Abstract

We continue the work by Huck et al. [ J. Opt. Soc. Am. A 2, 1644 ( 1985)] to assess image gathering and processing in terms of the information density of the acquired signal and of the fidelity of representations reproduced from this signal. The assessment is constrained by the assumptions that the system is linear and isoplanatic and that the signal and noise amplitudes are Gaussian, wide-sense stationary, and statistically independent. Within these constraints, it is found that (1) the combined process of image gathering and reconstruction (which is intended to reproduce the output of the image-gathering system) behaves as optical, or photographic, image formation in that the informationally optimized design of the image-gathering system ordinarily does not maximize the fidelity of the reconstructed image; (2) the combined process of image gathering and restoration (which is intended to reproduce the input to the image-gathering system) behaves more as a communication channel in that the informationally optimized design of the image-gathering system tends to maximize the fidelity of a variety of optimally restored representations ranging from images to edges; and (3) there exists an intuitively satisfying relationship among the informationally optimized design of image-gathering systems, the response and sensitivity of natural vision, and the reliable detection of edges.

© 1988 Optical Society of America

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References

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  1. F. O. Huck, C. L. Fales, N. Halyo, R. W. Samms, K. Stacy, “Image gathering and processing: information and fidelity,” J. Opt. Soc. Am. A 2, 1644–1666 (1985).
    [CrossRef] [PubMed]
  2. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423;Bell Syst. Tech. J. 28, 623–656 (1948);C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, Ill., 1964).
  3. P. B. Fellgett, E. H. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London 247, 369–407 (1955).
    [CrossRef]
  4. H. B. Barlow, “Critical limiting factors in the design of the eye and visual cortex,” Proc. R. Soc. London Ser. B 212, 1 (1981).
    [CrossRef]
  5. F. O. Huck, S. K. Park, “Optical-mechanical line-scan imaging process: its information capacity and efficiency,” Appl. Opt. 14, 2508–2520 (1975).
    [CrossRef] [PubMed]
  6. C. L. Fales, F. O. Huck, R. W. Samms, “Imaging system design for improved information capacity,” Appl. Opt. 23, 872–888 (1984).
    [CrossRef] [PubMed]
  7. C. L. Fales, F. O. Huck, J. A. McCormick, S. K. Park, “Wiener restoration of sampled image data: end-to-end analysis,” J. Opt. Soc. Am. A 5, 300–314 (1988).
    [CrossRef]
  8. Y. Itakura, S. Tsutsumi, T. Takagi, “Statistical properties of the background noise for the atmospheric windows in the intermediate infrared region,” Infrared Phys. 14, 17–29 (1974).
    [CrossRef]
  9. M. Kass, J. Hughes, “A stochastic image model for AI,” in Proceedings of IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 369–372.
  10. C. B. Johnson, “A method for characterizing electro-optical device modulation transfer functions,” Photogr. Sci. Eng. 15, 413–415 (1970).
  11. P. Mertz, F. Gray, “Theory of scanning and its relation to the characteristics of the transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 494–515 (1934).
  12. W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rush-forth, “Image gathering and processing for enhanced resolution,” J. Opt. Soc. Am. A 1, 241–250 (1984).
    [CrossRef]
  13. S. K. Park, R. A. Schowengerdt, “Image reconstruction by parametric cubic convolution,” Comput. Vision Graphics Image Process. 23, 258–272 (1983).
    [CrossRef]
  14. D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
    [CrossRef]
  15. D. Marr, Vision (Freeman, San Francisco, 1982).
  16. F. O. Huck, C. L. Fales, S. K. Park, D. J. Jobson, R. W. Samms, “Image-plane processing of visual information,” Appl. Opt. 23, 3160–3167 (1984).
    [CrossRef] [PubMed]
  17. W. F. Schreiber, D. E. Troxel, “Transformation between continuous and discrete representations of images, a perceptual approach,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 178–186 (1985).
    [CrossRef]
  18. W. F. Schreiber, Fundamentals of Electronic Imaging Systems (Springer-Verlag, Berlin, 1986).
    [CrossRef]

1988

1985

F. O. Huck, C. L. Fales, N. Halyo, R. W. Samms, K. Stacy, “Image gathering and processing: information and fidelity,” J. Opt. Soc. Am. A 2, 1644–1666 (1985).
[CrossRef] [PubMed]

W. F. Schreiber, D. E. Troxel, “Transformation between continuous and discrete representations of images, a perceptual approach,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 178–186 (1985).
[CrossRef]

1984

1983

S. K. Park, R. A. Schowengerdt, “Image reconstruction by parametric cubic convolution,” Comput. Vision Graphics Image Process. 23, 258–272 (1983).
[CrossRef]

1981

H. B. Barlow, “Critical limiting factors in the design of the eye and visual cortex,” Proc. R. Soc. London Ser. B 212, 1 (1981).
[CrossRef]

1980

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[CrossRef]

1975

1974

Y. Itakura, S. Tsutsumi, T. Takagi, “Statistical properties of the background noise for the atmospheric windows in the intermediate infrared region,” Infrared Phys. 14, 17–29 (1974).
[CrossRef]

1970

C. B. Johnson, “A method for characterizing electro-optical device modulation transfer functions,” Photogr. Sci. Eng. 15, 413–415 (1970).

1955

P. B. Fellgett, E. H. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London 247, 369–407 (1955).
[CrossRef]

1934

P. Mertz, F. Gray, “Theory of scanning and its relation to the characteristics of the transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 494–515 (1934).

Barlow, H. B.

H. B. Barlow, “Critical limiting factors in the design of the eye and visual cortex,” Proc. R. Soc. London Ser. B 212, 1 (1981).
[CrossRef]

Cathey, W. T.

Fales, C. L.

Fellgett, P. B.

P. B. Fellgett, E. H. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London 247, 369–407 (1955).
[CrossRef]

Frieden, B. R.

Gray, F.

P. Mertz, F. Gray, “Theory of scanning and its relation to the characteristics of the transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 494–515 (1934).

Halyo, N.

Hildreth, E.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[CrossRef]

Huck, F. O.

Hughes, J.

M. Kass, J. Hughes, “A stochastic image model for AI,” in Proceedings of IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 369–372.

Itakura, Y.

Y. Itakura, S. Tsutsumi, T. Takagi, “Statistical properties of the background noise for the atmospheric windows in the intermediate infrared region,” Infrared Phys. 14, 17–29 (1974).
[CrossRef]

Jobson, D. J.

Johnson, C. B.

C. B. Johnson, “A method for characterizing electro-optical device modulation transfer functions,” Photogr. Sci. Eng. 15, 413–415 (1970).

Kass, M.

M. Kass, J. Hughes, “A stochastic image model for AI,” in Proceedings of IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 369–372.

Linfoot, E. H.

P. B. Fellgett, E. H. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London 247, 369–407 (1955).
[CrossRef]

Marr, D.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[CrossRef]

D. Marr, Vision (Freeman, San Francisco, 1982).

McCormick, J. A.

Mertz, P.

P. Mertz, F. Gray, “Theory of scanning and its relation to the characteristics of the transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 494–515 (1934).

Park, S. K.

Rhodes, W. T.

Rush-forth, C. K.

Samms, R. W.

Schowengerdt, R. A.

S. K. Park, R. A. Schowengerdt, “Image reconstruction by parametric cubic convolution,” Comput. Vision Graphics Image Process. 23, 258–272 (1983).
[CrossRef]

Schreiber, W. F.

W. F. Schreiber, D. E. Troxel, “Transformation between continuous and discrete representations of images, a perceptual approach,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 178–186 (1985).
[CrossRef]

W. F. Schreiber, Fundamentals of Electronic Imaging Systems (Springer-Verlag, Berlin, 1986).
[CrossRef]

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423;Bell Syst. Tech. J. 28, 623–656 (1948);C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, Ill., 1964).

Stacy, K.

Takagi, T.

Y. Itakura, S. Tsutsumi, T. Takagi, “Statistical properties of the background noise for the atmospheric windows in the intermediate infrared region,” Infrared Phys. 14, 17–29 (1974).
[CrossRef]

Troxel, D. E.

W. F. Schreiber, D. E. Troxel, “Transformation between continuous and discrete representations of images, a perceptual approach,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 178–186 (1985).
[CrossRef]

Tsutsumi, S.

Y. Itakura, S. Tsutsumi, T. Takagi, “Statistical properties of the background noise for the atmospheric windows in the intermediate infrared region,” Infrared Phys. 14, 17–29 (1974).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423;Bell Syst. Tech. J. 28, 623–656 (1948);C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, Ill., 1964).

P. Mertz, F. Gray, “Theory of scanning and its relation to the characteristics of the transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 494–515 (1934).

Comput. Vision Graphics Image Process.

S. K. Park, R. A. Schowengerdt, “Image reconstruction by parametric cubic convolution,” Comput. Vision Graphics Image Process. 23, 258–272 (1983).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

W. F. Schreiber, D. E. Troxel, “Transformation between continuous and discrete representations of images, a perceptual approach,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 178–186 (1985).
[CrossRef]

Infrared Phys.

Y. Itakura, S. Tsutsumi, T. Takagi, “Statistical properties of the background noise for the atmospheric windows in the intermediate infrared region,” Infrared Phys. 14, 17–29 (1974).
[CrossRef]

J. Opt. Soc. Am. A

Philos. Trans. R. Soc. London

P. B. Fellgett, E. H. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London 247, 369–407 (1955).
[CrossRef]

Photogr. Sci. Eng.

C. B. Johnson, “A method for characterizing electro-optical device modulation transfer functions,” Photogr. Sci. Eng. 15, 413–415 (1970).

Proc. R. Soc. London Ser. B

H. B. Barlow, “Critical limiting factors in the design of the eye and visual cortex,” Proc. R. Soc. London Ser. B 212, 1 (1981).
[CrossRef]

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[CrossRef]

Other

D. Marr, Vision (Freeman, San Francisco, 1982).

M. Kass, J. Hughes, “A stochastic image model for AI,” in Proceedings of IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 369–372.

W. F. Schreiber, Fundamentals of Electronic Imaging Systems (Springer-Verlag, Berlin, 1986).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Critical design constraints.

Fig. 2
Fig. 2

Model of image gathering and processing.

Fig. 3
Fig. 3

Wiener spectra of the radiance field as characterized by the mean spatial detail μr and the frequency exponent m.

Fig. 4
Fig. 4

Random targets with the mean spatial detail μr and the sampling lattice. The Wiener spectrum of these targets is closely approximated by Eq. (4) for m = 3.

Fig. 5
Fig. 5

Responses of the image-gathering system as characterized by the shape index n and frequency index ρc.

Fig. 6
Fig. 6

Spatial-frequency responses of three data-processing algorithms. The radiance-field parameters are m = 3 and μr = 1; the SNR is LN= 32.

Fig. 7
Fig. 7

Blurring, aliasing, noise, and fidelity versus the frequency index ρc of the image-gathering response τ ̂ g ( ν , ω ) (see Fig. 5). The curves for the image reconstruction are given for three shape indexes n and one SNR LN, and the curves for the image restoration and the smoothed radiance-field transitions, or edge enhancements, are given for one shape index n and five SNR’s LN.

Fig. 8
Fig. 8

Information density h versus the frequency index ρc of the image-gathering response τ ̂ g ( ν , ω ) [Fig. 5(c)]. The results are given for several SNR’s. The radiance-field parameters are m = 3 and μr = 1; the image-gathering response shape index is n = 2.

Fig. 9
Fig. 9

Information density h versus SNR LN. Results are given for the theoretically maximum (but unrealizable) information density h0 and for the realizable information density h. The difference represents the information lost by aliasing and blurring in the presence of noise. The radiance-field parameters are m = 3 or 6 and μr = 1 (see Fig. 3).

Fig. 10
Fig. 10

(a)–(c) Information density h and maximum realizable fidelities fm and fem versus the frequency index ρc of the image-gathering response τ ̂ g ( ν , ω ) (see Fig. 5). The fidelity fm assesses the quality of images restored with the Wiener filter, and the fidelity fem assesses the quality of the smoothed radiance-field transition, or edges, extracted with the Wiener–Laplacian filter. The results are given for several shape indexes n and SNR’s LN; the radiance-field parameters are m = 3 and μr = 1. (d)–(f) The results are given for several radiance-field exponents m and SNR’s LN. The mean spatial detail is μr = 1, and the shape index is n = 2.

Fig. 11
Fig. 11

Image restoration, edge enhancement, and zero-crossing detection for two image-gathering responses τ ̂ g ( ν , ω ) as characterized by the frequency constant ρc [Fig. 5(c)]. The shape index is n = 2, the SNR is LN = 128, and the mean spatial detail is μr = 9. The images are 62 × 62 pixels. The original pictures are produced with 8×8 display elements per pixel to minimize the effects of the image-display process; however, some of the sharpness, resolution, and artifacts are lost in the graphic printing and reduction process.

Fig. 12
Fig. 12

Information density h and image fidelity f versus the frequency index ρc of the image-gathering response τ ̂ g ( ν , ω ) [see Fig. 5(c)] for three SNR’s LN. Results for image fidelity are given for the matched and several mismatched Wiener restorations and for the cubic-convolution reconstruction. The shape index is n = 2, and the mean spatial detail is μr = 1.

Fig. 13
Fig. 13

Matched and mismatched image restorations and image reconstructions for two image-gathering responses τ ̂ g ( ν , ω ) as characterized by the frequency constant ρc [see Fig. 5(c)]. The shape index is n = 2, the SNR is LN = 64, and the mean spatial detail is μr = 3. The detail μr is erroneously assumed to be 1/3 instead of 3 for the mismatched restorations.

Fig. 14
Fig. 14

Image restorations for four SNR’s LN. The image-gathering response shape index is n = 2, the frequency constant is ρc = 0.4, and the mean spatial detail is μr = 3.

Equations (16)

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ĥ ( ν , ω ) = log 2 [ 1 + Φ L ( ν , ω ) | τ ̂ g ( ν , ω ) | 2 Φ ̂ L ( ν , ω ) | τ ̂ g ( ν , ω ) | 2 * ̲ ̂ 0 , 0 ( ν , ω ) + ( K σ L σ N ) 2 ] ,
h = ½ B ̂ ĥ ( ν , ω ) d ν d ω ,
h 0 = ½ B ̂ log 2 [ 1 + ( K σ L σ N ) 2 Φ ̂ L ( ν , ω ) ] d ν d ω . .
h 0 m = ½ | B ̂ | log 2 [ 1 + ( SNR ) 2 ] ,
Φ ̂ L ( ν , ω ) | τ ̂ g ( ν , ω ) 2 | = { 1 ( ν , ω ) B ̂ 0 ( ν , ω ) B ̂ .
f c = 1 σ L ( c ) 2 A | L c ( x , y ) R ( x , y ) | 2 ¯ d x d y ,
Ψ ̂ c ( ν , ω ) = Φ ̂ L ( ν , ω ) τ ̂ g * ( ν , ω ) τ ̂ c ( ν , ω ) Φ ̂ L ( ν , ω ) | τ ̂ g ( ν , ω ) | 2 * ̲ ̂ ( ν , ω ) + ( K σ L σ N ) 2 .
f c m = σ L 2 σ L ( c ) 2 Φ ̂ L ( ν , ω ) τ ̂ g ( ν , ω ) τ ̂ c * ( ν , ω ) Φ ̂ L ( ν , ω ) d ν d ω .
Ψ ̂ c ( ν , ω ) = τ ̂ c ( ν , ω ) τ ̂ g ( ν , ω ) [ 1 2 ĥ ( ν , ω ) ]
f c m = σ L ( c ) 2 ϕ ̂ L ( ν , ω ) | τ ̂ c ( ν , ω ) | 2 [ 1 2 ĥ ( ν , ω ) ] d ν d ω .
Φ ̂ L ( ν , ω ) = 2 π μ r 2 σ L 2 ( m 2 ) [ 1 + ( 2 π μ r ρ ) 2 ] m / 2 ,
τ ̂ g ( ν , ω ) = exp [ ( ρ / ρ c ) n ] .
f c = 1 ( b 2 + a 2 + n 2 ) ,
b 2 = σ L ( c ) 2 Φ ̂ L ( ν , ω ) | τ ̂ c ( ν , ω ) τ ̂ g ( ν , ω ) τ ̂ p ( ν , ω ) | 2 d ν d ω
a 2 = σ L ( c ) 2 [ Φ ̂ L ( ν , ω ) | τ ̂ P ( ν , ω ) | 2 * ̲ ̂ 0 , 0 ( ν , ω ) ] × | τ ̂ P ( ν , ω ) | 2 d ν d ω
a 2 = ( K σ L σ N ) 2 σ L 2 ( σ L ( c ) ) 2 | τ ̂ p ( ν , ω ) | 2 d ν d ω

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