Abstract

A tomographic reconstruction algorithm is developed for the nonoverlapping redundant array x-ray imaging system whereby the background contributions from out-of-focus planes can be eliminated. The algorithm makes use of two constraints derived from the physical characteristics of the nonoverlapping redundant array system in tandem with the correlation decoding process. It is simple, direct, and noniterative. Tomographic images of computer-generated planar and three-dimensional objects are provided to illustrate the effectiveness of the algorithm.

© 1993 Optical Society of America

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References

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  1. L. Yin, J. I. Trombka, S. M. Seltzer, M. J. Bielefeld, “X-ray imaging of extended objects using nonoverlapping redundant array,” Appl. Opt. 22, 2155–2160 (1983).
    [Crossref] [PubMed]
  2. L. Yin, M. J. Bielefeld, S. M. Seltzer, J. I. Trombka, “Tomographic and analog 3-D simulations using NORA,” Appl. Opt. 23, 2239–2241 (1984).
    [Crossref] [PubMed]
  3. L. T. Chang, B. Macdonald, V. Perez-Mendez, “Axial tomography and three-dimensional image reconstruction,” IEEE Trans. Nucl. Sci. NS-23, 568–572 (1976).
    [Crossref]
  4. M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Arendt, G. R. Gindi, “Three-dimensional radiographic imaging with a restricted view angle,” J. Opt. Soc. Am. 69, 1323–1333 (1979).
    [Crossref]
  5. R. A. Vogel, D. Kirch, M. LeFree, P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
    [PubMed]
  6. M. T. LeFree, R. A. Vogel, D. L. Kirch, P. P. Steele, “Seven-pinhole tomography: a technical description,” J. Nucl. Med. 22, 48–54 (1981).
    [PubMed]
  7. J. W. van Giessen, M. A. Viergever, C. N. de Graaf, “Improved tomographic reconstruction in seven-pinhole imaging,” IEEE Trans. Med. Imag. MI-4, 91–103 (1985).
    [Crossref]
  8. J. W. van Giessen, M. A. Viergever, C. N. de Graaf, “Seven-pinhole tomography of the thyroid,” IEEE Trans. Med. Imag. MI-5, 84–90 (1986).
    [Crossref]

1986 (1)

J. W. van Giessen, M. A. Viergever, C. N. de Graaf, “Seven-pinhole tomography of the thyroid,” IEEE Trans. Med. Imag. MI-5, 84–90 (1986).
[Crossref]

1985 (1)

J. W. van Giessen, M. A. Viergever, C. N. de Graaf, “Improved tomographic reconstruction in seven-pinhole imaging,” IEEE Trans. Med. Imag. MI-4, 91–103 (1985).
[Crossref]

1984 (1)

1983 (1)

1981 (1)

M. T. LeFree, R. A. Vogel, D. L. Kirch, P. P. Steele, “Seven-pinhole tomography: a technical description,” J. Nucl. Med. 22, 48–54 (1981).
[PubMed]

1979 (1)

1978 (1)

R. A. Vogel, D. Kirch, M. LeFree, P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
[PubMed]

1976 (1)

L. T. Chang, B. Macdonald, V. Perez-Mendez, “Axial tomography and three-dimensional image reconstruction,” IEEE Trans. Nucl. Sci. NS-23, 568–572 (1976).
[Crossref]

Arendt, J. W.

Barrett, H. H.

Bielefeld, M. J.

Chang, L. T.

L. T. Chang, B. Macdonald, V. Perez-Mendez, “Axial tomography and three-dimensional image reconstruction,” IEEE Trans. Nucl. Sci. NS-23, 568–572 (1976).
[Crossref]

Chiu, M. Y.

Chou, C.

de Graaf, C. N.

J. W. van Giessen, M. A. Viergever, C. N. de Graaf, “Seven-pinhole tomography of the thyroid,” IEEE Trans. Med. Imag. MI-5, 84–90 (1986).
[Crossref]

J. W. van Giessen, M. A. Viergever, C. N. de Graaf, “Improved tomographic reconstruction in seven-pinhole imaging,” IEEE Trans. Med. Imag. MI-4, 91–103 (1985).
[Crossref]

Gindi, G. R.

Kirch, D.

R. A. Vogel, D. Kirch, M. LeFree, P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
[PubMed]

Kirch, D. L.

M. T. LeFree, R. A. Vogel, D. L. Kirch, P. P. Steele, “Seven-pinhole tomography: a technical description,” J. Nucl. Med. 22, 48–54 (1981).
[PubMed]

LeFree, M.

R. A. Vogel, D. Kirch, M. LeFree, P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
[PubMed]

LeFree, M. T.

M. T. LeFree, R. A. Vogel, D. L. Kirch, P. P. Steele, “Seven-pinhole tomography: a technical description,” J. Nucl. Med. 22, 48–54 (1981).
[PubMed]

Macdonald, B.

L. T. Chang, B. Macdonald, V. Perez-Mendez, “Axial tomography and three-dimensional image reconstruction,” IEEE Trans. Nucl. Sci. NS-23, 568–572 (1976).
[Crossref]

Perez-Mendez, V.

L. T. Chang, B. Macdonald, V. Perez-Mendez, “Axial tomography and three-dimensional image reconstruction,” IEEE Trans. Nucl. Sci. NS-23, 568–572 (1976).
[Crossref]

Seltzer, S. M.

Simpson, R. G.

Steele, P.

R. A. Vogel, D. Kirch, M. LeFree, P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
[PubMed]

Steele, P. P.

M. T. LeFree, R. A. Vogel, D. L. Kirch, P. P. Steele, “Seven-pinhole tomography: a technical description,” J. Nucl. Med. 22, 48–54 (1981).
[PubMed]

Trombka, J. I.

van Giessen, J. W.

J. W. van Giessen, M. A. Viergever, C. N. de Graaf, “Seven-pinhole tomography of the thyroid,” IEEE Trans. Med. Imag. MI-5, 84–90 (1986).
[Crossref]

J. W. van Giessen, M. A. Viergever, C. N. de Graaf, “Improved tomographic reconstruction in seven-pinhole imaging,” IEEE Trans. Med. Imag. MI-4, 91–103 (1985).
[Crossref]

Viergever, M. A.

J. W. van Giessen, M. A. Viergever, C. N. de Graaf, “Seven-pinhole tomography of the thyroid,” IEEE Trans. Med. Imag. MI-5, 84–90 (1986).
[Crossref]

J. W. van Giessen, M. A. Viergever, C. N. de Graaf, “Improved tomographic reconstruction in seven-pinhole imaging,” IEEE Trans. Med. Imag. MI-4, 91–103 (1985).
[Crossref]

Vogel, R. A.

M. T. LeFree, R. A. Vogel, D. L. Kirch, P. P. Steele, “Seven-pinhole tomography: a technical description,” J. Nucl. Med. 22, 48–54 (1981).
[PubMed]

R. A. Vogel, D. Kirch, M. LeFree, P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
[PubMed]

Yin, L.

Appl. Opt. (2)

IEEE Trans. Med. Imag. (2)

J. W. van Giessen, M. A. Viergever, C. N. de Graaf, “Improved tomographic reconstruction in seven-pinhole imaging,” IEEE Trans. Med. Imag. MI-4, 91–103 (1985).
[Crossref]

J. W. van Giessen, M. A. Viergever, C. N. de Graaf, “Seven-pinhole tomography of the thyroid,” IEEE Trans. Med. Imag. MI-5, 84–90 (1986).
[Crossref]

IEEE Trans. Nucl. Sci. (1)

L. T. Chang, B. Macdonald, V. Perez-Mendez, “Axial tomography and three-dimensional image reconstruction,” IEEE Trans. Nucl. Sci. NS-23, 568–572 (1976).
[Crossref]

J. Nucl. Med. (2)

R. A. Vogel, D. Kirch, M. LeFree, P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
[PubMed]

M. T. LeFree, R. A. Vogel, D. L. Kirch, P. P. Steele, “Seven-pinhole tomography: a technical description,” J. Nucl. Med. 22, 48–54 (1981).
[PubMed]

J. Opt. Soc. Am. (1)

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Figures (7)

Fig. 1
Fig. 1

Simplified 1-D three-pinhole NORA x-ray imaging system. Point sources S1 and S2 are imaged onto the detector through the pinholes on the NORA mask. The positions of the detected photons are labeled at the far right for tracking the photon origins. In the tomographic reconstruction of the two planes with either the correlation or the back-projection method, each plane will contain the in-focus source as well as the out-of-focus contributions, marked by X’s, from the other source. However, with both the NORA constraints and the correlation process, the out-of-focus contributions can be eliminated to give background-free tomographic reconstructions of the sources alone with the correct intensities.

Fig. 2
Fig. 2

Illustration of the depth-resolution problem in a realistic NORA system, which is caused by the restricted angle of view. A uniformly emitting planar object with edges E1 and E2 is imaged with a NORA mask of three pinholes. (This could also be a 2-D object-and-imaging system viewed in profile.) Because of the limited perspectives that are defined by the outermost pinholes P1 and P3, the planar object is indistinguishable in the reconstruction process from a volume object whose profile is given by the shaded area, which is in the shape of a kite.

Fig. 3
Fig. 3

Reconstructed tomograms of computer-generated planar objects as imaged by a simulated NORA mask containing a square array of 25 (5 × 5) 0.8-mm-diameter pinholes placed 30 mm in front of a 240 mm × 240 mm planar imaging detector consisting of 337 × 337 pixels. The three objects (an X in the plane 30 mm from the mask, a diamond frame at 37.5 mm, and a square frame at 45 mm) are assumed to be uniformly emitting and are sampled according to Poisson statistics. The distances (in millimeters) of the tomographic planes from the mask are indicated on the tomograms: (a) tomographic images obtained by means of straightforward correlation; note the large amount of out-of-focus background in all the planes and the pronounced artifacts in some planes; (b) tomographic images obtained by invoking the NORA constraints in the correlation process.

Fig. 4
Fig. 4

Sketch of a large 3-D object used in the simulation of the NORA imaging system. The total object is 60 mm × 60 mm in the xy direction and 20 mm thick (in the z direction), with the latter dimension extending from 50 mm to 70 mm from the pinhole mask. The activity in the left half is 2 times higher than that in the right half. There is a 10 mm × 10 mm square void through the left side and a 10 mm × 10 mm × 10 mm hot region with 4 times the background intensity on the right side, the latter of which starts at a distance halfway through the object’s thickness.

Fig. 5
Fig. 5

Mini-images formed on the detector from the 3-D object of Fig. 4 under the following conditions: (a) without background noise; the highest detected count per pixel is 121 and the average count is 10 per pixel; (b) with random noise sampled from a Poisson distribution having a mean of 12 counts per pixel added to the entire detector; (c) with Poisson noise having a mean of 60 counts per pixel added to the entire detector.

Fig. 6
Fig. 6

Reconstructed tomographic images, obtained with the NORA constraints, of the 3-D object of Fig. 4. The distance (in millimeters) of the tomographic planes from the mask are indicated on the tomograms. The object extends from 50 to 70 mm: (a) reconstructions from the mini-images shown in Fig. 5(a), (b) reconstructions from Fig. 5(b) obtained by subtracting a uniform background of 12 counts from all the pixels before applying the NORA algorithm, (c) Reconstructions from Fig. 5(c) obtained by subtracting a uniform background of 60 counts from all the pixels before applying the NORA algorithm.

Fig. 7
Fig. 7

Reconstructed tomographic images of the 3-D object of Fig. 4 obtained without using the NORA constraints. The distances (in millimeters) of the tomographic planes from the mask are indicated on the tomograms. The object extends from 50 to 70 mm. Reconstructions, from the mini-images shown in Fig. 5(c), were obtained by subtracting a uniform background of 60 counts from all the pixels before applying direct correlation.

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