Abstract

Complex moments of the local power spectrum (CMP) are investigated in a multiscale context. The multiscale CMPs are shown to approximate well the 1D angular Fourier transform of the band in question. This observation is used to derive further properties of the power spectrum in terms of texture orientations or n-folded symmetry patterns. A method is presented to approximate the power spectrum using only separable filtering in the spatial domain. Interesting implications to the Gabor decomposition are shown. The number of orientations in the filter bank is related to the order of n-folded symmetry detectable. Furthermore, the multiscale CMPs can be estimated incrementally in the spatial domain, which is both fast and reliable. Experiments on power spectrum estimation, orientation estimation, and texture segmentation are presented.

© 2007 Optical Society of America

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  1. A. C. Bovik, M. Clark, and W. S. Geisler, "Multichannel texture analysis using localized spatial filters," IEEE Trans. Pattern Anal. Mach. Intell. 12, 55-73 (1990).
    [CrossRef]
  2. I. Fogel and D. Sagi, "Gabor filters as texture discriminator," Biol. Cybern. 61, 103-113 (1989).
    [CrossRef]
  3. A. K. Jain and F. Farrokhnia, "Unsupervised texture segmentation using Gabor filters," Pattern Recogn. 24, 1167-1186 (1991).
    [CrossRef]
  4. D. Gabor, "Theory of communication," J. IEE 93, 429-457 (1946).
  5. J. Bigun and J. M. H. du Buf, "N-folded symmetries by complex moments in Gabor space and their application to unsupervised texture segmentation," IEEE Trans. Pattern Anal. Mach. Intell. 16, 80-87 (1994).
    [CrossRef]
  6. R. B. H. Tootell, M. S. Silverman, E. Switkes, and R. L. de Valois, "Deoxyglucose analysis of retinotopic organization in primate striate cortex," Science 218, 902-904 (1982).
    [CrossRef] [PubMed]
  7. D. Field, "Relation between the statistics of natural images and the response properties of cortical cells," J. Opt. Soc. Am. A 4, 2379-2394 (1987).
    [CrossRef] [PubMed]
  8. H. Knutsson, "Filtering and reconstruction in image processing," Ph.D. dissertation (Linköping University, 1982).
  9. J. Bigun and G. H. Granlund, "Optimal orientation detection of linear symmetry," in First International Conference on Computer Vision (IEEE, 1987), pp. 433-438.
  10. S. S. Reddi, "Radial and angular invariants for image identification," IEEE Trans. Pattern Anal. Mach. Intell. 3, 240-242 (1981).
    [CrossRef] [PubMed]
  11. M.-K. Hu, "Visual pattern recognition by moment invariants," IRE Trans. Inf. Theory 8, 179-187 (1962).
    [CrossRef]
  12. J. Bigun, T. Bigun, and K. Nilsson, "Recognition by symmetry derivatives and the generalized structure tensor," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1590-1605 (2004).
    [CrossRef] [PubMed]
  13. J. M. H. du Buf, "Abstract processes in texture discrimination," Spatial Vis. 6, 221-242 (1992).
    [CrossRef]
  14. G. Cristobal and J. Hormigo, "Texture segmentation through eigen-analysis of the Pseudo-Wigner distribution," Pattern Recogn. Lett. 20, 337-345 (1999).
    [CrossRef]
  15. D. A. Clausi and H. Deng, "Design-based texture feature fusion using Gabor filters and co-occurrence probabilities," IEEE Trans. Image Process. 14, 925-936 (2005).
    [CrossRef] [PubMed]
  16. R. Duda and P. Hart, Pattern Classification and Scene Analysis (Wiley, 1973).
  17. P. Schroeter and J. Bigun, "Hierarchical image segmentation by multi-dimensional clustering and orientation-adaptive boundary refinement," Pattern Recogn. 28, 695-709 (1995).
    [CrossRef]
  18. M. Spann and R. Wilson, "A quad-tree approach to image segmentation which combines statistical and spatial information," Pattern Recogn. 18, 257-269 (1985).
    [CrossRef]
  19. J. C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithm (Plenum, 1981).
  20. S. Karlsson and R. Poomari, "Unsupervised texture segmentation using multi-scale complex moments," M.S. thesis (Halmstad University, School of Information Science Computer and Electrical Engineering, 2005).
  21. W. Pomwenger, "Texture classification by high order symmetry derivatives of Gaussians," M.S. thesis (Halmstad University, School of Information Science Computer and Electrical Engineering, May 2003).

2005

D. A. Clausi and H. Deng, "Design-based texture feature fusion using Gabor filters and co-occurrence probabilities," IEEE Trans. Image Process. 14, 925-936 (2005).
[CrossRef] [PubMed]

2004

J. Bigun, T. Bigun, and K. Nilsson, "Recognition by symmetry derivatives and the generalized structure tensor," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1590-1605 (2004).
[CrossRef] [PubMed]

1999

G. Cristobal and J. Hormigo, "Texture segmentation through eigen-analysis of the Pseudo-Wigner distribution," Pattern Recogn. Lett. 20, 337-345 (1999).
[CrossRef]

1995

P. Schroeter and J. Bigun, "Hierarchical image segmentation by multi-dimensional clustering and orientation-adaptive boundary refinement," Pattern Recogn. 28, 695-709 (1995).
[CrossRef]

1994

J. Bigun and J. M. H. du Buf, "N-folded symmetries by complex moments in Gabor space and their application to unsupervised texture segmentation," IEEE Trans. Pattern Anal. Mach. Intell. 16, 80-87 (1994).
[CrossRef]

1992

J. M. H. du Buf, "Abstract processes in texture discrimination," Spatial Vis. 6, 221-242 (1992).
[CrossRef]

1991

A. K. Jain and F. Farrokhnia, "Unsupervised texture segmentation using Gabor filters," Pattern Recogn. 24, 1167-1186 (1991).
[CrossRef]

1990

A. C. Bovik, M. Clark, and W. S. Geisler, "Multichannel texture analysis using localized spatial filters," IEEE Trans. Pattern Anal. Mach. Intell. 12, 55-73 (1990).
[CrossRef]

1989

I. Fogel and D. Sagi, "Gabor filters as texture discriminator," Biol. Cybern. 61, 103-113 (1989).
[CrossRef]

1987

1985

M. Spann and R. Wilson, "A quad-tree approach to image segmentation which combines statistical and spatial information," Pattern Recogn. 18, 257-269 (1985).
[CrossRef]

1982

R. B. H. Tootell, M. S. Silverman, E. Switkes, and R. L. de Valois, "Deoxyglucose analysis of retinotopic organization in primate striate cortex," Science 218, 902-904 (1982).
[CrossRef] [PubMed]

1981

S. S. Reddi, "Radial and angular invariants for image identification," IEEE Trans. Pattern Anal. Mach. Intell. 3, 240-242 (1981).
[CrossRef] [PubMed]

1962

M.-K. Hu, "Visual pattern recognition by moment invariants," IRE Trans. Inf. Theory 8, 179-187 (1962).
[CrossRef]

1946

D. Gabor, "Theory of communication," J. IEE 93, 429-457 (1946).

Bezdek, J. C.

J. C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithm (Plenum, 1981).

Bigun, J.

J. Bigun, T. Bigun, and K. Nilsson, "Recognition by symmetry derivatives and the generalized structure tensor," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1590-1605 (2004).
[CrossRef] [PubMed]

P. Schroeter and J. Bigun, "Hierarchical image segmentation by multi-dimensional clustering and orientation-adaptive boundary refinement," Pattern Recogn. 28, 695-709 (1995).
[CrossRef]

J. Bigun and J. M. H. du Buf, "N-folded symmetries by complex moments in Gabor space and their application to unsupervised texture segmentation," IEEE Trans. Pattern Anal. Mach. Intell. 16, 80-87 (1994).
[CrossRef]

J. Bigun and G. H. Granlund, "Optimal orientation detection of linear symmetry," in First International Conference on Computer Vision (IEEE, 1987), pp. 433-438.

Bigun, T.

J. Bigun, T. Bigun, and K. Nilsson, "Recognition by symmetry derivatives and the generalized structure tensor," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1590-1605 (2004).
[CrossRef] [PubMed]

Bovik, A. C.

A. C. Bovik, M. Clark, and W. S. Geisler, "Multichannel texture analysis using localized spatial filters," IEEE Trans. Pattern Anal. Mach. Intell. 12, 55-73 (1990).
[CrossRef]

Clark, M.

A. C. Bovik, M. Clark, and W. S. Geisler, "Multichannel texture analysis using localized spatial filters," IEEE Trans. Pattern Anal. Mach. Intell. 12, 55-73 (1990).
[CrossRef]

Clausi, D. A.

D. A. Clausi and H. Deng, "Design-based texture feature fusion using Gabor filters and co-occurrence probabilities," IEEE Trans. Image Process. 14, 925-936 (2005).
[CrossRef] [PubMed]

Cristobal, G.

G. Cristobal and J. Hormigo, "Texture segmentation through eigen-analysis of the Pseudo-Wigner distribution," Pattern Recogn. Lett. 20, 337-345 (1999).
[CrossRef]

de Valois, R. L.

R. B. H. Tootell, M. S. Silverman, E. Switkes, and R. L. de Valois, "Deoxyglucose analysis of retinotopic organization in primate striate cortex," Science 218, 902-904 (1982).
[CrossRef] [PubMed]

Deng, H.

D. A. Clausi and H. Deng, "Design-based texture feature fusion using Gabor filters and co-occurrence probabilities," IEEE Trans. Image Process. 14, 925-936 (2005).
[CrossRef] [PubMed]

du Buf, J. M. H.

J. Bigun and J. M. H. du Buf, "N-folded symmetries by complex moments in Gabor space and their application to unsupervised texture segmentation," IEEE Trans. Pattern Anal. Mach. Intell. 16, 80-87 (1994).
[CrossRef]

J. M. H. du Buf, "Abstract processes in texture discrimination," Spatial Vis. 6, 221-242 (1992).
[CrossRef]

Duda, R.

R. Duda and P. Hart, Pattern Classification and Scene Analysis (Wiley, 1973).

Farrokhnia, F.

A. K. Jain and F. Farrokhnia, "Unsupervised texture segmentation using Gabor filters," Pattern Recogn. 24, 1167-1186 (1991).
[CrossRef]

Field, D.

Fogel, I.

I. Fogel and D. Sagi, "Gabor filters as texture discriminator," Biol. Cybern. 61, 103-113 (1989).
[CrossRef]

Gabor, D.

D. Gabor, "Theory of communication," J. IEE 93, 429-457 (1946).

Geisler, W. S.

A. C. Bovik, M. Clark, and W. S. Geisler, "Multichannel texture analysis using localized spatial filters," IEEE Trans. Pattern Anal. Mach. Intell. 12, 55-73 (1990).
[CrossRef]

Granlund, G. H.

J. Bigun and G. H. Granlund, "Optimal orientation detection of linear symmetry," in First International Conference on Computer Vision (IEEE, 1987), pp. 433-438.

Hart, P.

R. Duda and P. Hart, Pattern Classification and Scene Analysis (Wiley, 1973).

Hormigo, J.

G. Cristobal and J. Hormigo, "Texture segmentation through eigen-analysis of the Pseudo-Wigner distribution," Pattern Recogn. Lett. 20, 337-345 (1999).
[CrossRef]

Hu, M.-K.

M.-K. Hu, "Visual pattern recognition by moment invariants," IRE Trans. Inf. Theory 8, 179-187 (1962).
[CrossRef]

Jain, A. K.

A. K. Jain and F. Farrokhnia, "Unsupervised texture segmentation using Gabor filters," Pattern Recogn. 24, 1167-1186 (1991).
[CrossRef]

Karlsson, S.

S. Karlsson and R. Poomari, "Unsupervised texture segmentation using multi-scale complex moments," M.S. thesis (Halmstad University, School of Information Science Computer and Electrical Engineering, 2005).

Knutsson, H.

H. Knutsson, "Filtering and reconstruction in image processing," Ph.D. dissertation (Linköping University, 1982).

Nilsson, K.

J. Bigun, T. Bigun, and K. Nilsson, "Recognition by symmetry derivatives and the generalized structure tensor," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1590-1605 (2004).
[CrossRef] [PubMed]

Pomwenger, W.

W. Pomwenger, "Texture classification by high order symmetry derivatives of Gaussians," M.S. thesis (Halmstad University, School of Information Science Computer and Electrical Engineering, May 2003).

Poomari, R.

S. Karlsson and R. Poomari, "Unsupervised texture segmentation using multi-scale complex moments," M.S. thesis (Halmstad University, School of Information Science Computer and Electrical Engineering, 2005).

Reddi, S. S.

S. S. Reddi, "Radial and angular invariants for image identification," IEEE Trans. Pattern Anal. Mach. Intell. 3, 240-242 (1981).
[CrossRef] [PubMed]

Sagi, D.

I. Fogel and D. Sagi, "Gabor filters as texture discriminator," Biol. Cybern. 61, 103-113 (1989).
[CrossRef]

Schroeter, P.

P. Schroeter and J. Bigun, "Hierarchical image segmentation by multi-dimensional clustering and orientation-adaptive boundary refinement," Pattern Recogn. 28, 695-709 (1995).
[CrossRef]

Silverman, M. S.

R. B. H. Tootell, M. S. Silverman, E. Switkes, and R. L. de Valois, "Deoxyglucose analysis of retinotopic organization in primate striate cortex," Science 218, 902-904 (1982).
[CrossRef] [PubMed]

Spann, M.

M. Spann and R. Wilson, "A quad-tree approach to image segmentation which combines statistical and spatial information," Pattern Recogn. 18, 257-269 (1985).
[CrossRef]

Switkes, E.

R. B. H. Tootell, M. S. Silverman, E. Switkes, and R. L. de Valois, "Deoxyglucose analysis of retinotopic organization in primate striate cortex," Science 218, 902-904 (1982).
[CrossRef] [PubMed]

Tootell, R. B. H.

R. B. H. Tootell, M. S. Silverman, E. Switkes, and R. L. de Valois, "Deoxyglucose analysis of retinotopic organization in primate striate cortex," Science 218, 902-904 (1982).
[CrossRef] [PubMed]

Wilson, R.

M. Spann and R. Wilson, "A quad-tree approach to image segmentation which combines statistical and spatial information," Pattern Recogn. 18, 257-269 (1985).
[CrossRef]

Biol. Cybern.

I. Fogel and D. Sagi, "Gabor filters as texture discriminator," Biol. Cybern. 61, 103-113 (1989).
[CrossRef]

IEEE Trans. Image Process.

D. A. Clausi and H. Deng, "Design-based texture feature fusion using Gabor filters and co-occurrence probabilities," IEEE Trans. Image Process. 14, 925-936 (2005).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell.

A. C. Bovik, M. Clark, and W. S. Geisler, "Multichannel texture analysis using localized spatial filters," IEEE Trans. Pattern Anal. Mach. Intell. 12, 55-73 (1990).
[CrossRef]

J. Bigun and J. M. H. du Buf, "N-folded symmetries by complex moments in Gabor space and their application to unsupervised texture segmentation," IEEE Trans. Pattern Anal. Mach. Intell. 16, 80-87 (1994).
[CrossRef]

S. S. Reddi, "Radial and angular invariants for image identification," IEEE Trans. Pattern Anal. Mach. Intell. 3, 240-242 (1981).
[CrossRef] [PubMed]

J. Bigun, T. Bigun, and K. Nilsson, "Recognition by symmetry derivatives and the generalized structure tensor," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1590-1605 (2004).
[CrossRef] [PubMed]

IRE Trans. Inf. Theory

M.-K. Hu, "Visual pattern recognition by moment invariants," IRE Trans. Inf. Theory 8, 179-187 (1962).
[CrossRef]

J. IEE

D. Gabor, "Theory of communication," J. IEE 93, 429-457 (1946).

J. Opt. Soc. Am. A

Pattern Recogn.

A. K. Jain and F. Farrokhnia, "Unsupervised texture segmentation using Gabor filters," Pattern Recogn. 24, 1167-1186 (1991).
[CrossRef]

P. Schroeter and J. Bigun, "Hierarchical image segmentation by multi-dimensional clustering and orientation-adaptive boundary refinement," Pattern Recogn. 28, 695-709 (1995).
[CrossRef]

M. Spann and R. Wilson, "A quad-tree approach to image segmentation which combines statistical and spatial information," Pattern Recogn. 18, 257-269 (1985).
[CrossRef]

Pattern Recogn. Lett.

G. Cristobal and J. Hormigo, "Texture segmentation through eigen-analysis of the Pseudo-Wigner distribution," Pattern Recogn. Lett. 20, 337-345 (1999).
[CrossRef]

Science

R. B. H. Tootell, M. S. Silverman, E. Switkes, and R. L. de Valois, "Deoxyglucose analysis of retinotopic organization in primate striate cortex," Science 218, 902-904 (1982).
[CrossRef] [PubMed]

Spatial Vis.

J. M. H. du Buf, "Abstract processes in texture discrimination," Spatial Vis. 6, 221-242 (1992).
[CrossRef]

Other

J. C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithm (Plenum, 1981).

S. Karlsson and R. Poomari, "Unsupervised texture segmentation using multi-scale complex moments," M.S. thesis (Halmstad University, School of Information Science Computer and Electrical Engineering, 2005).

W. Pomwenger, "Texture classification by high order symmetry derivatives of Gaussians," M.S. thesis (Halmstad University, School of Information Science Computer and Electrical Engineering, May 2003).

H. Knutsson, "Filtering and reconstruction in image processing," Ph.D. dissertation (Linköping University, 1982).

J. Bigun and G. H. Granlund, "Optimal orientation detection of linear symmetry," in First International Conference on Computer Vision (IEEE, 1987), pp. 433-438.

R. Duda and P. Hart, Pattern Classification and Scene Analysis (Wiley, 1973).

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Figures (8)

Fig. 1
Fig. 1

Example of rosettelike partitioning of Gabor filters in the Fourier domain. The center of the graph is the dc component. The position and relative scope of the filters are illustrated by the ellipses.

Fig. 2
Fig. 2

Example of texture with six-folded symmetry. (a) Texture in the spatial domain, (b) power spectrum, black indicates high energy, (c) same spectrum magnified with one circular band explicitly illustrated within gray, dashed curves [as partitioned, e.g., by Eq. (8)].

Fig. 3
Fig. 3

Test images used in the experiments.

Fig. 4
Fig. 4

Segmentation results for multiscale CMPs, top row: p = 1 , bottom row: p { 1 , 2 } .

Fig. 5
Fig. 5

Segmentation results for the Gabor power spectrum, top row: three orientations, bottom row: five orientations.

Fig. 6
Fig. 6

Left, a FM-test image, the axes of which are marked with fractions of π representing the spatial frequency. Right, the same image but bandpass filtered corresponding to one level of a Laplacian pyramid. Note that only one circular band remains of the original.

Fig. 7
Fig. 7

Graphs represent the estimated [ arg ( I 2 , 0 ( R ) ) ] (solid curves) as well as the accurate direction angle (dashed lines) on a ring in the FM-test image. Left, estimated with Gabor spectrum of six orientations; right, estimated by spatial approach.

Fig. 8
Fig. 8

Estimated power spectrum along one narrow band using Gabor filters (dotted) and CMPs (solid curve). Image used to estimate is the p1 image of Fig. 3.

Tables (1)

Tables Icon

Table 1 Segmentation Performance in Percentage Correctly Classified Pixels

Equations (16)

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F x 0 ( ω ) = F { f ( x 0 x ) w σ ( x ) } = f ( x 0 x ) w σ ( x ) exp ( j ω T x ) d x ,
Ψ r o ϕ o ( r , ϕ ) = exp ( ( log ( r r o ) ) 2 2 ( log ( σ r r o ) ) 2 ) exp ( ( ϕ ϕ o ) 2 2 σ ϕ 2 ) ,
I a , b { f } = ( x + j y ) a ( x j y ) b f ( x , y ) d x d y ,
I 2 p , 0 = ( u + j v ) 2 p ρ ( u , v ) d u d v = ( λ p max λ p min ) exp ( j 2 φ p min ) ,
I p , p = u + j v 2 p ρ ( u , v ) d u d v = λ p max + λ p min ,
I 2 p , 0 = ( ( D x + j D y ) p f ) 2 d x ,
I p , p = ( D x + j D y ) p f 2 d x .
ρ R ( r , ϕ ) = ρ ( r , ϕ ) g ( r R , σ ) ,
I 2 p , 0 ( R ) = ( u + j v ) 2 p ρ R ( u , v ) d u d v = r 2 p + 1 ρ R ( r , ϕ ) exp ( j ϕ 2 p ) d r d ϕ .
I 2 p , 0 ( R ) = C p [ F 1 { ρ ( R , ϕ ) } ( 2 p ) ] ,
I p , p ( R ) = C p [ F 1 { ρ ( R , ϕ ) } ( 0 ) ] .
ξ ( p ) = [ F 1 { ρ ( R , ϕ ) } ( 2 p ) ] = lim σ 0 I 2 p , 0 ( R ) C p .
ξ ( p ) = k = 0 q 1 ρ ( R , k π q ) exp ( j k π 2 p q ) = lim σ 0 I 2 p , 0 ( R ) C p .
2 p < q p q 1 2 p max = q 1 2 ,
n max = 2 p max = q 1 mod ( q + 1 , 2 ) .
ξ ( p ) = { I N , 0 * C N ; ; I 2 , 0 * C 1 ; I 1 , 1 C 1 ; I 2 , 0 C 1 ; ; I 2 N , 0 C N }

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