.

Olivo, A.

Orion, I.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Parham, Ch.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, Ch. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

Pfeiffer, F.

F. Pfeiffer, C. Kottler, O. Bunk, and C. Davis, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

T. Weitkamp, A. Diaz, Ch. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express. 13, 6296–6304 (2005).
[Crossref] [PubMed]

Pinzer, B. R.

T. Thuering, P. Modregger, B. R. Pinzer, Z. Wang, and M. Stampanoni, “Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiography,” Opt. Express 19, 25542–25558 (2011).

Pisano, E.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, Ch. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C, 2 (Cambridge University, 1992, pp. 871–872).

Ren, B.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Rigon, L.

Sergeev, A.A.

V.A. Bushuev and A.A. Sergeev, “Inverse problem in the X-ray phase contrast method,” Technical Phys. Lett. 25, 83–85 (1999).
[Crossref]

Shashkov, M.

J. M. Hyman and M. Shashkov, ”Natural discretizations for the divergence, gradient, and curl on logically rectangular grids”, Computers Math. Applic. 33, 81–104 (1997).
[Crossref]

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging(IEEE Press, 1988, Chap. 5).

Speller, R.D.

Stampanoni, M.

T. Thuering, P. Modregger, B. R. Pinzer, Z. Wang, and M. Stampanoni, “Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiography,” Opt. Express 19, 25542–25558 (2011).

T. Weitkamp, A. Diaz, Ch. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express. 13, 6296–6304 (2005).
[Crossref] [PubMed]

Sugiyama, H.

Suortti, P.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol 58, R1–R35 (2013).
[Crossref]

Teboulle, M.

A. Beck and M. Teboulle, “A fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems,” SIAM J. Imaging Sciences 2, 183–202 (2009).
[Crossref]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C, 2 (Cambridge University, 1992, pp. 871–872).

Thomlinson, W. C.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Thuering, T.

T. Thuering, P. Modregger, B. R. Pinzer, Z. Wang, and M. Stampanoni, “Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiography,” Opt. Express 19, 25542–25558 (2011).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C, 2 (Cambridge University, 1992, pp. 871–872).

Wang, Z.

T. Thuering, P. Modregger, B. R. Pinzer, Z. Wang, and M. Stampanoni, “Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiography,” Opt. Express 19, 25542–25558 (2011).

Weitkamp, T.

T. Weitkamp, A. Diaz, Ch. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express. 13, 6296–6304 (2005).
[Crossref] [PubMed]

Wernick, M. N.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, Ch. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

White, D. R.

G. R. Hammerstein, D. W. Miller, D. R. White, M. E. Masterson, H. Q. Woodard, and J. S. Laughlin, “Absorbed radiation dose in mammography,” Radiology 130, 485–491 (1979).
[PubMed]

Woodard, H. Q.

G. R. Hammerstein, D. W. Miller, D. R. White, M. E. Masterson, H. Q. Woodard, and J. S. Laughlin, “Absorbed radiation dose in mammography,” Radiology 130, 485–491 (1979).
[PubMed]

Wu, X. Y.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Yang, Y.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, Ch. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

Yausa, T.

A. Maksimenko, M. Ando, S. Hiroshi, and T. Yausa, “Computed tomographic reconstruction based on x-ray refraction contrast,” Appl. Phys. Lett. 86, 124105 (2005).
[Crossref]

Yuasa, T.

Zhong, Z.

M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, Ch. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Ziegler, E.

T. Weitkamp, A. Diaz, Ch. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express. 13, 6296–6304 (2005).
[Crossref] [PubMed]

Zienkiewicz, O. C.

O. C. Zienkiewicz and K. Morgan, Finite Elements and Approximation (Dover Pubn. Inc., 2006, Chap. 3).

Appl. Opt. (1)

Appl. Phys. Lett. (2)

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Figures (4)

Fig. 1
Fig. 1 A chart illustrating several possible ways to perform refractive index CT in the X-ray spectral range. The method adapted in this work is highlighted by the dashed line.

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Fig. 2
Fig. 2 Description of the coordinate systems used in the formalism.

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Fig. 3
Fig. 3 Reconstruction of the materials density in the test phantom obtained with: (a) FBP algorithm for gradient projection, (b) presented method, (c) phase-retrieval based approach, and (d) FBP algorithm for gradient projection applied to the analytically calculated deflection angles. Panel (e) shows profiles of the material density taken along vertical line crossing the center of each image: case (a) is shown by the sparsely dashed blue line, (b) - black dotted line, (c) - dash-dotted red line, (d) - solid magenta line; cyan dashed line indicates the tabulated values.

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Fig. 4
Fig. 4 Images of the refractive index decrement δ in a slice of a formalin-fixed human breast sample obtained using (a) FBP algorithm for gradient projections, (b) the adapted vectorial approach, and (c) by means of the phase-retrieval-based method. In all images the darker gray levels correspond to smaller values of δ (darkest parts are the adipose tissue, this area is indicated with the “+” sign in the panel (b)) and the brighter gray represents larger values of δ (almost white parts are the skin, indicated with the “o” sign in panel (b); the formalin and glandular tissues are correspondingly indicated with “*” and “Δ” signs). Images (d), (g), and (j) represent the fraction of materials in the corresponding slices versus the material density derived from δ (number of bins is 255 for the density interval 0.8–1.2 g/cm3). Insets (e), (h), and (k) show a group of microcalcifications (bright white spots) in the dotted rectangle (1) at the left side of the corresponding slices (note that there are possibly two closely spaced calcifications). A magnified region (marked by dotted rectangles (2) in each slice) that contains soft tissues is presented in panels (f), (i), and (l). Strands of fat within the skin as well as fibrous strands within the fatty tissue beneath are most accurately depicted in image (i).

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Equations (12)

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d d s [ n ( r ) t ( r ) ] = ∇ n ( r ) ,
d n d s = ∇ n ⋅ t ,     d α d s = ∇ n ⋅ v ,
α ( y ′ , θ ) sin θ = ∫ ∂ n ( y ′ , z ′ ) ∂ z ′ d z ′ ; α ( y ′ , θ ) cos θ = ∫ ∂ n ( y ′ , z ′ ) ∂ y ′ d z ′ ,
∇ n = f ,
Δ n = div f ,
α ≈ ∫ ∂ n ( y , z ) ∂ y d z ,
α ( y ′ , θ ) ≈ ∂ ∂ y ′ ∫ n ( y ′ , z ′ ) d z ′ .
δ ˜ ( y , z ) = ∫ 0 π [ α ( y ′ , θ ) * g ( y ′ ) ] y ′ = − z sin θ + y cos θ d θ ,
ϕ ( y ′ , θ ) = k ⋅ ∫ δ ( y ′ , z ′ ) d z ′
∂ ϕ ( x , y ′ , θ ) ∂ y ′ = k α ( x , y ′ , θ ) ,
∂ 2 ϕ ( x , y ′ , θ ) ∂ 2 y ′ = k ∂ α ( x , y ′ , θ ) ∂ y ′ ,
∂ 2 ϕ ( x , y ′ , θ ) ∂ 2 y ′ + γ ∂ 2 ϕ ( x , y ′ , θ ) ∂ 2 x = k ∂ α ( x , y ′ , θ ) ∂ y ′ .

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