Abstract

A new class of random field models, called generalized circular autoregressive (GCAR) models, is introduced. GCAR models have noncausal neighbors that have the same autoregressive parameter values if they are on the same circle or ellipse and that have circular or elliptical correlation structures. This model is better for modeling isotropic or anisotropic natural textures than earlier approaches to modeling of isotropic textures and can represent complex textures with a small number of parameters. Parameter estimation is also considered, and a multistep estimation algorithm is presented. Properties of estimators of GCAR models are also investigated. The efficacy of GCAR models in modeling real textures is demonstrated by synthesizing images resembling real textures by use of parameters estimated from textures selected from the Brodatz texture album. Limitations of GCAR models are also discussed.

© 2001 Optical Society of America

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References

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  1. T. R. Reed, J. M. H. du Buf, “A review of recent texture segmentation and feature extraction techniques,” CVGIP Image Understand. 57, 359–372 (1993).
    [CrossRef]
  2. R. L. Kashyap, R. Chellappa, “Estimation and choice of neighbors in spatial-interaction models of images,” IEEE Trans. Inf. Theory 29, 60–69 (1983).
    [CrossRef]
  3. R. L. Kashyap, K. B. Eom, “Texture boundary detection based on long correlation model,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 58–67 (1989).
    [CrossRef]
  4. I. M. Elfadel, R. W. Picard, “Gibbs random fields, co-occurrences, and texture modeling,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 24–37 (1994).
    [CrossRef]
  5. A. Speis, G. Healey, “An analytical and experimental study of the performance of Markov random fields applied to textured images using small samples,” IEEE Trans. Image Process. 5, 447–458 (1996).
    [CrossRef] [PubMed]
  6. J. M. Francos, A. Z. Meiri, B. Porat, “A unified texture model based on a 2-D Wold-like decomposition,” IEEE Trans. Signal Process. 41, 2665–2677 (1993).
    [CrossRef]
  7. D. J. Heeger, J. R. Bergen, “Pyramid-based texture analysis/synthesis,” Comput. Graph.229–238 (1995).
  8. R. L. Kashyap, A. Khotanzad, “A model-based method for rotation invariant texture classification,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 472–481 (1986).
    [CrossRef]
  9. K. B. Eom, “2D moving average models for texture synthesis and analysis,” IEEE Trans. Signal Process. 7, 1741–1746 (1998).
  10. P. Brodatz, Textures: A Photographic Album for Artist and Designers (Dover, Mineola, N.Y., 1966).
  11. P. J. Davis, Circulant Matrices (Wiley, New York, 1979), Chap. 5.
  12. A. Rosenfeld, A. Kak, Digital Picture Processing, 2nd ed. (Academic, New York, 1982), Vol. 1.
  13. G. E. P. Box, G. M. Jenkins, G. C. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed. (Prentice-Hall, Upper Saddle River, N.J., 1994).
  14. D. R. Brillinger, Time series, Data Analysis and Theory, expanded ed. (Holden Day, San Francisco, Calif., 1981).
  15. P. J. Bickel, K. A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics (Holden-Day, San Francisco, Calif., 1977).

1998

K. B. Eom, “2D moving average models for texture synthesis and analysis,” IEEE Trans. Signal Process. 7, 1741–1746 (1998).

1996

A. Speis, G. Healey, “An analytical and experimental study of the performance of Markov random fields applied to textured images using small samples,” IEEE Trans. Image Process. 5, 447–458 (1996).
[CrossRef] [PubMed]

1995

D. J. Heeger, J. R. Bergen, “Pyramid-based texture analysis/synthesis,” Comput. Graph.229–238 (1995).

1994

I. M. Elfadel, R. W. Picard, “Gibbs random fields, co-occurrences, and texture modeling,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 24–37 (1994).
[CrossRef]

1993

T. R. Reed, J. M. H. du Buf, “A review of recent texture segmentation and feature extraction techniques,” CVGIP Image Understand. 57, 359–372 (1993).
[CrossRef]

J. M. Francos, A. Z. Meiri, B. Porat, “A unified texture model based on a 2-D Wold-like decomposition,” IEEE Trans. Signal Process. 41, 2665–2677 (1993).
[CrossRef]

1989

R. L. Kashyap, K. B. Eom, “Texture boundary detection based on long correlation model,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 58–67 (1989).
[CrossRef]

1986

R. L. Kashyap, A. Khotanzad, “A model-based method for rotation invariant texture classification,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 472–481 (1986).
[CrossRef]

1983

R. L. Kashyap, R. Chellappa, “Estimation and choice of neighbors in spatial-interaction models of images,” IEEE Trans. Inf. Theory 29, 60–69 (1983).
[CrossRef]

Bergen, J. R.

D. J. Heeger, J. R. Bergen, “Pyramid-based texture analysis/synthesis,” Comput. Graph.229–238 (1995).

Bickel, P. J.

P. J. Bickel, K. A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics (Holden-Day, San Francisco, Calif., 1977).

Box, G. E. P.

G. E. P. Box, G. M. Jenkins, G. C. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed. (Prentice-Hall, Upper Saddle River, N.J., 1994).

Brillinger, D. R.

D. R. Brillinger, Time series, Data Analysis and Theory, expanded ed. (Holden Day, San Francisco, Calif., 1981).

Brodatz, P.

P. Brodatz, Textures: A Photographic Album for Artist and Designers (Dover, Mineola, N.Y., 1966).

Chellappa, R.

R. L. Kashyap, R. Chellappa, “Estimation and choice of neighbors in spatial-interaction models of images,” IEEE Trans. Inf. Theory 29, 60–69 (1983).
[CrossRef]

Davis, P. J.

P. J. Davis, Circulant Matrices (Wiley, New York, 1979), Chap. 5.

Doksum, K. A.

P. J. Bickel, K. A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics (Holden-Day, San Francisco, Calif., 1977).

du Buf, J. M. H.

T. R. Reed, J. M. H. du Buf, “A review of recent texture segmentation and feature extraction techniques,” CVGIP Image Understand. 57, 359–372 (1993).
[CrossRef]

Elfadel, I. M.

I. M. Elfadel, R. W. Picard, “Gibbs random fields, co-occurrences, and texture modeling,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 24–37 (1994).
[CrossRef]

Eom, K. B.

K. B. Eom, “2D moving average models for texture synthesis and analysis,” IEEE Trans. Signal Process. 7, 1741–1746 (1998).

R. L. Kashyap, K. B. Eom, “Texture boundary detection based on long correlation model,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 58–67 (1989).
[CrossRef]

Francos, J. M.

J. M. Francos, A. Z. Meiri, B. Porat, “A unified texture model based on a 2-D Wold-like decomposition,” IEEE Trans. Signal Process. 41, 2665–2677 (1993).
[CrossRef]

Healey, G.

A. Speis, G. Healey, “An analytical and experimental study of the performance of Markov random fields applied to textured images using small samples,” IEEE Trans. Image Process. 5, 447–458 (1996).
[CrossRef] [PubMed]

Heeger, D. J.

D. J. Heeger, J. R. Bergen, “Pyramid-based texture analysis/synthesis,” Comput. Graph.229–238 (1995).

Jenkins, G. M.

G. E. P. Box, G. M. Jenkins, G. C. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed. (Prentice-Hall, Upper Saddle River, N.J., 1994).

Kak, A.

A. Rosenfeld, A. Kak, Digital Picture Processing, 2nd ed. (Academic, New York, 1982), Vol. 1.

Kashyap, R. L.

R. L. Kashyap, K. B. Eom, “Texture boundary detection based on long correlation model,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 58–67 (1989).
[CrossRef]

R. L. Kashyap, A. Khotanzad, “A model-based method for rotation invariant texture classification,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 472–481 (1986).
[CrossRef]

R. L. Kashyap, R. Chellappa, “Estimation and choice of neighbors in spatial-interaction models of images,” IEEE Trans. Inf. Theory 29, 60–69 (1983).
[CrossRef]

Khotanzad, A.

R. L. Kashyap, A. Khotanzad, “A model-based method for rotation invariant texture classification,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 472–481 (1986).
[CrossRef]

Meiri, A. Z.

J. M. Francos, A. Z. Meiri, B. Porat, “A unified texture model based on a 2-D Wold-like decomposition,” IEEE Trans. Signal Process. 41, 2665–2677 (1993).
[CrossRef]

Picard, R. W.

I. M. Elfadel, R. W. Picard, “Gibbs random fields, co-occurrences, and texture modeling,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 24–37 (1994).
[CrossRef]

Porat, B.

J. M. Francos, A. Z. Meiri, B. Porat, “A unified texture model based on a 2-D Wold-like decomposition,” IEEE Trans. Signal Process. 41, 2665–2677 (1993).
[CrossRef]

Reed, T. R.

T. R. Reed, J. M. H. du Buf, “A review of recent texture segmentation and feature extraction techniques,” CVGIP Image Understand. 57, 359–372 (1993).
[CrossRef]

Reinsel, G. C.

G. E. P. Box, G. M. Jenkins, G. C. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed. (Prentice-Hall, Upper Saddle River, N.J., 1994).

Rosenfeld, A.

A. Rosenfeld, A. Kak, Digital Picture Processing, 2nd ed. (Academic, New York, 1982), Vol. 1.

Speis, A.

A. Speis, G. Healey, “An analytical and experimental study of the performance of Markov random fields applied to textured images using small samples,” IEEE Trans. Image Process. 5, 447–458 (1996).
[CrossRef] [PubMed]

Comput. Graph.

D. J. Heeger, J. R. Bergen, “Pyramid-based texture analysis/synthesis,” Comput. Graph.229–238 (1995).

CVGIP Image Understand.

T. R. Reed, J. M. H. du Buf, “A review of recent texture segmentation and feature extraction techniques,” CVGIP Image Understand. 57, 359–372 (1993).
[CrossRef]

IEEE Trans. Image Process.

A. Speis, G. Healey, “An analytical and experimental study of the performance of Markov random fields applied to textured images using small samples,” IEEE Trans. Image Process. 5, 447–458 (1996).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory

R. L. Kashyap, R. Chellappa, “Estimation and choice of neighbors in spatial-interaction models of images,” IEEE Trans. Inf. Theory 29, 60–69 (1983).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

R. L. Kashyap, K. B. Eom, “Texture boundary detection based on long correlation model,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 58–67 (1989).
[CrossRef]

I. M. Elfadel, R. W. Picard, “Gibbs random fields, co-occurrences, and texture modeling,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 24–37 (1994).
[CrossRef]

R. L. Kashyap, A. Khotanzad, “A model-based method for rotation invariant texture classification,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 472–481 (1986).
[CrossRef]

IEEE Trans. Signal Process.

K. B. Eom, “2D moving average models for texture synthesis and analysis,” IEEE Trans. Signal Process. 7, 1741–1746 (1998).

J. M. Francos, A. Z. Meiri, B. Porat, “A unified texture model based on a 2-D Wold-like decomposition,” IEEE Trans. Signal Process. 41, 2665–2677 (1993).
[CrossRef]

Other

P. Brodatz, Textures: A Photographic Album for Artist and Designers (Dover, Mineola, N.Y., 1966).

P. J. Davis, Circulant Matrices (Wiley, New York, 1979), Chap. 5.

A. Rosenfeld, A. Kak, Digital Picture Processing, 2nd ed. (Academic, New York, 1982), Vol. 1.

G. E. P. Box, G. M. Jenkins, G. C. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed. (Prentice-Hall, Upper Saddle River, N.J., 1994).

D. R. Brillinger, Time series, Data Analysis and Theory, expanded ed. (Holden Day, San Francisco, Calif., 1981).

P. J. Bickel, K. A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics (Holden-Day, San Francisco, Calif., 1977).

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Figures (6)

Fig. 1
Fig. 1

A set of four neighbors equidistant from the center pixel is represented by distance r and angle θ. All neighbors at the same distance are decomposed into one or more four-neighbors sets as shown above.

Fig. 2
Fig. 2

Synthesized textures with the same GCAR parameters except different elongation and orientation parameters. The elongation parameters are α=1,2,3,4 (top to bottom), and the orientation parameters are θ=0,π/4,π/2,3π/4 (left to right).

Fig. 3
Fig. 3

Top, original raffia woven with cotton threads texture (D51); bottom, texture synthesized in two different orientations with the parameters estimated from the original.

Fig. 4
Fig. 4

Left, original isotropic textures; right, textures synthesized at double size with parameters estimated from the originals. The original textures used are (top to bottom) beach sand (D29), water (D38), cloud (D90), and pigskin (D92).

Fig. 5
Fig. 5

Left, original textures with directional patterns; right, textures synthesized at double size with parameters estimated from the originals. The original textures used are (top to bottom) pressed cork (D04), grass lawn (D09), pressed calf leather (D24), and wood grain (D68).

Fig. 6
Fig. 6

Left, original textures with structured patterns; right, textures synthesized at double size with parameters estimated from the originals. The original textures used are straw matting (D55), cotton canvas (D77), and raffia looped to a high pile (D84).

Tables (1)

Tables Icon

Table 1 Estimated Parameters Used to Synthesize Textures in Figs. 36

Equations (69)

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y(s)=rKgary(sΘr)+ρw(s),
ar=at,ifr=t.
y(s)=i=1paij=1qiTij[y(s+(0, 1))+y(s+(0, -1))+y(s+(1, 0))+y(s+(-1, 0))]+ρw(s),
Tij[y(s+t)]=y(s+riRθijt),
Rθij=cos θijsin θij-sin θijcos θij.
y=[y(0, 0)  y(0, N-1)|y(1, 0)  y(1,N-1)|..|y×(N-1, 0)  y(N-1 , N-1)]T,
w=[w(0, 0)  w(0, N-1)|w(1, 0)  w(1,N-1)|..×|w(N-1, 0)  w(N-1,N-1)|]T.
By=ρw,
fk,l=col{1 exp(i2πk/N)  exp[i2πk(N-1)/N]}col{1 exp(i2πl/N)  exp[i2πl(N-1)/N]},
fork, l=0, 1 ,, N-1,
λ(k, l)=1-i=1paij=1qiλij(k, l),
λij(k, l)
=2 cos(2πrik/N)+2 cos(2πril/N)ifθi=04 cos(2πrik cos θij/N)cos(2πril sin θij/N)otherwise.
λij(k, l)=exp[i2πril/N]+exp[i2πrik/N]+exp[-i2πril/N]+exp[-i2πrik/N]=2 cos(2πril/N)+2 cos(2πrik/N).
λij(k, l)=exp[i2π(ck+sl)/N]+exp[i2π(-ck+sl)/N]+exp[i2πa(-ck-sl)/N]+exp[i2πa(ck-sl)/N]=4 cos(2πrik cos θij/N)cos(2πril sin θij/N).
B=F-1diag[λ(0, 0)  λ(N-1,N-1)]F.
ΛY=ρW,
y(s)=rKgary(sΘr)+ρw(s),
ar=at,ifAr=At,
A=α cos θ-α sin θsin θcos θ
y(s)=i=1paij=1qiTij[y(s+(1, 0))+y(s+(-1, 0))]+ρw(s),
Tij[y(s+t)]=y(s+rijRθijt),
Rθij=cos θijsin θij-sin θijcos θij
rij=(i2+j2)1/2,θij=tan-1(j/i).
(α2i2+j2)cos θ+(i2+α2j2)sin2 θ+2ij cos θ sin θ(1
-α2)=r2 foracertainr>0.
By=ρw,
λ(k, l)=1-i=1paij=1qiλij(k, l),
λij(k, l)=2 cos(2πrijk/N)ifθij=02 cos[2πrij(k cos θij+l sin θij)/N]otherwise.
j=1qiTij[y(s+(1, 0))+y(s-(1, 0))]
=j=1qi[y(s+rij(cos θij, sin θij))+y(s-rij(cos θij, sin θij))].
y(s)=i=1pj=1qi[y(s+rij(cos θij, sin θij))+y(s-rij(cos θij, sin θij))]+ρw(s)
By=ρw.
λ(k, l)=1-i=1paij=1qiλij(k, l),
λij(k, l)
=2 cos(2πrijk/N),if θij=02 cos[2πrij(k cos θij+l sin θij)/N],otherwise.
Y(u)=sΩy(s)exp-j2πsTuN,
y(s)=1N2uΩY(u)expj2πsTuN.
ΛY=ρW,
Y(k, l)=ρW(k, l)/λ(k, l),k, l=0, 1,, N-1.
a=[a1a2  ap]T,
z(s)=[z1(s)z2(s)  zp(s)]T,
zi(s)=j=1qi[y(s+ri(cos θij, sin θij))+y(s-ri(cos θij, sin θij))+y(s+ri(cos θij, -sin θij))+y(s-ri(cos θij, -sin θij))]
zi(s)=j=1qi[y(s+rij(cos θij, sin θij))+y(s-rij(cos θij, sin θij))]
y(s)=aTz(s)+ρw(s).
p(y|a, ρ, θ, α)=|B|(2πρ)N2/2×exp-12ρsΩ[y(s)-aTz(s)]2.
a^t+1=sΩ[z(s)zT(s)+ρ^tψ(s)ψT(s)]-1×sΩ[z(s)y(s)-ρ^tψ(s)],
ρ^t=1N2sΩ[y(s)-a^tTz(s)]2,
ψ(s)=j=1q1λ1j(s) ,, j=1qpλpj(s)T.
log p(y|a, ρ, θ, α)=log(|B|)-N22log(2πρ)-12ρsΩ[y(s)-aTz(s)]2,
log(|B|)=sΩlog[λ(s)]=sΩlog1-i=1paij=1qiλij(s).
log(|B|)=sΩlog[1-aTψ(s)]-sΩaTψ(s)+12aTψ(s)ψT(s)a,
log p(y|a, ρ, θ, α)-N22log(2πρ)-12ρsΩ{y2(s)-2aT[z(s)y(s)-ρψ(s)]+aT[z(s)zT(s)+ρψ(s)ψT]a}.
alog p(y|a, ρ, θ, α)
=1ρsΩ{[z(s)y(s)+ρψ(s)]-[z(s)zT(s)+ρψ(s)ψT(s)]a}=0,
 log p(y|a, ρ, θ, α)ρ
=-N22ρ+12ρ2sΩ[y(s)-aTz(s)]2=0
a^t+1=sΩ[z(s)zT(s)+ρ^tψ(s)ψT(s)]-1×sΩ[z(s)y(s)-ρ^tψ(s)],
ρ^t=1N2sΩ[y(s)-a^tTz(s)]2.
log p(y|Φ, p, θ, α)
log p(y|Φˆ, p, θ, α)+12 (Φ-Φˆ)ΦΦlog p(y|Φˆ, p, θ, α)(Φ-Φˆ)T,
p(y|p, θ, α)=p(y|Φ, p, θ, α)p(Φ)dΦ.
p(y|p, θ, α)|p(y|Φˆ, p, θ, α)
exp-12ρ^2 (Φ-Φˆ)TS00N2/2(Φ-Φˆ)
[p(Φˆ)+Φp(Φˆ)(Φ-Φˆ)+higher-orderterms]dΦ
|p(y|Φˆ, p, θ, α)p(Φˆ)(2πρ^2)(p+1)/22N|S|-1/2,
S=sΩ[z(s)zT(s)+ρˆψ(s)ψT(s)].
log p(y|p, θ, α)=sΩlog(1-a^Tψ(s))+log p(Φˆ)-N22+-N2+p+12log(2π)+-N22+p+1log ρˆ-12logN22-12log|S|.
J(p, θ, α)=sΩlog(1-a^Tψ(s))-p2log(2π)+N22-p-1log ρˆ+12log |S|.

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