Abstract

The human visual system combines a wide field of view with a high-resolution fovea and uses eye, head, and body movements to direct the fovea to potentially relevant locations in the visual scene. This strategy is sensible for a visual system with limited neural resources. However, for this strategy to be effective, the visual system needs sophisticated central mechanisms that efficiently exploit the varying spatial resolution of the retina. To gain insight into some of the design requirements of these central mechanisms, we have analyzed the effects of variable spatial resolution on local contrast in 300 calibrated natural images. Specifically, for each retinal eccentricity (which produces a certain effective level of blur), and for each value of local contrast observed at that eccentricity, we measured the probability distribution of the local contrast in the unblurred image. These conditional probability distributions can be regarded as posterior probability distributions for the “true” unblurred contrast, given an observed contrast at a given eccentricity. We find that these conditional probability distributions are adequately described by a few simple formulas. To explore how these statistics might be exploited by central perceptual mechanisms, we consider the task of selecting successive fixation points, where the goal on each fixation is to maximize total contrast information gained about the image (i.e., minimize total contrast uncertainty). We derive an entropy minimization algorithm and find that it performs optimally at reducing total contrast uncertainty and that it also works well at reducing the mean squared error between the original image and the image reconstructed from the multiple fixations. Our results show that measurements of local contrast alone could efficiently drive the scan paths of the eye when the goal is to gain as much information about the spatial structure of a scene as possible.

© 2005 Optical Society of America

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2005 (1)

J. Najemnik, W. S. Geisler, “Optimal eye movement strategies in visual search,” Nature 434, 387–391 (2005).
[CrossRef] [PubMed]

2003 (2)

P. L. Clatworthy, M. Chirimuuta, J. S. Lauritzen, D. J. Tolhurst, “Coding of the contrasts in natural images by populations of neurons in primary visual cortex (VI),” Vision Res. 43, 1983–2001 (2003).
[CrossRef] [PubMed]

R. M. Balboa, N. M. Grzywacz, “Power spectra and distribution of contrasts of natural images from different habitats,” Vision Res. 43, 2527–2537 (2003).
[CrossRef] [PubMed]

2002 (2)

W. S. Geisler, R. Diehl, “Bayesian natural selection and the evolution of perceptual systems,” Philos. Trans. R. Soc. London, Ser. B 357, 419–448 (2002).
[CrossRef] [PubMed]

G. E. Legge, T. A. Hooven, T. S. Klitz, J. G. Mansfield, B. S. Tjan, “Mr. Chips 2002: new insights from an ideal observer model of reading,” Vision Res. 42, 2219–2234 (2002).
[CrossRef] [PubMed]

2001 (3)

W. S. Geisler, J. S. Perry, B. J. Super, D. P. Gallogly, “Edge co-occurrence in natural images predicts contour grouping performance,” Vision Res. 41, 711–724 (2001).
[CrossRef] [PubMed]

E. P. Simoncelli, B. A. Olshausen, “Natural image statistics and neural representation,” Annu. Rev. Neurosci. 24, 1193–1215 (2001).
[CrossRef] [PubMed]

O. Schwartz, E. P. Simoncelli, “Natural signal statistics and sensory gain control,” Nat. Neurosci. 4, 819–825 (2001).
[CrossRef] [PubMed]

2000 (2)

Y. Tadmor, D. J. Tolhurst, “Calculating the contrasts that retinal ganglion cells and LGN neurones encounter in natural scenes,” Vision Res. 40, 3145–3157 (2000).
[CrossRef] [PubMed]

N. Brady, D. J. Field, “Local contrast in natural images: normalization and coding efficiency,” Perception 29, 1041–1055 (2000).
[CrossRef]

1999 (1)

P. Reinagel, A. M. Zador, “Natural scene statistics at the centre of gaze,” Network Comput. Neural Syst. 10, 1–10 (1999).

1998 (2)

J. H. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London, Ser. B 265, 359–366 (1998).
[CrossRef]

W. S. Geisler, J. S. Perry, “A real-time foveated multi-resolution system for low-bandwidth video communication,” Proc. SPIE 3299, 294–305 (1998).
[CrossRef]

1997 (2)

A. J. Bell, T. J. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
[CrossRef]

B. A. Olshausen, D. J. Field, “Sparse coding with an overcomplete basis set: a strategy by V1?” Vision Res. 37, 3311–3325 (1997).
[CrossRef]

1996 (2)

D. Geman, B. Jedynak, “An active testing model for tracking roads in satellite images,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1–14 (1996).
[CrossRef]

T. L. Arnow, W. S. Geisler, “Visual detection following retinal damage: predictions of an inhomogeneous retino-cortical model,” Proc. SPIE 2674, 119–130 (1996).
[CrossRef]

1994 (1)

D. L. Ruderman, “The statistics of natural images,” Network Comput. Neural Syst. 5, 517–548 (1994).
[CrossRef]

1992 (3)

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

J. J. Atick, A. N. Redlich, “What does the retina know about natural scenes?” Neural Comput. 4, 196–210 (1992).
[CrossRef]

J. H. van Hateren, “Real and optimal neural images in early vision,” Nature 360, 68–70 (1992).
[CrossRef] [PubMed]

1991 (1)

1987 (1)

1981 (2)

S. B. Laughlin, “A simple coding procedure enhances a neuron’s information capacity,” Z. Naturforsch. C 36, 910–912 (1981).
[PubMed]

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 21, 409–418 (1981).
[CrossRef] [PubMed]

Anderson, S. J.

Arnow, T. L.

T. L. Arnow, W. S. Geisler, “Visual detection following retinal damage: predictions of an inhomogeneous retino-cortical model,” Proc. SPIE 2674, 119–130 (1996).
[CrossRef]

Atick, J. J.

J. J. Atick, A. N. Redlich, “What does the retina know about natural scenes?” Neural Comput. 4, 196–210 (1992).
[CrossRef]

Balboa, R. M.

R. M. Balboa, N. M. Grzywacz, “Power spectra and distribution of contrasts of natural images from different habitats,” Vision Res. 43, 2527–2537 (2003).
[CrossRef] [PubMed]

Banks, M. S.

Bell, A. J.

A. J. Bell, T. J. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
[CrossRef]

Bovik, A. C.

U. Rajashekar, L. K. Cormack, A. C. Bovik, “Visual search: structure from noise,” in Proceedings of Eye Tracking Research & Applications, ACM SIGGRAPH2002, A. T. Duchowski, ed. pp. 119–123 (www.siggrraph.org).

Brady, N.

N. Brady, D. J. Field, “Local contrast in natural images: normalization and coding efficiency,” Perception 29, 1041–1055 (2000).
[CrossRef]

Chao, T.

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

Chirimuuta, M.

P. L. Clatworthy, M. Chirimuuta, J. S. Lauritzen, D. J. Tolhurst, “Coding of the contrasts in natural images by populations of neurons in primary visual cortex (VI),” Vision Res. 43, 1983–2001 (2003).
[CrossRef] [PubMed]

Clatworthy, P. L.

P. L. Clatworthy, M. Chirimuuta, J. S. Lauritzen, D. J. Tolhurst, “Coding of the contrasts in natural images by populations of neurons in primary visual cortex (VI),” Vision Res. 43, 1983–2001 (2003).
[CrossRef] [PubMed]

Cormack, L. K.

U. Rajashekar, L. K. Cormack, A. C. Bovik, “Visual search: structure from noise,” in Proceedings of Eye Tracking Research & Applications, ACM SIGGRAPH2002, A. T. Duchowski, ed. pp. 119–123 (www.siggrraph.org).

Coughlan, J.

L. W. Renninger, J. Coughlan, P. Verghese, J. Malik, “An information maximization model of eye movements,” in Advances in Neural Information Processing Systems 17, L. K. Saul, Y. Weiss, and L. Bottou, eds. (MIT Press, 2005), pp. 1121–1128.

Cover, T.

T. Cover, J. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Diehl, R.

W. S. Geisler, R. Diehl, “Bayesian natural selection and the evolution of perceptual systems,” Philos. Trans. R. Soc. London, Ser. B 357, 419–448 (2002).
[CrossRef] [PubMed]

Field, D. J.

N. Brady, D. J. Field, “Local contrast in natural images: normalization and coding efficiency,” Perception 29, 1041–1055 (2000).
[CrossRef]

B. A. Olshausen, D. J. Field, “Sparse coding with an overcomplete basis set: a strategy by V1?” Vision Res. 37, 3311–3325 (1997).
[CrossRef]

D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
[CrossRef] [PubMed]

Finkelstein, M. A.

D. C. Hood, M. A. Finkelstein, “Sensitivity to light,” in Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, and J. P. Thomas, eds. (Wiley, 1986), Vol. 1.

Gallogly, D. P.

W. S. Geisler, J. S. Perry, B. J. Super, D. P. Gallogly, “Edge co-occurrence in natural images predicts contour grouping performance,” Vision Res. 41, 711–724 (2001).
[CrossRef] [PubMed]

Geisler, W. S.

J. Najemnik, W. S. Geisler, “Optimal eye movement strategies in visual search,” Nature 434, 387–391 (2005).
[CrossRef] [PubMed]

W. S. Geisler, R. Diehl, “Bayesian natural selection and the evolution of perceptual systems,” Philos. Trans. R. Soc. London, Ser. B 357, 419–448 (2002).
[CrossRef] [PubMed]

W. S. Geisler, J. S. Perry, B. J. Super, D. P. Gallogly, “Edge co-occurrence in natural images predicts contour grouping performance,” Vision Res. 41, 711–724 (2001).
[CrossRef] [PubMed]

W. S. Geisler, J. S. Perry, “A real-time foveated multi-resolution system for low-bandwidth video communication,” Proc. SPIE 3299, 294–305 (1998).
[CrossRef]

T. L. Arnow, W. S. Geisler, “Visual detection following retinal damage: predictions of an inhomogeneous retino-cortical model,” Proc. SPIE 2674, 119–130 (1996).
[CrossRef]

Geman, D.

D. Geman, B. Jedynak, “An active testing model for tracking roads in satellite images,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1–14 (1996).
[CrossRef]

Graham, N.

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 21, 409–418 (1981).
[CrossRef] [PubMed]

Grzywacz, N. M.

R. M. Balboa, N. M. Grzywacz, “Power spectra and distribution of contrasts of natural images from different habitats,” Vision Res. 43, 2527–2537 (2003).
[CrossRef] [PubMed]

Hood, D. C.

D. C. Hood, M. A. Finkelstein, “Sensitivity to light,” in Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, and J. P. Thomas, eds. (Wiley, 1986), Vol. 1.

Hooven, T. A.

G. E. Legge, T. A. Hooven, T. S. Klitz, J. G. Mansfield, B. S. Tjan, “Mr. Chips 2002: new insights from an ideal observer model of reading,” Vision Res. 42, 2219–2234 (2002).
[CrossRef] [PubMed]

Jedynak, B.

D. Geman, B. Jedynak, “An active testing model for tracking roads in satellite images,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1–14 (1996).
[CrossRef]

Klitz, T. S.

G. E. Legge, T. A. Hooven, T. S. Klitz, J. G. Mansfield, B. S. Tjan, “Mr. Chips 2002: new insights from an ideal observer model of reading,” Vision Res. 42, 2219–2234 (2002).
[CrossRef] [PubMed]

Laughlin, S. B.

S. B. Laughlin, “A simple coding procedure enhances a neuron’s information capacity,” Z. Naturforsch. C 36, 910–912 (1981).
[PubMed]

Lauritzen, J. S.

P. L. Clatworthy, M. Chirimuuta, J. S. Lauritzen, D. J. Tolhurst, “Coding of the contrasts in natural images by populations of neurons in primary visual cortex (VI),” Vision Res. 43, 1983–2001 (2003).
[CrossRef] [PubMed]

Lee, T. S.

T. S. Lee, S. Yu, “An information-theoretic framework for understanding saccadic behaviors,” in Advances in Neural Information Processing Systems, S. A. Solla, T. K. Leen, and K.-R. Muller, eds. (MIT Press, 2000) Vol. 12, pp. 834–840.

Legge, G. E.

G. E. Legge, T. A. Hooven, T. S. Klitz, J. G. Mansfield, B. S. Tjan, “Mr. Chips 2002: new insights from an ideal observer model of reading,” Vision Res. 42, 2219–2234 (2002).
[CrossRef] [PubMed]

Lotto, R. B.

D. Purves, R. B. Lotto, Why We See What We Do: An Empirical Theory of Vision (Sinauer, 2003).

Malik, J.

L. W. Renninger, J. Coughlan, P. Verghese, J. Malik, “An information maximization model of eye movements,” in Advances in Neural Information Processing Systems 17, L. K. Saul, Y. Weiss, and L. Bottou, eds. (MIT Press, 2005), pp. 1121–1128.

Mansfield, J. G.

G. E. Legge, T. A. Hooven, T. S. Klitz, J. G. Mansfield, B. S. Tjan, “Mr. Chips 2002: new insights from an ideal observer model of reading,” Vision Res. 42, 2219–2234 (2002).
[CrossRef] [PubMed]

Najemnik, J.

J. Najemnik, W. S. Geisler, “Optimal eye movement strategies in visual search,” Nature 434, 387–391 (2005).
[CrossRef] [PubMed]

Olshausen, B. A.

E. P. Simoncelli, B. A. Olshausen, “Natural image statistics and neural representation,” Annu. Rev. Neurosci. 24, 1193–1215 (2001).
[CrossRef] [PubMed]

B. A. Olshausen, D. J. Field, “Sparse coding with an overcomplete basis set: a strategy by V1?” Vision Res. 37, 3311–3325 (1997).
[CrossRef]

Perry, J. S.

W. S. Geisler, J. S. Perry, B. J. Super, D. P. Gallogly, “Edge co-occurrence in natural images predicts contour grouping performance,” Vision Res. 41, 711–724 (2001).
[CrossRef] [PubMed]

W. S. Geisler, J. S. Perry, “A real-time foveated multi-resolution system for low-bandwidth video communication,” Proc. SPIE 3299, 294–305 (1998).
[CrossRef]

Purves, D.

D. Purves, R. B. Lotto, Why We See What We Do: An Empirical Theory of Vision (Sinauer, 2003).

Rajashekar, U.

U. Rajashekar, L. K. Cormack, A. C. Bovik, “Visual search: structure from noise,” in Proceedings of Eye Tracking Research & Applications, ACM SIGGRAPH2002, A. T. Duchowski, ed. pp. 119–123 (www.siggrraph.org).

Redlich, A. N.

J. J. Atick, A. N. Redlich, “What does the retina know about natural scenes?” Neural Comput. 4, 196–210 (1992).
[CrossRef]

Reinagel, P.

P. Reinagel, A. M. Zador, “Natural scene statistics at the centre of gaze,” Network Comput. Neural Syst. 10, 1–10 (1999).

Renninger, L. W.

L. W. Renninger, J. Coughlan, P. Verghese, J. Malik, “An information maximization model of eye movements,” in Advances in Neural Information Processing Systems 17, L. K. Saul, Y. Weiss, and L. Bottou, eds. (MIT Press, 2005), pp. 1121–1128.

Robson, J. G.

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 21, 409–418 (1981).
[CrossRef] [PubMed]

Ruderman, D. L.

D. L. Ruderman, “The statistics of natural images,” Network Comput. Neural Syst. 5, 517–548 (1994).
[CrossRef]

Schwartz, O.

O. Schwartz, E. P. Simoncelli, “Natural signal statistics and sensory gain control,” Nat. Neurosci. 4, 819–825 (2001).
[CrossRef] [PubMed]

Sejnowski, T. J.

A. J. Bell, T. J. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
[CrossRef]

Sekuler, A. B.

Simoncelli, E. P.

O. Schwartz, E. P. Simoncelli, “Natural signal statistics and sensory gain control,” Nat. Neurosci. 4, 819–825 (2001).
[CrossRef] [PubMed]

E. P. Simoncelli, B. A. Olshausen, “Natural image statistics and neural representation,” Annu. Rev. Neurosci. 24, 1193–1215 (2001).
[CrossRef] [PubMed]

Super, B. J.

W. S. Geisler, J. S. Perry, B. J. Super, D. P. Gallogly, “Edge co-occurrence in natural images predicts contour grouping performance,” Vision Res. 41, 711–724 (2001).
[CrossRef] [PubMed]

Tadmor, Y.

Y. Tadmor, D. J. Tolhurst, “Calculating the contrasts that retinal ganglion cells and LGN neurones encounter in natural scenes,” Vision Res. 40, 3145–3157 (2000).
[CrossRef] [PubMed]

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

Thomas, J.

T. Cover, J. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Tjan, B. S.

G. E. Legge, T. A. Hooven, T. S. Klitz, J. G. Mansfield, B. S. Tjan, “Mr. Chips 2002: new insights from an ideal observer model of reading,” Vision Res. 42, 2219–2234 (2002).
[CrossRef] [PubMed]

Tolhurst, D. J.

P. L. Clatworthy, M. Chirimuuta, J. S. Lauritzen, D. J. Tolhurst, “Coding of the contrasts in natural images by populations of neurons in primary visual cortex (VI),” Vision Res. 43, 1983–2001 (2003).
[CrossRef] [PubMed]

Y. Tadmor, D. J. Tolhurst, “Calculating the contrasts that retinal ganglion cells and LGN neurones encounter in natural scenes,” Vision Res. 40, 3145–3157 (2000).
[CrossRef] [PubMed]

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

van der Schaaf, A.

J. H. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London, Ser. B 265, 359–366 (1998).
[CrossRef]

van Hateren, J. H.

J. H. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London, Ser. B 265, 359–366 (1998).
[CrossRef]

J. H. van Hateren, “Real and optimal neural images in early vision,” Nature 360, 68–70 (1992).
[CrossRef] [PubMed]

Verghese, P.

L. W. Renninger, J. Coughlan, P. Verghese, J. Malik, “An information maximization model of eye movements,” in Advances in Neural Information Processing Systems 17, L. K. Saul, Y. Weiss, and L. Bottou, eds. (MIT Press, 2005), pp. 1121–1128.

Yarbus, A. L.

A. L. Yarbus, Eye Movements and Vision (Plenum, 1967).
[CrossRef]

Yu, S.

T. S. Lee, S. Yu, “An information-theoretic framework for understanding saccadic behaviors,” in Advances in Neural Information Processing Systems, S. A. Solla, T. K. Leen, and K.-R. Muller, eds. (MIT Press, 2000) Vol. 12, pp. 834–840.

Zador, A. M.

P. Reinagel, A. M. Zador, “Natural scene statistics at the centre of gaze,” Network Comput. Neural Syst. 10, 1–10 (1999).

Annu. Rev. Neurosci. (1)

E. P. Simoncelli, B. A. Olshausen, “Natural image statistics and neural representation,” Annu. Rev. Neurosci. 24, 1193–1215 (2001).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

D. Geman, B. Jedynak, “An active testing model for tracking roads in satellite images,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1–14 (1996).
[CrossRef]

J. Opt. Soc. Am. A (2)

Nat. Neurosci. (1)

O. Schwartz, E. P. Simoncelli, “Natural signal statistics and sensory gain control,” Nat. Neurosci. 4, 819–825 (2001).
[CrossRef] [PubMed]

Nature (2)

J. Najemnik, W. S. Geisler, “Optimal eye movement strategies in visual search,” Nature 434, 387–391 (2005).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Probability distributions of local rms contrast for various levels of blur based on the human contrast sensitivity function at different retinal eccentricities. These distributions were obtained by randomly sampling small patches from 300 calibrated natural images.

Fig. 2
Fig. 2

These plots show examples of the conditional probability distributions of local rms contrast in unblurred images, given the local rms contrast in the blurred versions of the images (columns) and given the retinal eccentricity (rows). The solid symbols are empirical histograms computed from 300 natural images that contained no man-made objects. The smooth curves are the best-fitting skewed Gaussian distribution (a Gaussian with different standard deviations above and below the mode).

Fig. 3
Fig. 3

Modes and average standard deviations of the conditional probability densities are plotted as a function of blurred image contrast and retinal eccentricity. The average standard deviation is the average of the two standard deviation parameters in the skewed Gaussian distribution. See Fig. 2 for examples of the conditional densities and fits of the skewed Gaussian distribution. The curves are best-fitting straight lines through the origin.

Fig. 4
Fig. 4

Slopes of the linear functions in Fig. 3. A, Slope of the contrast versus mode plot as a function of retinal eccentricity. B, Slope of the contrast versus average standard deviation plot as a function of retinal eccentricity. The curves show the predictions of the linear model: c ̂ = k ε c + c and σ ¯ = k ε c , where k = 0.105 .

Fig. 5
Fig. 5

Images used to test a fixation selection algorithm based on the principle of minimizing contrast entropy.

Fig. 6
Fig. 6

Fixation points selected by the principle of minimizing total contrast entropy (contrast uncertainty), using the average local contrast statistics of natural images. A, Sequence of nine fixations (eight saccades) for a distant image containing sky, ground, and trees. B, Relative contrast entropy as a function of fixation number for the image in A (open circles), predicted relative contrast entropy before the fixation was made (solid circles), and optimal relative contrast entropy that could be obtained (open triangles). C, Sequence of nine fixations (eight saccades) for a close-up image containing foliage. D, Same type of plot shown in B.

Fig. 7
Fig. 7

Average fixation selection performance for the 16 test images in Fig. 5. A, Relative contrast entropy as a function of fixation number (open circles), predicted relative contrast entropy before the fixation was made (solid circles), and optimal relative contrast entropy that could be obtained (open triangles). B, Ratio of the optimal contrast entropy that could be obtained to the contrast entropy that was obtained: CEM algorithm (solid circles), tiling algorithm (open circles), random algorithm (open triangles). C, Relative mean squared error (MSE) between the original (unblurred) image and the image reconstructed from the fixations up to and including the fixation number given on the horizontal axis: CEM algorithm (solid circles), optimal (open circles). D, Ratio of optimal MSE that could be obtained to the MSE that was obtained: CEM algorithm (solid circles), tiling algorithm (open circles), random algorithm (open triangles).

Equations (26)

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c ̂ = k c ε + c ,
σ ¯ 2 = ( k c ε ) 2 + σ 0 2 ,
h = 1 2 log 2 ( 2 π e σ ¯ 2 ) .
C ( f , ε ) = C 0 exp ( α f ε 2 + ε ε 2 ) ,
w i = 0.5 { cos [ π p ( x i x c ) 2 + ( y i y c ) 2 ] + 1 } ,
c = 1 i = 1 N w i i = 1 N w i ( L i L ) 2 ( L + L 0 ) 2 ,
L = 1 i = 1 N w i i = 1 N w i L i ;
F ( f x , f y , ε ) = exp ( α f x 2 + f y 2 ε 2 + ε ε 2 ) .
r ( ε ) ε 2 ε + ε 2 .
g ( x ; u , σ l , σ h ) = { 1 2 π ( σ l + σ h 2 ) exp [ ( x u ) 2 2 σ l 2 ] x u 1 2 π ( σ l + σ h 2 ) exp [ ( x u ) 2 2 σ h 2 ] x > u } .
h ( p ) = p ( x ) ln [ p ( x ) ] d x .
h ( g ) = 2 σ l ( σ l + σ h ) u ϕ l ( x ) { ln [ ( σ l + σ h ) 2 σ l ] + ln ( 2 π σ l ) + ( x u ) 2 2 σ l 2 } d x + 2 σ h ( σ l + σ h ) u ϕ h ( x ) { ln [ ( σ l + σ h ) 2 σ h ] + ln ( 2 π σ h ) + ( x u ) 2 2 σ h 2 } d x ,
h ( g ) = σ l ( σ l + σ h ) { ln [ ( σ l + σ h ) 2 σ l ] + ln ( 2 π σ l 2 ) + 1 2 } + σ h ( σ l + σ h ) { ln [ ( σ l + σ h ) 2 σ h ] + ln ( 2 π σ h 2 ) + 1 2 } ,
h ( g ) = σ l ( σ l + σ h ) { ln [ 2 π ( σ l + σ h ) 2 ] + 1 2 } + σ h ( σ l + σ h ) { ln [ 2 π ( σ l + σ h ) 2 ] + 1 2 } ,
h ( g ) = ln [ 2 π ( σ l + σ h ) 2 ] + 1 2 ,
h ( g ) = 1 2 log 2 ( 2 π e σ ¯ 2 ) .
ε i t = ( x i x t ) 2 + ( y i y t ) 2 .
ε i ( T ) = min t T ε i t .
h i ( T ) = 1 2 log 2 ( 2 π e { [ k ε i ( T ) c i ( T ) ] 2 + σ 0 2 } ) .
U ( T ) = i = 1 n h i ( T ) .
( x ̂ T + 1 , y ̂ T + 1 ) = arg max x T + 1 , y T + 1 [ U ̂ ( T + 1 , x T + 1 , y T + 1 ) ] ,
U ̂ ( T + 1 , x T + 1 , y T + 1 ) = i = 1 n h ̂ i ( T + 1 , x T + 1 , y T + 1 ) ,
h ̂ i ( T + 1 , x T + 1 , y T + 1 ) = 1 2 log 2 ( 2 π e { [ k ε i ( T + 1 , x T + 1 , y T + 1 ) c ̂ i ( T + 1 , x T + 1 , y T + 1 ) ] 2 + σ 0 2 } ) .
C ̂ i ( T ) = k c i ( T ) ε i ( T ) + c i ( T ) .
C ̂ i ( T ) k c i ( T + 1 , x T + 1 , y T + 1 ) ε i ( T + 1 , x T + 1 , y T + 1 ) + c i ( T + 1 , x T + 1 , y T + 1 ) ,
c ̂ i ( T + 1 , x T + 1 , y T + 1 ) = C ̂ i ( T ) k ε i ( T + 1 , x T + 1 , y T + 1 ) + 1 .

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