# American Institute of Mathematical Sciences

• 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
May  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). Google Scholar [2] J. Brunol, "Reconstruction d'Images Tomographiques en Médecine Nucléaire,", Thèse d'état, (1979). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] N. de Beaucoudrey, L. Garnero and J.-P. Hugonin, Imagerie tomographique par codage et reconstruction,, Traitement du signal, 5 (1988), 209. Google Scholar [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. Google Scholar [8] J-F. Crouzet, "La Gammagraphie par Ouverture de Codage,", Ph.D thesis, (1996). Google Scholar [9] J-F.Crouzet, Radon transform over cones and related deconvolution problems,, J. Integral Equations Appl., 13 (2001), 311. doi: 10.1216/jiea/1020254808. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [15] J. Illingworth and J. Kittler, A survey of the hough transform,, Computer vision, 44 (1988). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] P. Mandrou and al., "Astronomy and Astrophysics,", \textbf{Suppl. 97, Suppl. 97, 1 (1993). Google Scholar [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. Google Scholar [22] F. Natterer, "The Mathematics of Computerized Tomography,'', John Wiley & Sons, (1986). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

show all references

##### References:
 [1] D. Barret and al., "Astrophysical Journal Letters,", \textbf{405, 405, L59 (1993). Google Scholar [2] J. Brunol, "Reconstruction d'Images Tomographiques en Médecine Nucléaire,", Thèse d'état, (1979). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] N. de Beaucoudrey, L. Garnero and J.-P. Hugonin, Imagerie tomographique par codage et reconstruction,, Traitement du signal, 5 (1988), 209. Google Scholar [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. Google Scholar [8] J-F. Crouzet, "La Gammagraphie par Ouverture de Codage,", Ph.D thesis, (1996). Google Scholar [9] J-F.Crouzet, Radon transform over cones and related deconvolution problems,, J. Integral Equations Appl., 13 (2001), 311. doi: 10.1216/jiea/1020254808. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [15] J. Illingworth and J. Kittler, A survey of the hough transform,, Computer vision, 44 (1988). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] P. Mandrou and al., "Astronomy and Astrophysics,", \textbf{Suppl. 97, Suppl. 97, 1 (1993). Google Scholar [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. Google Scholar [22] F. Natterer, "The Mathematics of Computerized Tomography,'', John Wiley & Sons, (1986). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
 [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] Felix Lucka, Katharina Proksch, Christoph Brune, Nicolai Bissantz, Martin Burger, Holger Dette, Frank Wübbeling. Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations. Inverse Problems & Imaging, 2018, 12 (5) : 1121-1155. doi: 10.3934/ipi.2018047 [6] Olha P. Kupenko, Rosanna Manzo. On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1363-1393. doi: 10.3934/dcdsb.2018155 [7] 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 [8] 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 [9] 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 [10] 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 [11] Guozhi Dong, Bert Jüttler, Otmar Scherzer, Thomas Takacs. Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces. Inverse Problems & Imaging, 2017, 11 (2) : 221-246. doi: 10.3934/ipi.2017011 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [18] Lianwang Deng. Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 145-174. doi: 10.3934/cpaa.2020009 [19] Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016 [20] 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

2018 Impact Factor: 1.469