Abstract

Digitized images can be decomposed into a sum of images by use of a multiresolution wavelet expansion. Each image of the expansion can be analyzed with a parametric statistical model for the histogram associated with each expansion component. The statistical analysis of the individual expansion components is relatively simple, whereas the analysis of the original image is complicated. We show that the probability model for each expansion component can be approximated with a Laplace probability-density function for some important applications in digital mammography. An approach to random variable analysis based on the predominant low-frequency characteristics of the random fields provides theoretical support for the approximation. The theoretical framework of multiresolution analysis provides a natural extension for modeling many levels of image detail simultaneously for special cases. We demonstrate this with a noise field simulation and a mammographic application, where the expansion components are treated as independent random variables.

© 1998 Optical Society of America

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References

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  1. J. J. Heine, “Multiresolution statistical analysis of high resolution digitized mammograms and other gray scaled images,” Ph.D. dissertation (University of South Florida, Tampa, Fla., 1996).
  2. J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar, L. P. Clarke, “Multiresolution statistical analysis of high-resolution digital mammograms,” IEEE Trans. Med. Imaging 16, 503–515 (1997).
    [CrossRef] [PubMed]
  3. J. G. Jones, R. W. Thomas, P. G. Earwicker, S. Addison, “Multiresolution statistical analysis of computer-generated fractal imagery,” CVGIP Graph. Models Image Process. 53, 349–363 (1991).
    [CrossRef]
  4. G. H. Watson, S. K. Watson, “Detection of unusual events in intermittent non-Gaussian images using multiresolution background models,” Opt. Eng. 35, 3159–3171 (1996).
    [CrossRef]
  5. D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
    [CrossRef] [PubMed]
  6. D. I. Barnea, H. F. Silverman, “A class of algorithms for fast digital image registration,” IEEE Trans. Comput. C-21, 179–186 (1972).
    [CrossRef]
  7. S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern. Anal. Mach. Intell. 11, 674–693 (1989).
    [CrossRef]
  8. M. Antonini, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
    [CrossRef] [PubMed]
  9. K. C. Chou, S. Golden, A. S. Willsky, “Modeling and estimation of multiscale stochastic processes,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing ’91 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1991), pp. 1709–1712.
  10. V. V. Digalakis, K. C. Chou, “Maximum likelihood identification of multiscale stochastic models using the wavelet transform and the EM algorithm,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing ’93 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1993), pp. 93–96.
  11. J.-L. Chen, A. Kundu, “Rotation and gray scale transform invariant texture identification using wavelet decomposition and hidden Markov model,” IEEE Trans. Pattern. Anal. Mach. Intell. 16, 208–214 (1994).
    [CrossRef]
  12. G. Wornell, Signal Processing with Fractals: A Wavelet-Based Approach (Prentice-Hall, Englewood Cliffs, N.J., 1996).
  13. B. Gold, G. O. Young, “The response of linear systems to non-Gaussian noise,” IRE Trans. Inf. Theory PGIT 2-4, 63–67 (1953/54).
  14. L. Andersson, N. Hall, B. Jawerth, G. Peters, “Wavelets on closed subsets of the real line,” in Recent Advances in Wavelet Analysis, L. L. Schumaker, G. Webb, eds. (Academic, Boston, Mass., 1994).
  15. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
  16. R. Bracewell, “The Fourier Transform and Its Applications,” 2nd ed. (McGraw-Hill, New York, 1986).
  17. N. L. Johnson, S. Kotz, Continuous Univariant Distributions, 2nd ed. (Wiley, New York, 1995), Vol. 2.
  18. C. W. Helstrom, Probability and Stochastic Processes for Engineers, 2nd ed. (Macmillan, New York, 1991).
  19. D. M. Titterington, A. F. M. Smith, U. E. Makov, Chichester, Statistical Analysis of Finite Mixture Distributions (Wiley, New York, 1985).

1997

J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar, L. P. Clarke, “Multiresolution statistical analysis of high-resolution digital mammograms,” IEEE Trans. Med. Imaging 16, 503–515 (1997).
[CrossRef] [PubMed]

1996

G. H. Watson, S. K. Watson, “Detection of unusual events in intermittent non-Gaussian images using multiresolution background models,” Opt. Eng. 35, 3159–3171 (1996).
[CrossRef]

1994

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

J.-L. Chen, A. Kundu, “Rotation and gray scale transform invariant texture identification using wavelet decomposition and hidden Markov model,” IEEE Trans. Pattern. Anal. Mach. Intell. 16, 208–214 (1994).
[CrossRef]

1992

M. Antonini, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
[CrossRef] [PubMed]

1991

J. G. Jones, R. W. Thomas, P. G. Earwicker, S. Addison, “Multiresolution statistical analysis of computer-generated fractal imagery,” CVGIP Graph. Models Image Process. 53, 349–363 (1991).
[CrossRef]

1989

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern. Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

1972

D. I. Barnea, H. F. Silverman, “A class of algorithms for fast digital image registration,” IEEE Trans. Comput. C-21, 179–186 (1972).
[CrossRef]

Addison, S.

J. G. Jones, R. W. Thomas, P. G. Earwicker, S. Addison, “Multiresolution statistical analysis of computer-generated fractal imagery,” CVGIP Graph. Models Image Process. 53, 349–363 (1991).
[CrossRef]

Andersson, L.

L. Andersson, N. Hall, B. Jawerth, G. Peters, “Wavelets on closed subsets of the real line,” in Recent Advances in Wavelet Analysis, L. L. Schumaker, G. Webb, eds. (Academic, Boston, Mass., 1994).

Antonini, M.

M. Antonini, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
[CrossRef] [PubMed]

Barlaud, M.

M. Antonini, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
[CrossRef] [PubMed]

Barnea, D. I.

D. I. Barnea, H. F. Silverman, “A class of algorithms for fast digital image registration,” IEEE Trans. Comput. C-21, 179–186 (1972).
[CrossRef]

Bialek, W.

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

Bracewell, R.

R. Bracewell, “The Fourier Transform and Its Applications,” 2nd ed. (McGraw-Hill, New York, 1986).

Chen, J.-L.

J.-L. Chen, A. Kundu, “Rotation and gray scale transform invariant texture identification using wavelet decomposition and hidden Markov model,” IEEE Trans. Pattern. Anal. Mach. Intell. 16, 208–214 (1994).
[CrossRef]

Chichester,

D. M. Titterington, A. F. M. Smith, U. E. Makov, Chichester, Statistical Analysis of Finite Mixture Distributions (Wiley, New York, 1985).

Chou, K. C.

V. V. Digalakis, K. C. Chou, “Maximum likelihood identification of multiscale stochastic models using the wavelet transform and the EM algorithm,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing ’93 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1993), pp. 93–96.

K. C. Chou, S. Golden, A. S. Willsky, “Modeling and estimation of multiscale stochastic processes,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing ’91 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1991), pp. 1709–1712.

Clarke, L. P.

J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar, L. P. Clarke, “Multiresolution statistical analysis of high-resolution digital mammograms,” IEEE Trans. Med. Imaging 16, 503–515 (1997).
[CrossRef] [PubMed]

Cullers, D. K.

J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar, L. P. Clarke, “Multiresolution statistical analysis of high-resolution digital mammograms,” IEEE Trans. Med. Imaging 16, 503–515 (1997).
[CrossRef] [PubMed]

Daubechies, I.

M. Antonini, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
[CrossRef] [PubMed]

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Deans, S. R.

J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar, L. P. Clarke, “Multiresolution statistical analysis of high-resolution digital mammograms,” IEEE Trans. Med. Imaging 16, 503–515 (1997).
[CrossRef] [PubMed]

Digalakis, V. V.

V. V. Digalakis, K. C. Chou, “Maximum likelihood identification of multiscale stochastic models using the wavelet transform and the EM algorithm,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing ’93 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1993), pp. 93–96.

Earwicker, P. G.

J. G. Jones, R. W. Thomas, P. G. Earwicker, S. Addison, “Multiresolution statistical analysis of computer-generated fractal imagery,” CVGIP Graph. Models Image Process. 53, 349–363 (1991).
[CrossRef]

Gold, B.

B. Gold, G. O. Young, “The response of linear systems to non-Gaussian noise,” IRE Trans. Inf. Theory PGIT 2-4, 63–67 (1953/54).

Golden, S.

K. C. Chou, S. Golden, A. S. Willsky, “Modeling and estimation of multiscale stochastic processes,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing ’91 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1991), pp. 1709–1712.

Hall, N.

L. Andersson, N. Hall, B. Jawerth, G. Peters, “Wavelets on closed subsets of the real line,” in Recent Advances in Wavelet Analysis, L. L. Schumaker, G. Webb, eds. (Academic, Boston, Mass., 1994).

Heine, J. J.

J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar, L. P. Clarke, “Multiresolution statistical analysis of high-resolution digital mammograms,” IEEE Trans. Med. Imaging 16, 503–515 (1997).
[CrossRef] [PubMed]

J. J. Heine, “Multiresolution statistical analysis of high resolution digitized mammograms and other gray scaled images,” Ph.D. dissertation (University of South Florida, Tampa, Fla., 1996).

Helstrom, C. W.

C. W. Helstrom, Probability and Stochastic Processes for Engineers, 2nd ed. (Macmillan, New York, 1991).

Jawerth, B.

L. Andersson, N. Hall, B. Jawerth, G. Peters, “Wavelets on closed subsets of the real line,” in Recent Advances in Wavelet Analysis, L. L. Schumaker, G. Webb, eds. (Academic, Boston, Mass., 1994).

Johnson, N. L.

N. L. Johnson, S. Kotz, Continuous Univariant Distributions, 2nd ed. (Wiley, New York, 1995), Vol. 2.

Jones, J. G.

J. G. Jones, R. W. Thomas, P. G. Earwicker, S. Addison, “Multiresolution statistical analysis of computer-generated fractal imagery,” CVGIP Graph. Models Image Process. 53, 349–363 (1991).
[CrossRef]

Kotz, S.

N. L. Johnson, S. Kotz, Continuous Univariant Distributions, 2nd ed. (Wiley, New York, 1995), Vol. 2.

Kundu, A.

J.-L. Chen, A. Kundu, “Rotation and gray scale transform invariant texture identification using wavelet decomposition and hidden Markov model,” IEEE Trans. Pattern. Anal. Mach. Intell. 16, 208–214 (1994).
[CrossRef]

Makov, U. E.

D. M. Titterington, A. F. M. Smith, U. E. Makov, Chichester, Statistical Analysis of Finite Mixture Distributions (Wiley, New York, 1985).

Mallat, S. G.

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern. Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

Mathieu, P.

M. Antonini, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
[CrossRef] [PubMed]

Peters, G.

L. Andersson, N. Hall, B. Jawerth, G. Peters, “Wavelets on closed subsets of the real line,” in Recent Advances in Wavelet Analysis, L. L. Schumaker, G. Webb, eds. (Academic, Boston, Mass., 1994).

Ruderman, D. L.

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

Silverman, H. F.

D. I. Barnea, H. F. Silverman, “A class of algorithms for fast digital image registration,” IEEE Trans. Comput. C-21, 179–186 (1972).
[CrossRef]

Smith, A. F. M.

D. M. Titterington, A. F. M. Smith, U. E. Makov, Chichester, Statistical Analysis of Finite Mixture Distributions (Wiley, New York, 1985).

Stauduhar, R.

J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar, L. P. Clarke, “Multiresolution statistical analysis of high-resolution digital mammograms,” IEEE Trans. Med. Imaging 16, 503–515 (1997).
[CrossRef] [PubMed]

Thomas, R. W.

J. G. Jones, R. W. Thomas, P. G. Earwicker, S. Addison, “Multiresolution statistical analysis of computer-generated fractal imagery,” CVGIP Graph. Models Image Process. 53, 349–363 (1991).
[CrossRef]

Titterington, D. M.

D. M. Titterington, A. F. M. Smith, U. E. Makov, Chichester, Statistical Analysis of Finite Mixture Distributions (Wiley, New York, 1985).

Watson, G. H.

G. H. Watson, S. K. Watson, “Detection of unusual events in intermittent non-Gaussian images using multiresolution background models,” Opt. Eng. 35, 3159–3171 (1996).
[CrossRef]

Watson, S. K.

G. H. Watson, S. K. Watson, “Detection of unusual events in intermittent non-Gaussian images using multiresolution background models,” Opt. Eng. 35, 3159–3171 (1996).
[CrossRef]

Willsky, A. S.

K. C. Chou, S. Golden, A. S. Willsky, “Modeling and estimation of multiscale stochastic processes,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing ’91 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1991), pp. 1709–1712.

Wornell, G.

G. Wornell, Signal Processing with Fractals: A Wavelet-Based Approach (Prentice-Hall, Englewood Cliffs, N.J., 1996).

Young, G. O.

B. Gold, G. O. Young, “The response of linear systems to non-Gaussian noise,” IRE Trans. Inf. Theory PGIT 2-4, 63–67 (1953/54).

CVGIP Graph. Models Image Process.

J. G. Jones, R. W. Thomas, P. G. Earwicker, S. Addison, “Multiresolution statistical analysis of computer-generated fractal imagery,” CVGIP Graph. Models Image Process. 53, 349–363 (1991).
[CrossRef]

IEEE Trans. Comput.

D. I. Barnea, H. F. Silverman, “A class of algorithms for fast digital image registration,” IEEE Trans. Comput. C-21, 179–186 (1972).
[CrossRef]

IEEE Trans. Image Process.

M. Antonini, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging

J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar, L. P. Clarke, “Multiresolution statistical analysis of high-resolution digital mammograms,” IEEE Trans. Med. Imaging 16, 503–515 (1997).
[CrossRef] [PubMed]

IEEE Trans. Pattern. Anal. Mach. Intell.

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern. Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

J.-L. Chen, A. Kundu, “Rotation and gray scale transform invariant texture identification using wavelet decomposition and hidden Markov model,” IEEE Trans. Pattern. Anal. Mach. Intell. 16, 208–214 (1994).
[CrossRef]

IRE Trans. Inf. Theory

B. Gold, G. O. Young, “The response of linear systems to non-Gaussian noise,” IRE Trans. Inf. Theory PGIT 2-4, 63–67 (1953/54).

Opt. Eng.

G. H. Watson, S. K. Watson, “Detection of unusual events in intermittent non-Gaussian images using multiresolution background models,” Opt. Eng. 35, 3159–3171 (1996).
[CrossRef]

Phys. Rev. Lett.

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

Other

J. J. Heine, “Multiresolution statistical analysis of high resolution digitized mammograms and other gray scaled images,” Ph.D. dissertation (University of South Florida, Tampa, Fla., 1996).

K. C. Chou, S. Golden, A. S. Willsky, “Modeling and estimation of multiscale stochastic processes,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing ’91 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1991), pp. 1709–1712.

V. V. Digalakis, K. C. Chou, “Maximum likelihood identification of multiscale stochastic models using the wavelet transform and the EM algorithm,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing ’93 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1993), pp. 93–96.

L. Andersson, N. Hall, B. Jawerth, G. Peters, “Wavelets on closed subsets of the real line,” in Recent Advances in Wavelet Analysis, L. L. Schumaker, G. Webb, eds. (Academic, Boston, Mass., 1994).

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

R. Bracewell, “The Fourier Transform and Its Applications,” 2nd ed. (McGraw-Hill, New York, 1986).

N. L. Johnson, S. Kotz, Continuous Univariant Distributions, 2nd ed. (Wiley, New York, 1995), Vol. 2.

C. W. Helstrom, Probability and Stochastic Processes for Engineers, 2nd ed. (Macmillan, New York, 1991).

D. M. Titterington, A. F. M. Smith, U. E. Makov, Chichester, Statistical Analysis of Finite Mixture Distributions (Wiley, New York, 1985).

G. Wornell, Signal Processing with Fractals: A Wavelet-Based Approach (Prentice-Hall, Englewood Cliffs, N.J., 1996).

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Figures (6)

Fig. 1
Fig. 1

(a) Raw f0 image, 256 × 256 pixel ROI (left), and empirical histogram (right). (b) f1 lower-resolution image (left) and empirical histogram (right). (c) f2 lower-resolution image (left) and empirical histogram (right). Note that there is very little change in the histograms as the image is blurred to the next-coarser resolution.

Fig. 2
Fig. 2

Fine-to-coarse detail representation with empirical histograms and theoretical pdf’s d1d6 in (a) upper left and right through (f) lower left and right. The images are represented with a 256 × 256 pixel ROI. The empirical histograms (solid curves) are to be compared with the theoretical estimates (dashed curves). The probability models are presented with the absolute value data.

Fig. 3
Fig. 3

Simulated noise field, 256 × 256 pixel ROI, and fine-to-coarse detail representation with histograms and theoretical pdf’s, where applicable, d1d5 in (a) upper left and right through (f) lower left and right. The empirical histograms (solid curves) are to be compared with the theoretical ones (diamonds). Points have been skipped on the theoretical plots for clarity because of the close agreement. The detail fields are all modeled by normal distributions.

Fig. 4
Fig. 4

(a) Total detail representation sum of d1d5 with histogram (solid curve) and theoretical pdf (diamonds), upper left and right, and (b) f5 low-resolution image with histogram, bottom left and right. (a) Detail images can be treated as rv’s and combined according to theory; (b) shows that sometimes it is possible to remove the irregular background. Points have been skipped on the theoretical plots for clarity.

Fig. 5
Fig. 5

(a) Simulation noise field d6 image and histogram, top left and right, and (b) f6 image with histogram, bottom left and right, illustrating that at a certain resolution the background will mix with the detail. Both histograms appear to be tending toward a normal distribution.

Fig. 6
Fig. 6

d1+d2 image and associated empirical histogram (solid curve) compared with the theoretical pdf (dashed curve), again illustrating that the detail images can be treated as rv’s. The histogram is obtained from the absolute value data.

Equations (24)

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

f0=(f0-f1)+(f1-f2)+(f2-f3)++(fJ-1-fJ)+fJ,
f0=d1+d2+d3++dJ+fJ.
f0=d1+f1.
the pdf governing f0the pdf governing the distribution
of the local average.
s1(x,κ)=s0(x+κ)g(κ), 0κ<.
s1(x)0[s0(x)+κs0(x)]g(κ)dκ,
s1(x)s0(x)+τs0(x),
s0=s1(x) * p1(x).
s0(x)[s0(x)+τs0(x)] * p1(x).
S0(ω)S0(ω)(1+iωτ)P1(ω),
P1(ω)=11+iωτ.
p1(x)1τexp-xτ, 0x.
P1(ω)11-iωτ,
p1(x)1τexpxτ, 0>x-.
p1(x)12τexp-|x|τ.
p(x; τj)=1τjexp-xτj for x0.
P(ω)=Pj(ω)Pk(ω).
Pd(ω)=j=1JPj(ω)
σT=jσj21/2.
F[pj(y; kj)]=pj(y; kj) exp(-iωy)dy=kj2ω2+kj2,
pm,n(r)=12(kn2-km2)  ×[kn2km exp(-km|r|)-km2kn exp(-kn|r|)].
pm,n(z)=1(kn2-km2)  ×[kn2km exp(-kmz)-km2kn exp(-knz)].
0xmaxs0(x)dx=s0(xmax)-s0(0)=0.

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