Abstract

In tomographic scanning (TOSCA) imaging, light from a scene is focused onto a reticle mask using conical scan optics, and collected on a single element detector. Alternatively, one or several detectors replace the reticle. Tomographic processing techniques are then applied to the one-dimensional signal to reproduce a two-dimensional image. The TOSCA technique is presented in detail, including its mathematical foundations and some of its limitations. It is shown how TOSCA imaging can be used in a multispectral configuration, and compares well with more conventional alternatives both in simplicity and performance. Examples of image reconstruction using TOSCA techniques are shown.

© 2009 Optical Society of America

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  1. Federation of American Scientists web page, http://www.fas.org.
  2. R. G. Driggers, C. E. Halford, and G. D. Boreman, "Marriage of Frequency-Modulation Reticles to Focal Plane Arrays," Opt. Eng. 30, 1516-1521 (1991).
    [CrossRef]
  3. H. K. Hong, S. H. Han, and J. S. Choi, "Improved reticle seeker using the segmented focal plane array," Proc. SPIE 2744, 433-440 (1996).
    [CrossRef]
  4. J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, "Imaging with Frequency-Modulated Reticles," Opt. Eng. 30, 1720-1724 (1991).
    [CrossRef]
  5. M. R. Wellfare, "Two-dimensional encoding of images using discrete reticles," Proc. SPIE 1478, 33-40 (1991).
    [CrossRef]
  6. J. S. Sanders, and C. E. Halford, "Multispectral imaging with frequency-modulated reticles," Proc. SPIE 1478, 52-63 (1991).
    [CrossRef]
  7. J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, "Imaging system using frequency modulation time division multiplexing hybrid reticle," Opt. Eng. 37, 2119-2123 (1998).
    [CrossRef]
  8. H. H. Szu, I. Kopriva, and A. Persin, "Independent component analysis approach to resolve the multi-source limitation of the nutating rising-sun reticle based optical trackers," Opt. Commun. 176, 77-89 (2000).
    [CrossRef]
  9. I. Kopriva, H. Szu, and A. Persin, "Optical reticle trackers with the multi-source discrimination capability by using independent component analysis," Opt. Commun. 203, 197-211 (2002).
    [CrossRef]
  10. I. Kopriva, and A. Persin, "Discrimination of optical sources by use of adaptive blind source separation theory," Appl. Opt. 38, 1115-1126 (1999).
    [CrossRef]
  11. C. Jutten, and J. Herault, "Blind Separation of Sources," Signal Process. 24, 1, 1-10 (1991).
    [CrossRef]
  12. H. Hovland, "Tomographic scanning imaging seeker," Proc. SPIE 5430, 76-85 (2004).
    [CrossRef]
  13. H. Hovland, "Specialized tomographic scanning imaging seeker," Proc. SPIE 5778, 725-731 (2005).
    [CrossRef]
  14. H. Hovland, "Optimization of the tomographic scanning (TOSCA) imager," Proc. SPIE 1478, 65690I-656910 (2007).
    [CrossRef]
  15. J. Radon, "Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten" Ber. Verh. S/ichs. Akad. Wiss. Leipzig, Math.-Nat. KI. 69 (1917), 262-277
  16. G. N. Hounsfield, "A method of and apparatus for examination of a body by radiation such as X- or gamma-radiation." UK Patent 1283915 (1972).
  17. A. C. Kak and M. Slaney, Principles of computerized tomographic imaging, (IEEE Press, New York, 1988). http://www.slaney.org/pct/pct-toc.html
  18. S. R. Deans, The Radon Transform and Some of Its Application, (Dover Publications Co., 1983)
  19. F. Natterer, The Mathematics of Computerized Tomography, (Wiley, New York, 1986).
  20. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "The focus of light - theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B-Lasers and Optics 72, 109-113 (2001).
  21. G. N. Ramachandran, and A. V. Lakshminarayanan, "3-Dimensional Reconstruction from Radiographs and Electron Micrographs - Application of Convolutions Instead of Fourier Transforms," Proc. Natl. Acad. Sci. U. S. A. 68, 2236-2240 (1971).
    [CrossRef] [PubMed]
  22. R. N. Bracewell, and A. C. Riddle, "Inversion of Fan-Beam Scans in Radio Astronomy," Astrophys. J. 150, 427-434 (1967).
    [CrossRef]
  23. J. Hsieh, Computed tomography principles, design, artefacts, and recent advances, (SPIE Optical Engineering Press, Bellingham, WA, 2003).
  24. P. Mourolis, R. O. Green, and T. G. Chrien, "Design of pushbroom imaging spectrometers for optimum recovery of spectroscopic and spatial information," Appl. Opt. 39, 2210-2220 (2000).
    [CrossRef]
  25. J. B. Pendry, "Negative refraction makes a perfect lens". Phys. Rev. Lett. 85, 3966-3969 (2000)
    [CrossRef] [PubMed]
  26. D. O. S. Melville, and R. J. Blaikie, "Super-resolution imaging through a planar silver layer," Opt. Express 13, 2127-2134 (2005)
    [CrossRef] [PubMed]
  27. N. Fang, H. Lee, C. Sun, and X. Zhang "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
    [CrossRef] [PubMed]

2007

H. Hovland, "Optimization of the tomographic scanning (TOSCA) imager," Proc. SPIE 1478, 65690I-656910 (2007).
[CrossRef]

2005

H. Hovland, "Specialized tomographic scanning imaging seeker," Proc. SPIE 5778, 725-731 (2005).
[CrossRef]

N. Fang, H. Lee, C. Sun, and X. Zhang "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

D. O. S. Melville, and R. J. Blaikie, "Super-resolution imaging through a planar silver layer," Opt. Express 13, 2127-2134 (2005)
[CrossRef] [PubMed]

2004

H. Hovland, "Tomographic scanning imaging seeker," Proc. SPIE 5430, 76-85 (2004).
[CrossRef]

2002

I. Kopriva, H. Szu, and A. Persin, "Optical reticle trackers with the multi-source discrimination capability by using independent component analysis," Opt. Commun. 203, 197-211 (2002).
[CrossRef]

2001

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "The focus of light - theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B-Lasers and Optics 72, 109-113 (2001).

2000

J. B. Pendry, "Negative refraction makes a perfect lens". Phys. Rev. Lett. 85, 3966-3969 (2000)
[CrossRef] [PubMed]

H. H. Szu, I. Kopriva, and A. Persin, "Independent component analysis approach to resolve the multi-source limitation of the nutating rising-sun reticle based optical trackers," Opt. Commun. 176, 77-89 (2000).
[CrossRef]

P. Mourolis, R. O. Green, and T. G. Chrien, "Design of pushbroom imaging spectrometers for optimum recovery of spectroscopic and spatial information," Appl. Opt. 39, 2210-2220 (2000).
[CrossRef]

1999

1998

J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, "Imaging system using frequency modulation time division multiplexing hybrid reticle," Opt. Eng. 37, 2119-2123 (1998).
[CrossRef]

1996

H. K. Hong, S. H. Han, and J. S. Choi, "Improved reticle seeker using the segmented focal plane array," Proc. SPIE 2744, 433-440 (1996).
[CrossRef]

1991

J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, "Imaging with Frequency-Modulated Reticles," Opt. Eng. 30, 1720-1724 (1991).
[CrossRef]

M. R. Wellfare, "Two-dimensional encoding of images using discrete reticles," Proc. SPIE 1478, 33-40 (1991).
[CrossRef]

J. S. Sanders, and C. E. Halford, "Multispectral imaging with frequency-modulated reticles," Proc. SPIE 1478, 52-63 (1991).
[CrossRef]

R. G. Driggers, C. E. Halford, and G. D. Boreman, "Marriage of Frequency-Modulation Reticles to Focal Plane Arrays," Opt. Eng. 30, 1516-1521 (1991).
[CrossRef]

C. Jutten, and J. Herault, "Blind Separation of Sources," Signal Process. 24, 1, 1-10 (1991).
[CrossRef]

1971

G. N. Ramachandran, and A. V. Lakshminarayanan, "3-Dimensional Reconstruction from Radiographs and Electron Micrographs - Application of Convolutions Instead of Fourier Transforms," Proc. Natl. Acad. Sci. U. S. A. 68, 2236-2240 (1971).
[CrossRef] [PubMed]

1967

R. N. Bracewell, and A. C. Riddle, "Inversion of Fan-Beam Scans in Radio Astronomy," Astrophys. J. 150, 427-434 (1967).
[CrossRef]

1917

J. Radon, "Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten" Ber. Verh. S/ichs. Akad. Wiss. Leipzig, Math.-Nat. KI. 69 (1917), 262-277

Bae, J. K.

J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, "Imaging system using frequency modulation time division multiplexing hybrid reticle," Opt. Eng. 37, 2119-2123 (1998).
[CrossRef]

Blaikie, R. J.

Boreman, G. D.

R. G. Driggers, C. E. Halford, and G. D. Boreman, "Marriage of Frequency-Modulation Reticles to Focal Plane Arrays," Opt. Eng. 30, 1516-1521 (1991).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, and A. C. Riddle, "Inversion of Fan-Beam Scans in Radio Astronomy," Astrophys. J. 150, 427-434 (1967).
[CrossRef]

Choi, J. S.

H. K. Hong, S. H. Han, and J. S. Choi, "Improved reticle seeker using the segmented focal plane array," Proc. SPIE 2744, 433-440 (1996).
[CrossRef]

Chrien, T. G.

Doh, Y. H.

J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, "Imaging system using frequency modulation time division multiplexing hybrid reticle," Opt. Eng. 37, 2119-2123 (1998).
[CrossRef]

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "The focus of light - theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B-Lasers and Optics 72, 109-113 (2001).

Driggers, R. G.

J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, "Imaging with Frequency-Modulated Reticles," Opt. Eng. 30, 1720-1724 (1991).
[CrossRef]

R. G. Driggers, C. E. Halford, and G. D. Boreman, "Marriage of Frequency-Modulation Reticles to Focal Plane Arrays," Opt. Eng. 30, 1516-1521 (1991).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "The focus of light - theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B-Lasers and Optics 72, 109-113 (2001).

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Glockl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "The focus of light - theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B-Lasers and Optics 72, 109-113 (2001).

Green, R. O.

Griffin, S. T.

J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, "Imaging with Frequency-Modulated Reticles," Opt. Eng. 30, 1720-1724 (1991).
[CrossRef]

Halford, C. E.

R. G. Driggers, C. E. Halford, and G. D. Boreman, "Marriage of Frequency-Modulation Reticles to Focal Plane Arrays," Opt. Eng. 30, 1516-1521 (1991).
[CrossRef]

J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, "Imaging with Frequency-Modulated Reticles," Opt. Eng. 30, 1720-1724 (1991).
[CrossRef]

J. S. Sanders, and C. E. Halford, "Multispectral imaging with frequency-modulated reticles," Proc. SPIE 1478, 52-63 (1991).
[CrossRef]

Han, S. H.

H. K. Hong, S. H. Han, and J. S. Choi, "Improved reticle seeker using the segmented focal plane array," Proc. SPIE 2744, 433-440 (1996).
[CrossRef]

Herault, J.

C. Jutten, and J. Herault, "Blind Separation of Sources," Signal Process. 24, 1, 1-10 (1991).
[CrossRef]

Hong, H. K.

H. K. Hong, S. H. Han, and J. S. Choi, "Improved reticle seeker using the segmented focal plane array," Proc. SPIE 2744, 433-440 (1996).
[CrossRef]

Hovland, H.

H. Hovland, "Optimization of the tomographic scanning (TOSCA) imager," Proc. SPIE 1478, 65690I-656910 (2007).
[CrossRef]

H. Hovland, "Specialized tomographic scanning imaging seeker," Proc. SPIE 5778, 725-731 (2005).
[CrossRef]

H. Hovland, "Tomographic scanning imaging seeker," Proc. SPIE 5430, 76-85 (2004).
[CrossRef]

Jutten, C.

C. Jutten, and J. Herault, "Blind Separation of Sources," Signal Process. 24, 1, 1-10 (1991).
[CrossRef]

Kim, S. J.

J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, "Imaging system using frequency modulation time division multiplexing hybrid reticle," Opt. Eng. 37, 2119-2123 (1998).
[CrossRef]

Kopriva, I.

I. Kopriva, H. Szu, and A. Persin, "Optical reticle trackers with the multi-source discrimination capability by using independent component analysis," Opt. Commun. 203, 197-211 (2002).
[CrossRef]

H. H. Szu, I. Kopriva, and A. Persin, "Independent component analysis approach to resolve the multi-source limitation of the nutating rising-sun reticle based optical trackers," Opt. Commun. 176, 77-89 (2000).
[CrossRef]

I. Kopriva, and A. Persin, "Discrimination of optical sources by use of adaptive blind source separation theory," Appl. Opt. 38, 1115-1126 (1999).
[CrossRef]

Lakshminarayanan, A. V.

G. N. Ramachandran, and A. V. Lakshminarayanan, "3-Dimensional Reconstruction from Radiographs and Electron Micrographs - Application of Convolutions Instead of Fourier Transforms," Proc. Natl. Acad. Sci. U. S. A. 68, 2236-2240 (1971).
[CrossRef] [PubMed]

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Leuchs, G.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "The focus of light - theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B-Lasers and Optics 72, 109-113 (2001).

Melville, D. O. S.

Mourolis, P.

Noh, D. S.

J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, "Imaging system using frequency modulation time division multiplexing hybrid reticle," Opt. Eng. 37, 2119-2123 (1998).
[CrossRef]

Pendry, J. B.

J. B. Pendry, "Negative refraction makes a perfect lens". Phys. Rev. Lett. 85, 3966-3969 (2000)
[CrossRef] [PubMed]

Persin, A.

I. Kopriva, H. Szu, and A. Persin, "Optical reticle trackers with the multi-source discrimination capability by using independent component analysis," Opt. Commun. 203, 197-211 (2002).
[CrossRef]

H. H. Szu, I. Kopriva, and A. Persin, "Independent component analysis approach to resolve the multi-source limitation of the nutating rising-sun reticle based optical trackers," Opt. Commun. 176, 77-89 (2000).
[CrossRef]

I. Kopriva, and A. Persin, "Discrimination of optical sources by use of adaptive blind source separation theory," Appl. Opt. 38, 1115-1126 (1999).
[CrossRef]

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "The focus of light - theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B-Lasers and Optics 72, 109-113 (2001).

Radon, J.

J. Radon, "Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten" Ber. Verh. S/ichs. Akad. Wiss. Leipzig, Math.-Nat. KI. 69 (1917), 262-277

Ramachandran, G. N.

G. N. Ramachandran, and A. V. Lakshminarayanan, "3-Dimensional Reconstruction from Radiographs and Electron Micrographs - Application of Convolutions Instead of Fourier Transforms," Proc. Natl. Acad. Sci. U. S. A. 68, 2236-2240 (1971).
[CrossRef] [PubMed]

Riddle, A. C.

R. N. Bracewell, and A. C. Riddle, "Inversion of Fan-Beam Scans in Radio Astronomy," Astrophys. J. 150, 427-434 (1967).
[CrossRef]

Sanders, J. S.

J. S. Sanders, and C. E. Halford, "Multispectral imaging with frequency-modulated reticles," Proc. SPIE 1478, 52-63 (1991).
[CrossRef]

J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, "Imaging with Frequency-Modulated Reticles," Opt. Eng. 30, 1720-1724 (1991).
[CrossRef]

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Szu, H.

I. Kopriva, H. Szu, and A. Persin, "Optical reticle trackers with the multi-source discrimination capability by using independent component analysis," Opt. Commun. 203, 197-211 (2002).
[CrossRef]

Szu, H. H.

H. H. Szu, I. Kopriva, and A. Persin, "Independent component analysis approach to resolve the multi-source limitation of the nutating rising-sun reticle based optical trackers," Opt. Commun. 176, 77-89 (2000).
[CrossRef]

Wellfare, M. R.

M. R. Wellfare, "Two-dimensional encoding of images using discrete reticles," Proc. SPIE 1478, 33-40 (1991).
[CrossRef]

Zhang, X.

N. Fang, H. Lee, C. Sun, and X. Zhang "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. B-Lasers and Optics

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "The focus of light - theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B-Lasers and Optics 72, 109-113 (2001).

Astrophys. J.

R. N. Bracewell, and A. C. Riddle, "Inversion of Fan-Beam Scans in Radio Astronomy," Astrophys. J. 150, 427-434 (1967).
[CrossRef]

Nat. KI.

J. Radon, "Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten" Ber. Verh. S/ichs. Akad. Wiss. Leipzig, Math.-Nat. KI. 69 (1917), 262-277

Opt. Commun.

H. H. Szu, I. Kopriva, and A. Persin, "Independent component analysis approach to resolve the multi-source limitation of the nutating rising-sun reticle based optical trackers," Opt. Commun. 176, 77-89 (2000).
[CrossRef]

I. Kopriva, H. Szu, and A. Persin, "Optical reticle trackers with the multi-source discrimination capability by using independent component analysis," Opt. Commun. 203, 197-211 (2002).
[CrossRef]

Opt. Eng.

J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, "Imaging system using frequency modulation time division multiplexing hybrid reticle," Opt. Eng. 37, 2119-2123 (1998).
[CrossRef]

R. G. Driggers, C. E. Halford, and G. D. Boreman, "Marriage of Frequency-Modulation Reticles to Focal Plane Arrays," Opt. Eng. 30, 1516-1521 (1991).
[CrossRef]

J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, "Imaging with Frequency-Modulated Reticles," Opt. Eng. 30, 1720-1724 (1991).
[CrossRef]

Opt. Express

Phys. Rev. Lett.

J. B. Pendry, "Negative refraction makes a perfect lens". Phys. Rev. Lett. 85, 3966-3969 (2000)
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U. S. A.

G. N. Ramachandran, and A. V. Lakshminarayanan, "3-Dimensional Reconstruction from Radiographs and Electron Micrographs - Application of Convolutions Instead of Fourier Transforms," Proc. Natl. Acad. Sci. U. S. A. 68, 2236-2240 (1971).
[CrossRef] [PubMed]

Proc. SPIE

H. Hovland, "Tomographic scanning imaging seeker," Proc. SPIE 5430, 76-85 (2004).
[CrossRef]

H. Hovland, "Specialized tomographic scanning imaging seeker," Proc. SPIE 5778, 725-731 (2005).
[CrossRef]

H. Hovland, "Optimization of the tomographic scanning (TOSCA) imager," Proc. SPIE 1478, 65690I-656910 (2007).
[CrossRef]

M. R. Wellfare, "Two-dimensional encoding of images using discrete reticles," Proc. SPIE 1478, 33-40 (1991).
[CrossRef]

J. S. Sanders, and C. E. Halford, "Multispectral imaging with frequency-modulated reticles," Proc. SPIE 1478, 52-63 (1991).
[CrossRef]

H. K. Hong, S. H. Han, and J. S. Choi, "Improved reticle seeker using the segmented focal plane array," Proc. SPIE 2744, 433-440 (1996).
[CrossRef]

Science

N. Fang, H. Lee, C. Sun, and X. Zhang "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Signal Process.

C. Jutten, and J. Herault, "Blind Separation of Sources," Signal Process. 24, 1, 1-10 (1991).
[CrossRef]

Other

Federation of American Scientists web page, http://www.fas.org.

G. N. Hounsfield, "A method of and apparatus for examination of a body by radiation such as X- or gamma-radiation." UK Patent 1283915 (1972).

A. C. Kak and M. Slaney, Principles of computerized tomographic imaging, (IEEE Press, New York, 1988). http://www.slaney.org/pct/pct-toc.html

S. R. Deans, The Radon Transform and Some of Its Application, (Dover Publications Co., 1983)

F. Natterer, The Mathematics of Computerized Tomography, (Wiley, New York, 1986).

J. Hsieh, Computed tomography principles, design, artefacts, and recent advances, (SPIE Optical Engineering Press, Bellingham, WA, 2003).

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Figures (8)

Fig. 1.
Fig. 1.

Schematics and possible realization of con-scan based sensors. (a) Schematics of classical con-scan sensor. (b) Schematics of TOSCA reticle based sensor. (c) Possible realization of classical con-scan sensor. (d) Possible realization of TOSCA reticle based sensor.

Fig. 2.
Fig. 2.

TOSCA scan principle. The target orientation remains constant, whereas the knife edges have a regular angular distribution. The scan follows a circular path.

Fig. 3.
Fig. 3.

Geometrical considerations. (a) Scene masked by circular aperture and knife-edge. The latter is defined by the line Li and the normal unit vector k i . Also indicated is r i , a point along Li . The indicated scan velocity gives a positive scan speed and an increasing signal. The terms in equation (4) are due to scene variations and the moving knife-edge. (b) When reconstructing an image using discrete samples, the frequencies of a shaded area are represented by the value of its centre point. The sampled value is therefore multiplied by the area it represents in Fourier space. The two darkest patches are represented by the origin in the given scan. The angles are in the range [0,π[, and the frequency values include both positive and negative values.

Fig. 4.
Fig. 4.

Possible reticle and detector array configurations. Dark sectors are transparent regions, light grey circles show the maximum size of the moving circular aperture. Slashed circles show the scan circle. Thick lines in (a) and (b) indicates (some of the) redundancies due to parallel knife edges. (a) Radial spoke knife-edge reticle. (b) Compact knife-edge reticle. (c) Thin slit reticle configuration. (d) Circular detector array configuration, enabling a smaller scan radius.

Fig. 5.
Fig. 5.

Monochrome “Lena” (a), reconstructed using 51 (b) and 501 (c) independent scans.

Fig. 6.
Fig. 6.

Normalized detector signal from knife-edge reticle based TOSCA configuration with 51 independent scans. (a) Signal from integer frame (b) Detail from 4 angular scans.

Fig. 7.
Fig. 7.

Normalized detector signal from narrow-slit reticle based TOSCA configuration with 51 independent scans. (a) Signal from integer frame (b) Detail from 4 angular scans.

Fig. 8.
Fig. 8.

Colour “Lena” (a), reconstructed using 51 (b) and 501 (c) independent scans.

Tables (1)

Tables Icon

Table 1. General parameters governing noise performance of various detector configurations for a single image frame, normalized to the 2-dimensional array detector. A channel is here a pixel in the reconstructed image, and n denotes the array length or diameter, as well as the number of independent angular scans.

Equations (54)

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IA(r,t)=I (r,t)A(r),A(r)={1,rr00,r>r0
K(r,t,i)=u(ki(rri(t))),u(x)={1,x>00,x<0
S(t,i)=rPIA(r,t)K(r,t,i)dr=rPIA(r,t)u(ki(rri(t)))dr
ddtS(t,i)=rP(ddtIA(r,t))u(ki(rri(t)))dr
+[ki(ddtri(t))] rPIA(r,t)δ(ki(rri(t)))dr
ddtS(t,i)=[ki(ddtri((t)))]rPIA(r,t)δ(ki(rri(t)))dr
Li={ri=(xi,yi),xicosθi+yisinθi=τitVC,i(t)dt}
VC,i(t)=ki(ddtri(t))
ddtS(t,i)=VC,i(t)IA(x,y)δ(xcosθi+ysinθiτitVC,i(τ)dτ)dxdy
Ti(t)=τitVC,i(τ)dτ
dTidt=VC,i(t)
Pθi(T)=[IA(x,y)]=IA(x,y)δ(xcosθi+ysinθiT)dxdy
Pθi(Ti)=1dTi(dTidt)Pθi(Ti)dt=dS(t,i)dTi
K(r,t,i)=δ (Ki(rri(t)))
S (t,i)=rPIA(r,t)K(r,t,i)dr=rPIA(r,t)δ(ki(rri(t)))dr
Pθi(Ti)=S (t,i)
ΔS(tj,i)Δt=rPΔIA(r,tj)Δtu(ki(rri(tj)))dr
+ki(Δri(tj)Δt)rPIA(r,tj)δ(ki(rri(tj)))dr
Pθi(Ti,j)=ΔS(tj,i)ΔTi,j
h(U)={U,U>0ΔU4,U=0
h(U)={U,UCU>0ΔU4,U=00,U>UC
NMax πarcsin(rApertureRScancircle)
NEff,1=NMax2π2arcsin(rApertureRScancircle)
NEff,2 NMax 1
NEff,2 < πarcsin(rAperture/Rscancircle) 1
VC,i=RScancircleωcos(ω(tτi)+φi)
VC,i,minVC,i,max=1(rAperture/RScancircle)2
VC,i,min/VC,i,max=1(2rAperture/RScancircle)2
RScan circle=ns (12+12π)=ns(π+1)2π
NSample=NA1/2φ
S=nSSample
N=n1/2NSample
S/N=n1/2Ssample/Nsample
BNarrowlinereticle=n2FF2πRScan circleS
BNarrowlinedetector=F2πRScan circles =F n (π+1)
Di(U)=FT[Pθi(T)] =12πPθi(T)ej2πUTdT
=12π(IA(x,y)δ(xcosθi+ysinθiTi)dxdy)ej2πUTidTi
=12πIA(x,y)ej2πU(xcosθi+ysinθi)dxdy
F(u,v)=FT (IA(x,y)) =12πIA(x,y)ej2π(ux+vy)dxdy
Di(U)=2πF(Ucosθi,Usinθi)
Di(U)=Di(U)
Di(U)=Di(U)
Di(U)UdθidU=2πF(Ucosθi,Usinθi)UdθidU=2πF(X,Y)dXdY
g(k)=FT1(Di(U)U)=12πDi(U)Uej2πUkdU
IA(x,y)=FT1(F(X,Y))
=(2π)1F(X,Y)ej2π(xX+yY)dXdY
=(2π)3/20π0Di(U)Uej2πU(xcos(θi)+ysin(θi))dUdθi
+(2π)3/2 π2π 0 Di (U)Uej2πU(xcos(θi)+ysin(θi))dUdθi
IA(x,y)=(2π)3/20π0Di(U)Uej2πU(xcos(θi)+ysin(θi))dUdθi
+(2π)3/20π0Di(U)Uej2πU(xcos(θi)+ysin(θi))dUdθi
=(2π)3/20π(Di(U)Uej2πU(xcos(θi)+ysin(θi)))dUdθi
=(2π)3/20πDi(U)U(δ(xcosθi+ysinθik)ej2πUkdk)dUdθi
=12π0πδ(xcosθi+ysinθik)g(k)dkdθi
=12π0πg(xcosθi+ysinθi)dθi

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