Abstract

In this paper we formulate and use information and fidelity criteria to assess image gathering and processing, combining optical design with image-forming and edge-detection algorithms. The optical design of the image-gathering system revolves around the relationship among sampling passband, spatial response, and signal-to-noise ratio (SNR). Our formulations of information, fidelity, and optimal (Wiener) restoration account for the insufficient sampling (i.e., aliasing) common in image gathering as well as for the blurring and noise that conventional formulations account for. Performance analyses and simulations for ordinary optical-design constraints and random scenes indicate that (1) different image-forming algorithms prefer different optical designs; (2) informationally optimized designs maximize the robustness of optimal image restorations and lead to the highest-spatial-frequency channel (relative to the sampling passband) for which edge detection is reliable (if the SNR is sufficiently high); and (3) combining the informationally optimized design with a 3 by 3 lateral-inhibitory image-plane-processing algorithm leads to a spatial-response shape that approximates the optimal edge-detection response of (Marr’s model of) human vision and thus reduces the data preprocessing and transmission required for machine vision.

© 1985 Optical Society of America

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1984 (4)

1983 (3)

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

M. Kass, J. Hughes, “A stochastic image model for AI,” IEEE Proc. Syst. Man Cybern. Conf. X, 369–372 (1983).

E. C. Hildreth, “The detection of intensity changes by computer and biological vision systems,” Comput. Vision Graphics Image Process. 22, 1–27 (1983).
[CrossRef]

1982 (2)

1981 (3)

W. L. Eversole, J. F. Salzman, F. V. Taylor, W. L. Harland, “Programmable image processing element,” SPIE Proc. 301, 66–77 (1981).
[CrossRef]

M. E. Jernigan, R. W. Wardell, “Does the eye contain optimal edge detection mechanisms?” IEEE Trans. Syst. Man Cybern. SMC-11, 441–444 (1981).

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

1980 (6)

J. P. Carroll, “Apodization model of the Stiles–Crawford effect,” J. Opt. Soc. Am. 70, 1155–1156 (1980).
[CrossRef] [PubMed]

D. Marr, T. Poggio, E. Hildreth, “Smallest channel in early human vision,” J. Opt. Soc. Am. 70, 868–870 (1980).
[CrossRef] [PubMed]

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

J. E. Hall, J. D. Awtrey, “Real-time image enhancement using 3 by 3 pixel neighborhood operator functions,” Opt. Eng. 19, 421–424 (1980).
[CrossRef]

L. S. Davis, “Computer architectures for image processing,” IEEE Comput. Soc. Workshop Proc. CH1530-5, 249–254 (1980).

F. O. Huck, N. Halyo, S. K. Park, “Aliasing and blurring in 2-D sampled imagery,” Appl. Opt. 19, 2174–2181 (1980).
[CrossRef] [PubMed]

1979 (1)

1977 (1)

J. W. Modestino, R. W. Fries, “Edge detection in noisy images using recursive digital filtering,” Comput. Graphics Image Process. 6, 409–433 (1977).
[CrossRef]

1975 (1)

1974 (1)

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]

1973 (2)

1972 (1)

M. M. Sondhi, “Image restoration: the removal of spatially invariant degradations,” IEEE Proc. 60, 842–852 (1972).
[CrossRef]

1970 (3)

1968 (1)

1967 (1)

1965 (1)

1963 (1)

1958 (2)

E. H. Linfoot, “Quality evaluations of optical systems,” Opt. Acta 5, 1–14 (1958).
[CrossRef]

E. H. Linfoot, “Optical image evaluation from the standpoint of communication theory,” Physica XXIV, 476–494 (1958).
[CrossRef]

1956 (1)

1955 (2)

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

E. H. Linfoot, “Information theory and optical images,” J. Opt. Soc. Am. 45, 808–819 (1955).
[CrossRef]

1951 (1)

O. H. Schade, “Image gradation, graininess and sharpness in television and motion-picture systems,” J. Soc. Motion Pict. Telev. Eng. 56, 137–171 (1951);J. Soc. Motion Pict. Telev. Eng. 58, 181–222 (1952);J. Soc. Motion Pict. Telev. Eng. 61, 97–164 (1953);J. Soc. Motion Pict. Telev. Eng. 64, 593–617 (1955).

1934 (1)

P. Mertz, F. Gray, “Theory of scanning and its relation to characteristics of transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 464–515 (July1934).
[CrossRef]

Andrews, H. C.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Awtrey, J. D.

J. E. Hall, J. D. Awtrey, “Real-time image enhancement using 3 by 3 pixel neighborhood operator functions,” Opt. Eng. 19, 421–424 (1980).
[CrossRef]

Ballard, D. H.

D. H. Ballard, C. M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, N.J., 1982).

Barlow, H. B.

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

Brady, M.

M. Brady, “Computational approaches to image understanding,” Comput. Surv. 14, 3–71 (1982).
[CrossRef]

Brown, C. M.

D. H. Ballard, C. M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, N.J., 1982).

Brown, W. M.

Callahan, L. G.

Carroll, J. P.

Cathey, W. T.

Davis, L. S.

L. S. Davis, “Computer architectures for image processing,” IEEE Comput. Soc. Workshop Proc. CH1530-5, 249–254 (1980).

Eversole, W. L.

W. L. Eversole, J. F. Salzman, F. V. Taylor, W. L. Harland, “Programmable image processing element,” SPIE Proc. 301, 66–77 (1981).
[CrossRef]

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.

Fries, R. W.

J. W. Modestino, R. W. Fries, “Edge detection in noisy images using recursive digital filtering,” Comput. Graphics Image Process. 6, 409–433 (1977).
[CrossRef]

Gray, F.

P. Mertz, F. Gray, “Theory of scanning and its relation to characteristics of transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 464–515 (July1934).
[CrossRef]

Gray, R.

R. A. Schowengerdt, S. K. Park, R. Gray, “Topics in the two-dimensional sampling and reconstruction of images,” Int. J. Remote Sensing 5(2), 333–347 (1984).
[CrossRef]

Hall, J. E.

J. E. Hall, J. D. Awtrey, “Real-time image enhancement using 3 by 3 pixel neighborhood operator functions,” Opt. Eng. 19, 421–424 (1980).
[CrossRef]

Halyo, N.

Harland, W. L.

W. L. Eversole, J. F. Salzman, F. V. Taylor, W. L. Harland, “Programmable image processing element,” SPIE Proc. 301, 66–77 (1981).
[CrossRef]

Helstrom, C. W.

Hildreth, E.

D. Marr, T. Poggio, E. Hildreth, “Smallest channel in early human vision,” J. Opt. Soc. Am. 70, 868–870 (1980).
[CrossRef] [PubMed]

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

Hildreth, E. C.

E. C. Hildreth, “The detection of intensity changes by computer and biological vision systems,” Comput. Vision Graphics Image Process. 22, 1–27 (1983).
[CrossRef]

Huck, F. O.

Hughes, J.

M. Kass, J. Hughes, “A stochastic image model for AI,” IEEE Proc. Syst. Man Cybern. Conf. X, 369–372 (1983).

Hunt, B. R.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

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]

Jernigan, M. E.

M. E. Jernigan, R. W. Wardell, “Does the eye contain optimal edge detection mechanisms?” IEEE Trans. Syst. Man Cybern. SMC-11, 441–444 (1981).

Jobson, D. J.

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1982).

Kass, M.

M. Kass, J. Hughes, “A stochastic image model for AI,” IEEE Proc. Syst. Man Cybern. Conf. X, 369–372 (1983).

Katzberg, S. J.

Lee, Y. L.

Y. L. Lee, Statistical Theory of Communications (Wiley, New York, 1964).

Linfoot, E. H.

E. H. Linfoot, “Quality evaluations of optical systems,” Opt. Acta 5, 1–14 (1958).
[CrossRef]

E. H. Linfoot, “Optical image evaluation from the standpoint of communication theory,” Physica XXIV, 476–494 (1958).
[CrossRef]

E. H. Linfoot, “Transmission factors and optical design,” J. Opt. Soc. Am. 46, 740–752 (1956).
[CrossRef]

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

E. H. Linfoot, “Information theory and optical images,” J. Opt. Soc. Am. 45, 808–819 (1955).
[CrossRef]

Macovski, A.

Marr, D.

Mertz, P.

P. Mertz, F. Gray, “Theory of scanning and its relation to characteristics of transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 464–515 (July1934).
[CrossRef]

Metcalf, H. J.

Mino, M.

Modestino, J. W.

J. W. Modestino, R. W. Fries, “Edge detection in noisy images using recursive digital filtering,” Comput. Graphics Image Process. 6, 409–433 (1977).
[CrossRef]

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Okano, Y.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York; 1965).

Park, S. K.

Poggio, T.

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

Rhodes, W. T.

Robinson, A. H.

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1982).

Rushforth, C. K.

Salzman, J. F.

W. L. Eversole, J. F. Salzman, F. V. Taylor, W. L. Harland, “Programmable image processing element,” SPIE Proc. 301, 66–77 (1981).
[CrossRef]

Samms, R. W.

Schade, O. H.

O. H. Schade, “Image gradation, graininess and sharpness in television and motion-picture systems,” J. Soc. Motion Pict. Telev. Eng. 56, 137–171 (1951);J. Soc. Motion Pict. Telev. Eng. 58, 181–222 (1952);J. Soc. Motion Pict. Telev. Eng. 61, 97–164 (1953);J. Soc. Motion Pict. Telev. Eng. 64, 593–617 (1955).

Schowengerdt, R. A.

R. A. Schowengerdt, S. K. Park, R. Gray, “Topics in the two-dimensional sampling and reconstruction of images,” Int. J. Remote Sensing 5(2), 333–347 (1984).
[CrossRef]

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

S. K. Park, R. A. Schowengerdt, “Image sampling, reconstruction, and the effect of sample-scene phasing,” Appl. Opt. 21, 3142–3151 (1982).
[CrossRef] [PubMed]

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).

Sondhi, M. M.

M. M. Sondhi, “Image restoration: the removal of spatially invariant degradations,” IEEE Proc. 60, 842–852 (1972).
[CrossRef]

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]

Taylor, F. V.

W. L. Eversole, J. F. Salzman, F. V. Taylor, W. L. Harland, “Programmable image processing element,” SPIE Proc. 301, 66–77 (1981).
[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]

Ullman, S.

Wall, S. D.

Wardell, R. W.

M. E. Jernigan, R. W. Wardell, “Does the eye contain optimal edge detection mechanisms?” IEEE Trans. Syst. Man Cybern. SMC-11, 441–444 (1981).

Wiener, N.

N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Times Series (Wiley, New York, 1949).

Appl. Opt. (10)

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]

L. G. Callahan, W. M. Brown, “One- and two-dimensional processing in line scanning systems,” Appl. Opt. 2, 401–407 (1963).
[CrossRef]

A. Macovski, “Spatial and temporal analysis of scanned systems,” Appl. Opt. 9, 1906–1910 (1970).
[PubMed]

S. J. Katzberg, F. O. Huck, S. D. Wall, “Photosensor aperture shaping to reduce aliasing in optical-mechanical line-scan imaging systems,” Appl. Opt. 12, 1054–1060 (1973).
[CrossRef] [PubMed]

A. H. Robinson, “Multidimensional Fourier transforms and image processing with finite scanning apertures,” Appl. Opt. 12, 2344–2352 (1973).
[CrossRef] [PubMed]

F. O. Huck, N. Halyo, S. K. Park, “Aliasing and blurring in 2-D sampled imagery,” Appl. Opt. 19, 2174–2181 (1980).
[CrossRef] [PubMed]

C. L. Fales, F. O. Huck, R. W. Samms, “Imaging system design for improved information capacity,” Appl. Opt. 23, 872–888 (1984).
[CrossRef] [PubMed]

S. K. Park, R. A. Schowengerdt, “Image sampling, reconstruction, and the effect of sample-scene phasing,” Appl. Opt. 21, 3142–3151 (1982).
[CrossRef] [PubMed]

M. Mino, Y. Okano, “Improvement in the OTF of a defocused optical system through the use of shaded apertures,” Appl. Opt. 10, 2219–2225 (1970).
[CrossRef]

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]

Bell Syst. Tech. J. (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).

P. Mertz, F. Gray, “Theory of scanning and its relation to characteristics of transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 464–515 (July1934).
[CrossRef]

Comput. Graphics Image Process. (1)

J. W. Modestino, R. W. Fries, “Edge detection in noisy images using recursive digital filtering,” Comput. Graphics Image Process. 6, 409–433 (1977).
[CrossRef]

Comput. Surv. (1)

M. Brady, “Computational approaches to image understanding,” Comput. Surv. 14, 3–71 (1982).
[CrossRef]

Comput. Vision Graphics Image Process. (2)

E. C. Hildreth, “The detection of intensity changes by computer and biological vision systems,” Comput. Vision Graphics Image Process. 22, 1–27 (1983).
[CrossRef]

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

IEEE Comput. Soc. Workshop Proc. (1)

L. S. Davis, “Computer architectures for image processing,” IEEE Comput. Soc. Workshop Proc. CH1530-5, 249–254 (1980).

IEEE Proc. (1)

M. M. Sondhi, “Image restoration: the removal of spatially invariant degradations,” IEEE Proc. 60, 842–852 (1972).
[CrossRef]

IEEE Proc. Syst. Man Cybern. Conf. (1)

M. Kass, J. Hughes, “A stochastic image model for AI,” IEEE Proc. Syst. Man Cybern. Conf. X, 369–372 (1983).

IEEE Trans. Syst. Man Cybern. (1)

M. E. Jernigan, R. W. Wardell, “Does the eye contain optimal edge detection mechanisms?” IEEE Trans. Syst. Man Cybern. SMC-11, 441–444 (1981).

Infrared Phys. (1)

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]

Int. J. Remote Sensing (1)

R. A. Schowengerdt, S. K. Park, R. Gray, “Topics in the two-dimensional sampling and reconstruction of images,” Int. J. Remote Sensing 5(2), 333–347 (1984).
[CrossRef]

J. Opt. Soc. Am. (9)

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

J. Soc. Motion Pict. Telev. Eng. (1)

O. H. Schade, “Image gradation, graininess and sharpness in television and motion-picture systems,” J. Soc. Motion Pict. Telev. Eng. 56, 137–171 (1951);J. Soc. Motion Pict. Telev. Eng. 58, 181–222 (1952);J. Soc. Motion Pict. Telev. Eng. 61, 97–164 (1953);J. Soc. Motion Pict. Telev. Eng. 64, 593–617 (1955).

Opt. Acta (1)

E. H. Linfoot, “Quality evaluations of optical systems,” Opt. Acta 5, 1–14 (1958).
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Opt. Eng. (1)

J. E. Hall, J. D. Awtrey, “Real-time image enhancement using 3 by 3 pixel neighborhood operator functions,” Opt. Eng. 19, 421–424 (1980).
[CrossRef]

Philos. Trans. R. Soc. London (1)

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

Physica (1)

E. H. Linfoot, “Optical image evaluation from the standpoint of communication theory,” Physica XXIV, 476–494 (1958).
[CrossRef]

Proc. R. Soc. London Ser. B (2)

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

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

SPIE Proc. (1)

W. L. Eversole, J. F. Salzman, F. V. Taylor, W. L. Harland, “Programmable image processing element,” SPIE Proc. 301, 66–77 (1981).
[CrossRef]

Other (12)

J. M. Enoch, F. L. Tobey, ed., Vertebrate Photoreceptor Optics (Springer-Verlag, Berlin, 1981).
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Y. L. Lee, Statistical Theory of Communications (Wiley, New York, 1964).

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

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

D. H. Ballard, C. M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, N.J., 1982).

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N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Times Series (Wiley, New York, 1949).

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

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T. S. Huang, ed., Picture Processing and Digital Filtering (Springer-Verlag, Berlin, 1979).
[CrossRef]

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1982).

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Figures (21)

Fig. 1
Fig. 1

Model of image gathering and processing.

Fig. 2
Fig. 2

Sensor-array image-gathering system with lateral-inhibitory signal processing.

Fig. 3
Fig. 3

Responses of the image-gathering system.

Fig. 4
Fig. 4

Wiener spectra of the radiance field.

Fig. 5
Fig. 5

Spectral information density ĥ(υ, ω) for (a) the theoretically optimal spatial-frequency response, (b) a typical low-pass spatial-frequency response, and (c) a typical bandpass spatial-frequency response. The log curves are given for five mean spatial details μr, and the linear three-dimensional perspectives are given for μr = 1. Sampling passband,; SNR, LN = 32.

Fig. 6
Fig. 6

Information density h versus SNR LN for the theoretically optimal spatial-frequency response and for the typically realizable spatial-frequency responses shown in Fig. 3. The difference represents the information lost by aliasing and blurring in the presence of noise. The mean spatial detail μr = 1.

Fig. 7
Fig. 7

Information density h versus SNR LN) and mean spatial detail μr for (a) the theoretically optimal, spatial-frequency response, (b) a typical low-pass spatial-frequency response, and (c) a typical bandpass spatial-frequency response.

Fig. 8
Fig. 8

Information density h and image fidelity f versus optical-design parameter β [Fig. 3(b)] for three SNR’s LN. Image fidelity is given for the matched and the two unmatched Wiener restorations and for the sinc, the cubic, and the linear interpolations. The assumed mean spatial detail for the unmatched restoration is μr = 1/9, which approximates the scene spectrum as white noise within the sampling passband. One of the two unmatched restorations is done with the parametric Wiener filter, letting γ = 1/4.

Fig. 9
Fig. 9

Typical response of the image-gathering system, the Wiener restoration, and the combined image-gathering and -restoring process. The sampling intervals, X = Y = 1; the optical-design parameter, β = 0.6; the neighborhood-weighting parameter, W = 0; the SNR, LN = 32; and the mean spatial detail, μr = 1.

Fig. 10
Fig. 10

Responses of the reconstruction algorithms.

Fig. 11
Fig. 11

One-dimensional simulations for the matched Wiener restoration and the cubic and the linear interpolations. L(x) is the input radiance field; s(x) is the sampled signal; and R(x) is the representation constructed from s(x). The mean spatial detail μr = 1, and the SNR = 32.

Fig. 12
Fig. 12

One-dimensional simulations for the matched and the two unmatched Wiener restorations using the same conditions as for Fig. 11. The sampling-frequency artifact in the mismatched Wiener restorations is ordinarily blurred in practice by the two-dimensional image-reconstruction mechanism.

Fig. 13
Fig. 13

Responses of the Wiener restoration filters for the one-dimensional simulations shown in Fig. 12.

Fig. 14
Fig. 14

Information density h and image fidelity f versus optical-design parameters β for three SNR’s LN. Neighborhood-weighting parameter, W = 0.8; otherwise the conditions are the same as for Fig. 8.

Fig. 15
Fig. 15

One-dimensional simulations for the matched and the two unmatched Wiener restorations. Neighborhood-weighting parameter, W = 0.8; otherwise the conditions are the same as for Figs. 11 and 12.

Fig. 16
Fig. 16

Standard deviation of the signal versus lateral-inhibitory weighting W. The optical-design parameter, β = 0.6.

Fig. 17
Fig. 17

Information density h and edge fidelity fe versus optical-design parameter β [Fig. 3(c)] for three SNR’s LN Edge fidelity is given for the Wiener edge restoration and three interpolations. The ideal edge-detection response is assumed to be.

Fig. 18
Fig. 18

One-dimensional simulations for the Wiener edge filter and the cubic and linear interpolations. L(x) is the input radiance field, s(x) is the sampled signal, L(x) * τe(x; β) is the ideal representation, and R(x) is the representation constructed from s(x). Mean spatial detail, μr = 1; SNR, LN = 32. [The magnitude of L(x) * τ(x), s(x), and R(x) has been amplified by a factor of 3 relative to L(x) for easier visual comparison.]

Fig. 19
Fig. 19

One-dimensional simulations for six optical-design parameters β and the Wiener edge filter using the same conditions as those for Fig. 18.

Fig. 20
Fig. 20

Combining optical design with sensor-array lateral-inhibitory processing to approximate the DOG function.

Fig. 21
Fig. 21

Comparison of image-gathering response with DOG function.

Equations (81)

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s ( x , y ) = [ K L ( x , y ) * τ g ( x , y ) + N ( x , y ) ] ̲ ( x , y )
R ( x , y ) = ( x , y ) * K 1 τ p ( x , y )
= { [ L ( x , y ) * τ g ( x , y ) + K 1 N ( x , y ) ] ̲ ( x , y ) } * τ p ( x , y ) ,
R ̂ ( υ , ω ) = { [ L ̂ ( υ , ω ) τ ̂ g ( υ , ω ) + K 1 N ̂ ( υ , ω ) ] * ̲ ̂ ( υ , ω ) } τ ̂ p ( υ , ω ) ,
̲ ̂ ( υ , ω ) = m = n = δ ( υ m X , ω n Y ) = δ ( υ , ω ) + ̲ ̂ 0 , 0 ( υ , ω ) ,
B ̂ = { ( υ , ω ) , | υ | < 1 / 2 X , | ω | < 1 / 2 Y } ,
| B ̂ | = 1 / X Y .
R ̂ ( υ , ω ) = R ̂ s ( υ , ω ) + R ̂ a ( υ , ω ) + R ̂ n ( υ , ω ) ,
R ̂ s ( υ , ω ) = L ̂ ( υ , ω ) τ ̂ g ( υ , ω ) τ ̂ p ( υ , ω ) , R ̂ a ( υ , ω ) = [ L ̂ ( υ , ω ) τ ̂ g ( υ , ω ) * ̲ ̂ 0 , 0 ( υ , ω ) ] τ ̂ p ( υ , ω ) , R ̂ n ( υ , ω ) = K 1 [ N ̂ ( υ , ω ) * ̲ ̂ ( υ , ω ) ] τ ̂ p ( υ , ω ) .
R ( x , y ) = R s ( x , y ) + R a ( x , y ) + R n ( x , y ) .
Φ ̂ L ( υ , ω ) - 1 | A | | L ̂ ( υ , ω ) | 2 ¯ ,
σ L 2 = Φ ̂ L ( υ , ω ) d υ d ω .
Φ ̂ R ( υ , ω ) = 1 | A | | R ̂ ( υ , ω ) | 2 ¯ ,
Φ ̂ R ( υ , ω ) = { [ Φ ̂ L ( υ , ω ) | τ ̂ g ( υ , ω ) | 2 + K 2 Φ ̂ N ( υ , ω ) ] * ̲ ̂ ( υ , ω ) } | τ ̂ p ( υ , ω ) | 2
= Φ ̂ s ( υ , ω ) + Φ ̂ a ( υ , ω ) + Φ ̂ n ( υ , ω ) ,
Φ ̂ s ( υ , ω ) = Φ ̂ L ( υ , ω ) | τ ̂ g ( υ , ω ) τ ̂ p ( υ , ω ) | 2 , Φ ̂ a ( υ , ω ) = [ Φ ̂ L ( υ , ω ) | τ ̂ g ( υ , ω ) | 2 * ̲ ̂ 0 , 0 ( υ , ω ) ] | τ ̂ p ( υ , ω ) | 2 , Φ ̂ n ( υ , ω ) = K 2 [ Φ ̂ N ( υ , ω ) * ̲ ̂ ( υ , ω ) ] | τ ̂ p ( υ , ω ) | 2 .
Φ ̂ n ( υ , ω ) = | B ̂ | 1 K 2 σ N 2 | τ ̂ p ( υ , ω ) | 2 ,
σ N 2 = 1 | A | | N ̂ ( υ , ω ) | 2 ¯ d υ d ω
τ g ( x , y ) = 1 2 π β 2 { exp ( r 2 / 2 β 2 ) W α 2 exp [ r 2 / 2 ( α β ) 2 ] } ,
τ ̂ g ( υ , ω ) = exp [ 2 ( π β ρ ) 2 ] W exp [ 2 ( π α β ρ ) 2 ] ,
Φ L ( x , y ) = σ L 2 e r / μ r .
Φ ̂ L ( υ , ω ) = 2 π μ r 2 σ L 2 [ 1 + ( 2 π μ r ρ ) 3 ] 3 / 2 .
ĥ ( υ , ω ) = log 2 [ 1 + Φ ̂ s ( υ , ω ) Φ ̂ a ( υ , ω ) + Φ ̂ n ( υ , ω ) ] .
h = 1 2 B ĥ ( υ , ω ) d υ d ω = 1 2 B ̂ log 2 [ 1 + Φ ̂ s ( υ , ω ) Φ ̂ a ( υ , ω ) + Φ ̂ n ( υ , ω ) ] d υ d ω
= 1 2 B log 2 [ 1 + Φ ̂ L ( υ , ω ) | τ ̂ g ( υ , ω ) | 2 Φ ̂ L ( υ , ω ) | τ ̂ g ( υ , ω ) | 2 * ̲ ̂ 0 , 0 ( υ , ω ) + | B ̂ | 1 ( K σ L σ N ) 2 ] d υ d ω ,
τ ̂ g o ( υ , ω ) = { 1 , ( υ , ω ) B ̂ 0 , elsewhere .
h o = 1 2 B ̂ log 2 [ 1 + ( K σ L σ N ) 2 Φ ̂ L ( υ , ω ) ] d υ d ω .
h 0 = 1 2 | B ̂ | log 2 [ 1 + ( SNR ) 2 ] .
f = 1 A | L ( x , y ) R ( x , y ) | 2 ¯ d x d y A | L ( x , y ) | 2 ¯ d x d y
= 1 1 | A | | L ̂ ( υ , ω ) R ̂ ( υ , ω ) | 2 ¯ d υ d ω 1 | A | | L ̂ ( υ , ω ) | 2 ¯ d υ d ω ,
f = 1 2 / σ L 2 .
τ ̂ p ( υ , ω ) = Ψ ̂ ( υ , ω ) = Φ ̂ L ( υ , ω ) τ ̂ g * ( υ , ω ) [ Φ ̂ L ( υ , ω ) | τ ̂ g ( υ , ω ) | 2 + K 2 Φ ̂ N ( υ , ω ) ] * ̲ ̂ ( υ , ω )
= Φ ̂ L ( υ , ω ) τ ̂ g * ( υ , ω ) Φ ̂ L ( υ , ω ) | τ ̂ g ( υ , ω ) | 2 * ̲ ̂ ( υ , ω ) + | B ̂ | 1 ( K σ L σ N ) 2 ,
f m = ω Φ ̂ L ( υ , ω ) τ ̂ g ( υ , ω ) Ψ ̂ ( υ , ω ) d υ d ω .
ĥ ( υ , ω ) = log 2 [ Φ ̂ R ( υ , ω ) Φ ̂ a ( υ , ω ) + Φ ̂ n ( υ , ω ) ]
Ψ ̂ ( υ , ω ) = 1 τ ̂ g ( υ , ω ) [ Φ ̂ s ( υ , ω ) Φ ̂ R ( υ , ω ) ] = 1 τ ̂ g ( υ , ω ) [ 1 Φ ̂ a ( υ , ω ) + Φ ̂ n ( υ , ω ) Φ ̂ R ( υ , ω ) ] .
Ψ ̂ ( υ , ω ) = 1 τ ̂ g ( υ , ω ) [ 1 2 ĥ ( υ , ω ) ] ,
ĥ ( υ , ω ) = log 2 [ 1 1 τ ̂ g ( υ , ω ) Ψ ̂ ( υ , ω ) ]
2 = Φ ̂ L ( υ , ω ) 2 ĥ ( υ , ω ) d υ d ω .
f m = Φ ̂ L ( υ , ω ) [ 1 2 ĥ ( υ , ω ) ] d υ d ω .
σ s 2 ( W ) = Φ ̂ L ( υ , ω ) | τ ̂ g ( υ , ω ) | 2 d υ d ω .
f e = 1 A | L e ( x , y ) R ( x , y ) | 2 ¯ d x d y A | L e ( x , y ) | 2 ¯ d x d y
= 1 1 | A | | L ̂ e ( υ , ω ) R ̂ ( υ , ω ) | 2 ¯ d υ d ω 1 | A | | L ̂ e ( υ , ω ) | 2 ¯ d υ d ω ( 17 b )
= 1 e 2 / σ L ( e ) 2 ,
τ ̂ p ( υ , ω ) = Ψ ̂ e ( υ , ω ) = Ψ ̂ ( υ , ω ) τ ̂ e ( υ , ω ) ,
f e m = σ L 2 σ L ( e ) 2 Φ ̂ L ( υ , ω ) | τ ̂ e ( υ , ω ) | 2 τ ̂ g ( υ , ω ) Ψ ̂ ( υ , ω ) d υ d ω .
Ψ ̂ e ( υ , ω ) = τ ̂ e ( υ , ω ) τ ̂ g ( υ , ω ) [ 1 2 ĥ ( υ , ω ) ] ,
f e m = σ L 2 σ L ( e ) 2 Φ ̂ L ( υ , ω ) | τ ̂ e ( υ , ω ) | 2 [ 1 2 ĥ ( υ , ω ) ] d υ d ω .
R ( x , y ) = p , q R ̂ p q exp [ i 2 π ( υ p x + ω q y ) ] ,
R ̂ p q = 1 | A | R ̂ A ( υ p , ω q ) ,
R ̂ ( υ p , ω q ) = m , n R ̂ m n ( υ p , ω q ) ,
R ̂ m n ( υ p , ω q ) = L ̂ ( υ p + m X , ω q + n Y ) × τ ̂ g ( υ p + m X , ω q + n Y ) τ ̂ p ( υ p , ω q )
E { R ̂ m n ( υ p , ω q ) R ̂ m n * ( υ p , ω q ) } = E { L ̂ ( υ p + m X , ω q + n Y ) × L ̂ * ( υ p + m X , ω q + n Y ) } × τ ̂ g ( υ p + m X , ω q + n Y ) τ ̂ g * ( υ p + m X , ω q + n Y ) × τ ̂ p ( υ p , ω q ) τ ̂ p * ( υ p , ω q ) ,
E { L ̂ p q L ̂ p q * } E { | L ̂ p q | 2 } δ p , p ; q , q .
E { | L ̂ p q | 2 } = 1 | A | 2 E { | L ̂ ( υ p , ω q ) | 2 } .
Φ ̂ L ( υ p , ω q ) = 1 Δ υ p Δ ω q E { | L ̂ p q | 2 } = 1 | A | E { | L ̂ ( υ p , ω q ) | 2 } .
E { L ̂ ( υ p , ω q ) L ̂ * ( υ p , ω q ) } = Φ ̂ L ( υ p , ω q ) | A | δ p , p ; q , q ,
E { R ̂ m n ( υ p , ω q ) R ̂ m n * ( υ p , ω q ) } = | A | Φ ̂ L ( υ p + m X , ω q + n Y ) × δ υ p + m X , υ p + m X ; ω q + n Y , ω q + n Y | τ ̂ g ( υ p + m X , ω q + n Y ) | 2 × τ ̂ p ( υ p , ω q ) τ ̂ p * ( υ p + m m X , ω q + n n Y ) .
R m n ( x , y ) 1 | A | p q R ̂ m n ( υ p , ω q ) exp [ i 2 π ( υ p x + ω q y ) ]
τ ̂ p ( υ p , ω q ) τ ̂ p * ( υ p + m m X , ω q + n n Y ) = 0
2 E { 1 | A | A | ( x , y ) | 2 d x d y } ,
( x , y ) = p , q ̂ p q exp [ i 2 π ( υ p x + ω q y ) ] ,
̂ p q 1 | A | ̂ A ( υ p , ω q )
1 | A | | ( x , y ) | 2 d x d y = p , q | ̂ p q | 2 .
2 = 1 | A | 2 E { p , q | L ̂ ( υ p , ω q ) [ 1 τ ̂ g ( υ p , ω q ) τ ̂ p ( υ p , ω q ) ] [ m , n 0 L ̂ ( ( υ p + m X , ω q + n Y ) × τ ̂ g ( υ p + m X , ω q + n Y ) ] τ ̂ p ( υ p , ω q ) | 2 } .
E { L ̂ ( υ p + m X , ω q + n Y ) L ̂ * ( υ p + m X , ω q + n Y ) } = | A | Φ ̂ L ( υ p + m X , ω q + n Y ) δ m , m ; n , n ,
2 = b 2 + a 2 ,
b 2 = Φ ̂ L ( υ , ω ) | 1 τ ̂ g ( υ , ω ) τ ̂ p ( υ , ω ) | 2 d υ d ω
a 2 = [ Φ ̂ L ( υ , ω ) | τ ̂ g ( υ , ω ) | 2 * ̲ ̂ 0 , 0 ( υ , ω ) ] × | τ ̂ p ( υ , ω ) | 2 d υ d ω
2 = b 2 + a 2 + n 2 ,
n 2 = K 2 [ Φ ̂ N ( υ , ω ) * ̲ ̂ ( υ , ω ) ] | τ ̂ p ( υ , ω ) | 2 d υ d ω
f = 1 ( b 2 + a 2 + n 2 ) / σ L 2 .
τ s ( x , y ) = Π ( x , y ) + a [ Π ( x / 3 , y ) + Π ( x , y / 3 ) 2 Π ( x , y ) ] + b [ Π ( x / 3 , y / 3 ) Π ( x / 3 , y ) Π ( x , y / 3 ) + Π ( x , y ) ] ,
Π ( x , y ) = { 1 , | x | , | y | 1 / 2 0 , elsewhere
τ ̂ s ( υ , ω ) = ( 1 2 a + b ) sinc υ sinc ω + 3 ( a b ) [ sinc 3 υ sinc ω + sinc υ sinc 3 ω ] + 9 b sinc 3 υ sinc 3 ω .
Φ ̂ L ( υ ) = 2 μ r σ L 2 1 + ( 2 π μ r υ ) 2 ;
τ g ( x ) = 1 2 π β { exp [ x 2 / 2 β 2 ] W α [ x 2 / 2 ( α β ) 2 ] } τ ̂ g ( υ ) = exp [ 2 ( π β υ ) 2 ] W exp [ 2 ( π α β υ ) 2 ] ,
R ̂ ( υ ) = { [ L ̂ ( υ ) τ ̂ g ( υ ) * ̲ ̂ ( υ ) } τ ̂ p ( υ ) = L ̂ ( υ ) τ ̂ g ( υ ) τ ̂ p ( υ ) + L ̂ ( υ 1 ) τ ̂ g ( υ 1 ) τ ̂ p ( υ ) + L ̂ ( υ + 1 ) τ ̂ g ( υ + 1 ) τ ̂ p ( υ ) +
R ( x ) = R 0 ( x ) + 2 R e { e i 2 π x R 1 ( x ) } + ,
R 0 ( x ) = L ̂ ( υ ) τ ̂ g ( υ ) τ ̂ p ( υ ) exp ( i 2 π x υ ) d υ ,
R 1 ( x ) = L ̂ ( υ ) τ ̂ g ( υ ) τ ̂ p ( υ + 1 ) exp ( i 2 π x υ ) d υ .

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