Abstract

The circularly averaged power spectra of natural image ensembles tend to have a power-law dependence on spatial frequency with an exponent of approximately 2. This phenomenon has been attributed to object occlusion, the presence of edges, and scaling of object sizes (self-similarity) in natural scenes, although the relative importance of these properties is still unclear. A detailed examination of the effects of occlusion, edges, and self-similarity on the behavior of the power spectrum is conducted using a simple model of natural images. Numerical simulations show that edges and self-similarity are necessary for a power-law power spectrum over a wide range of spatial frequencies. Object occlusion is not an essential factor. A theoretical analysis for images containing nonoccluding objects supports these results.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
    [CrossRef] [PubMed]
  3. A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
    [CrossRef] [PubMed]
  4. D. L. Ruderman, “Origins of scaling in natural images,” Vision Res. 37, 3385–3398 (1997).
    [CrossRef]
  5. D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3383 (1997).
    [CrossRef]
  6. D. J. Field, “What the statistics of natural images tell us about visual coding,” Proc. SPIE1077, 269–276 (1989).
  7. D. C. Knill, D. J. Field, D. Kersten, “Human discrimination of fractal images,” J. Opt. Soc. Am. A 7, 1113–1123 (1990).
    [CrossRef] [PubMed]
  8. D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
    [CrossRef] [PubMed]
  9. R. M. Balboa, C. W. Tyles, N. M. Grzywacz, “Occlusions contribute to scaling in natural images,” Vision Res. 41, 955–964 (2001).
    [CrossRef] [PubMed]
  10. R. P. Millane, S. Alzaidi, W. H. Hsiao, “Scaling and power spectra of natural images,” in Proceedings of Image and Vision Computing New Zealand 2003, D. G. Bailey, ed. (Massey University, Palmerston North, New Zealand, 2003), 148–153).
  11. D. L. Ruderman, “Letter to the editor,” Vision Res. 42, 2799–2801 (2002).
    [CrossRef]
  12. N. M. Grzywacz, R. M. Balboa, C. W. Tyler, “Response to letter to the editor,” Vision Res. 42, 2803–2805 (2002).
    [CrossRef]
  13. I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, Table of Integrals, Series, and Products (Academic, 1980).
  14. See http://www.wolfram.com/.
  15. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, 1964).

2002 (2)

D. L. Ruderman, “Letter to the editor,” Vision Res. 42, 2799–2801 (2002).
[CrossRef]

N. M. Grzywacz, R. M. Balboa, C. W. Tyler, “Response to letter to the editor,” Vision Res. 42, 2803–2805 (2002).
[CrossRef]

2001 (1)

R. M. Balboa, C. W. Tyles, N. M. Grzywacz, “Occlusions contribute to scaling in natural images,” Vision Res. 41, 955–964 (2001).
[CrossRef] [PubMed]

1997 (2)

D. L. Ruderman, “Origins of scaling in natural images,” Vision Res. 37, 3385–3398 (1997).
[CrossRef]

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3383 (1997).
[CrossRef]

1996 (1)

A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

1994 (1)

D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

1992 (1)

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

1990 (1)

1987 (1)

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, 1964).

Alzaidi, S.

R. P. Millane, S. Alzaidi, W. H. Hsiao, “Scaling and power spectra of natural images,” in Proceedings of Image and Vision Computing New Zealand 2003, D. G. Bailey, ed. (Massey University, Palmerston North, New Zealand, 2003), 148–153).

Balboa, R. M.

N. M. Grzywacz, R. M. Balboa, C. W. Tyler, “Response to letter to the editor,” Vision Res. 42, 2803–2805 (2002).
[CrossRef]

R. M. Balboa, C. W. Tyles, N. M. Grzywacz, “Occlusions contribute to scaling in natural images,” Vision Res. 41, 955–964 (2001).
[CrossRef] [PubMed]

Bialek, W.

D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

Brady, N.

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3383 (1997).
[CrossRef]

Chao, T.

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

Field, D. J.

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3383 (1997).
[CrossRef]

D. C. Knill, D. J. Field, D. Kersten, “Human discrimination of fractal images,” J. Opt. Soc. Am. A 7, 1113–1123 (1990).
[CrossRef] [PubMed]

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]

D. J. Field, “What the statistics of natural images tell us about visual coding,” Proc. SPIE1077, 269–276 (1989).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, Table of Integrals, Series, and Products (Academic, 1980).

Grzywacz, N. M.

N. M. Grzywacz, R. M. Balboa, C. W. Tyler, “Response to letter to the editor,” Vision Res. 42, 2803–2805 (2002).
[CrossRef]

R. M. Balboa, C. W. Tyles, N. M. Grzywacz, “Occlusions contribute to scaling in natural images,” Vision Res. 41, 955–964 (2001).
[CrossRef] [PubMed]

Hsiao, W. H.

R. P. Millane, S. Alzaidi, W. H. Hsiao, “Scaling and power spectra of natural images,” in Proceedings of Image and Vision Computing New Zealand 2003, D. G. Bailey, ed. (Massey University, Palmerston North, New Zealand, 2003), 148–153).

Jeffrey, A.

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, Table of Integrals, Series, and Products (Academic, 1980).

Kersten, D.

Knill, D. C.

Millane, R. P.

R. P. Millane, S. Alzaidi, W. H. Hsiao, “Scaling and power spectra of natural images,” in Proceedings of Image and Vision Computing New Zealand 2003, D. G. Bailey, ed. (Massey University, Palmerston North, New Zealand, 2003), 148–153).

Ruderman, D. L.

D. L. Ruderman, “Letter to the editor,” Vision Res. 42, 2799–2801 (2002).
[CrossRef]

D. L. Ruderman, “Origins of scaling in natural images,” Vision Res. 37, 3385–3398 (1997).
[CrossRef]

D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, Table of Integrals, Series, and Products (Academic, 1980).

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, 1964).

Tadmor, Y.

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

Tolhurst, D. J.

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

Tyler, C. W.

N. M. Grzywacz, R. M. Balboa, C. W. Tyler, “Response to letter to the editor,” Vision Res. 42, 2803–2805 (2002).
[CrossRef]

Tyles, C. W.

R. M. Balboa, C. W. Tyles, N. M. Grzywacz, “Occlusions contribute to scaling in natural images,” Vision Res. 41, 955–964 (2001).
[CrossRef] [PubMed]

van der Schaaf, A.

A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

van Hateren, J. H.

A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

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

Ophthalmic Physiol. Opt. (1)

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

Phys. Rev. Lett. (1)

D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

Vision Res. (6)

R. M. Balboa, C. W. Tyles, N. M. Grzywacz, “Occlusions contribute to scaling in natural images,” Vision Res. 41, 955–964 (2001).
[CrossRef] [PubMed]

A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

D. L. Ruderman, “Origins of scaling in natural images,” Vision Res. 37, 3385–3398 (1997).
[CrossRef]

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3383 (1997).
[CrossRef]

D. L. Ruderman, “Letter to the editor,” Vision Res. 42, 2799–2801 (2002).
[CrossRef]

N. M. Grzywacz, R. M. Balboa, C. W. Tyler, “Response to letter to the editor,” Vision Res. 42, 2803–2805 (2002).
[CrossRef]

Other (5)

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, Table of Integrals, Series, and Products (Academic, 1980).

See http://www.wolfram.com/.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, 1964).

D. J. Field, “What the statistics of natural images tell us about visual coding,” Proc. SPIE1077, 269–276 (1989).

R. P. Millane, S. Alzaidi, W. H. Hsiao, “Scaling and power spectra of natural images,” in Proceedings of Image and Vision Computing New Zealand 2003, D. G. Bailey, ed. (Massey University, Palmerston North, New Zealand, 2003), 148–153).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Circularly averaged power spectrum (thick lines) and a linear fit (thin lines) for (a) single images and (b) an ensemble over 40 images. The slopes are 2.3, 2.5, and 2.9 in (a) and 2.3 in (b).

Fig. 2
Fig. 2

Amplitude function for variable amplitude objects.

Fig. 3
Fig. 3

Constant amplitude (a) occluding and (b) nonoccluding disks, and variable amplitude (c) occluding and (d) nonoccluding disks.

Fig. 4
Fig. 4

Model images of each type as labeled and described in Table 1. The columns correspond to α = 1 or d = 0.04 (left), α = 2 or d = 0.02 (center), and α = 3 or d = 0.006 (right).

Fig. 5
Fig. 5

Ensemble-averaged power spectra (thick lines) for (a) type A, (b) type B, (c) type C, (d) type D, (e) type E, and (f) type F images for α = 1 (top curves), α = 2 (middle curves), and α = 3 (bottom curves) (for image types A, C, E, and F) and for d = 0.04 (top curves), d = 0.02 (middle curves), and d = 0.006 (bottom curves) (for image types B and D). Plots have been shifted vertically to eliminate overlap. Linear fits as described in the text are shown by the thin lines.

Fig. 6
Fig. 6

Circularly averaged power spectra for images containing nonoccluding disks with a power-law size distribution calculated using Eq. (14) for α = 1 (top curve), α = 2 (middle curve), and α = 3 (bottom curve). Thin lines show regression lines.

Fig. 7
Fig. 7

Circularly averaged power spectra for images with exponentially distributed disk sizes for d = 0.04 (top curve), d = 0.02 (middle curve), and d = 0.006 (bottom curve) calculated numerically as described in the text. These curves show asymptotic linear fits with slopes of 3 .

Fig. 8
Fig. 8

M ( ρ ) versus ρ on a log–log plot.

Tables (2)

Tables Icon

Table 1 Image Types as Described in the Text

Tables Icon

Table 2 Slopes of Log–Log Power Spectra

Equations (39)

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

l ( r ) = L 2 [ 1 + cos ( π r a ) ] ,
P ( a ) = k p ( α , a 1 , a 2 ) a α for a 1 < a < a 2 ,
= 0 otherwise ,
k p ( α , a 1 , a 2 ) = α 1 a 1 α + 1 a 2 α + 1 for α 1 ,
= 1 ln ( a 2 ) ln ( a 1 ) for α = 1 ,
P ( a ) = k e ( d , a 1 , a 2 ) exp ( a d ) for a 1 < a < a 2 ,
= 0          otherwise ,
k e ( d , a 1 , a 2 ) = { d [ exp ( a 1 d ) exp ( a 2 d ) ] } 1
S ( ρ ) = S i ( ρ ) i = F i ( ρ , ϕ ) 2 ϕ i ,
S ( ρ ) = B ρ γ
f ( x , y ) = m = 1 N A m disk [ ( x x m ) 2 + ( y y m ) 2 a m ] ,
F ( u , v ) = m = 1 N A m a m 2 jinc ( a m u 2 + v 2 ) exp ( i 2 π ( u x m + v y m ) ) ,
F ( ρ , ϕ ) = m = 1 N A m a m 2 jinc ( ρ a m ) exp ( i 2 π ρ ( x m cos ϕ + y m sin ϕ ) ) ,
F ( ρ , ϕ ) 2 = m = 1 N n = 1 N A m A n a m 2 a n 2 jinc ( ρ a m ) jinc ( ρ a n ) exp ( i 2 π ρ C m n [ cos ( ϕ θ m n ) ] ) ,
C m n = ( x m x n ) 2 + ( y m y n ) 2
1 2 π 0 2 π F ( ρ , ϕ ) 2 d ϕ
= m = 1 N n = 1 N A m A n a m 2 a n 2 jinc ( ρ a m ) jinc ( ρ a n ) J 0 ( 2 π ρ C m n ) .
S ( ρ ) = 1 2 π N 0 2 π F ( ρ ; ϕ ) 2 d ϕ = A 2 a 4 jinc 2 ( ρ a ) + N A 2 a 2 jinc ( ρ a ) 2 M ( ρ ) ,
M ( ρ ) = J 0 ( 2 π ρ C m n ) k 1 ρ 7 2 ,
a 4 jinc 2 ( ρ a ) p k 2 ρ α 5 , for 2 < α < 3 ,
a 2 jinc ( ρ a ) p 2 k 3 ρ 2 α 6 , for 1 2 < α < 3 ,
S p ( ρ ) ρ α 5 .
a 4 jinc 2 ( ρ a ) e = ( d 2 π 2 ) ρ 3 for ρ 1 4 π d ,
a 2 jinc ( ρ a ) e 2 = ( 16 π 4 d 2 ) ρ 6 for ρ 1 2 π d .
M ( ρ ) = J 0 ( 2 π ρ C m n ) = 0 2 J 0 ( 2 π ρ C ) P ( C ) d C ,
P ( C ) = 2 π C 8 C 2 + 2 C 3 for 0 C 1 ,
= 4 [ sin 1 ( 2 C 2 1 ) 1 ] C + 8 C C 2 1 2 C 3 for 1 C 2 .
a 4 jinc 2 ( ρ a ) p = k p a 1 a 2 a 4 jinc 2 ( ρ a ) a α d a .
a 4 jinc 2 ( ρ a ) p k p ρ 2 0 J 1 2 ( 2 π ρ a ) a 2 α d a = k p Γ ( 5 α 2 ) 2 5 π 3 α ρ 5 α Γ ( α 1 2 ) F [ 5 α 2 , 3 2 ; 3 ; 1 ] = ( k p Γ ( α ) ( α 3 5 α 2 + 7 α 3 ) 2 5 π 2 α ( α 2 ) cos ( π α 2 ) [ Γ ( ( α + 1 ) 2 ) ] 4 ) ρ α 5 for 2 < α < 3 ,
a 2 jinc ( ρ a ) p = k p a 1 a 2 a 2 jinc ( ρ a ) a α d a .
a 2 jinc ( ρ a ) p 2 [ k p ρ 0 J 1 ( 2 π ρ a ) a 1 α d a ] 2 = ( k p [ ( 1 α ) π α 1 ] 4 cos ( π α 2 ) [ Γ ( ( 1 + α ) 2 ) ] 2 ) 2 ρ 2 α 6 for 1 2 < α < 3 .
a 4 jinc 2 ( ρ a ) e = k e a 1 a 2 a 4 jinc 2 ( ρ a ) exp ( a d ) d a .
a 4 jinc 4 ( ρ a ) e = 1 d ρ 2 0 a 2 J 1 2 ( 2 π ρ a ) exp ( a d ) d a = 3 π d 0 π 2 cos 2 θ ( 1 4 d 2 + 4 π 2 ρ 2 cos 2 θ ) 5 2 d θ .
a 4 jinc 2 ( ρ a ) e = 2 d 2 π ( 1 + x 2 ) 5 2 ρ 2 [ ( x 4 1 ) E ( x 2 1 + x 2 ) + ( 1 + x 2 ) K ( x 2 1 + x 2 ) ] ,
K ( x 2 1 + x 2 ) 2 ln ( 2 ) + ln ( x ) for x .
a 4 jinc 2 ( ρ a ) e = ( d 2 π 2 ) ρ 3 for ρ 1∕4 π d .
a 2 jinc ( ρ a ) e = k e a 1 a 2 a 2 jinc ( ρ a ) exp ( a d ) d a .
a 2 jinc ( ρ a ) e 2 = [ 1 d ρ 0 a J 1 ( ρ a ) exp ( a d ) d a ] 2 = [ 2 π d ( 1 d 2 + 4 π 2 ρ 2 ) 3 2 ] 2 = ( 16 π 4 d 2 ) 1 ρ 6 for ρ 1 2 π d .
γ = 5 α .

Metrics