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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

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