Abstract

We present a local Fourier slice equation that enables local and sparse projection of a signal. Our result exploits that a slice in frequency space is an iso-parameter set in spherical coordinates. The projection of suitable wavelets defined separably in these coordinates can therefore be computed analytically, yielding a sequence of wavelets closed under projection. Our local Fourier slice equation then realizes projection as reconstruction with “sliced” wavelets with computational costs that scale linearly in the complexity of the projected signal. We numerically evaluate the performance of our local Fourier slice equation for synthetic test data and tomographic reconstruction, demonstrating that locality and sparsity can significantly reduce computation times and memory requirements.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, mathematisch-physikalische Klasse 69, 262–277 (1917).
  2. R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198 (1956).
    [Crossref]
  3. R. N. Bracewell, “Numerical transforms,” Science 248, 697–704 (1990).
    [Crossref] [PubMed]
  4. D. H. Garces, W. T. Rhodes, and N. M. Peña, “Projection-slice theorem: a compact notation,” J. Opt. Soc. Am. A 28, 766 (2011).
    [Crossref]
  5. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
    [Crossref]
  6. G. T. Herman, Fundamentals of Computerized Tomography, Advances in Pattern Recognition (SpringerLondon, 2009).
    [Crossref]
  7. C. L. Epstein, Introduction to the Mathematics of Medical Imaging (SIAM, 2007).
    [Crossref]
  8. R. Ng, “Fourier slice photography,” ACM Transactions on Graph. 24, 735 (2005).
    [Crossref]
  9. R. N. Bracewell and A. C. Riddle, “Inversion of fan-beam scans in radio astronomy,” The Astrophys. J. 150, 427 (1967).
    [Crossref]
  10. R. A. Crowther, D. J. DeRosier, and A. Klug, “The Reconstruction of a Three-Dimensional Structure from Projections and its Application to Electron Microscopy,” Proc. Royal Soc. A: Math. Phys. Eng. Sci. 317, 319–340 (1970).
    [Crossref]
  11. M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light Field Microscopy,” in ACM Trans. Graph. (Proceedings of ACM SIGGRAPH 2006), vol. 25 (ACM Press, 2006), p. 924.
    [Crossref]
  12. E. J. Candès and D. L. Donoho, “Curvelets and reconstruction of images from noisy radon data,” (International Society for Optics and Photonics, 2000), p. 108.
  13. J. Frikel, “Sparse regularization in limited angle tomography,” Appl. Comput. Harmon. Analysis 34, 117–141 (2013).
    [Crossref]
  14. E. Garduño, G. T. Herman, and R. Davidi, “Reconstruction from a few projections by ℓ1-minimization of the Haar transform,” Inverse Probl. 27055006 (2011).
    [Crossref]
  15. E. Garduño and G. T. Herman, “Computerized tomography with total variation and with shearlets,” Inverse Probl. 33, 044011 (2017).
    [Crossref]
  16. B. Vandeghinste, B. Goossens, R. Van Holen, C. Vanhove, A. Pizurica, S. Vandenberghe, and S. Staelens, “Iterative CT reconstruction using shearlet-based regularization,” IEEE Transactions on Nucl. Sci. 60, 3305–3317 (2013).
    [Crossref]
  17. M. V. de Hoop, H. Smith, G. Uhlmann, and R. D. van der Hilst, “Seismic imaging with the generalized Radon transform: a curvelet transform perspective,” Inverse Probl. 25, 025005 (2009).
    [Crossref]
  18. J. S. Jørgensen and E. Y. Sidky, “How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray computed tomography,” Philos. transactions. Ser. A, Math. physical, engineering sciences 373, 20140387 (2015).
    [Crossref]
  19. J. S. Jørgensen, E. Y. Sidky, P. C. Hansen, and X. Pan, “Empirical average-case relation between undersampling and sparsity in X-ray CT,” Inverse Probl. Imaging 9, 431–446 (2015).
    [Crossref]
  20. E. P. Simoncelli and W. T. Freeman, “The steerable pyramid: a flexible architecture for multi-scale derivative computation,” in Proceedings., International Conference on Image Processing, vol. 3 (IEEE Comput. Soc. Press, 1995), pp. 444–447.
  21. J. Portilla and E. P. Simoncelli, “A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients,” Int. J. Comput. Vis. 40, 49–70 (2000).
    [Crossref]
  22. E. J. Candès and D. L. Donoho, Curvelets: A surprisingly effective nonadaptive representation of objects with edges (Vanderbilt University, 1999).
  23. E. J. Candès and D. L. Donoho, “Continuous curvelet transform: I. Resolution of the Wavefront Set,” Appl. Comput. Harmon. Analysis 19, 162–197 (2005).
    [Crossref]
  24. M. Unser and D. Van De Ville, “Wavelet Steerability and the Higher-Order Riesz Transform,” IEEE Transactions on Image Process. 19, 636–652 (2010).
    [Crossref]
  25. M. Unser, N. Chenouard, and D. Van De Ville, “Steerable Pyramids and Tight Wavelet Frames in L2(Rd),” IEEE Transactions on Image Process. 20, 2705–2721 (2011).
    [Crossref]
  26. J. P. Ward and M. Unser, “Harmonic singular integrals and steerable wavelets in L2(Rd),” Appl. Comput. Harmon. Analysis 36, 183–197 (2014).
    [Crossref]
  27. C. Lessig, “Polar Wavelets in Space,” Submitt. to IEEE Signal Process. Lett. (2018).
  28. M. Unser and N. Chenouard, “A Unifying Parametric Framework for 2D Steerable Wavelet Transforms,” SIAM J. on Imaging Sci. 6, 102–135 (2013).
    [Crossref]
  29. R. Azencott, B. G. Bodmann, and M. Papadakis, “Steerlets: a novel approach to rigid-motion covariant multiscale transforms,” (2009), p. 74460A.
  30. M. Do and M. Vetterli, “The contourlet transform: an efficient directional multiresolution image representation,” IEEE Transactions on Image Process. 14, 2091–2106 (2005).
    [Crossref]
  31. D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse Multidimensional Representation using Shearlets,” in Wavelets XI, M. Papadakis, A. F. Laine, and M. A. Unser, eds. (International Society for Optics and Photonics, 2005), pp. 254–262.
  32. E. T. Quinto, “Singularities of the X-Ray Transform and Limited Data Tomography in R2 and R3,” SIAM J. on Math. Analysis 24, 1215–1225 (1993).
    [Crossref]
  33. E. T. Quinto, “Local algorithms in exterior tomography,” J. Comput. Appl. Math. 199, 141–148 (2007).
    [Crossref]
  34. C. Lessig, “Code accompanying ‘a local fourier slice equation’,” Code 1 https://figshare.com/s/e96444f24305771d51c4 (2018).
  35. F. Natterer, The Mathematics of Computerized Tomography (SIAM, 2001).
    [Crossref]
  36. L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
    [Crossref]
  37. Y. Meyer, Wavelets and Operators, vol. 37 of Cambridge Studies in Advanced Mathematics (Cambridge University, 1992), translation ed.
  38. I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
    [Crossref]
  39. S. G. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way (Academic Press, 2009), 3rd ed.
  40. M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. medica 28, 94–108 (2012).
    [Crossref]
  41. E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” Inf. Theory, IEEE Transactions on 52, 489–509 (2006).
    [Crossref]
  42. D. L. Donoho, “Compressed Sensing,” IEEE Transactions on Inf. Theory 52, 1289–1306 (2006).
    [Crossref]
  43. G. G. Walter and L. Cai, “Periodic wavelets from scratch,” J. Comput. Analysis Appl. 1, 25–41 (1999).
  44. J. D. McEwen, C. Durastanti, and Y. Wiaux, “Localisation of directional scale-discretised wavelets on the sphere,” Appl. Comput. Harmon. Analysis (2016).

2017 (1)

E. Garduño and G. T. Herman, “Computerized tomography with total variation and with shearlets,” Inverse Probl. 33, 044011 (2017).
[Crossref]

2015 (2)

J. S. Jørgensen and E. Y. Sidky, “How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray computed tomography,” Philos. transactions. Ser. A, Math. physical, engineering sciences 373, 20140387 (2015).
[Crossref]

J. S. Jørgensen, E. Y. Sidky, P. C. Hansen, and X. Pan, “Empirical average-case relation between undersampling and sparsity in X-ray CT,” Inverse Probl. Imaging 9, 431–446 (2015).
[Crossref]

2014 (1)

J. P. Ward and M. Unser, “Harmonic singular integrals and steerable wavelets in L2(Rd),” Appl. Comput. Harmon. Analysis 36, 183–197 (2014).
[Crossref]

2013 (3)

M. Unser and N. Chenouard, “A Unifying Parametric Framework for 2D Steerable Wavelet Transforms,” SIAM J. on Imaging Sci. 6, 102–135 (2013).
[Crossref]

B. Vandeghinste, B. Goossens, R. Van Holen, C. Vanhove, A. Pizurica, S. Vandenberghe, and S. Staelens, “Iterative CT reconstruction using shearlet-based regularization,” IEEE Transactions on Nucl. Sci. 60, 3305–3317 (2013).
[Crossref]

J. Frikel, “Sparse regularization in limited angle tomography,” Appl. Comput. Harmon. Analysis 34, 117–141 (2013).
[Crossref]

2012 (1)

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. medica 28, 94–108 (2012).
[Crossref]

2011 (3)

M. Unser, N. Chenouard, and D. Van De Ville, “Steerable Pyramids and Tight Wavelet Frames in L2(Rd),” IEEE Transactions on Image Process. 20, 2705–2721 (2011).
[Crossref]

E. Garduño, G. T. Herman, and R. Davidi, “Reconstruction from a few projections by ℓ1-minimization of the Haar transform,” Inverse Probl. 27055006 (2011).
[Crossref]

D. H. Garces, W. T. Rhodes, and N. M. Peña, “Projection-slice theorem: a compact notation,” J. Opt. Soc. Am. A 28, 766 (2011).
[Crossref]

2010 (1)

M. Unser and D. Van De Ville, “Wavelet Steerability and the Higher-Order Riesz Transform,” IEEE Transactions on Image Process. 19, 636–652 (2010).
[Crossref]

2009 (1)

M. V. de Hoop, H. Smith, G. Uhlmann, and R. D. van der Hilst, “Seismic imaging with the generalized Radon transform: a curvelet transform perspective,” Inverse Probl. 25, 025005 (2009).
[Crossref]

2007 (1)

E. T. Quinto, “Local algorithms in exterior tomography,” J. Comput. Appl. Math. 199, 141–148 (2007).
[Crossref]

2006 (2)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” Inf. Theory, IEEE Transactions on 52, 489–509 (2006).
[Crossref]

D. L. Donoho, “Compressed Sensing,” IEEE Transactions on Inf. Theory 52, 1289–1306 (2006).
[Crossref]

2005 (3)

E. J. Candès and D. L. Donoho, “Continuous curvelet transform: I. Resolution of the Wavefront Set,” Appl. Comput. Harmon. Analysis 19, 162–197 (2005).
[Crossref]

M. Do and M. Vetterli, “The contourlet transform: an efficient directional multiresolution image representation,” IEEE Transactions on Image Process. 14, 2091–2106 (2005).
[Crossref]

R. Ng, “Fourier slice photography,” ACM Transactions on Graph. 24, 735 (2005).
[Crossref]

2003 (1)

L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
[Crossref]

2000 (1)

J. Portilla and E. P. Simoncelli, “A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients,” Int. J. Comput. Vis. 40, 49–70 (2000).
[Crossref]

1999 (1)

G. G. Walter and L. Cai, “Periodic wavelets from scratch,” J. Comput. Analysis Appl. 1, 25–41 (1999).

1993 (1)

E. T. Quinto, “Singularities of the X-Ray Transform and Limited Data Tomography in R2 and R3,” SIAM J. on Math. Analysis 24, 1215–1225 (1993).
[Crossref]

1990 (1)

R. N. Bracewell, “Numerical transforms,” Science 248, 697–704 (1990).
[Crossref] [PubMed]

1970 (1)

R. A. Crowther, D. J. DeRosier, and A. Klug, “The Reconstruction of a Three-Dimensional Structure from Projections and its Application to Electron Microscopy,” Proc. Royal Soc. A: Math. Phys. Eng. Sci. 317, 319–340 (1970).
[Crossref]

1967 (1)

R. N. Bracewell and A. C. Riddle, “Inversion of fan-beam scans in radio astronomy,” The Astrophys. J. 150, 427 (1967).
[Crossref]

1956 (1)

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198 (1956).
[Crossref]

1917 (1)

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, mathematisch-physikalische Klasse 69, 262–277 (1917).

Adams, A.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light Field Microscopy,” in ACM Trans. Graph. (Proceedings of ACM SIGGRAPH 2006), vol. 25 (ACM Press, 2006), p. 924.
[Crossref]

Azencott, R.

R. Azencott, B. G. Bodmann, and M. Papadakis, “Steerlets: a novel approach to rigid-motion covariant multiscale transforms,” (2009), p. 74460A.

Beister, M.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. medica 28, 94–108 (2012).
[Crossref]

Bellet, D.

L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
[Crossref]

Blandin, J. J.

L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
[Crossref]

Bodmann, B. G.

R. Azencott, B. G. Bodmann, and M. Papadakis, “Steerlets: a novel approach to rigid-motion covariant multiscale transforms,” (2009), p. 74460A.

Boller, E.

L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
[Crossref]

Bracewell, R. N.

R. N. Bracewell, “Numerical transforms,” Science 248, 697–704 (1990).
[Crossref] [PubMed]

R. N. Bracewell and A. C. Riddle, “Inversion of fan-beam scans in radio astronomy,” The Astrophys. J. 150, 427 (1967).
[Crossref]

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198 (1956).
[Crossref]

Buffière, J. Y.

L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
[Crossref]

Cai, L.

G. G. Walter and L. Cai, “Periodic wavelets from scratch,” J. Comput. Analysis Appl. 1, 25–41 (1999).

Candès, E. J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” Inf. Theory, IEEE Transactions on 52, 489–509 (2006).
[Crossref]

E. J. Candès and D. L. Donoho, “Continuous curvelet transform: I. Resolution of the Wavefront Set,” Appl. Comput. Harmon. Analysis 19, 162–197 (2005).
[Crossref]

E. J. Candès and D. L. Donoho, Curvelets: A surprisingly effective nonadaptive representation of objects with edges (Vanderbilt University, 1999).

E. J. Candès and D. L. Donoho, “Curvelets and reconstruction of images from noisy radon data,” (International Society for Optics and Photonics, 2000), p. 108.

Chenouard, N.

M. Unser and N. Chenouard, “A Unifying Parametric Framework for 2D Steerable Wavelet Transforms,” SIAM J. on Imaging Sci. 6, 102–135 (2013).
[Crossref]

M. Unser, N. Chenouard, and D. Van De Ville, “Steerable Pyramids and Tight Wavelet Frames in L2(Rd),” IEEE Transactions on Image Process. 20, 2705–2721 (2011).
[Crossref]

Cloetens, P.

L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
[Crossref]

Crowther, R. A.

R. A. Crowther, D. J. DeRosier, and A. Klug, “The Reconstruction of a Three-Dimensional Structure from Projections and its Application to Electron Microscopy,” Proc. Royal Soc. A: Math. Phys. Eng. Sci. 317, 319–340 (1970).
[Crossref]

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
[Crossref]

Davidi, R.

E. Garduño, G. T. Herman, and R. Davidi, “Reconstruction from a few projections by ℓ1-minimization of the Haar transform,” Inverse Probl. 27055006 (2011).
[Crossref]

de Hoop, M. V.

M. V. de Hoop, H. Smith, G. Uhlmann, and R. D. van der Hilst, “Seismic imaging with the generalized Radon transform: a curvelet transform perspective,” Inverse Probl. 25, 025005 (2009).
[Crossref]

DeRosier, D. J.

R. A. Crowther, D. J. DeRosier, and A. Klug, “The Reconstruction of a Three-Dimensional Structure from Projections and its Application to Electron Microscopy,” Proc. Royal Soc. A: Math. Phys. Eng. Sci. 317, 319–340 (1970).
[Crossref]

Do, M.

M. Do and M. Vetterli, “The contourlet transform: an efficient directional multiresolution image representation,” IEEE Transactions on Image Process. 14, 2091–2106 (2005).
[Crossref]

Donoho, D. L.

D. L. Donoho, “Compressed Sensing,” IEEE Transactions on Inf. Theory 52, 1289–1306 (2006).
[Crossref]

E. J. Candès and D. L. Donoho, “Continuous curvelet transform: I. Resolution of the Wavefront Set,” Appl. Comput. Harmon. Analysis 19, 162–197 (2005).
[Crossref]

E. J. Candès and D. L. Donoho, Curvelets: A surprisingly effective nonadaptive representation of objects with edges (Vanderbilt University, 1999).

E. J. Candès and D. L. Donoho, “Curvelets and reconstruction of images from noisy radon data,” (International Society for Optics and Photonics, 2000), p. 108.

Durastanti, C.

J. D. McEwen, C. Durastanti, and Y. Wiaux, “Localisation of directional scale-discretised wavelets on the sphere,” Appl. Comput. Harmon. Analysis (2016).

Epstein, C. L.

C. L. Epstein, Introduction to the Mathematics of Medical Imaging (SIAM, 2007).
[Crossref]

Footer, M.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light Field Microscopy,” in ACM Trans. Graph. (Proceedings of ACM SIGGRAPH 2006), vol. 25 (ACM Press, 2006), p. 924.
[Crossref]

Freeman, W. T.

E. P. Simoncelli and W. T. Freeman, “The steerable pyramid: a flexible architecture for multi-scale derivative computation,” in Proceedings., International Conference on Image Processing, vol. 3 (IEEE Comput. Soc. Press, 1995), pp. 444–447.

Frikel, J.

J. Frikel, “Sparse regularization in limited angle tomography,” Appl. Comput. Harmon. Analysis 34, 117–141 (2013).
[Crossref]

Garces, D. H.

Garduño, E.

E. Garduño and G. T. Herman, “Computerized tomography with total variation and with shearlets,” Inverse Probl. 33, 044011 (2017).
[Crossref]

E. Garduño, G. T. Herman, and R. Davidi, “Reconstruction from a few projections by ℓ1-minimization of the Haar transform,” Inverse Probl. 27055006 (2011).
[Crossref]

Goossens, B.

B. Vandeghinste, B. Goossens, R. Van Holen, C. Vanhove, A. Pizurica, S. Vandenberghe, and S. Staelens, “Iterative CT reconstruction using shearlet-based regularization,” IEEE Transactions on Nucl. Sci. 60, 3305–3317 (2013).
[Crossref]

Hansen, P. C.

J. S. Jørgensen, E. Y. Sidky, P. C. Hansen, and X. Pan, “Empirical average-case relation between undersampling and sparsity in X-ray CT,” Inverse Probl. Imaging 9, 431–446 (2015).
[Crossref]

Herman, G. T.

E. Garduño and G. T. Herman, “Computerized tomography with total variation and with shearlets,” Inverse Probl. 33, 044011 (2017).
[Crossref]

E. Garduño, G. T. Herman, and R. Davidi, “Reconstruction from a few projections by ℓ1-minimization of the Haar transform,” Inverse Probl. 27055006 (2011).
[Crossref]

G. T. Herman, Fundamentals of Computerized Tomography, Advances in Pattern Recognition (SpringerLondon, 2009).
[Crossref]

Horowitz, M.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light Field Microscopy,” in ACM Trans. Graph. (Proceedings of ACM SIGGRAPH 2006), vol. 25 (ACM Press, 2006), p. 924.
[Crossref]

Jørgensen, J. S.

J. S. Jørgensen and E. Y. Sidky, “How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray computed tomography,” Philos. transactions. Ser. A, Math. physical, engineering sciences 373, 20140387 (2015).
[Crossref]

J. S. Jørgensen, E. Y. Sidky, P. C. Hansen, and X. Pan, “Empirical average-case relation between undersampling and sparsity in X-ray CT,” Inverse Probl. Imaging 9, 431–446 (2015).
[Crossref]

Josserond, C.

L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
[Crossref]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[Crossref]

Kalender, W. A.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. medica 28, 94–108 (2012).
[Crossref]

Klug, A.

R. A. Crowther, D. J. DeRosier, and A. Klug, “The Reconstruction of a Three-Dimensional Structure from Projections and its Application to Electron Microscopy,” Proc. Royal Soc. A: Math. Phys. Eng. Sci. 317, 319–340 (1970).
[Crossref]

Kolditz, D.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. medica 28, 94–108 (2012).
[Crossref]

Kutyniok, G.

D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse Multidimensional Representation using Shearlets,” in Wavelets XI, M. Papadakis, A. F. Laine, and M. A. Unser, eds. (International Society for Optics and Photonics, 2005), pp. 254–262.

Labate, D.

D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse Multidimensional Representation using Shearlets,” in Wavelets XI, M. Papadakis, A. F. Laine, and M. A. Unser, eds. (International Society for Optics and Photonics, 2005), pp. 254–262.

Lessig, C.

C. Lessig, “Polar Wavelets in Space,” Submitt. to IEEE Signal Process. Lett. (2018).

C. Lessig, “Code accompanying ‘a local fourier slice equation’,” Code 1 https://figshare.com/s/e96444f24305771d51c4 (2018).

Levoy, M.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light Field Microscopy,” in ACM Trans. Graph. (Proceedings of ACM SIGGRAPH 2006), vol. 25 (ACM Press, 2006), p. 924.
[Crossref]

Lim, W.-Q.

D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse Multidimensional Representation using Shearlets,” in Wavelets XI, M. Papadakis, A. F. Laine, and M. A. Unser, eds. (International Society for Optics and Photonics, 2005), pp. 254–262.

Ludwig, W.

L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
[Crossref]

Maire, E.

L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
[Crossref]

Mallat, S. G.

S. G. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way (Academic Press, 2009), 3rd ed.

McEwen, J. D.

J. D. McEwen, C. Durastanti, and Y. Wiaux, “Localisation of directional scale-discretised wavelets on the sphere,” Appl. Comput. Harmon. Analysis (2016).

Meyer, Y.

Y. Meyer, Wavelets and Operators, vol. 37 of Cambridge Studies in Advanced Mathematics (Cambridge University, 1992), translation ed.

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (SIAM, 2001).
[Crossref]

Ng, R.

R. Ng, “Fourier slice photography,” ACM Transactions on Graph. 24, 735 (2005).
[Crossref]

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light Field Microscopy,” in ACM Trans. Graph. (Proceedings of ACM SIGGRAPH 2006), vol. 25 (ACM Press, 2006), p. 924.
[Crossref]

Pan, X.

J. S. Jørgensen, E. Y. Sidky, P. C. Hansen, and X. Pan, “Empirical average-case relation between undersampling and sparsity in X-ray CT,” Inverse Probl. Imaging 9, 431–446 (2015).
[Crossref]

Papadakis, M.

R. Azencott, B. G. Bodmann, and M. Papadakis, “Steerlets: a novel approach to rigid-motion covariant multiscale transforms,” (2009), p. 74460A.

Peña, N. M.

Pizurica, A.

B. Vandeghinste, B. Goossens, R. Van Holen, C. Vanhove, A. Pizurica, S. Vandenberghe, and S. Staelens, “Iterative CT reconstruction using shearlet-based regularization,” IEEE Transactions on Nucl. Sci. 60, 3305–3317 (2013).
[Crossref]

Portilla, J.

J. Portilla and E. P. Simoncelli, “A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients,” Int. J. Comput. Vis. 40, 49–70 (2000).
[Crossref]

Quinto, E. T.

E. T. Quinto, “Local algorithms in exterior tomography,” J. Comput. Appl. Math. 199, 141–148 (2007).
[Crossref]

E. T. Quinto, “Singularities of the X-Ray Transform and Limited Data Tomography in R2 and R3,” SIAM J. on Math. Analysis 24, 1215–1225 (1993).
[Crossref]

Radon, J.

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, mathematisch-physikalische Klasse 69, 262–277 (1917).

Rhodes, W. T.

Riddle, A. C.

R. N. Bracewell and A. C. Riddle, “Inversion of fan-beam scans in radio astronomy,” The Astrophys. J. 150, 427 (1967).
[Crossref]

Romberg, J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” Inf. Theory, IEEE Transactions on 52, 489–509 (2006).
[Crossref]

Salvo, L.

L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
[Crossref]

Sidky, E. Y.

J. S. Jørgensen, E. Y. Sidky, P. C. Hansen, and X. Pan, “Empirical average-case relation between undersampling and sparsity in X-ray CT,” Inverse Probl. Imaging 9, 431–446 (2015).
[Crossref]

J. S. Jørgensen and E. Y. Sidky, “How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray computed tomography,” Philos. transactions. Ser. A, Math. physical, engineering sciences 373, 20140387 (2015).
[Crossref]

Simoncelli, E. P.

J. Portilla and E. P. Simoncelli, “A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients,” Int. J. Comput. Vis. 40, 49–70 (2000).
[Crossref]

E. P. Simoncelli and W. T. Freeman, “The steerable pyramid: a flexible architecture for multi-scale derivative computation,” in Proceedings., International Conference on Image Processing, vol. 3 (IEEE Comput. Soc. Press, 1995), pp. 444–447.

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[Crossref]

Smith, H.

M. V. de Hoop, H. Smith, G. Uhlmann, and R. D. van der Hilst, “Seismic imaging with the generalized Radon transform: a curvelet transform perspective,” Inverse Probl. 25, 025005 (2009).
[Crossref]

Staelens, S.

B. Vandeghinste, B. Goossens, R. Van Holen, C. Vanhove, A. Pizurica, S. Vandenberghe, and S. Staelens, “Iterative CT reconstruction using shearlet-based regularization,” IEEE Transactions on Nucl. Sci. 60, 3305–3317 (2013).
[Crossref]

Tao, T.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” Inf. Theory, IEEE Transactions on 52, 489–509 (2006).
[Crossref]

Uhlmann, G.

M. V. de Hoop, H. Smith, G. Uhlmann, and R. D. van der Hilst, “Seismic imaging with the generalized Radon transform: a curvelet transform perspective,” Inverse Probl. 25, 025005 (2009).
[Crossref]

Unser, M.

J. P. Ward and M. Unser, “Harmonic singular integrals and steerable wavelets in L2(Rd),” Appl. Comput. Harmon. Analysis 36, 183–197 (2014).
[Crossref]

M. Unser and N. Chenouard, “A Unifying Parametric Framework for 2D Steerable Wavelet Transforms,” SIAM J. on Imaging Sci. 6, 102–135 (2013).
[Crossref]

M. Unser, N. Chenouard, and D. Van De Ville, “Steerable Pyramids and Tight Wavelet Frames in L2(Rd),” IEEE Transactions on Image Process. 20, 2705–2721 (2011).
[Crossref]

M. Unser and D. Van De Ville, “Wavelet Steerability and the Higher-Order Riesz Transform,” IEEE Transactions on Image Process. 19, 636–652 (2010).
[Crossref]

Van De Ville, D.

M. Unser, N. Chenouard, and D. Van De Ville, “Steerable Pyramids and Tight Wavelet Frames in L2(Rd),” IEEE Transactions on Image Process. 20, 2705–2721 (2011).
[Crossref]

M. Unser and D. Van De Ville, “Wavelet Steerability and the Higher-Order Riesz Transform,” IEEE Transactions on Image Process. 19, 636–652 (2010).
[Crossref]

van der Hilst, R. D.

M. V. de Hoop, H. Smith, G. Uhlmann, and R. D. van der Hilst, “Seismic imaging with the generalized Radon transform: a curvelet transform perspective,” Inverse Probl. 25, 025005 (2009).
[Crossref]

Van Holen, R.

B. Vandeghinste, B. Goossens, R. Van Holen, C. Vanhove, A. Pizurica, S. Vandenberghe, and S. Staelens, “Iterative CT reconstruction using shearlet-based regularization,” IEEE Transactions on Nucl. Sci. 60, 3305–3317 (2013).
[Crossref]

Vandeghinste, B.

B. Vandeghinste, B. Goossens, R. Van Holen, C. Vanhove, A. Pizurica, S. Vandenberghe, and S. Staelens, “Iterative CT reconstruction using shearlet-based regularization,” IEEE Transactions on Nucl. Sci. 60, 3305–3317 (2013).
[Crossref]

Vandenberghe, S.

B. Vandeghinste, B. Goossens, R. Van Holen, C. Vanhove, A. Pizurica, S. Vandenberghe, and S. Staelens, “Iterative CT reconstruction using shearlet-based regularization,” IEEE Transactions on Nucl. Sci. 60, 3305–3317 (2013).
[Crossref]

Vanhove, C.

B. Vandeghinste, B. Goossens, R. Van Holen, C. Vanhove, A. Pizurica, S. Vandenberghe, and S. Staelens, “Iterative CT reconstruction using shearlet-based regularization,” IEEE Transactions on Nucl. Sci. 60, 3305–3317 (2013).
[Crossref]

Vetterli, M.

M. Do and M. Vetterli, “The contourlet transform: an efficient directional multiresolution image representation,” IEEE Transactions on Image Process. 14, 2091–2106 (2005).
[Crossref]

Walter, G. G.

G. G. Walter and L. Cai, “Periodic wavelets from scratch,” J. Comput. Analysis Appl. 1, 25–41 (1999).

Ward, J. P.

J. P. Ward and M. Unser, “Harmonic singular integrals and steerable wavelets in L2(Rd),” Appl. Comput. Harmon. Analysis 36, 183–197 (2014).
[Crossref]

Weiss, G.

D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse Multidimensional Representation using Shearlets,” in Wavelets XI, M. Papadakis, A. F. Laine, and M. A. Unser, eds. (International Society for Optics and Photonics, 2005), pp. 254–262.

Wiaux, Y.

J. D. McEwen, C. Durastanti, and Y. Wiaux, “Localisation of directional scale-discretised wavelets on the sphere,” Appl. Comput. Harmon. Analysis (2016).

Zabler, S.

L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
[Crossref]

ACM Transactions on Graph. (1)

R. Ng, “Fourier slice photography,” ACM Transactions on Graph. 24, 735 (2005).
[Crossref]

Appl. Comput. Harmon. Analysis (3)

J. Frikel, “Sparse regularization in limited angle tomography,” Appl. Comput. Harmon. Analysis 34, 117–141 (2013).
[Crossref]

E. J. Candès and D. L. Donoho, “Continuous curvelet transform: I. Resolution of the Wavefront Set,” Appl. Comput. Harmon. Analysis 19, 162–197 (2005).
[Crossref]

J. P. Ward and M. Unser, “Harmonic singular integrals and steerable wavelets in L2(Rd),” Appl. Comput. Harmon. Analysis 36, 183–197 (2014).
[Crossref]

Aust. J. Phys. (1)

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198 (1956).
[Crossref]

Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, mathematisch-physikalische Klasse (1)

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, mathematisch-physikalische Klasse 69, 262–277 (1917).

IEEE Transactions on Image Process. (3)

M. Do and M. Vetterli, “The contourlet transform: an efficient directional multiresolution image representation,” IEEE Transactions on Image Process. 14, 2091–2106 (2005).
[Crossref]

M. Unser and D. Van De Ville, “Wavelet Steerability and the Higher-Order Riesz Transform,” IEEE Transactions on Image Process. 19, 636–652 (2010).
[Crossref]

M. Unser, N. Chenouard, and D. Van De Ville, “Steerable Pyramids and Tight Wavelet Frames in L2(Rd),” IEEE Transactions on Image Process. 20, 2705–2721 (2011).
[Crossref]

IEEE Transactions on Inf. Theory (1)

D. L. Donoho, “Compressed Sensing,” IEEE Transactions on Inf. Theory 52, 1289–1306 (2006).
[Crossref]

IEEE Transactions on Nucl. Sci. (1)

B. Vandeghinste, B. Goossens, R. Van Holen, C. Vanhove, A. Pizurica, S. Vandenberghe, and S. Staelens, “Iterative CT reconstruction using shearlet-based regularization,” IEEE Transactions on Nucl. Sci. 60, 3305–3317 (2013).
[Crossref]

Inf. Theory, IEEE Transactions on (1)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” Inf. Theory, IEEE Transactions on 52, 489–509 (2006).
[Crossref]

Int. J. Comput. Vis. (1)

J. Portilla and E. P. Simoncelli, “A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients,” Int. J. Comput. Vis. 40, 49–70 (2000).
[Crossref]

Inverse Probl. (3)

M. V. de Hoop, H. Smith, G. Uhlmann, and R. D. van der Hilst, “Seismic imaging with the generalized Radon transform: a curvelet transform perspective,” Inverse Probl. 25, 025005 (2009).
[Crossref]

E. Garduño, G. T. Herman, and R. Davidi, “Reconstruction from a few projections by ℓ1-minimization of the Haar transform,” Inverse Probl. 27055006 (2011).
[Crossref]

E. Garduño and G. T. Herman, “Computerized tomography with total variation and with shearlets,” Inverse Probl. 33, 044011 (2017).
[Crossref]

Inverse Probl. Imaging (1)

J. S. Jørgensen, E. Y. Sidky, P. C. Hansen, and X. Pan, “Empirical average-case relation between undersampling and sparsity in X-ray CT,” Inverse Probl. Imaging 9, 431–446 (2015).
[Crossref]

J. Comput. Analysis Appl. (1)

G. G. Walter and L. Cai, “Periodic wavelets from scratch,” J. Comput. Analysis Appl. 1, 25–41 (1999).

J. Comput. Appl. Math. (1)

E. T. Quinto, “Local algorithms in exterior tomography,” J. Comput. Appl. Math. 199, 141–148 (2007).
[Crossref]

J. Opt. Soc. Am. A (1)

Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms (1)

L. Salvo, P. Cloetens, E. Maire, S. Zabler, J. J. Blandin, J. Y. Buffière, W. Ludwig, E. Boller, D. Bellet, and C. Josserond, “X-ray micro-tomography an attractive characterisation technique in materials science,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 200, 273–286 (2003).
[Crossref]

Philos. transactions. Ser. A, Math. physical, engineering sciences (1)

J. S. Jørgensen and E. Y. Sidky, “How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray computed tomography,” Philos. transactions. Ser. A, Math. physical, engineering sciences 373, 20140387 (2015).
[Crossref]

Phys. medica (1)

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. medica 28, 94–108 (2012).
[Crossref]

Proc. Royal Soc. A: Math. Phys. Eng. Sci. (1)

R. A. Crowther, D. J. DeRosier, and A. Klug, “The Reconstruction of a Three-Dimensional Structure from Projections and its Application to Electron Microscopy,” Proc. Royal Soc. A: Math. Phys. Eng. Sci. 317, 319–340 (1970).
[Crossref]

Science (1)

R. N. Bracewell, “Numerical transforms,” Science 248, 697–704 (1990).
[Crossref] [PubMed]

SIAM J. on Imaging Sci. (1)

M. Unser and N. Chenouard, “A Unifying Parametric Framework for 2D Steerable Wavelet Transforms,” SIAM J. on Imaging Sci. 6, 102–135 (2013).
[Crossref]

SIAM J. on Math. Analysis (1)

E. T. Quinto, “Singularities of the X-Ray Transform and Limited Data Tomography in R2 and R3,” SIAM J. on Math. Analysis 24, 1215–1225 (1993).
[Crossref]

The Astrophys. J. (1)

R. N. Bracewell and A. C. Riddle, “Inversion of fan-beam scans in radio astronomy,” The Astrophys. J. 150, 427 (1967).
[Crossref]

Other (16)

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[Crossref]

G. T. Herman, Fundamentals of Computerized Tomography, Advances in Pattern Recognition (SpringerLondon, 2009).
[Crossref]

C. L. Epstein, Introduction to the Mathematics of Medical Imaging (SIAM, 2007).
[Crossref]

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light Field Microscopy,” in ACM Trans. Graph. (Proceedings of ACM SIGGRAPH 2006), vol. 25 (ACM Press, 2006), p. 924.
[Crossref]

E. J. Candès and D. L. Donoho, “Curvelets and reconstruction of images from noisy radon data,” (International Society for Optics and Photonics, 2000), p. 108.

E. P. Simoncelli and W. T. Freeman, “The steerable pyramid: a flexible architecture for multi-scale derivative computation,” in Proceedings., International Conference on Image Processing, vol. 3 (IEEE Comput. Soc. Press, 1995), pp. 444–447.

D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse Multidimensional Representation using Shearlets,” in Wavelets XI, M. Papadakis, A. F. Laine, and M. A. Unser, eds. (International Society for Optics and Photonics, 2005), pp. 254–262.

Y. Meyer, Wavelets and Operators, vol. 37 of Cambridge Studies in Advanced Mathematics (Cambridge University, 1992), translation ed.

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
[Crossref]

S. G. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way (Academic Press, 2009), 3rd ed.

R. Azencott, B. G. Bodmann, and M. Papadakis, “Steerlets: a novel approach to rigid-motion covariant multiscale transforms,” (2009), p. 74460A.

C. Lessig, “Polar Wavelets in Space,” Submitt. to IEEE Signal Process. Lett. (2018).

E. J. Candès and D. L. Donoho, Curvelets: A surprisingly effective nonadaptive representation of objects with edges (Vanderbilt University, 1999).

C. Lessig, “Code accompanying ‘a local fourier slice equation’,” Code 1 https://figshare.com/s/e96444f24305771d51c4 (2018).

F. Natterer, The Mathematics of Computerized Tomography (SIAM, 2001).
[Crossref]

J. D. McEwen, C. Durastanti, and Y. Wiaux, “Localisation of directional scale-discretised wavelets on the sphere,” Appl. Comput. Harmon. Analysis (2016).

Supplementary Material (1)

NameDescription
» Code 1       Code and data for generating experimental results presented in Sec. 3

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Figures (9)

Fig. 1
Fig. 1 Left: Conceptual depiction of our construction. Middle and right: Directional polar wavelet ψ s 2 ( x ) in two dimensions (middle) and its “sliced” counter-part ψ 0 , 0 , 0 1 ( x 1 ) = h 1 ( | x 1 | ) obtained by projecting along the x2-axis (right).
Fig. 2
Fig. 2 Directional polar wavelet ψ s 3 ( x ) in ℝ3 and its projection ψ s 2 ( x 12 ) along the x3 axis, which is a two-dimensional polar wavelet. Note how the orientation of ψ s 3 ( x ) is essentially preserved under projection. This is critical for the conservation of sparsity.
Fig. 3
Fig. 3 Left: Original signal and its projections onto the x1-axis (horizontally aligned). Center columns: Basis function coefficients (top) and resulting contribution to the projected signal (bottom) for horizontal and vertical orientation of the wavelet (insets) Right, top: Coefficients f j , k 1 , t 1 of the projected signal. They are small away from the signal, providing an example for the conservation of sparsity under “slicing”. Right, bottom: projected signal obtained using the local Fourier slice equation with all (dotted, “ours”) and only the 5% largest coefficients (dashed, “ours, th”).
Fig. 4
Fig. 4 Local Fourier slice equation applied to a 2D Gaussian over the entire x1 axis and localized only over the positive one (blue, reference solution; yellow, local Fourier slice equation). The local reconstruction required 55% of the time for the one over the entire axis.
Fig. 5
Fig. 5 Relative L1, L2 and L error (w.r.t. the respective norm of the original signal) of the Fourier slice equation for the box signal (left) as a function of the projection direction (projected signals as insets). Although the error fluctuates, it is overall independent of the direction. Variations result from changes in the regularity of the projection.
Fig. 6
Fig. 6 Top: Reconstruction (blue) and 5X-magnified error (rainbow colored) for the projection (black, dashed) of the signal in Fig. 7 for different hard thresholds. Bottom: Corresponding relative error rates (left) with respect to the wavelet coefficients and execution time and non-zero elements (right) as a function of the threshold.
Fig. 7
Fig. 7 Tomographic reconstruction of the Shepp-Logan-like signal on the left for j ≤ 2, 3, 4 levels (from left to right) with 196 × 256 measurements and basis functions in [−5, 5]2. The right plot shows the relative error for j = 4.
Fig. 8
Fig. 8 Sparse tomographic reconstruction (middle) of the Shepp-Logan-like signal (left) using the local Fourier slice equation for a sparse set of basis functions concentrated around the features, see Fig. 9. The right plot shows the relative L error.
Fig. 9
Fig. 9 Left: Sparse set of basis functions used for the reconstruction in Fig. 8. Right: L2 error (blue), computation time (yellow, in seconds), and memory (green, in MBs) as a function of the sparsity (relative, 1.0 corresponds to the results in Fig. 8).

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

f ν ( y ) = ν f ( x ) d ν = P ν 1 ( f ^ | P ν ) ,
f ( x ) = s f s ψ s n ( x ) ,
proj . ψ j , k , t n ( x ) proj . ψ j , k ν , t ν n 1 ( x ) proj .
f ν ( y ) = ν f ( x ) d ν = s f s ψ s n 1 , ν ( y )
ψ ^ s ( ξ ) ψ ^ j k t ( ξ ) = ( n β j , n t e i n θ ξ ) h ^ ( 2 j | ξ | ) e i ξ , 2 j k
ψ s ( x ) ψ j k t ( x ) = 2 j 2 π n i n β j , n t e i n θ x h n ( 2 j | x k | )
f ( x ) = s f ( y ) , ψ s ( y ) ψ s ( x ) = j = 1 k 2 t = 1 N j f ( y ) , ψ j k t ( y ) ψ j k t ( x )
ψ ^ j , k , t ( ξ ) = γ ^ j , t ( ξ ¯ ) h ^ ( 2 j | ξ | ) e i ξ , 2 j k = l , m κ l m j t y l m ( ξ ¯ ) h ^ ( 2 j | ξ | ) e i ξ , 2 j k
f 2 ( x 1 ) = 1 2 π x 2 ξ 2 f ^ ( ξ 1 , ξ 2 ) e i ξ , x d ξ d x 2 .
f 2 ( x 1 ) = 1 2 π ξ 1 f ^ ( ξ 1 , 0 ) e i ξ 1 , x 1 d ξ 1
f 2 ( x 1 ) = 1 2 π ξ 1 ( s f s ψ ^ s ( ξ 1 , 0 ) ) e i ξ 1 , x 1 d ξ 1 .
f 2 ( x 1 ) = 1 2 π s f s γ ^ s ( 0 ) ξ 1 h ^ ( 2 j s | ξ | ) e i ξ 1 , k 1 s e i ξ 1 , x 1 d ξ 1
ψ s 1 ( x 1 ) = 1 2 π γ ^ s ( 0 ) ξ 1 h ^ ( 2 j s | ξ | ) e i ξ 1 , k 1 s e i ξ 1 , x 1 d ξ 1 h 1 ( 2 j x 1 k 1 s )
h 1 ( x ) = i π 8 ( E c ( z ) + E c ( z ) + 4 E c ( 4 z ) + 4 E c ( z ) )
ψ s 1 , ν ( x ν ) = ψ j s , k s ν 1 , ν ( x ν 1 ) = 1 2 π γ ^ ( θ ν ) h 1 ( 2 j x ν k s ν )
ψ s 2 , ν ( x ν ) = 1 ( 2 π ) 3 / 2 m i m β m j t , ν e i m ϕ x ν h m ( 2 j | x ν k s ν | )
β m j t , ν = l C l m ( m W l m m ( ν ) κ l m j t ) P l m ( π / 2 )
ψ s 1 , ν ( x ν ) = 1 ( 2 π ) 3 / 2 γ ^ s ( R ν ξ 3 ) h 1 ( 2 j x ν k s ν )
f 1 ( x 1 ) = s f s ψ s 1 ( x 1 ) = j , k 1 , t ( k 2 f j k t ) ψ j , k 1 , t 1 ( x 1 ) = j , k 1 , t f j , k 1 , t 1 ψ j , k 1 , t 1 ( x 1 )
m ν ( y ) = log ( I in I out ) = ν ϱ ( x ) d x n
m ν ( y ) = s ϱ s ψ s n 1 , ν ( y )
ϱ ( x ) = s ϱ s ψ s n ( x ) .
m ν ( λ i a j ) = s ϱ s ψ s n 1 , ν a ( λ i ν j ) .
( f ) ( ξ ) = f ^ ( ξ ) = 1 ( 2 π ) n / 2 x n f ( x ) e i x , ξ d x .
f ( ω ) = l = 0 m = l l f ( η ) , y l m ( η ) y l m ( ω ) = l = 0 m = l l f l m y l m ( ω )
y l m ( ω ) = y l m ( θ , ϕ ) = C l m P l m ( cos θ ) e i m ϕ
f 12 ( x 12 ) = 1 ( 2 π ) 3 / 2 ξ 12 2 f ^ ( ξ 12 , 0 ) e i ξ 12 , x 12 d ξ 12
f 12 ( x 12 ) = 1 ( 2 π ) 3 / 2 s f s ξ 12 2 ψ ^ s ( x 12 , 0 ) e i ξ 12 , x 12 d ξ 12 .
ψ ^ s 12 ( ξ 12 , 0 ) = l , m C l m κ l m j t P l m ( π / 2 ) e i m ϕ ξ 12 h ^ ( 2 j | ξ 12 | ) e i ξ 12 , k 12 .
ψ ^ s ( ξ 12 , 0 ) = m ( l C l m κ l m j t P l m ( π / 2 ) ) β m j t e i m ϕ ξ 12 h ^ ( 2 j | ξ 12 | ) e i ξ 12 , k 12
= m β m j t e i m ϕ ξ 12 h ^ ( 2 j | ξ 12 | ) e i ξ 12 , k 12 .
ψ ^ s ν ( ξ ν ) ψ ^ s ( R ν ξ 12 ) = γ ^ ( R ν ξ 12 ) h ^ ( 2 j s | ξ | ) e i R ν ξ 12 , 2 j s k s
= l , m ( m W l m m ( ν ) κ l m j s t s ) y l m ( π / 2 , ϕ ξ ) h ^ ( 2 j s | ξ | ) e i R ν ξ 12 , 2 j s k s
f 3 ( x 3 ) = R x 1 2 2 f ( x ) d x 12 = 1 ( 2 π ) 3 / 2 x 3 f ^ ( 0 , 0 , x 3 ) e i x 3 , ξ 3 d ξ 3 .
f 3 ( x 3 ) = 1 ( 2 π ) 3 / 2 s f s x 3 ψ ^ s ( 0 , 0 , x 3 ) e i x 3 , ξ 3 d ξ 3 .
ψ ^ s 3 ( 0 , 0 , x 3 ) = ( l κ l j s t s y l ( 0 , 0 ) ) γ ^ j s , t s ( 0 , 0 ) h ^ ( 2 j s | ξ | ) e i ξ 1 , 2 j s k 1 s
ψ ^ s 3 ( x 3 ) = γ ^ j s , t s ( 0 , 0 ) h ^ ( 2 j s | ξ | ) e i ξ 1 , 2 j s k 1 s .

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