Abstract

Visual communication with retinex coding seeks to suppress the spatial variation of the irradiance (e.g., shadows) across natural scenes and preserve only the spatial detail and the reflectance (or the lightness) of the surface itself. The separation of reflectance from irradiance begins with nonlinear retinex coding that sharply and clearly enhances edges and preserves their contrast, and it ends with a Wiener filter that restores images from this edge and contrast information. An approximate small-signal model of image gathering with retinex coding is found to consist of the familiar difference-of-Gaussian bandpass filter and a locally adaptive automatic-gain control. A linear representation of this model is used to develop expressions within the small-signal constraint for the information rate and the theoretical minimum data rate of the retinex-coded signal and for the maximum-realizable fidelity of the images restored from this signal. Extensive computations and simulations demonstrate that predictions based on these figures of merit correlate closely with perceptual and measured performance. Hence these predictions can serve as a general guide for the design of visual communication channels that produce images with a visual quality that consistently approaches the best possible sharpness, clarity, and reflectance constancy, even for nonuniform irradiances. The suppression of shadows in the restored image is found to be constrained inherently more by the sharpness of their penumbra than by their depth.

© 2000 Optical Society of America

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    [CrossRef]

1999

F. O. Huck, C. L. Fales, R. Alter-Gartenberg, S. K. Park, Z. Rahman, “Information-theoretic assessment of sampled imaging systems,” Opt. Eng. 38, 742–762 (1999).
[CrossRef]

1997

D. J. Jobson, Z. Rahman, G. A. Woodell, “Properties and performance of a center/surround retinex,” IEEE Trans. Image Process. 6, 451–462 (1997).
[CrossRef] [PubMed]

D. J. Jobson, Z. Rahman, G. A. Woodell, “A multiscale retinex for bridging the gap between color images and the human observation of scenes,” IEEE Trans. Image Process. 6, 965–976 (1997).
[CrossRef] [PubMed]

1996

C. L. Fales, F. O. Huck, R. Alter-Gartenberg, Z. Rahman, “Image gathering and digital restoration,” Philos. Trans. R. Soc. London A 354, 2249–2287 (1996).
[CrossRef]

F. O. Huck, C. L. Fales, Z. Rahman, “An information theory of visual communication,” Philos. Trans. R. Soc. London A 354, 2193–2248 (1996).
[CrossRef]

S. Najand, D. Blough, G. Healey, “Forward and inverse model for the intensity-dependent spread filter,” J. Opt. Soc. Am. A 13, 1305–1314 (1996).
[CrossRef]

1995

J. Villasenor, B. Belzer, J. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. Image Process. 4, 1053–1060 (1995).
[CrossRef] [PubMed]

1994

Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the second-order statistics of natural and synthetic images,” Vision Res. 34, 541–554 (1994).
[CrossRef] [PubMed]

1993

J. M. Shapiro, “Embedded image coding using zero trees of wavelet coefficients,” special issue on Wavelets and Signal Processing, IEEE Trans. Signal Process. 41, 3445–3462 (1993).

K. Ramchandran, M. Vetterli, “Best wavelet packet bases in a rate-distortion sense,” IEEE Trans. Image Process. 2, 160–175 (1993).
[CrossRef] [PubMed]

1992

R. R. Coifman, M. V. Wickerhauser, “Entropy-based algorithms for best basis selection,” special issue on Wavelet Transforms and Multiresolution Signal Analysis, IEEE Trans. Inf. Theory 38, 713–718 (1992).

J. J. Atick, “Could information theory provide an ecological theory of sensory processing?” Network 3, 213–251 (1992).
[CrossRef]

1991

C. L. Fales, F. O. Huck, “An information theory of image gathering,” Inf. Sci. 57–58, 245–285 (1991).
[CrossRef]

1990

J. J. Atick, A. N. Redlich, “Towards a theory of early visual processing,” Neural Comp. 2, 308–320 (1990).
[CrossRef]

A. Valberg, B. Lange-Malecki, “‘Color constancy’ in Mondrian patterns: a partial cancellation of physical chromaticity shifts by simultaneous contrast,” Vision Res. 30, 371–380 (1990).
[CrossRef]

R. Alter-Gartenberg, F. O. Huck, N. Narayanswamy, “Image recovery from edge primitives,” J. Opt. Soc. Am. A 7, 898–911 (1990).
[CrossRef]

1987

1986

1985

T. N. Cornsweet, J. I. Yellott, “Intensity-dependent spatial summation,” J. Opt. Soc. Am. A 2, 1769–1789 (1985).
[CrossRef] [PubMed]

T. N. Cornsweet, “A simple retinal mechanism that has complex and profound effects on perception,” Am. J. Optom. Physiol. Opt. 62, 427–438 (1985).
[CrossRef] [PubMed]

1983

E. H. Land, “Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image,” Proc. Natl. Acad. Sci. USA 80, 5163–5169 (1983).
[CrossRef] [PubMed]

1980

J. W. Modestino, R. W. Fries, “Construction and properties of a useful two-dimensional random field,” IEEE Trans. Inf. Theory IT-26, 44–50 (1980).
[CrossRef]

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

1978

W. F. Schreiber, “Image processing for quality improvement,” Proc. IEEE 60, 1640–1651 (1978).
[CrossRef]

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]

B. K. P. Horn, “Determining lightness from an image,” Comput. Graphics Image Process. 3, 277–299 (1974).
[CrossRef]

1971

1959

E. H. Land, “Color vision and the natural image,” Proc. Natl. Acad. Sci. USA 45, 115–129 (1959).
[CrossRef]

1948

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

Alter-Gartenberg, R.

F. O. Huck, C. L. Fales, R. Alter-Gartenberg, S. K. Park, Z. Rahman, “Information-theoretic assessment of sampled imaging systems,” Opt. Eng. 38, 742–762 (1999).
[CrossRef]

C. L. Fales, F. O. Huck, R. Alter-Gartenberg, Z. Rahman, “Image gathering and digital restoration,” Philos. Trans. R. Soc. London A 354, 2249–2287 (1996).
[CrossRef]

R. Alter-Gartenberg, F. O. Huck, N. Narayanswamy, “Image recovery from edge primitives,” J. Opt. Soc. Am. A 7, 898–911 (1990).
[CrossRef]

Atick, J. J.

J. J. Atick, “Could information theory provide an ecological theory of sensory processing?” Network 3, 213–251 (1992).
[CrossRef]

J. J. Atick, A. N. Redlich, “Towards a theory of early visual processing,” Neural Comp. 2, 308–320 (1990).
[CrossRef]

Belzer, B.

J. Villasenor, B. Belzer, J. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. Image Process. 4, 1053–1060 (1995).
[CrossRef] [PubMed]

Blake, A.

A. Blake, “On lightness computation in the Mondrian world,” in Central and Peripheral Mechanisms of Colour Vision, T. Ottoson, S. Zeki, eds. (Macmillan, New York, 1985), pp. 45–49.

Blough, D.

Coifman, R. R.

R. R. Coifman, M. V. Wickerhauser, “Entropy-based algorithms for best basis selection,” special issue on Wavelet Transforms and Multiresolution Signal Analysis, IEEE Trans. Inf. Theory 38, 713–718 (1992).

R. R. Coifman, Y. Meyer, M. V. Wickerhauser, “Wavelet analysis and signal processing,” in Wavelets and Their Applications, M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, eds. (Jones and Bartlett, Boston, 1992), pp. 153–178.

Cornsweet, T. N.

T. N. Cornsweet, “A simple retinal mechanism that has complex and profound effects on perception,” Am. J. Optom. Physiol. Opt. 62, 427–438 (1985).
[CrossRef] [PubMed]

T. N. Cornsweet, J. I. Yellott, “Intensity-dependent spatial summation,” J. Opt. Soc. Am. A 2, 1769–1789 (1985).
[CrossRef] [PubMed]

T. N. Cornsweet, Visual Perception (Academic, New York, 1970).

Fales, C. L.

F. O. Huck, C. L. Fales, R. Alter-Gartenberg, S. K. Park, Z. Rahman, “Information-theoretic assessment of sampled imaging systems,” Opt. Eng. 38, 742–762 (1999).
[CrossRef]

C. L. Fales, F. O. Huck, R. Alter-Gartenberg, Z. Rahman, “Image gathering and digital restoration,” Philos. Trans. R. Soc. London A 354, 2249–2287 (1996).
[CrossRef]

F. O. Huck, C. L. Fales, Z. Rahman, “An information theory of visual communication,” Philos. Trans. R. Soc. London A 354, 2193–2248 (1996).
[CrossRef]

C. L. Fales, F. O. Huck, “An information theory of image gathering,” Inf. Sci. 57–58, 245–285 (1991).
[CrossRef]

F. O. Huck, C. L. Fales, Z. Rahman, Visual Communication: An Information Theory Approach (Kluwer, Boston, 1997).

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. A5, 300–315 (1988); C. L. Fales, F. O. Huck, J. A. McCormick, S. K. Park, “Wiener restoration of sampled image data: end-to-end analysis,” in Selected Papers on Digital Image Restoration, M. Ibrahim Sezan, ed., Vol. MS47 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1992), pp. 229–243.

Field, D. J.

Fries, R. W.

J. W. Modestino, R. W. Fries, “Construction and properties of a useful two-dimensional random field,” IEEE Trans. Inf. Theory IT-26, 44–50 (1980).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez, R. E. Wood, Digital Image Processing, 3rd ed. (Addison-Wesley, Reading, Mass., 1992).

Hawken, M. J.

M. J. Hawken, A. J. Parker, “Spatial receptive field organization in monkey V1 and its relationship to the cone mosaic,” in Computational Models of Visual Processing, M. S. Landy, J. A. Morshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 83–93.

Healey, G.

Hildreth, E.

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

Horn, B. K. P.

B. K. P. Horn, “Determining lightness from an image,” Comput. Graphics Image Process. 3, 277–299 (1974).
[CrossRef]

B. K. P. Horn, Robotic Vision (MIT Press, Cambridge, Mass., and McGraw-Hill, New York, 1985).

Huck, F. O.

F. O. Huck, C. L. Fales, R. Alter-Gartenberg, S. K. Park, Z. Rahman, “Information-theoretic assessment of sampled imaging systems,” Opt. Eng. 38, 742–762 (1999).
[CrossRef]

F. O. Huck, C. L. Fales, Z. Rahman, “An information theory of visual communication,” Philos. Trans. R. Soc. London A 354, 2193–2248 (1996).
[CrossRef]

C. L. Fales, F. O. Huck, R. Alter-Gartenberg, Z. Rahman, “Image gathering and digital restoration,” Philos. Trans. R. Soc. London A 354, 2249–2287 (1996).
[CrossRef]

C. L. Fales, F. O. Huck, “An information theory of image gathering,” Inf. Sci. 57–58, 245–285 (1991).
[CrossRef]

R. Alter-Gartenberg, F. O. Huck, N. Narayanswamy, “Image recovery from edge primitives,” J. Opt. Soc. Am. A 7, 898–911 (1990).
[CrossRef]

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. A5, 300–315 (1988); C. L. Fales, F. O. Huck, J. A. McCormick, S. K. Park, “Wiener restoration of sampled image data: end-to-end analysis,” in Selected Papers on Digital Image Restoration, M. Ibrahim Sezan, ed., Vol. MS47 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1992), pp. 229–243.

F. O. Huck, C. L. Fales, Z. Rahman, Visual Communication: An Information Theory Approach (Kluwer, Boston, 1997).

Hughes, J.

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

Hulbert, A. C.

A. C. Hulbert, “The computation of color,” Ph.D. dissertation, MIT Tech. Rep.1154 (MIT Press, Cambridge, Mass., 1989).

Hurlbert, A. C.

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]

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Jobson, D. J.

D. J. Jobson, Z. Rahman, G. A. Woodell, “Properties and performance of a center/surround retinex,” IEEE Trans. Image Process. 6, 451–462 (1997).
[CrossRef] [PubMed]

D. J. Jobson, Z. Rahman, G. A. Woodell, “A multiscale retinex for bridging the gap between color images and the human observation of scenes,” IEEE Trans. Image Process. 6, 965–976 (1997).
[CrossRef] [PubMed]

Kass, M.

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

Land, E. H.

E. H. Land, “Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image,” Proc. Natl. Acad. Sci. USA 80, 5163–5169 (1983).
[CrossRef] [PubMed]

E. H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).
[CrossRef] [PubMed]

E. H. Land, “Color vision and the natural image,” Proc. Natl. Acad. Sci. USA 45, 115–129 (1959).
[CrossRef]

Lange-Malecki, B.

A. Valberg, B. Lange-Malecki, “‘Color constancy’ in Mondrian patterns: a partial cancellation of physical chromaticity shifts by simultaneous contrast,” Vision Res. 30, 371–380 (1990).
[CrossRef]

Liao, J.

J. Villasenor, B. Belzer, J. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. Image Process. 4, 1053–1060 (1995).
[CrossRef] [PubMed]

Mallat, S.

S. Mallat, S. Zhong, “Wavelet maxima representation,” in Wavelets and Applications, Y. Meyer, ed. (Masson, Paris, 1991), pp. 207–284.

Marr, D.

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

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

McCann, J. J.

McCormick, J. A.

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. A5, 300–315 (1988); C. L. Fales, F. O. Huck, J. A. McCormick, S. K. Park, “Wiener restoration of sampled image data: end-to-end analysis,” in Selected Papers on Digital Image Restoration, M. Ibrahim Sezan, ed., Vol. MS47 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1992), pp. 229–243.

Mead, C.

C. Mead, Analog VLSI and Neural Systems (Addison-Wesley, Reading, Mass., 1989).
[CrossRef]

Meyer, Y.

R. R. Coifman, Y. Meyer, M. V. Wickerhauser, “Wavelet analysis and signal processing,” in Wavelets and Their Applications, M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, eds. (Jones and Bartlett, Boston, 1992), pp. 153–178.

Modestino, J. W.

J. W. Modestino, R. W. Fries, “Construction and properties of a useful two-dimensional random field,” IEEE Trans. Inf. Theory IT-26, 44–50 (1980).
[CrossRef]

Najand, S.

Narayanswamy, N.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Park, S. K.

F. O. Huck, C. L. Fales, R. Alter-Gartenberg, S. K. Park, Z. Rahman, “Information-theoretic assessment of sampled imaging systems,” Opt. Eng. 38, 742–762 (1999).
[CrossRef]

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. A5, 300–315 (1988); C. L. Fales, F. O. Huck, J. A. McCormick, S. K. Park, “Wiener restoration of sampled image data: end-to-end analysis,” in Selected Papers on Digital Image Restoration, M. Ibrahim Sezan, ed., Vol. MS47 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1992), pp. 229–243.

Parker, A. J.

M. J. Hawken, A. J. Parker, “Spatial receptive field organization in monkey V1 and its relationship to the cone mosaic,” in Computational Models of Visual Processing, M. S. Landy, J. A. Morshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 83–93.

Rahman, Z.

F. O. Huck, C. L. Fales, R. Alter-Gartenberg, S. K. Park, Z. Rahman, “Information-theoretic assessment of sampled imaging systems,” Opt. Eng. 38, 742–762 (1999).
[CrossRef]

D. J. Jobson, Z. Rahman, G. A. Woodell, “Properties and performance of a center/surround retinex,” IEEE Trans. Image Process. 6, 451–462 (1997).
[CrossRef] [PubMed]

D. J. Jobson, Z. Rahman, G. A. Woodell, “A multiscale retinex for bridging the gap between color images and the human observation of scenes,” IEEE Trans. Image Process. 6, 965–976 (1997).
[CrossRef] [PubMed]

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Figures (16)

Fig. 1
Fig. 1

Model of the visual communication channel: (a) signal-flow diagram, (b) retinex transformation, (c) small-signal model.

Fig. 2
Fig. 2

Models of (a) the reflectance ρ(x, y) with μρ = 5, (b) the normalized irradiance ι(x, y) with μι = 10 and σ b = 10, and (c) their product, the radiance field l(x, y).

Fig. 3
Fig. 3

Normalized PSD’s Φ̂ρ(ν, ω) of (a) the reflectance with m = 3 and (b) the irradiance with σ b = 12.

Fig. 4
Fig. 4

SR’s and SFR’s of image gathering and the retinex transformation: (a) image-gathering response with ζ = 0.3, (b) surround-function response with ζ = 0.08, (c) high-pass response produced by the surround function, (d) DOG response produced by the image-gathering and the surround-function responses.

Fig. 5
Fig. 5

Features of the retinex-transformed signal s e (x, y) for a uniform (top two rows) and a nonuniform (bottom two rows) irradiance: (a) the irradiance ι(x, y), (b) the radiance field l(x, y), (c) the retinex-transformed signal s e (x, y) without perturbations in the image gathering, (d) the retinex-transformed signal s e (x, y) with blurring in the image gathering.

Fig. 6
Fig. 6

Information rate ℋ and the maximum-realizable fidelity ℱ for several SNR’s Kσ ρ p plotted as functions of (a) the optical response index σ c with μρ = 1, (b) the η-bit quantization with μρ = 1, (c) the mean spatial detail μρ with 0.1 < μρ < 50. The PSD Φ̂ρ(ν, ω) of the scene is given for m = 3, and the irradiance is uniform. The retinex coding uses the surround function with an index of σ e = 12.

Fig. 7
Fig. 7

Images restored from the retinex-coded signal for three informationally optimized designs of the image-gathering device, as specified by the optical response index σ c and the SNR Kσ ρ p . The scene has the PSD Φ̂ρ(ν, ω) with m = 3 and a mean spatial detail of μρ = 5. The retinex coding uses the surround function with an index of σ e = 12 and η-bit quantization. The large, medium, and small formats display the restored images with densities of 16, 32, and 64 pixels/cm, respectively: (a) σ c = 2.4, SNR = 16, η = 6. (b) σ c = 3.0, SNR = 64, η = 8. (c) σ c = 3.4, SNR = 256, η = 10.

Fig. 8
Fig. 8

SFR’s of three image-gathering devices and of their corresponding Wiener filters for the images shown in Fig. 6: (a) image gathering, (b) Wiener filter, (c) Wiener filter with σ c = 3.4, SNR = 256, η = 10.

Fig. 9
Fig. 9

Information rate ℋ plotted as a function of the optical response index σ c for a SNR of Kσ ρ p = 256 and several PSD’s Φ̂ρ(ν, ω) of the scene, as specified by the fall-off rate m and the mean spatial detail μρ relative to the sampling interval: (a) m = 3 and (b) m = 2.

Fig. 10
Fig. 10

Information rate ℋ and the maximum-realizable fidelity ℱ for nonuniform irradiances plotted as functions of the optical response index σ c . The irradiances are characterized by the mean spatial detail μι and the blur index σ b : (a) μι = 10 and (b) μι = 2. The mean spatial detail of the scene is μρ = 1, the acquired signal has a SNR of 256, and the retinex coding uses the surround function with an index of σ e = 12 and an η = 10-bit quantization.

Fig. 11
Fig. 11

Radiance fields captured by the image-gathering device (upper row) and images restored from the retinex-coded signal (lower rows) for three nonuniform irradiances. The irradiances differ in the sharpness of their shadows, as specified by the blur index σ b : (a) σ b = 10, (b) σ b = 5, (c) σ b = 3. The mean spatial detail of the scene is μρ = 5, and the mean spatial detail of the irradiance is μι = 10. The image-gathering device uses the optical response index of σ c = 3.4 and a SNR of Kσ ρ p = 256, and the retinex coding uses the surround function with an index of σ e = 12 and an η = 10-bit quantization. The large, medium, and small formats display the restored images with densities of 16, 32, and 64 pixels/cm, respectively.

Fig. 12
Fig. 12

SFR’s of the image-gathering device and of their corresponding Wiener filters for the images shown in Fig. 10: (a) image gathering, (b) Wiener filter, (c) Wiener filter with σ b = 10.

Fig. 13
Fig. 13

Radiance field captured by the image-gathering device (top) and images restored from the retinex-coded signal for three retinex-surround functions with an index of (a) σ e = 6, (b) σ e = 12, (c) σ e = 24. The upper row of three images contains Wiener restorations, and the lower row contains slightly blurred Wiener restorations. The mean spatial detail of the scene is σρ = 5, and the mean spatial detail and the sharpness of the irradiance are μι = 10 and σ b = 10, respectively. The informationally optimized design is given by ρ c = 3.4 and SNR = 256, and the retinex coding uses an η = 10-bit quantization.

Fig. 14
Fig. 14

Information rate ℋ, the theoretical minimum data rate ε, and the information efficiency ℋ/ε plotted as functions of the mean spatial detail μρ of the scene for (a) uniform irradiance and (b) nonuniform irradiance. The mean spatial detail and the sharpness of the nonuniform irradiance are μι = 10 and σ b = 10, respectively. The image-gathering device is specified by the optical response index ρ c and the SNR, and the retinex coding is done with the retinex-surround index of σ e = 12 and an η-bit quantization.

Fig. 15
Fig. 15

Comparison of the theoretical minimum data rate ε with the data rate E 3 plotted versus the mean spatial detail μρ of the scene for a uniform and a nonuniform irradiance. The mean spatial detail and the sharpness of the nonuniform irradiance are μι = 10 and σ b = 10, respectively. The image-gathering device is given by the optical response index of ρ c = 3.4 and SNR = 256, and the retinex coding is given by the surround index of σ c = 12 and an η-bit quantization.

Fig. 16
Fig. 16

Differential pulse-code modulation with entropy coding.

Equations (83)

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

sx, y=KcLx, y * τx, y|||¯+npx, y,
|||¯|||¯x, y=m,n δx-m, y-n
sx, y=x, y sx, yδx-x, y-y,
sx, y=KcLx, y * τx, y+npx, y.
sex, y; κ=sex, y+nqx, y; κ,
sex, y=logsx, y-logsx, y   τex, y
sex, y=logsx, ysx, y   τex, y=log1+sx, y   Δτex, ysx, y   τex, y,
Δτex, y=δx, y-τex, y
sx, y   Δτex, ysx, y   τex, y 1.
sex, y=gx, ysx, y   Δτex, y,
gx, y=sx, y   τex, y-1.
Lx, y=1π ρx, yIx, y,
ρx, y=Δρx, y+ρ¯,
Ix, y=I0 ιx, y=I0Διx, y+ι¯,
Lx, y=1π I0lx, y=1π I0ρx, yιx, y,
Φˆρν, ω=1|A||Δρˆν, ω|2¯, Φˆιν, ω=1|A||Διˆν, ω|2¯,
σρ2= Φˆρν, ωdν dω, σι2= Φˆιν, ωdν dω.
ΦˆLν, ω=Ki2ι¯2Φˆρν, ω+ρ¯2Φˆιν, ω+Φˆρν, ω * Φˆιν, ω,
σL2=Ki2ι¯2σρ2+ρ¯2σι2+σι2σρ2.
Φˆρν, ω=2πμρ2σρ2m-21+2πμρζ2m/2  m>2Kρ1+2πμρζ2m/2  m2,
Kρ=4πμρ2σρ21-m/21+2πμρζl21-m/2 m<24πμρ2σρ2ln1+2πμρζl2  m=2.
Φˆιν, ω=2πμι2σι21+2πμιζ23/2exp-2σbζ2,
τx, y=πσc-2 exp-πr/σc2,
τˆν, ω=exp-σcζ2,
τex, y=πσe-2 exp-πr/σe2,
τ˜eν, ω=τˆeν, ω * |||¯ˆexp-σeζ2 if τˆeν, ω is band limited,
sx, y=Δsx, y+s¯x, y,
Δsx, y=KΔρx, yιx, y * τx, y+npx, y,s¯x, y=Kρ¯ιx, y * τx, y,
Δsx, y=s¯x, yΔρx, y * τx, y+npx, y,
s¯x, y=Kρ¯ιx, y,
sex, y=Δsx, y   Δτex, y+s¯x, y   Δτex, yΔsx, y   τex, y+s¯x, y   τex, y.
sex, y=Δρx, y *  τx, y   Δτex, y+eιx, y+eιρx, y+npx, y   Δτex, y/Kρ¯ιx, y1+Δρx, y *  τx, y   τex, y-eιx, y+eιρx, y+npx, y   Δτex, y/Kρ¯ιx, y,
eιx, y=ιx, y   Δτex, y/ιx, y, eιρx, y=ιx, y×Δρx, y *  τx, y   Δτex, y/ιx, y-Δρx, y *  τx, y   Δτex, y=-ιx, yΔρx, y *  τx, y  τex, y/ιx, y-Δρx, y *  τx, y   τex, y.
ιx, y-1=ι¯+Διx, y-11ι¯1-Διx, yι¯,
sex, y=Δρx, y *  τx, y+Διx, y+npx, y   Δτex, y,
s˜eν, ω; κ=Δρˆν, ωτˆν, ω+Διˆν, ω *  |||¯ˆ+ñpν, ωΔτ˜eν, ω+ñqν, ω; κ,
|||¯ˆ|||¯ˆν, ω=m,n δν-m, ω-n =δν, ω+|||¯ˆsν, ω
Bˆ=ν, ω; |ν|1/2, |ω|1/2
s˜eν, ω; κ=Δρˆν, ωτˆgν, ω+Δι˜ν, ω+nˆaν, ω+ñpν, ωΔτ˜eν, ω+ñqν, ω; κ.
τˆgν, ω τˆν, ωΔτ˜eν, ω=τˆν, ω-τˆν, ωτ˜eν, ω=exp-σcζ2-exp-σgζ2,τgx, y=πσc-2 exp-πr/σc2-πσg-2 exp-πr/σg2,
nˆaν, ω=Δρˆν, ωτˆν, ω *  |||¯ˆsν, ω.
s˜eν, ω; κ=Δρˆν, ωτˆν, ω+nˆν, ω; κ,
nˆν, ω; κ=e˜ν, ω+nˆaν, ω+ñpν, ω×Δτ˜eν, ω+ñqν, ω; κ
e˜ν, ω=e˜ιν, ω+eˆρν, ω,
e˜ιν, ω=Δι˜ν, ωΔτ˜eν, ω, eˆρν, ω=-Δρˆν, ωτˆν, ωτ˜eν, ω-Δρˆν, ωτ˜eν, ω,
Φ˜seν, ω; κ=Φ˜seν, ω+Φ˜qν, ω; κ,
Φ˜seν, ω=Φˆρgν, ω+Φˆneν, ω
Φˆρgν, ω=ρ¯-2Φˆρν, ω|τˆgν, ω|2,
Φˆneν, ω=Φ˜eιν, ω+Φˆaeν, ω+Φ˜peν, ω.
Φ˜eιν, ω=ι¯-2Φ˜ιν, ω|Δτ˜eν, ω|2,
Φ˜aeν, ω=ρ¯-2Φˆρν, ω|τˆν, ω|2 *  |||¯ˆs|Δτ˜eν, ω|2,
Φ˜peν, ω=Kρ¯ι¯-2Φ˜pν, ω|Δτ˜eν, ω|2,
Φ˜qν, ω; κ1|A||ñqν, ω; κ|2¯, σq2=Bˆ  Φ¯qν, ω; κdν dω.
Φ˜qν, ω; κ=Φ¯qκ=σq2=13cσseκ2=σse2κ-2,
σse2=B˜  Φ˜seν, ωdνdω.
rx, y; κ=ρ¯1+Δrx, y; κ,
Δrˆx, y; κ=x, y sex, y; κΨx-x, y-y *  τdx, y
Δrν, ω; κ=s˜eν, ω; κΨˆν, ω; κ,
Ψˆν, ω; κ=ρ¯-2Φˆρν, ωτˆg*ν, ωΦ˜seν, ω+Φ˜qν, ω; κ,
Ψˆν, ω; κ=Φˆρν, ωτˆg*ν, ωΦ˜seν, ω+Φ˜qν, ω; κ,
Ψˆν, ω; κ=Φˆρν, ωτˆg*ν, ωΦˆρν, ω|τˆgν, ω|2+Φˆneν, ω+ρ¯σse/Kσρκ2,
Φ˜seν, ω=Φˆρν, ω|τˆgν, ω|2+Φˆneν, ω
Φˆneν, ω=Φˆρν, ω|τˆν, ω|2 *  |||¯ˆs+σp/Kι¯σρ2+ρ¯/ι¯2σι/σρ2Φ˜ιν, ω|Δτ˜eν, ω|2
=εs˜eν, ω; κ-εs˜eν, ω; κ|Δρˆν, ω.
=εs˜eν, ω; κ-εnˆeν, ω; κ,
=12Bˆ  log1+Φˆρgν, ωΦˆneν, ω+Φ˜qν, ω; κdνdω,
=12Bˆ  log1+Φˆρν, ω|τˆgν, ω|2Φˆneν, ω+ρ¯σse/Kσρκ2dνdω,
C=12 |Bˆ|log1+Kσρ/σp2.
Φˆρν, ω=σρ2ν, ω  Bˆ0elsewhere, τˆgν, ω=1ν, ω  Bˆ0elsewhere.
ε=εs˜eν, ω; κ-εs˜eν, ω; κ|s˜ν, ω,
ε=εs˜eν, ω; κ-εñqν, ω; κ.
ε=12Bˆ  log1+Φ˜seν, ωΦ˜qν, ω; κdνdω,
ε=12Bˆ  log1+Kσρκσseρ¯2Φ˜seν, ωdνdω,
ε==12Bˆ  log1+Φˆρgν, ωΦ˜qν, ω; κdνdω.
ε=C=12 |Bˆ|log1+Kσρ/σq2.
ε=C=12 |Bˆ|log1+3κ2c2|Bˆ|log3κc.
F=1-A|ρx, y-rx, y; κ|2¯dxdyA|Δρx, y|2¯dxdy=1-σρ-2eˆr2ν, ω; κdνdω.
ˆd2ν, ω; κ=Φˆρν, ω1-τˆgν, ωΨˆν, ω; κ,
= Φˆρν, ωτˆgν, ωΨˆν, ω; κdνdω.
ˆν, ω; κ=-log1-τˆgν, ωΨˆν, ω; κ, τˆgν, ωΨˆν, ω; κ=1-2-ˆν, ω; κ.
=Φˆρν, ω1-2-ˆν, ω; κdνdω.
En=-s0=0κ-1s1=0κ-1  sn=0κ-1 ps0, s1,, sn×log ps0|s1, , sn.
E0=-s0=0κ-1 ps0log ps0,

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