• Previous Article
    A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variation based image restoration
  • IPI Home
  • This Issue
  • Next Article
    Electrical impedance tomography using a point electrode inverse scheme for complete electrode data
2011, 5(2): 341-353. doi: 10.3934/ipi.2011.5.341

3D coded aperture imaging, ill-posedness and link with incomplete data radon transform

1. 

Institut de Mathématiques et de Modélisation de Montpellier, CNRS UMR 5149 Place Eugène Bataillon, 34095 Montpellier, France

Received  June 2010 Revised  December 2010 Published  May 2011

Coded Aperture Imaging is a cheap imaging process encountered in many fields of research like optics, medical imaging, astronomy, and that has led to several good results for two dimensional reconstruction methods. However, the three dimensional reconstruction problem remains nowadays severely ill-posed, and has not yet furnished satisfactory outcomes.
    In the present study, we propose an illustration of the poorness of the data in order to operate a good inversion in the 3D case. In the context of a far-field imaging, an inversion formula is derived when the detector screen can be widely translated. This reformulates the 3D inversion problem of coded aperture imaging in terms of classical Radon transform. In the sequel, we examine more accurately this reconstruction formula, and claim that it is equivalent to solve the limited angle Radon transform problem with very restricted data.
    We thus deduce that the performances of any numerical reconstruction will remain shrank, essentially because of the physical nature of the coding process, excepted when a very strong a priori knowledge is given for the 3D source.
Citation: Jean-François Crouzet. 3D coded aperture imaging, ill-posedness and link with incomplete data radon transform. Inverse Problems & Imaging, 2011, 5 (2) : 341-353. doi: 10.3934/ipi.2011.5.341
References:
[1]

D. Barret and al., "Astrophysical Journal Letters,", \textbf{405, 405, L59 (1993).

[2]

J. Brunol, "Reconstruction d'Images Tomographiques en Médecine Nucléaire,", Thèse d'état, (1979).

[3]

J. Brunol, N. de Beaucoudrey, J. Fonroget and S. Lowenthal, Imagerie tridimensionnelle en gammagraphie,, Optics Communications, 25 (1978), 163. doi: 10.1016/0030-4018(78)90297-3.

[4]

J. Brunol and J. Fonroget, Bruit multiplex en gammagraphie par codage,, Optics Communications, 22 (1977), 301. doi: 10.1016/S0030-4018(97)90015-8.

[5]

N. de Beaucoudrey and L. Garnero, Off-axis multi-slit coding for tomographic X-Ray imaging of microplasma,, Optics Communications, 49 (1984), 103. doi: 10.1016/0030-4018(84)90371-7.

[6]

N. de Beaucoudrey, L. Garnero and J.-P. Hugonin, Imagerie tomographique par codage et reconstruction,, Traitement du signal, 5 (1988), 209.

[7]

J. Brunol, R. Sauneuf and J.-P. Gex, Micro coded aperture imaging applied to laser plasma diagnosis,, Optics Communications, 31 (1979), 129. doi: 10.1016/0030-4018(79)90287-6.

[8]

J-F. Crouzet, "La Gammagraphie par Ouverture de Codage,", Ph.D thesis, (1996).

[9]

J-F.Crouzet, Radon transform over cones and related deconvolution problems,, J. Integral Equations Appl., 13 (2001), 311. doi: 10.1216/jiea/1020254808.

[10]

M.-E. Davison, The ill-conditionated nature of the limited angle tomography problem,, SIAM Journal of Applied Mathematics, 43 (1983), 428. doi: 10.1137/0143028.

[11]

J. Fonroget, Y. Belvaux and S. Lowenthal, Fonction de transfert de modulation d'un système de Gammagraphie holographique,, Optics Communications, 15 (1975), 76. doi: 10.1016/0030-4018(75)90187-X.

[12]

S. R. Gottesman and E. E. Fenimore, New family of binary arrays for coded aperture imaging,, Applied Optics, 28 (1989), 4344. doi: 10.1364/AO.28.004344.

[13]

G.-R. Gindi, R.-G. Paxman and H.-H. Barrett, Reconstruction of an object from its coded image and object constraints,, Applied Optics, 23 (1984), 851. doi: 10.1364/AO.23.000851.

[14]

B. Honga, Z. Mub and Y. Liu, A new approach of 3D spect reconstruction for near-field coded aperture imaging,, Proceedings of SPIE, 6142 (2006).

[15]

J. Illingworth and J. Kittler, A survey of the hough transform,, Computer vision, 44 (1988).

[16]

T.-P. Kohman, Coded-aperture $x$- or $\gamma$-ray telescope with least-squares image reconstruction. I. Design considerations,, Rev. Sci. Instrum., 60 (1989), 3396. doi: 10.1063/1.1140536.

[17]

T.-P. Kohman, Coded-aperture x-ray or gamma-ray telescope with least-squares image reconstruction. II. Computer simulation,, Rev. Sci. Instrum., 60 (1989), 3410. doi: 10.1063/1.1140537.

[18]

T.-P. Kohman, Coded-aperture x-ray or gamma-ray telescope with least-squares image reconstruction. III. Data acquisition and analysis enhancements,, Rev. Sci. Instrum., 68 (1997), 2404. doi: 10.1063/1.1148124.

[19]

A.-K. Louis, Incomplete data problems in X-Ray computerized tomography: Singular value decomposition of the limited angle transform,, Numerische Mathematik, 48 (1986), 251. doi: 10.1007/BF01389474.

[20]

P. Mandrou and al., "Astronomy and Astrophysics,", \textbf{Suppl. 97, Suppl. 97, 1 (1993).

[21]

R.-S. May, Z. Akcasu and G.-F. Knoll, Gamma-Ray imaging with stochastic apertures,, Applied Optics, 13 (1974), 2589. doi: 10.1364/AO.13.002589.

[22]

F. Natterer, "The Mathematics of Computerized Tomography,'', John Wiley & Sons, (1986).

[23]

N. Ohyama, T. Honda and J. Tsujiuchi, Tomogram reconstruction using advanced coded aperture imaging,, Optics Communications, 36 (1981), 434. doi: 10.1016/0030-4018(81)90184-X.

[24]

R. G. Paxman, W. E. Smith and H. H. Barrett, Two algorithms for use with an orthogonal-view coded-aperture system,, J. Nucl. Med., 25 (1984), 700.

[25]

C. Zhou and S. K. Nayar, "What are Good Apertures for Defocus Deblurring?,", IEEE International Conference on Computational Photography, (2009). doi: 10.1109/ICCPHOT.2009.5559018.

show all references

References:
[1]

D. Barret and al., "Astrophysical Journal Letters,", \textbf{405, 405, L59 (1993).

[2]

J. Brunol, "Reconstruction d'Images Tomographiques en Médecine Nucléaire,", Thèse d'état, (1979).

[3]

J. Brunol, N. de Beaucoudrey, J. Fonroget and S. Lowenthal, Imagerie tridimensionnelle en gammagraphie,, Optics Communications, 25 (1978), 163. doi: 10.1016/0030-4018(78)90297-3.

[4]

J. Brunol and J. Fonroget, Bruit multiplex en gammagraphie par codage,, Optics Communications, 22 (1977), 301. doi: 10.1016/S0030-4018(97)90015-8.

[5]

N. de Beaucoudrey and L. Garnero, Off-axis multi-slit coding for tomographic X-Ray imaging of microplasma,, Optics Communications, 49 (1984), 103. doi: 10.1016/0030-4018(84)90371-7.

[6]

N. de Beaucoudrey, L. Garnero and J.-P. Hugonin, Imagerie tomographique par codage et reconstruction,, Traitement du signal, 5 (1988), 209.

[7]

J. Brunol, R. Sauneuf and J.-P. Gex, Micro coded aperture imaging applied to laser plasma diagnosis,, Optics Communications, 31 (1979), 129. doi: 10.1016/0030-4018(79)90287-6.

[8]

J-F. Crouzet, "La Gammagraphie par Ouverture de Codage,", Ph.D thesis, (1996).

[9]

J-F.Crouzet, Radon transform over cones and related deconvolution problems,, J. Integral Equations Appl., 13 (2001), 311. doi: 10.1216/jiea/1020254808.

[10]

M.-E. Davison, The ill-conditionated nature of the limited angle tomography problem,, SIAM Journal of Applied Mathematics, 43 (1983), 428. doi: 10.1137/0143028.

[11]

J. Fonroget, Y. Belvaux and S. Lowenthal, Fonction de transfert de modulation d'un système de Gammagraphie holographique,, Optics Communications, 15 (1975), 76. doi: 10.1016/0030-4018(75)90187-X.

[12]

S. R. Gottesman and E. E. Fenimore, New family of binary arrays for coded aperture imaging,, Applied Optics, 28 (1989), 4344. doi: 10.1364/AO.28.004344.

[13]

G.-R. Gindi, R.-G. Paxman and H.-H. Barrett, Reconstruction of an object from its coded image and object constraints,, Applied Optics, 23 (1984), 851. doi: 10.1364/AO.23.000851.

[14]

B. Honga, Z. Mub and Y. Liu, A new approach of 3D spect reconstruction for near-field coded aperture imaging,, Proceedings of SPIE, 6142 (2006).

[15]

J. Illingworth and J. Kittler, A survey of the hough transform,, Computer vision, 44 (1988).

[16]

T.-P. Kohman, Coded-aperture $x$- or $\gamma$-ray telescope with least-squares image reconstruction. I. Design considerations,, Rev. Sci. Instrum., 60 (1989), 3396. doi: 10.1063/1.1140536.

[17]

T.-P. Kohman, Coded-aperture x-ray or gamma-ray telescope with least-squares image reconstruction. II. Computer simulation,, Rev. Sci. Instrum., 60 (1989), 3410. doi: 10.1063/1.1140537.

[18]

T.-P. Kohman, Coded-aperture x-ray or gamma-ray telescope with least-squares image reconstruction. III. Data acquisition and analysis enhancements,, Rev. Sci. Instrum., 68 (1997), 2404. doi: 10.1063/1.1148124.

[19]

A.-K. Louis, Incomplete data problems in X-Ray computerized tomography: Singular value decomposition of the limited angle transform,, Numerische Mathematik, 48 (1986), 251. doi: 10.1007/BF01389474.

[20]

P. Mandrou and al., "Astronomy and Astrophysics,", \textbf{Suppl. 97, Suppl. 97, 1 (1993).

[21]

R.-S. May, Z. Akcasu and G.-F. Knoll, Gamma-Ray imaging with stochastic apertures,, Applied Optics, 13 (1974), 2589. doi: 10.1364/AO.13.002589.

[22]

F. Natterer, "The Mathematics of Computerized Tomography,'', John Wiley & Sons, (1986).

[23]

N. Ohyama, T. Honda and J. Tsujiuchi, Tomogram reconstruction using advanced coded aperture imaging,, Optics Communications, 36 (1981), 434. doi: 10.1016/0030-4018(81)90184-X.

[24]

R. G. Paxman, W. E. Smith and H. H. Barrett, Two algorithms for use with an orthogonal-view coded-aperture system,, J. Nucl. Med., 25 (1984), 700.

[25]

C. Zhou and S. K. Nayar, "What are Good Apertures for Defocus Deblurring?,", IEEE International Conference on Computational Photography, (2009). doi: 10.1109/ICCPHOT.2009.5559018.

[1]

Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems & Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033

[2]

Matthew A. Fury. Estimates for solutions of nonautonomous semilinear ill-posed problems. Conference Publications, 2015, 2015 (special) : 479-488. doi: 10.3934/proc.2015.0479

[3]

Paola Favati, Grazia Lotti, Ornella Menchi, Francesco Romani. An inner-outer regularizing method for ill-posed problems. Inverse Problems & Imaging, 2014, 8 (2) : 409-420. doi: 10.3934/ipi.2014.8.409

[4]

Matthew A. Fury. Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space. Conference Publications, 2013, 2013 (special) : 259-272. doi: 10.3934/proc.2013.2013.259

[5]

Sergiy Zhuk. Inverse problems for linear ill-posed differential-algebraic equations with uncertain parameters. Conference Publications, 2011, 2011 (Special) : 1467-1476. doi: 10.3934/proc.2011.2011.1467

[6]

Eliane Bécache, Laurent Bourgeois, Lucas Franceschini, Jérémi Dardé. Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case. Inverse Problems & Imaging, 2015, 9 (4) : 971-1002. doi: 10.3934/ipi.2015.9.971

[7]

Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609

[8]

Markus Haltmeier, Richard Kowar, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear ill-posed equations II: Applications. Inverse Problems & Imaging, 2007, 1 (3) : 507-523. doi: 10.3934/ipi.2007.1.507

[9]

Youri V. Egorov, Evariste Sanchez-Palencia. Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1293-1305. doi: 10.3934/dcds.2011.31.1293

[10]

Johann Baumeister, Barbara Kaltenbacher, Antonio Leitão. On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations. Inverse Problems & Imaging, 2010, 4 (3) : 335-350. doi: 10.3934/ipi.2010.4.335

[11]

Alfredo Lorenzi, Luca Lorenzi. A strongly ill-posed integrodifferential singular parabolic problem in the unit cube of $\mathbb{R}^n$. Evolution Equations & Control Theory, 2014, 3 (3) : 499-524. doi: 10.3934/eect.2014.3.499

[12]

Faker Ben Belgacem. Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters. Inverse Problems & Imaging, 2012, 6 (2) : 163-181. doi: 10.3934/ipi.2012.6.163

[13]

Markus Haltmeier, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear ill-posed equations I: convergence analysis. Inverse Problems & Imaging, 2007, 1 (2) : 289-298. doi: 10.3934/ipi.2007.1.289

[14]

Adriano De Cezaro, Johann Baumeister, Antonio Leitão. Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations. Inverse Problems & Imaging, 2011, 5 (1) : 1-17. doi: 10.3934/ipi.2011.5.1

[15]

Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems & Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649

[16]

Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems & Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879

[17]

Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029

[18]

T. Varslo, C E Yarman, M. Cheney, B Yazıcı. A variational approach to waveform design for synthetic-aperture imaging. Inverse Problems & Imaging, 2007, 1 (3) : 577-592. doi: 10.3934/ipi.2007.1.577

[19]

Venkateswaran P. Krishnan, Eric Todd Quinto. Microlocal aspects of common offset synthetic aperture radar imaging. Inverse Problems & Imaging, 2011, 5 (3) : 659-674. doi: 10.3934/ipi.2011.5.659

[20]

Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721

2016 Impact Factor: 1.094

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]