## Tomographic scanning imager

Optics Express, Vol. 17, Issue 14, pp. 11371-11387 (2009)

http://dx.doi.org/10.1364/OE.17.011371

Acrobat PDF (3782 KB)

### 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

## 1. Introduction

### 1.1 Background

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

12. H. Hovland, “Tomographic scanning imaging seeker,” Proc. SPIE **5430**, 76–85 (2004).
[CrossRef]

14. H. Hovland, “Optimization of the tomographic scanning (TOSCA) imager,” Proc. SPIE **1478**, 65690I–656910 (2007).
[CrossRef]

## 2. Theory

### 2.1 Tomography

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

### 2.2 The con-scan reticle sensor

### 2.3 The TOSCA principle

13. H. Hovland, “Specialized tomographic scanning imaging seeker,” Proc. SPIE **5778**, 725–731 (2005).
[CrossRef]

### 2.4 Reconstruction in the continuous case

*I*(

**r**,

*t*), the “apertured” image becomes:

**r**is a image plane vector and

*A*(

**r**) is the transmission of the aperture with radius

*r*. Due to the aperture, the reticle transmission can be modeled as a sequence of unit step functions:

_{0}*i*denotes the knife-edge index,

**k**

*=(cos*

_{i}*θ*, sin

_{i}*θ*) denotes a unit vector normal to the knife-edge and oriented towards the transmitting region,

_{i}**r**

*i*(

*t*) is a point on the knife-edge, and u(

*x*) is a unit step function. The signal as the scene is scanned across the knife-edges becomes:

*P*is the image plane. The first order time derivative of this signal is given by:

*L*as:

_{i}*t*,

*θ*is the angle of the

_{i}*i*

^{th}knife-edge normal, and

*τ*is the time at which the knife-edge crosses the origin.

_{i}*V*is defined as the instantaneous scan speed:

_{C,i}*V*(

_{C,i}*t*) does not change its sign during the knife-edge scan, we can define an effective scan distance

*T*:

_{i}*τ*is the time at which the knife edge crosses the origin. The time derivative of

_{i}*T*is:

_{i}*I*(

_{A}*x,y*) as:

### 2.5 Narrow slit reticle

### 2.6 Reconstruction algorithm

### 2.7 Reconstruction in the discrete case

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

*t*is the sampling time interval, and the other increments, annotated with a Δ, correspond to the value differences between samples. The time

*t*is subscripted with a

*j*to indicate the sampling times, whereas the subscript

*i*denotes the

*i*

^{th}knife-edge scan. Spatial information is still continuous, as it corresponds to the physical integration.

*T*of the scan distance

_{i,j}*T*. The Radon transform is defined as before, but the sampled values

_{i}*T*replace the continuous values. Equation (12) then becomes:

_{i,j}*T*and

_{i}*t*by

*T*and

_{i,j}*t*.

_{j}*T*. By resampling the signals obtained in this known geometry, it is possible to find the values for constant scan distances. We may then follow the method proposed in for example [17

_{i,j}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

**1988**). http://www.slaney.org/pct/pct-toc.html

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]

## 3. TOSCA optimization and improvements

### 3.1 Reticle shapes

*r*and the scan circle radius

_{Aperture}*R*. The maximum number of non-overlapping knife-edge scans is:

_{Scan circle}*π*/(

*N*-1).

*V*varies as:

_{C,i}*ω*is the angular scan speed, and

*φ*is the angle between the scan line normal and the tangent of the scan circle where it crosses the scan line. The scan speed variation for the radial configuration is identical for all knife edges, symmetrical around each scan midpoint and approximately parabolic. The ratio between the lowest and the highest scan speed is given by:

_{i}### 3.2 Circular detector array configuration

*s*is the detector element width,

*n*being the number of elements (of length

*ns*). Compared to the smallest reticle based configuration, this corresponds to a scan circle radius reduction by a factor

*n*/(

*π*+1). With

*n*=129 and

*s*=30 µm, the scan radius is reduced to 2.6 mm. As the scan speed is proportional to the required frame rate and the scan radius, it can be reduced, along with the required detector bandwidth, leading to less detector and background noise. This increases the scan speed variations accordingly, but these variations can be accounted for in the re-sampling and/or filtering calculations. Another advantage is that a larger fill factor can be achieved, leading to a higher sensor responsivity.

## 4. Noise considerations

*n*×

*n*pixels, and it is assumed that

*n*angular line scans are required to reconstruct the image with sufficient fidelity. The spot size and the frame rate are denoted

*s*and

*F*, respectively.

*n*

^{2}

*s*

^{2}.

*n*

^{1/2}.

*N*is the active area normalized by the spot size (i.e. the number of pixels covered by the active area) and 〈φ〉 is the expected sample noise level produced in each pixel, the expected sample noise level 〈

_{A}*N*〉 in the whole active area, where the spatial noise adds incoherently, is:

_{Sample}*S*〉. The total signal contribution 〈

_{Sample}*S*〉 from one pixel in a entire frame is the coherent linear addition from all the line scans, thus we get a total frame pixel signal to be:

*N*〉 to be:

## 5. Simulations

## 6. Discussion

### 6.1 Noise performance comparison of TOSCA imagers and alternative techniques

### 6.2 Multispectral imaging

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]

### 6.3 Other issues

25. J. B. Pendry, “Negative refraction makes a perfect lens”. Phys. Rev. Lett. **85**, 3966–3969 (2000)
[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]

## 7. Conclusion

## Appendix: Development of the continuous case reconstruction

*I*(

_{A}*x,y*) can be defined as:

*θ*by

_{i}*θ*’=

_{i}*θ*+

_{i}*π*, and using (A3), we have:

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

*U*,

*θ*) and Cartesian (

_{i}*X,Y*) coordinates is found using the Jacobian, and we can rewrite (A3) to get:

*g*(

*k*) as an inverse Fourier transform of

*D*(

_{i}*U*)|

*U*|:

## Acknowledgements

## References and links

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. |

3. | H. K. Hong, S. H. Han, and J. S. Choi, “Improved reticle seeker using the segmented focal plane array,” Proc. SPIE |

4. | J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, “Imaging with Frequency-Modulated Reticles,” Opt. Eng. |

5. | M. R. Wellfare, “Two-dimensional encoding of images using discrete reticles,” Proc. SPIE |

6. | J. S. Sanders and C. E. Halford, “Multispectral imaging with frequency-modulated reticles,” Proc. SPIE |

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. |

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. |

9. | I. Kopriva, H. Szu, and A. Persin, “Optical reticle trackers with the multi-source discrimination capability by using independent component analysis,” Opt. Commun. |

10. | I. Kopriva and A. Persin, “Discrimination of optical sources by use of adaptive blind source separation theory,” App. Opt. |

11. | C. Jutten and J. Herault, “Blind Separation of Sources,” Signal Process. |

12. | H. Hovland, “Tomographic scanning imaging seeker,” Proc. SPIE |

13. | H. Hovland, “Specialized tomographic scanning imaging seeker,” Proc. SPIE |

14. | H. Hovland, “Optimization of the tomographic scanning (TOSCA) imager,” Proc. SPIE |

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 , |

18. | S. R. Deans, |

19. | F. Natterer, |

20. | S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” App. Phys. B-Lasers and Optics |

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. |

22. | R. N. Bracewell and A. C. Riddle, “Inversion of Fan-Beam Scans in Radio Astronomy,” Astrophys. J. |

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. |

25. | J. B. Pendry, “Negative refraction makes a perfect lens”. Phys. Rev. Lett. |

26. | D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express |

27. | N. Fang, H. Lee, C. Sun, and X. Zhang “Sub-Diffraction-Limited Optical Imaging with a Silver Superlens,” Science |

**OCIS Codes**

(100.3020) Image processing : Image reconstruction-restoration

(100.6950) Image processing : Tomographic image processing

(110.0110) Imaging systems : Imaging systems

(110.6960) Imaging systems : Tomography

(110.4234) Imaging systems : Multispectral and hyperspectral imaging

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Image Processing

**History**

Original Manuscript: March 11, 2009

Revised Manuscript: May 22, 2009

Manuscript Accepted: June 15, 2009

Published: June 23, 2009

**Citation**

Harald Hovland, "Tomographic scanning imager," Opt. Express **17**, 11371-11387 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11371

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### References

- Federation of American Scientists web page, http://www.fas.org.
- 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]
- 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]
- 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]
- 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]
- 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]
- I. Kopriva, and A. Persin, "Discrimination of optical sources by use of adaptive blind source separation theory," Appl. Opt. 38, 1115-1126 (1999). [CrossRef]
- C. Jutten, and J. Herault, "Blind Separation of Sources," Signal Process. 24, 1, 1-10 (1991). [CrossRef]
- 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]
- 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
- 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).
- 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).
- 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]
- R. N. Bracewell, and A. C. Riddle, "Inversion of Fan-Beam Scans in Radio Astronomy," Astrophys. J. 150, 427-434 (1967). [CrossRef]
- J. Hsieh, Computed tomography principles, design, artefacts, and recent advances, (SPIE Optical Engineering Press, Bellingham, WA, 2003).
- 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]
- J. B. Pendry, "Negative refraction makes a perfect lens". Phys. Rev. Lett. 85, 3966-3969 (2000) [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]
- 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]

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