Abstract

We describe a new paradigm for designing hybrid imaging systems. These imaging systems use optics with a special aspheric surface to code the image so that the point-spread function or the modulation transfer function has specified characteristics. Signal processing then decodes the detected image. The coding can be done so that the depth of focus can be extended. This allows the manufacturing tolerance to be reduced, focus-related aberrations to be controlled, and imaging systems to be constructed with only one optical element plus some signal processing.

© 2002 Optical Society of America

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References

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  1. W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rushforth, “Image gathering and processing for enhanced resolution,” J. Opt. Soc. Am. A 1, 241–249 (1984).
    [CrossRef]
  2. J. Ojeda-Castañeda, E. Tepichin, A. Diaz, “Arbitrary high focal depth with quasioptimum real and positive transmittance apodizer,” Appl. Opt. 28, 2666–2670 (1989).
    [CrossRef]
  3. G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
    [CrossRef]
  4. E. R. Dowski, W. T. Cathey, “Single-lens, single-image, incoherent passive ranging systems,” Appl. Opt. 33, 6762–6773 (1994).
    [CrossRef] [PubMed]
  5. G. E. Johnson, “Passive ranging systems using orthogonal encoding,” Ph.D. dissertation (University of ColoradoBoulder, Colo., 2000).
  6. E. R. Dowski, W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [CrossRef] [PubMed]
  7. H. B. Wach, E. R. Dowski, W. T. Cathey, “Control of chromatic focal shift through wave-front coding,” Appl. Opt. 37, 5359–5367 (1998).
    [CrossRef]
  8. P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, New York, 1953).
  9. K.-H. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
    [CrossRef]
  10. A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, New York, 1969).
  11. A. Papoulis, “Ambiguity function in fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  12. A. R. FitzGerrell, E. R. Dowski, W. T. Cathey, “Defocus transfer function for circularly symmetric pupils,” Appl. Opt. 36, 5796–5804 (1997).
    [CrossRef] [PubMed]
  13. E. R. Dowski, “An information theory approach to incoherent information processing systems,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 106–108.
  14. J. van der Gracht, G. W. Euliss, “Information-optimized extended depth-of-field imaging systems,” in Visual Information Processing X, S. K. Park, Z. Rahman, R. A. Schowengerdt, eds., Proc. SPIE4388, 103–112 (2001).
    [CrossRef]
  15. M. Roberts, “Athermalization of infrared optics: a review,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop II, R. E. Fischer, R. C. Juergens, eds., Proc. SPIE1049, 72–81 (1989).
    [CrossRef]
  16. T. H. Jamieson, “Thermal effects in optical systems,” Opt. Eng. 20, 156–160 (1981).
    [CrossRef]
  17. D. S. Grey, “Athermalization of optical systems,” J. Opt. Soc. Am. 38, 542–546 (1948).
    [CrossRef] [PubMed]
  18. E. R. Dowski, R. H. Cormack, S. D. Sarama, “Wavefront coding: jointly optimized optical and digital imaging systems,” in Visual Information Processing IX, S. K. Park, Z. Rahman, eds., Proc. SPIE4041, 114–120 (2000).
    [CrossRef]
  19. G. P. Berhmann, J. P. Bowen, “Influence of temperature on diffractive lens performance,” Appl. Opt. 32, 2483–2489 (1993).
    [CrossRef]

1998 (1)

1997 (1)

1995 (1)

1994 (1)

1993 (1)

1989 (1)

1984 (1)

1983 (1)

K.-H. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

1981 (1)

T. H. Jamieson, “Thermal effects in optical systems,” Opt. Eng. 20, 156–160 (1981).
[CrossRef]

1974 (1)

1972 (1)

G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

1948 (1)

Berhmann, G. P.

Bowen, J. P.

Brenner, K.-H.

K.-H. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Cathey, W. T.

Cormack, R. H.

E. R. Dowski, R. H. Cormack, S. D. Sarama, “Wavefront coding: jointly optimized optical and digital imaging systems,” in Visual Information Processing IX, S. K. Park, Z. Rahman, eds., Proc. SPIE4041, 114–120 (2000).
[CrossRef]

Diaz, A.

Dowski, E. R.

H. B. Wach, E. R. Dowski, W. T. Cathey, “Control of chromatic focal shift through wave-front coding,” Appl. Opt. 37, 5359–5367 (1998).
[CrossRef]

A. R. FitzGerrell, E. R. Dowski, W. T. Cathey, “Defocus transfer function for circularly symmetric pupils,” Appl. Opt. 36, 5796–5804 (1997).
[CrossRef] [PubMed]

E. R. Dowski, W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
[CrossRef] [PubMed]

E. R. Dowski, W. T. Cathey, “Single-lens, single-image, incoherent passive ranging systems,” Appl. Opt. 33, 6762–6773 (1994).
[CrossRef] [PubMed]

E. R. Dowski, “An information theory approach to incoherent information processing systems,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 106–108.

E. R. Dowski, R. H. Cormack, S. D. Sarama, “Wavefront coding: jointly optimized optical and digital imaging systems,” in Visual Information Processing IX, S. K. Park, Z. Rahman, eds., Proc. SPIE4041, 114–120 (2000).
[CrossRef]

Euliss, G. W.

J. van der Gracht, G. W. Euliss, “Information-optimized extended depth-of-field imaging systems,” in Visual Information Processing X, S. K. Park, Z. Rahman, R. A. Schowengerdt, eds., Proc. SPIE4388, 103–112 (2001).
[CrossRef]

FitzGerrell, A. R.

Frieden, B. R.

Grey, D. S.

Häusler, G.

G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Jamieson, T. H.

T. H. Jamieson, “Thermal effects in optical systems,” Opt. Eng. 20, 156–160 (1981).
[CrossRef]

Johnson, G. E.

G. E. Johnson, “Passive ranging systems using orthogonal encoding,” Ph.D. dissertation (University of ColoradoBoulder, Colo., 2000).

Lohmann, A.

K.-H. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Ojeda-Castañeda, J.

J. Ojeda-Castañeda, E. Tepichin, A. Diaz, “Arbitrary high focal depth with quasioptimum real and positive transmittance apodizer,” Appl. Opt. 28, 2666–2670 (1989).
[CrossRef]

K.-H. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Papoulis, A.

Rhodes, W. T.

Rihaczek, A. W.

A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, New York, 1969).

Roberts, M.

M. Roberts, “Athermalization of infrared optics: a review,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop II, R. E. Fischer, R. C. Juergens, eds., Proc. SPIE1049, 72–81 (1989).
[CrossRef]

Rushforth, C. K.

Sarama, S. D.

E. R. Dowski, R. H. Cormack, S. D. Sarama, “Wavefront coding: jointly optimized optical and digital imaging systems,” in Visual Information Processing IX, S. K. Park, Z. Rahman, eds., Proc. SPIE4041, 114–120 (2000).
[CrossRef]

Tepichin, E.

van der Gracht, J.

J. van der Gracht, G. W. Euliss, “Information-optimized extended depth-of-field imaging systems,” in Visual Information Processing X, S. K. Park, Z. Rahman, R. A. Schowengerdt, eds., Proc. SPIE4388, 103–112 (2001).
[CrossRef]

Wach, H. B.

Woodward, P. M.

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, New York, 1953).

Appl. Opt. (6)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

K.-H. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Opt. Eng. (1)

T. H. Jamieson, “Thermal effects in optical systems,” Opt. Eng. 20, 156–160 (1981).
[CrossRef]

Other (7)

G. E. Johnson, “Passive ranging systems using orthogonal encoding,” Ph.D. dissertation (University of ColoradoBoulder, Colo., 2000).

A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, New York, 1969).

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, New York, 1953).

E. R. Dowski, R. H. Cormack, S. D. Sarama, “Wavefront coding: jointly optimized optical and digital imaging systems,” in Visual Information Processing IX, S. K. Park, Z. Rahman, eds., Proc. SPIE4041, 114–120 (2000).
[CrossRef]

E. R. Dowski, “An information theory approach to incoherent information processing systems,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 106–108.

J. van der Gracht, G. W. Euliss, “Information-optimized extended depth-of-field imaging systems,” in Visual Information Processing X, S. K. Park, Z. Rahman, R. A. Schowengerdt, eds., Proc. SPIE4388, 103–112 (2001).
[CrossRef]

M. Roberts, “Athermalization of infrared optics: a review,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop II, R. E. Fischer, R. C. Juergens, eds., Proc. SPIE1049, 72–81 (1989).
[CrossRef]

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Figures (20)

Fig. 1
Fig. 1

Rectangularly separable cubic phase surface described by Eq. (1).

Fig. 2
Fig. 2

Rays of a point source focused by a one-dimensional lens (A) without and (B) with a cubic phase plate of Eq. (1) incorporated. The detail near the normal image plane is shown in (C) and (D).

Fig. 3
Fig. 3

PSFs associated with the rays of Fig. 2. The PSFs for a normal system are shown for (A) in focus and (B) out of focus. The PSFs for a coded system are shown (C) in the normal region of focus and (D) in the out-of-focus region.

Fig. 4
Fig. 4

Image of a bar pattern that was tilted at 60° to the image plane. (A) Image from a conventional system, (B) image from a coded system after signal processing, and (C) traces through both images.

Fig. 5
Fig. 5

MTFs corresponding with the PSFs of Fig. 3 for a conventional image in and out of focus and a coded image for the same misfocus values.

Fig. 6
Fig. 6

PSFs after filtering. The wave-front-coded PSFs of Figs. 3(C) and 3(D), after digital filtering with a single digital filter, are shown in (A) and (B). The in-focus filtered PSF is given in (A). The greatly out-of-focus PSF is given in (B). These two PSFs are essentially the same and nearly identical to the in-focus PSF from the traditional imaging system shown in Fig. 3(A).

Fig. 7
Fig. 7

Using depth of focus to control effects of field curvature. Images are formed on curved surfaces with field curvature. When a large depth of focus is available, the detector plane can fit within the extended depth of focus volume. All parts of the image within this volume will image clearly.

Fig. 8
Fig. 8

Using depth of focus to control axial chromatic aberration. With small depth of focus only a small region of the spectrum is imaged clearly. Other colors form blurred images. With a large depth of focus all colors can form clear sharp images.

Fig. 9
Fig. 9

Example of large depth of focus to control axial chromatic aberration. The object is from a small section of a U.S. Air Force resolution target. An imaging system with a large amount of axial chromatic aberration produces badly blurred images in red and green, whereas the blue image is sharply focused. When modified with wave-front-coding optics and signal processing, the resulting images are sharp and clear in all colors, producing a black-and-white three-color image of the black and white object.

Fig. 10
Fig. 10

Woodward function for a traditional lens with no aberrations is shown in (A). The trace through the Woodward function along the horizontal axis represents the in-focus MTF. The trace along the inclined line describes an out-of-focus MTF. Both the in-focus and the out-of-focus MTFs are shown in a traditional manner in (B).

Fig. 11
Fig. 11

Woodward function for a one-dimensional lens modified by a cubic phase function, ϕ = αx 3, is shown in (A). The optical power represented in the Woodward function for the cubic phase system is much broader than that of the traditional system. Traces of optical power through this Woodward function through the horizontal axis and at an inclined angle are displayed in the traditional manner in (B). The MTFs differ little with misfocus, in contrast to those of the traditional system of Fig. 10.

Fig. 12
Fig. 12

Image information in traditional versus wave-front-coded images with magnification of 100× and NA of 1.3. A traditional image of a leaf with oil bubbles is given in (A). The leaf occupies a small volume relative to the oil bubbles. The bubbles are so large that all are poorly imaged, and a number in the upper right-hand corner are not imaged at all. After the optics are modified with a simple cubic phase element, the image of (B) results. Note that all parts of the image are blurred equally. Even with the blur, many aspects of the object are recognizable. Note that objects in the upper right-hand corner are clearly visible in this image, whereas they were not imaged at all with the traditional system. After signal processing, the image of (C) is formed. The objects are sharp and clear. The depth of field is so large that perspective distortion is apparent.

Fig. 13
Fig. 13

IR imaging system. The fast optics, wide operating temperature, simple aluminum mounts, and lack of active thermal compensation make this a challenging design. The specifications are impossible to meet with traditional optics.

Fig. 14
Fig. 14

MTFs of (A) traditional and (B) wave-front-coded IR imaging system with temperature change. The wave-front-coded MTFs, over the same temperature range as the traditional system, are nearly identical. After signal processing, the MTFs within the passband of the digital detector closely match the ideal diffraction-limited MTF. MTFs include pixel MTFs.

Fig. 15
Fig. 15

Fast wide-field-of-view miniature imaging system with a single lens.

Fig. 16
Fig. 16

Traditional single-lens imaging systems experiences large aberrations with field angle. On-axis imaging has high MTFs, but off-axis imaging is badly degraded.

Fig. 17
Fig. 17

MTFs with field angle with wave-front coding. The MTFs even before filtering are essentially constant over the image field. After filtering, the MTFs are similar to the diffraction-limited MTF over the spatial passband of the 10-µm pixel detector. All MTFs include the detector pixel MTFs.

Fig. 18
Fig. 18

Rectangularly separable filtering of gray-scale images. Image (A) is the intermediate image after wave-front-coded optics but before digital processing. Image (B) is the intermediate image after filtering each column with a one-dimensional column filter. Note that this image has vertical resolution but little horizontal resolution. Image (C) is the resulting image after filtering with both a one-dimensional column and one-dimensional row filters.

Fig. 19
Fig. 19

Exaggerated representation of caustic in rectangularly separable cubic wave-front-coded imaging systems. During imaging, the caustic represents the movement of the centroid of the PSF with misfocus.

Fig. 20
Fig. 20

Example of filtered images and additive noise. Image (A) is a Bayer-detected three-color image made with a traditional system. The object was essentially perpendicular to the optical axis. Because of aberrations, the conventional image (A) has a soft blur. Note color aliasing of the left-hand side of George Washington in the conventional image. Image (B) is a wave-front-coded version of the same image that was time averaged to minimize additive noise. Note increased spatial resolution and reduction of color aliasing. Image (C) is a nonaveraged wave-front-coded image. The digital filter had a noise gain of approximately 5.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

z=αx3+y3
Au, v= Px+u/2P*x-u/2expj2νxdx,
Hu, ψ= Px+u/2expjx+u/22ψP*x-u/2exp-jx-u/22ψdx,
ψ=[πL2/4λ1/f-1/do-1/di=2πW20/λ=kW20,
Hu, ψ=Au, uψ/π.
log1+|Hu, ψ/σ|2,
Wo log1+So/σ2=Wo log1+SNRo,
MWo log1+So/sqrtMσ2=MWo×log1+SNRo/M.
SNRo  M,
z=αx3+y3

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