Abstract

The two-point probability density function (2P-PDF) gives a full description of the first- and second-order statistics of a random process. We propose a framework for texture classification based on a distance measure between 2P-PDF’s after equalization of first-order statistics. This framework allows extraction of the structural information of the process independently of the dynamic range of the image. We present two methods for estimating the 2P-PDF of texture images, and we establish some criteria for efficient computation. The theoretical framework for noise-free texture images is validated with four texture ensembles.

© 1999 Optical Society of America

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References

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  1. H. H. Barrett, J. Yao, J. P. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
    [CrossRef] [PubMed]
  2. J. P. Rolland, H. H. Barrett, “Model of background uncertainty and observers detection performance,” J. Opt. Soc. Am. A 9, 649–658 (1992).
    [CrossRef] [PubMed]
  3. G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiological abnormalities,” Invest. Radiol. 9, 479–486 (1974).
    [CrossRef] [PubMed]
  4. F. O. Bochud, F. R. Verdun, C. Hessler, J. F. Valley, “Detectability on radiological images: the influence of anatomical noise,” in Medical Imaging 1995: Image Perception, H. L. Kundel, ed., Proc. SPIE2436, 156–165 (1995).
    [CrossRef]
  5. J. W. Byng, M. J. Yaffe, G. A. Lockwood, L. E. Little, D. L. Tritchler, N. F. Boyd, “Automated analysis of mammographic densities and breast carcinoma risk,” Cancer (N.Y.) 80, 66–74 (1997).
    [CrossRef]
  6. J. P. Rolland, R. Strickland, “An approach to the synthesis of biological tissue,” Opt. Express 1, 414–423 (1997).
    [CrossRef] [PubMed]
  7. E. B. Gargill, “A mathematical liver model and its application to system optimization and texture analysis,” Ph.D dissertation (University of Arizona, Tucson, Ariz.1989).
  8. S. H. L. Hylen, “Image modifiers for use in photography,” U.S. patent5,649,259 (15July1997).
  9. J. P. Rolland, A. A. Goon, L. Yu, “Synthesis of textured complex backgrounds,” Opt. Eng. 37, 2055–2063 (1998).
    [CrossRef]
  10. J. P. Rolland, A. A. Goon, E. Clarkson, L. Yu, “Synthesis of biomedical tissue,” in Medical Imaging 1998: Image Perception, H. L. Kundel, ed., Proc. SPIE3340, 85–90 (1998).
    [CrossRef]
  11. R. Gonzalez, R. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), Chap. 8.
  12. K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979), Chap. 16.
  13. M. G. A. Thomson, D. H. Foster, “Role of second- and third-order statistics in the discriminability of natural images,” J. Opt. Soc. Am. A 14, 2081–2090 (1997).
    [CrossRef]
  14. A. Tremeau, J. Bousigue, B. Laget, “Co-occurrence shape descriptors applied to texture classification and segmentation,” in Machine Vision Applications in Industrial Inspection IV, A. R. Rao, N. Chang, eds., Proc. SPIE2665, 135–147 (1996).
    [CrossRef]
  15. R. S. Mia, M. H. Loew, K. A. Wear, R. F. Wagner, B. S. Garra, “Quantitative ultrasound tissue characterization using texture and cepstral features,” in Medical Imaging 1998: Image Processing, K. M. Hanson, ed., Proc. SPIE3338, 211–219 (1998).
    [CrossRef]
  16. R. M. Haralick, “Statistical and structural approaches to texture,” Proc. IEEE 67(5), 786–804 (1979).
    [CrossRef]
  17. A. Papoulis, Probability, Random Variables, and Statistical Processes (McGraw-Hill, New York, 1991).
  18. A. M. Mathai, S. B. Provost, Quadratic Forms in Random Variables (Marcel Dekker, New York, 1992), Chap. 3.2.

1998 (1)

J. P. Rolland, A. A. Goon, L. Yu, “Synthesis of textured complex backgrounds,” Opt. Eng. 37, 2055–2063 (1998).
[CrossRef]

1997 (3)

1993 (1)

H. H. Barrett, J. Yao, J. P. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[CrossRef] [PubMed]

1992 (1)

1979 (1)

R. M. Haralick, “Statistical and structural approaches to texture,” Proc. IEEE 67(5), 786–804 (1979).
[CrossRef]

1974 (1)

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiological abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[CrossRef] [PubMed]

Barrett, H. H.

H. H. Barrett, J. Yao, J. P. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[CrossRef] [PubMed]

J. P. Rolland, H. H. Barrett, “Model of background uncertainty and observers detection performance,” J. Opt. Soc. Am. A 9, 649–658 (1992).
[CrossRef] [PubMed]

Bochud, F. O.

F. O. Bochud, F. R. Verdun, C. Hessler, J. F. Valley, “Detectability on radiological images: the influence of anatomical noise,” in Medical Imaging 1995: Image Perception, H. L. Kundel, ed., Proc. SPIE2436, 156–165 (1995).
[CrossRef]

Bousigue, J.

A. Tremeau, J. Bousigue, B. Laget, “Co-occurrence shape descriptors applied to texture classification and segmentation,” in Machine Vision Applications in Industrial Inspection IV, A. R. Rao, N. Chang, eds., Proc. SPIE2665, 135–147 (1996).
[CrossRef]

Boyd, N. F.

J. W. Byng, M. J. Yaffe, G. A. Lockwood, L. E. Little, D. L. Tritchler, N. F. Boyd, “Automated analysis of mammographic densities and breast carcinoma risk,” Cancer (N.Y.) 80, 66–74 (1997).
[CrossRef]

Byng, J. W.

J. W. Byng, M. J. Yaffe, G. A. Lockwood, L. E. Little, D. L. Tritchler, N. F. Boyd, “Automated analysis of mammographic densities and breast carcinoma risk,” Cancer (N.Y.) 80, 66–74 (1997).
[CrossRef]

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979), Chap. 16.

Clarkson, E.

J. P. Rolland, A. A. Goon, E. Clarkson, L. Yu, “Synthesis of biomedical tissue,” in Medical Imaging 1998: Image Perception, H. L. Kundel, ed., Proc. SPIE3340, 85–90 (1998).
[CrossRef]

Foster, D. H.

Gargill, E. B.

E. B. Gargill, “A mathematical liver model and its application to system optimization and texture analysis,” Ph.D dissertation (University of Arizona, Tucson, Ariz.1989).

Garra, B. S.

R. S. Mia, M. H. Loew, K. A. Wear, R. F. Wagner, B. S. Garra, “Quantitative ultrasound tissue characterization using texture and cepstral features,” in Medical Imaging 1998: Image Processing, K. M. Hanson, ed., Proc. SPIE3338, 211–219 (1998).
[CrossRef]

Gonzalez, R.

R. Gonzalez, R. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), Chap. 8.

Goon, A. A.

J. P. Rolland, A. A. Goon, L. Yu, “Synthesis of textured complex backgrounds,” Opt. Eng. 37, 2055–2063 (1998).
[CrossRef]

J. P. Rolland, A. A. Goon, E. Clarkson, L. Yu, “Synthesis of biomedical tissue,” in Medical Imaging 1998: Image Perception, H. L. Kundel, ed., Proc. SPIE3340, 85–90 (1998).
[CrossRef]

Graber, M. A.

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiological abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[CrossRef] [PubMed]

Haralick, R. M.

R. M. Haralick, “Statistical and structural approaches to texture,” Proc. IEEE 67(5), 786–804 (1979).
[CrossRef]

Hessler, C.

F. O. Bochud, F. R. Verdun, C. Hessler, J. F. Valley, “Detectability on radiological images: the influence of anatomical noise,” in Medical Imaging 1995: Image Perception, H. L. Kundel, ed., Proc. SPIE2436, 156–165 (1995).
[CrossRef]

Hylen, S. H. L.

S. H. L. Hylen, “Image modifiers for use in photography,” U.S. patent5,649,259 (15July1997).

Kundel, H. L.

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiological abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[CrossRef] [PubMed]

Laget, B.

A. Tremeau, J. Bousigue, B. Laget, “Co-occurrence shape descriptors applied to texture classification and segmentation,” in Machine Vision Applications in Industrial Inspection IV, A. R. Rao, N. Chang, eds., Proc. SPIE2665, 135–147 (1996).
[CrossRef]

Little, L. E.

J. W. Byng, M. J. Yaffe, G. A. Lockwood, L. E. Little, D. L. Tritchler, N. F. Boyd, “Automated analysis of mammographic densities and breast carcinoma risk,” Cancer (N.Y.) 80, 66–74 (1997).
[CrossRef]

Lockwood, G. A.

J. W. Byng, M. J. Yaffe, G. A. Lockwood, L. E. Little, D. L. Tritchler, N. F. Boyd, “Automated analysis of mammographic densities and breast carcinoma risk,” Cancer (N.Y.) 80, 66–74 (1997).
[CrossRef]

Loew, M. H.

R. S. Mia, M. H. Loew, K. A. Wear, R. F. Wagner, B. S. Garra, “Quantitative ultrasound tissue characterization using texture and cepstral features,” in Medical Imaging 1998: Image Processing, K. M. Hanson, ed., Proc. SPIE3338, 211–219 (1998).
[CrossRef]

Mathai, A. M.

A. M. Mathai, S. B. Provost, Quadratic Forms in Random Variables (Marcel Dekker, New York, 1992), Chap. 3.2.

Mia, R. S.

R. S. Mia, M. H. Loew, K. A. Wear, R. F. Wagner, B. S. Garra, “Quantitative ultrasound tissue characterization using texture and cepstral features,” in Medical Imaging 1998: Image Processing, K. M. Hanson, ed., Proc. SPIE3338, 211–219 (1998).
[CrossRef]

Myers, K. J.

H. H. Barrett, J. Yao, J. P. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[CrossRef] [PubMed]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Statistical Processes (McGraw-Hill, New York, 1991).

Provost, S. B.

A. M. Mathai, S. B. Provost, Quadratic Forms in Random Variables (Marcel Dekker, New York, 1992), Chap. 3.2.

Revesz, G.

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiological abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[CrossRef] [PubMed]

Rolland, J. P.

J. P. Rolland, A. A. Goon, L. Yu, “Synthesis of textured complex backgrounds,” Opt. Eng. 37, 2055–2063 (1998).
[CrossRef]

J. P. Rolland, R. Strickland, “An approach to the synthesis of biological tissue,” Opt. Express 1, 414–423 (1997).
[CrossRef] [PubMed]

H. H. Barrett, J. Yao, J. P. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[CrossRef] [PubMed]

J. P. Rolland, H. H. Barrett, “Model of background uncertainty and observers detection performance,” J. Opt. Soc. Am. A 9, 649–658 (1992).
[CrossRef] [PubMed]

J. P. Rolland, A. A. Goon, E. Clarkson, L. Yu, “Synthesis of biomedical tissue,” in Medical Imaging 1998: Image Perception, H. L. Kundel, ed., Proc. SPIE3340, 85–90 (1998).
[CrossRef]

Strickland, R.

Thomson, M. G. A.

Tremeau, A.

A. Tremeau, J. Bousigue, B. Laget, “Co-occurrence shape descriptors applied to texture classification and segmentation,” in Machine Vision Applications in Industrial Inspection IV, A. R. Rao, N. Chang, eds., Proc. SPIE2665, 135–147 (1996).
[CrossRef]

Tritchler, D. L.

J. W. Byng, M. J. Yaffe, G. A. Lockwood, L. E. Little, D. L. Tritchler, N. F. Boyd, “Automated analysis of mammographic densities and breast carcinoma risk,” Cancer (N.Y.) 80, 66–74 (1997).
[CrossRef]

Valley, J. F.

F. O. Bochud, F. R. Verdun, C. Hessler, J. F. Valley, “Detectability on radiological images: the influence of anatomical noise,” in Medical Imaging 1995: Image Perception, H. L. Kundel, ed., Proc. SPIE2436, 156–165 (1995).
[CrossRef]

Verdun, F. R.

F. O. Bochud, F. R. Verdun, C. Hessler, J. F. Valley, “Detectability on radiological images: the influence of anatomical noise,” in Medical Imaging 1995: Image Perception, H. L. Kundel, ed., Proc. SPIE2436, 156–165 (1995).
[CrossRef]

Wagner, R. F.

R. S. Mia, M. H. Loew, K. A. Wear, R. F. Wagner, B. S. Garra, “Quantitative ultrasound tissue characterization using texture and cepstral features,” in Medical Imaging 1998: Image Processing, K. M. Hanson, ed., Proc. SPIE3338, 211–219 (1998).
[CrossRef]

Wear, K. A.

R. S. Mia, M. H. Loew, K. A. Wear, R. F. Wagner, B. S. Garra, “Quantitative ultrasound tissue characterization using texture and cepstral features,” in Medical Imaging 1998: Image Processing, K. M. Hanson, ed., Proc. SPIE3338, 211–219 (1998).
[CrossRef]

Woods, R.

R. Gonzalez, R. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), Chap. 8.

Yaffe, M. J.

J. W. Byng, M. J. Yaffe, G. A. Lockwood, L. E. Little, D. L. Tritchler, N. F. Boyd, “Automated analysis of mammographic densities and breast carcinoma risk,” Cancer (N.Y.) 80, 66–74 (1997).
[CrossRef]

Yao, J.

H. H. Barrett, J. Yao, J. P. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[CrossRef] [PubMed]

Yu, L.

J. P. Rolland, A. A. Goon, L. Yu, “Synthesis of textured complex backgrounds,” Opt. Eng. 37, 2055–2063 (1998).
[CrossRef]

J. P. Rolland, A. A. Goon, E. Clarkson, L. Yu, “Synthesis of biomedical tissue,” in Medical Imaging 1998: Image Perception, H. L. Kundel, ed., Proc. SPIE3340, 85–90 (1998).
[CrossRef]

Cancer (N.Y.) (1)

J. W. Byng, M. J. Yaffe, G. A. Lockwood, L. E. Little, D. L. Tritchler, N. F. Boyd, “Automated analysis of mammographic densities and breast carcinoma risk,” Cancer (N.Y.) 80, 66–74 (1997).
[CrossRef]

Invest. Radiol. (1)

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiological abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[CrossRef] [PubMed]

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

Opt. Eng. (1)

J. P. Rolland, A. A. Goon, L. Yu, “Synthesis of textured complex backgrounds,” Opt. Eng. 37, 2055–2063 (1998).
[CrossRef]

Opt. Express (1)

Proc. IEEE (1)

R. M. Haralick, “Statistical and structural approaches to texture,” Proc. IEEE 67(5), 786–804 (1979).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

H. H. Barrett, J. Yao, J. P. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[CrossRef] [PubMed]

Other (10)

A. Papoulis, Probability, Random Variables, and Statistical Processes (McGraw-Hill, New York, 1991).

A. M. Mathai, S. B. Provost, Quadratic Forms in Random Variables (Marcel Dekker, New York, 1992), Chap. 3.2.

J. P. Rolland, A. A. Goon, E. Clarkson, L. Yu, “Synthesis of biomedical tissue,” in Medical Imaging 1998: Image Perception, H. L. Kundel, ed., Proc. SPIE3340, 85–90 (1998).
[CrossRef]

R. Gonzalez, R. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), Chap. 8.

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979), Chap. 16.

A. Tremeau, J. Bousigue, B. Laget, “Co-occurrence shape descriptors applied to texture classification and segmentation,” in Machine Vision Applications in Industrial Inspection IV, A. R. Rao, N. Chang, eds., Proc. SPIE2665, 135–147 (1996).
[CrossRef]

R. S. Mia, M. H. Loew, K. A. Wear, R. F. Wagner, B. S. Garra, “Quantitative ultrasound tissue characterization using texture and cepstral features,” in Medical Imaging 1998: Image Processing, K. M. Hanson, ed., Proc. SPIE3338, 211–219 (1998).
[CrossRef]

F. O. Bochud, F. R. Verdun, C. Hessler, J. F. Valley, “Detectability on radiological images: the influence of anatomical noise,” in Medical Imaging 1995: Image Perception, H. L. Kundel, ed., Proc. SPIE2436, 156–165 (1995).
[CrossRef]

E. B. Gargill, “A mathematical liver model and its application to system optimization and texture analysis,” Ph.D dissertation (University of Arizona, Tucson, Ariz.1989).

S. H. L. Hylen, “Image modifiers for use in photography,” U.S. patent5,649,259 (15July1997).

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Figures (8)

Fig. 1
Fig. 1

Texture space.

Fig. 2
Fig. 2

ε(k) defined by Eq. (7) as a function of the number of moments used in the estimation of the first-order probability distribution for one realization of a texture image.

Fig. 3
Fig. 3

(a) 1P-PDF computed with the ME method. Note that the x axis corresponds to gray levels normalized to the range 0–1. (b) 1P-PDF computed with the relative-frequency method. This is also known as the gray level histogram of a texture image.

Fig. 4
Fig. 4

Illustration of the computation of the 2P-PDF (co-occurrence matrix) with the relative-frequency method.

Fig. 5
Fig. 5

(a) 2P-PDF calculated with the ME method. (b) 2P-PDF of a textured image calculated with the relative-frequency method.

Fig. 6
Fig. 6

(a) Space of the discrimination parameters. Points within boundaries 1, 2 and 3 satisfy Eq. (38). (b) Shaded regions show permitted points in space (d1, d2). Classes A and B are fully discriminable. dAB>r1A+r1B. (c) Classes A and B have common points and are not fully discriminable. dAB<r1A+r1B.

Fig. 7
Fig. 7

Error rate of the classifications between two ensembles with uniform distance distribution and the same radii. Point T(r1/dAB=0.5) corresponds to the moment when Eq. (1) becomes an equality.

Fig. 8
Fig. 8

Some realizations of the four texture ensembles used in the computations before equalization of their first-order statistics.

Tables (2)

Tables Icon

Table 1 Values of σ Computed with Either Eq. (36) or Direct Averaging

Tables Icon

Table 2 Values of Interdistances and Intradistances for Four Textures Classes

Equations (55)

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

dAB>r0.99A+r0.99B.
r0.99A>dAB+r0.99B.
d=Δr[ρa(g1, g2, Δr)-ρb(g1, g2, Δr)]2dg1dg21/2,
ρ(x)ln ρ(x)dx,
μi=xiρ(x)dx.
ρ(g)ln ρ(g)dgmax,giρ(g)dg=μi,
μi=E{gi(r)},i=0, N,
ε(k)=i=k+1m+k+1E{gi}-giρ(g)dg2.
μij(Δr)=E{gi(r)gj(r+Δr)},i, j=0,4,
ρ(g1, g2, Δr)ln[ρ(g1, g2, Δr)]dg1dg2max
g1ig2iρ(g1, g2, Δr)dg1dg2=μij(Δr),
i, j=0, N.
ρ(g1, g2, Δr)=expi,j=0Nβijg1ig2i.
σ=d2.
ρa(g1, g2, βij)=ρ(g1, g2, βij)+ijρ(g, g, βij)βij (Δβij),
d2=ijρ(g1, g2, βij)βij (Δβij)2dg1dg2=ijklΔβijΔβklAijkl,
Aijkl=ρ(g1, g2, βij)βij ρ(g1, g2, βkl)βkldg1dg2,
i, j, k, lN
ρ(g1, g2, Δr)βij=g1ig2iρ(g1, g2, Δr).
Aijkl=g1i+kg2j+lρ2(g1, g2, Δr)dg1dg2=gi+k(r)gj+l(r+Δr)ρ[g(r), g(r+Δr), Δr].
klg1ig2jρ(g1, g2, Δr)dg1dg2βkl Δβkl
=Δμij(Δr).
g1ig2jρ(g1, g2, Δr)dg1dg2βkl
=g1i+kg2j+lρ(g1, g2, Δr)dg1dg2
=μi+k,j+l(Δr)=Bijkl.
klBijklΔβkl=Δμij(Δr).
Δβkl=ij(B-1)klijΔμij(Δr).
d2=ijkl[(B-1)TAB-1]ijklΔμijΔμkl.
Cijkl=EΔμijΔμkl.
σ2=Ed2=ijkl[(B-1)TAB-1]ijklCijkl.
AijklAmn,BijklBmn,CijklCmn;
i, j, k, lN;m, nN2.
Ed2=tr[(B-1)TATB-1C]=σ2(Δr).
σ=Δrσ2(Δr)1/2=Δrtr{[B-1(Δr)]T[A(Δr)]TB-1(Δr)C(Δr)}1/2.
P{d2D2}=Sρ(μ01, μ02,,μnm)dμ01dμ02dμnn,
ijkl[(B-1)TAB-1]ijklΔμijΔμklD2.
ρ(μ00, μ01,, μnn)=ρ({μ}),
(σ)2=ijkl[(B-1)TAB-1]ijkl×(μija-μijb)(μkla-μklb)×ρ({μ}a)ρ({μ}b)d{μ}ad{μ}b,
σ2=ijkl[(B-1)TAB-1]ijkl(μij-μij)×(μkl-μkl)ρ({μ})d{μ}.
(μija-μijb)(μkla-μklb)ρ({μ}a)ρ({μ}b)d{μ}ad{μ}b
=(μijaμkla+μijbμklb-μijaμklb-μijbμkla)
×ρ({μ}a)ρ({μ}b)d{μ}ad{μ}b
=2(μijμkl-μijμkl).
(μij-μij)(μkl-μkl)ρ({μ})d{μ}
=(μijμkl+μijμkl-μijμkl-μklμij)
×ρ({μ})d{μ}=μijμkl-μijμkl.
σ=2σ.
σ=2Δrσ2(Δr)1/2=2Δrtr{[B-1(Δr)]T[A(Δr)]TB-1(Δr)C(Δr)}1/2.
d1=d(X, A),
d2=d(X, B),
dAB=d(A, B)
d1+d2dAB,
d1+dABd2,
d2+dABd1.
p(r)=1/r,rr10,r>r1.

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