Abstract

Many imaging applications deal with the detection of small targets or spots embedded within an inhomogeneous background. We present a method that accomplishes a multiresolution detection on the wavelet-transformed image. The targets are separated from the background by the exploitation of Renyi’s information, which is evaluated at the different decomposition levels of the wavelet transform. The scale-dependent candidate detections are successively combined by means of majority voting for final detection. Connections with results provided in different fields such as multifractal analysis, generalized information measures in scale-space, and cross-entropy analysis in fine-to-coarse transformations are discussed. Detection performance is investigated through an example from medical image analysis.

© 2000 Optical Society of America

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References

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  1. T. K. Bhattachary, S. Haykm, “ANN based adaptive radar detection scheme for small target in sea clutter,” Electron. Lett. 28, 1528–1536 (1992).
    [CrossRef]
  2. T. Soni, J. Zeidler, W. Ku, “Performance evaluation of 2-D adaptive prediction filters for detection of small objects in image data,” IEEE Trans. Image Process. 2, 327–340 (1993).
    [CrossRef] [PubMed]
  3. P. French, J. Zeidler, W. Ku, “Enhanced detectability of small objects in correlated clutter using an improved 2-D adaptive lattice algorithm,” IEEE Trans. Image Process. 6, 383–397 (1997).
    [CrossRef]
  4. R. N. Strickland, H. I. Han, “Wavelet transform for detecting microcalcifications in mammograms,” IEEE Trans. Med. Imaging 15, 218–229 (1996).
    [CrossRef]
  5. H. Chan, L. Niklason, D. Ikeda, K. Lam, D. Adler, “Digitization requirements in mammography: effects on computer aided detection of microcalcifications,” Med. Phys. 21, 1203–1211 (1994).
    [CrossRef] [PubMed]
  6. I. Brodie, R. Gutcheck, “Radiographic information theory and application to mammography,” Med. Phys. 9, 812–820 (1982).
    [CrossRef]
  7. K. White, T. Hutson, T. Hutchinson, “Modeling human eye behavior during mammographic scanning: preliminary results,” IEEE Trans. Syst. Man Cybern. 27, 494–505 (1997).
    [CrossRef]
  8. N. Karssemeijer, “Adaptive noise equalization and recognition of microcalcifications clusters in mammograms,” Int. J. Pattern Recognition Artif. Intell. 7, 1357–1377 (1993).
    [CrossRef]
  9. T. Cover, J. Thomas, Elements of Information Theory (Wiley, New York, 1991).
  10. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
  11. D. Donoho, I. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
    [CrossRef]
  12. S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1998).
  13. J. Sporring, J. Weickert, “Information measures in scale-spaces,” IEEE Trans. Inf. Theory 45, 1051–1058 (1999).
    [CrossRef]
  14. M. Ferraro, G. Boccignone, T. Caelli, “On the representation of image structures via scale-space entropy condition,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1199–1203 (1999).
    [CrossRef]
  15. T. Kailath, H. V. Poor, “Detection of stochastic processes,” IEEE Trans. Inf. Theory 44, 2230–2259 (1998).
    [CrossRef]
  16. D. Gabor, “Theory of communication,” J. IEE 93, 429–457 (1946).
  17. M. Antonini, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
    [CrossRef] [PubMed]
  18. S. Mallat, “A theory of multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
    [CrossRef]
  19. M. Shensa, “The discrete wavelet transform: wedding the à trous and Mallat algorithm,” IEEE Trans. Signal Process. 40, 2464–2482 (1992).
    [CrossRef]
  20. T. Takayasu, Fractals in Physical Sciences (Manchester U. Press, Manchester, UK, 1990).
  21. J. Kittler, M. Hatef, R. B. W. Duin, J. Matas, “On combining classifiers,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 226–239 (1998).
    [CrossRef]
  22. L. Huai, K. Liu, S. Lo, “Fractal modeling and segmentation for the enhancement of microcalcifications in digital mammograms,” IEEE Trans. Med. Imaging 16, 785–797 (1997).
    [CrossRef]
  23. B. Wandell, Foundations of Vision (Sinauer, Sunderland, Mass., 1995).
  24. M. Unser, A. Aldroubi, M. Eden, “On the asymptotic convergence of B-spline wavelets to Gabor functions,” IEEE Trans. Inf. Theory 38, 864–872 (1992).
    [CrossRef]
  25. P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, Toronto, 1953).
  26. D. Sheffer, D. Ingman, “The informational difference concept in analyzing target recognition issues,” J. Opt. Soc. Am. A 14, 1431–1438 (1997).
    [CrossRef]

1999 (2)

J. Sporring, J. Weickert, “Information measures in scale-spaces,” IEEE Trans. Inf. Theory 45, 1051–1058 (1999).
[CrossRef]

M. Ferraro, G. Boccignone, T. Caelli, “On the representation of image structures via scale-space entropy condition,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1199–1203 (1999).
[CrossRef]

1998 (2)

T. Kailath, H. V. Poor, “Detection of stochastic processes,” IEEE Trans. Inf. Theory 44, 2230–2259 (1998).
[CrossRef]

J. Kittler, M. Hatef, R. B. W. Duin, J. Matas, “On combining classifiers,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 226–239 (1998).
[CrossRef]

1997 (4)

L. Huai, K. Liu, S. Lo, “Fractal modeling and segmentation for the enhancement of microcalcifications in digital mammograms,” IEEE Trans. Med. Imaging 16, 785–797 (1997).
[CrossRef]

D. Sheffer, D. Ingman, “The informational difference concept in analyzing target recognition issues,” J. Opt. Soc. Am. A 14, 1431–1438 (1997).
[CrossRef]

P. French, J. Zeidler, W. Ku, “Enhanced detectability of small objects in correlated clutter using an improved 2-D adaptive lattice algorithm,” IEEE Trans. Image Process. 6, 383–397 (1997).
[CrossRef]

K. White, T. Hutson, T. Hutchinson, “Modeling human eye behavior during mammographic scanning: preliminary results,” IEEE Trans. Syst. Man Cybern. 27, 494–505 (1997).
[CrossRef]

1996 (1)

R. N. Strickland, H. I. Han, “Wavelet transform for detecting microcalcifications in mammograms,” IEEE Trans. Med. Imaging 15, 218–229 (1996).
[CrossRef]

1994 (2)

H. Chan, L. Niklason, D. Ikeda, K. Lam, D. Adler, “Digitization requirements in mammography: effects on computer aided detection of microcalcifications,” Med. Phys. 21, 1203–1211 (1994).
[CrossRef] [PubMed]

D. Donoho, I. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[CrossRef]

1993 (2)

N. Karssemeijer, “Adaptive noise equalization and recognition of microcalcifications clusters in mammograms,” Int. J. Pattern Recognition Artif. Intell. 7, 1357–1377 (1993).
[CrossRef]

T. Soni, J. Zeidler, W. Ku, “Performance evaluation of 2-D adaptive prediction filters for detection of small objects in image data,” IEEE Trans. Image Process. 2, 327–340 (1993).
[CrossRef] [PubMed]

1992 (4)

T. K. Bhattachary, S. Haykm, “ANN based adaptive radar detection scheme for small target in sea clutter,” Electron. Lett. 28, 1528–1536 (1992).
[CrossRef]

M. Shensa, “The discrete wavelet transform: wedding the à trous and Mallat algorithm,” IEEE Trans. Signal Process. 40, 2464–2482 (1992).
[CrossRef]

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

M. Unser, A. Aldroubi, M. Eden, “On the asymptotic convergence of B-spline wavelets to Gabor functions,” IEEE Trans. Inf. Theory 38, 864–872 (1992).
[CrossRef]

1989 (1)

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

1982 (1)

I. Brodie, R. Gutcheck, “Radiographic information theory and application to mammography,” Med. Phys. 9, 812–820 (1982).
[CrossRef]

1946 (1)

D. Gabor, “Theory of communication,” J. IEE 93, 429–457 (1946).

Adler, D.

H. Chan, L. Niklason, D. Ikeda, K. Lam, D. Adler, “Digitization requirements in mammography: effects on computer aided detection of microcalcifications,” Med. Phys. 21, 1203–1211 (1994).
[CrossRef] [PubMed]

Aldroubi, A.

M. Unser, A. Aldroubi, M. Eden, “On the asymptotic convergence of B-spline wavelets to Gabor functions,” IEEE Trans. Inf. Theory 38, 864–872 (1992).
[CrossRef]

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]

Bhattachary, T. K.

T. K. Bhattachary, S. Haykm, “ANN based adaptive radar detection scheme for small target in sea clutter,” Electron. Lett. 28, 1528–1536 (1992).
[CrossRef]

Boccignone, G.

M. Ferraro, G. Boccignone, T. Caelli, “On the representation of image structures via scale-space entropy condition,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1199–1203 (1999).
[CrossRef]

Brodie, I.

I. Brodie, R. Gutcheck, “Radiographic information theory and application to mammography,” Med. Phys. 9, 812–820 (1982).
[CrossRef]

Caelli, T.

M. Ferraro, G. Boccignone, T. Caelli, “On the representation of image structures via scale-space entropy condition,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1199–1203 (1999).
[CrossRef]

Chan, H.

H. Chan, L. Niklason, D. Ikeda, K. Lam, D. Adler, “Digitization requirements in mammography: effects on computer aided detection of microcalcifications,” Med. Phys. 21, 1203–1211 (1994).
[CrossRef] [PubMed]

Cover, T.

T. Cover, J. Thomas, Elements of Information Theory (Wiley, New York, 1991).

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).

Donoho, D.

D. Donoho, I. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[CrossRef]

Duin, R. B. W.

J. Kittler, M. Hatef, R. B. W. Duin, J. Matas, “On combining classifiers,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 226–239 (1998).
[CrossRef]

Eden, M.

M. Unser, A. Aldroubi, M. Eden, “On the asymptotic convergence of B-spline wavelets to Gabor functions,” IEEE Trans. Inf. Theory 38, 864–872 (1992).
[CrossRef]

Ferraro, M.

M. Ferraro, G. Boccignone, T. Caelli, “On the representation of image structures via scale-space entropy condition,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1199–1203 (1999).
[CrossRef]

Feshbach, H.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, Toronto, 1953).

French, P.

P. French, J. Zeidler, W. Ku, “Enhanced detectability of small objects in correlated clutter using an improved 2-D adaptive lattice algorithm,” IEEE Trans. Image Process. 6, 383–397 (1997).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of communication,” J. IEE 93, 429–457 (1946).

Gutcheck, R.

I. Brodie, R. Gutcheck, “Radiographic information theory and application to mammography,” Med. Phys. 9, 812–820 (1982).
[CrossRef]

Han, H. I.

R. N. Strickland, H. I. Han, “Wavelet transform for detecting microcalcifications in mammograms,” IEEE Trans. Med. Imaging 15, 218–229 (1996).
[CrossRef]

Hatef, M.

J. Kittler, M. Hatef, R. B. W. Duin, J. Matas, “On combining classifiers,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 226–239 (1998).
[CrossRef]

Haykm, S.

T. K. Bhattachary, S. Haykm, “ANN based adaptive radar detection scheme for small target in sea clutter,” Electron. Lett. 28, 1528–1536 (1992).
[CrossRef]

Huai, L.

L. Huai, K. Liu, S. Lo, “Fractal modeling and segmentation for the enhancement of microcalcifications in digital mammograms,” IEEE Trans. Med. Imaging 16, 785–797 (1997).
[CrossRef]

Hutchinson, T.

K. White, T. Hutson, T. Hutchinson, “Modeling human eye behavior during mammographic scanning: preliminary results,” IEEE Trans. Syst. Man Cybern. 27, 494–505 (1997).
[CrossRef]

Hutson, T.

K. White, T. Hutson, T. Hutchinson, “Modeling human eye behavior during mammographic scanning: preliminary results,” IEEE Trans. Syst. Man Cybern. 27, 494–505 (1997).
[CrossRef]

Ikeda, D.

H. Chan, L. Niklason, D. Ikeda, K. Lam, D. Adler, “Digitization requirements in mammography: effects on computer aided detection of microcalcifications,” Med. Phys. 21, 1203–1211 (1994).
[CrossRef] [PubMed]

Ingman, D.

Johnstone, I.

D. Donoho, I. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[CrossRef]

Kailath, T.

T. Kailath, H. V. Poor, “Detection of stochastic processes,” IEEE Trans. Inf. Theory 44, 2230–2259 (1998).
[CrossRef]

Karssemeijer, N.

N. Karssemeijer, “Adaptive noise equalization and recognition of microcalcifications clusters in mammograms,” Int. J. Pattern Recognition Artif. Intell. 7, 1357–1377 (1993).
[CrossRef]

Kittler, J.

J. Kittler, M. Hatef, R. B. W. Duin, J. Matas, “On combining classifiers,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 226–239 (1998).
[CrossRef]

Ku, W.

P. French, J. Zeidler, W. Ku, “Enhanced detectability of small objects in correlated clutter using an improved 2-D adaptive lattice algorithm,” IEEE Trans. Image Process. 6, 383–397 (1997).
[CrossRef]

T. Soni, J. Zeidler, W. Ku, “Performance evaluation of 2-D adaptive prediction filters for detection of small objects in image data,” IEEE Trans. Image Process. 2, 327–340 (1993).
[CrossRef] [PubMed]

Lam, K.

H. Chan, L. Niklason, D. Ikeda, K. Lam, D. Adler, “Digitization requirements in mammography: effects on computer aided detection of microcalcifications,” Med. Phys. 21, 1203–1211 (1994).
[CrossRef] [PubMed]

Liu, K.

L. Huai, K. Liu, S. Lo, “Fractal modeling and segmentation for the enhancement of microcalcifications in digital mammograms,” IEEE Trans. Med. Imaging 16, 785–797 (1997).
[CrossRef]

Lo, S.

L. Huai, K. Liu, S. Lo, “Fractal modeling and segmentation for the enhancement of microcalcifications in digital mammograms,” IEEE Trans. Med. Imaging 16, 785–797 (1997).
[CrossRef]

Mallat, S.

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

S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1998).

Matas, J.

J. Kittler, M. Hatef, R. B. W. Duin, J. Matas, “On combining classifiers,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 226–239 (1998).
[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]

Morse, P.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, Toronto, 1953).

Niklason, L.

H. Chan, L. Niklason, D. Ikeda, K. Lam, D. Adler, “Digitization requirements in mammography: effects on computer aided detection of microcalcifications,” Med. Phys. 21, 1203–1211 (1994).
[CrossRef] [PubMed]

Poor, H. V.

T. Kailath, H. V. Poor, “Detection of stochastic processes,” IEEE Trans. Inf. Theory 44, 2230–2259 (1998).
[CrossRef]

Sheffer, D.

Shensa, M.

M. Shensa, “The discrete wavelet transform: wedding the à trous and Mallat algorithm,” IEEE Trans. Signal Process. 40, 2464–2482 (1992).
[CrossRef]

Soni, T.

T. Soni, J. Zeidler, W. Ku, “Performance evaluation of 2-D adaptive prediction filters for detection of small objects in image data,” IEEE Trans. Image Process. 2, 327–340 (1993).
[CrossRef] [PubMed]

Sporring, J.

J. Sporring, J. Weickert, “Information measures in scale-spaces,” IEEE Trans. Inf. Theory 45, 1051–1058 (1999).
[CrossRef]

Strickland, R. N.

R. N. Strickland, H. I. Han, “Wavelet transform for detecting microcalcifications in mammograms,” IEEE Trans. Med. Imaging 15, 218–229 (1996).
[CrossRef]

Takayasu, T.

T. Takayasu, Fractals in Physical Sciences (Manchester U. Press, Manchester, UK, 1990).

Thomas, J.

T. Cover, J. Thomas, Elements of Information Theory (Wiley, New York, 1991).

Unser, M.

M. Unser, A. Aldroubi, M. Eden, “On the asymptotic convergence of B-spline wavelets to Gabor functions,” IEEE Trans. Inf. Theory 38, 864–872 (1992).
[CrossRef]

Wandell, B.

B. Wandell, Foundations of Vision (Sinauer, Sunderland, Mass., 1995).

Weickert, J.

J. Sporring, J. Weickert, “Information measures in scale-spaces,” IEEE Trans. Inf. Theory 45, 1051–1058 (1999).
[CrossRef]

White, K.

K. White, T. Hutson, T. Hutchinson, “Modeling human eye behavior during mammographic scanning: preliminary results,” IEEE Trans. Syst. Man Cybern. 27, 494–505 (1997).
[CrossRef]

Zeidler, J.

P. French, J. Zeidler, W. Ku, “Enhanced detectability of small objects in correlated clutter using an improved 2-D adaptive lattice algorithm,” IEEE Trans. Image Process. 6, 383–397 (1997).
[CrossRef]

T. Soni, J. Zeidler, W. Ku, “Performance evaluation of 2-D adaptive prediction filters for detection of small objects in image data,” IEEE Trans. Image Process. 2, 327–340 (1993).
[CrossRef] [PubMed]

Biometrika (1)

D. Donoho, I. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[CrossRef]

Electron. Lett. (1)

T. K. Bhattachary, S. Haykm, “ANN based adaptive radar detection scheme for small target in sea clutter,” Electron. Lett. 28, 1528–1536 (1992).
[CrossRef]

IEEE Trans. Image Process. (3)

T. Soni, J. Zeidler, W. Ku, “Performance evaluation of 2-D adaptive prediction filters for detection of small objects in image data,” IEEE Trans. Image Process. 2, 327–340 (1993).
[CrossRef] [PubMed]

P. French, J. Zeidler, W. Ku, “Enhanced detectability of small objects in correlated clutter using an improved 2-D adaptive lattice algorithm,” IEEE Trans. Image Process. 6, 383–397 (1997).
[CrossRef]

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. Inf. Theory (3)

J. Sporring, J. Weickert, “Information measures in scale-spaces,” IEEE Trans. Inf. Theory 45, 1051–1058 (1999).
[CrossRef]

T. Kailath, H. V. Poor, “Detection of stochastic processes,” IEEE Trans. Inf. Theory 44, 2230–2259 (1998).
[CrossRef]

M. Unser, A. Aldroubi, M. Eden, “On the asymptotic convergence of B-spline wavelets to Gabor functions,” IEEE Trans. Inf. Theory 38, 864–872 (1992).
[CrossRef]

IEEE Trans. Med. Imaging (2)

L. Huai, K. Liu, S. Lo, “Fractal modeling and segmentation for the enhancement of microcalcifications in digital mammograms,” IEEE Trans. Med. Imaging 16, 785–797 (1997).
[CrossRef]

R. N. Strickland, H. I. Han, “Wavelet transform for detecting microcalcifications in mammograms,” IEEE Trans. Med. Imaging 15, 218–229 (1996).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

M. Ferraro, G. Boccignone, T. Caelli, “On the representation of image structures via scale-space entropy condition,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1199–1203 (1999).
[CrossRef]

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

J. Kittler, M. Hatef, R. B. W. Duin, J. Matas, “On combining classifiers,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 226–239 (1998).
[CrossRef]

IEEE Trans. Signal Process. (1)

M. Shensa, “The discrete wavelet transform: wedding the à trous and Mallat algorithm,” IEEE Trans. Signal Process. 40, 2464–2482 (1992).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

K. White, T. Hutson, T. Hutchinson, “Modeling human eye behavior during mammographic scanning: preliminary results,” IEEE Trans. Syst. Man Cybern. 27, 494–505 (1997).
[CrossRef]

Int. J. Pattern Recognition Artif. Intell. (1)

N. Karssemeijer, “Adaptive noise equalization and recognition of microcalcifications clusters in mammograms,” Int. J. Pattern Recognition Artif. Intell. 7, 1357–1377 (1993).
[CrossRef]

J. IEE (1)

D. Gabor, “Theory of communication,” J. IEE 93, 429–457 (1946).

J. Opt. Soc. Am. A (1)

Med. Phys. (2)

H. Chan, L. Niklason, D. Ikeda, K. Lam, D. Adler, “Digitization requirements in mammography: effects on computer aided detection of microcalcifications,” Med. Phys. 21, 1203–1211 (1994).
[CrossRef] [PubMed]

I. Brodie, R. Gutcheck, “Radiographic information theory and application to mammography,” Med. Phys. 9, 812–820 (1982).
[CrossRef]

Other (6)

T. Cover, J. Thomas, Elements of Information Theory (Wiley, New York, 1991).

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

T. Takayasu, Fractals in Physical Sciences (Manchester U. Press, Manchester, UK, 1990).

S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1998).

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, Toronto, 1953).

B. Wandell, Foundations of Vision (Sinauer, Sunderland, Mass., 1995).

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Figures (6)

Fig. 1
Fig. 1

Example of an image (a mammogram) containing small targets, microcalcifications, embedded within an inhomogeneous background. A microcalcification is a tiny calcium deposit that has accumulated in the breast tissue, and it appears as a small bright spot in the mammogram.

Fig. 2
Fig. 2

Example of detection performed on the image represented in Fig. 1. Top image, the ground truth image; bottom image, the detection map.

Fig. 3
Fig. 3

Cluster detection performance done by varying the γ3 parameter (FROC curve).

Fig. 4
Fig. 4

Example of detection performed on the image represented in Fig. 1 but with γ3=0.56 and γ1=γ2=0.17.

Fig. 5
Fig. 5

Example of detection obtained with the discussed optimal settings. From top to bottom: the original image, the truth image, and the final map.

Fig. 6
Fig. 6

Another example of detection.

Equations (32)

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

f(n, m)=fτ (n, m)+fβ(n, m).
LSNR(dB)=10 log10k, j(n,m)×[fτ(n-k, m-j)-μβ]2/σβ2,
LCR(dB)=10 log10 N×f(n, m)/k, j(n,m)f(n-k, m-j),
Wsf(u)=1s -+f(x)ψ*x-usdx,
f(x)=ncL,n2-L/2ϕ(2-Lx-n)+j=-L2-l/2nwl,nψl,n(x),
f=l 12lkn,mf,ψl,n,mkψl,n,mk.
wlk(n, m)=wlk(n, m)τ+wlk(n, m)β.
wl(n, m)=k=13γk|wlk(n, m)|.
wl(n, m)=wl(n, m)τ+wl(n, m)β.
f, bv=fτ, bv+fβ, bv.
s=E(fτ-f˜τ2)= min(|fτ, bv|2, σ2).
f˜τ=v=0V-1t(f, bv)bv,
=E[|f, bv-t(f, bv)|2],
f˜τ=l 12lkn,mt{f,ψl,n,mk)ψl,n,mk.
xΩl[τl(x)-βl(x)]2=xΩlττl(x)2+xΩlββl(x)2-2xΩlτl(x)βl(x),
H=log xΩlτl(x)βl(x)
2Hlog xΩlττl(x)2+log xΩlββl(x)2.
IΩp(r)=11-r logxΩpp(x)r
H<-[IΩlτ (2)+IΩlβ(2)].
tl=arg max limr211-r logxΩlττl(x)r+11-r logxΩlββl(x)r.
σ2<cxΩlβl(x),
tl=σ arg mint 1c xΩlτl(x)1/2.
P(λ=Mk|x)=maxj=τ,β P(λ=Mj|x).
(1-L)P(Mk)l=1LP(λ=Mk|xl)
=maxj=τ,β(1-L)P(Mj)l=1LP(λ=Mj|xl).
Z(r, s)=x, yWs f (x, y)r,
λ(r)=lims0 log Z(r, s)log s.
rˆ(ω)=|ωx|nf^τ*(ωx).
Ws f(u)=snnxn(f * g¯s)(u),
cr[f(t)]=IΩf(r) log t,
H(f|f*)=x,yf(x, y, t) ln f(x, y, t)f(x, y),
H(f|f*)=log P--x,yf(x, y, t)log f(x, y, t).

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