Abstract

Multiresolution wavelet methods can be used to simplify the statistical analysis of highly correlated non-Gaussian random fields. Specific illustrations are given with the use of high-resolution digital mammograms. Fields of this kind are often difficult to analyze with parametric statistical methods. The introduction of a wavelet expansion simplifies the problem and permits the parametric analysis. The raw image is decomposed, and each expansion component is analyzed separately. The analysis applies to the individual components rather than the raw field. A suitable choice of probability function for modeling the individual components follows from the degree of detail information contained in each expansion component. The method is facilitated by using a linear operator approach. This method leads to a family of probability functions suitable for the random-field modeling of mammograms. In the limiting case the family of probability functions approaches a normal distribution. Generalizations are given that suggest possible extensions to other types of digitized image.

© 1999 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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1998 (1)

1997 (1)

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 (3)

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

V. Velanovich, “Fractal analysis of mammographic lesions: a feasibility study quantifying the difference between benign and malignant masses,” Am. J. Med. Sci. 311, 211–214 (1996).
[CrossRef] [PubMed]

J. F. Veenland, J. L. Grashuis, F. Van der Meer, A. L. D. Beckers, E. S. Gelsema, “Estimation of fractal dimension in radiographs,” Med. Phys. 23, 585–594 (1996).
[CrossRef] [PubMed]

1995 (2)

W. Qian, M. Kallergi, L. P. Clarke, H.-D. Li, P. Venugopal, D. Song, R. A. Clark, “Tree structured wavelet segmentation of microcalcifications in digital mammography,” Med. Phys. 22, 1247–1254 (1995).
[CrossRef] [PubMed]

F. Lefebvre, H. Benali, R. Gilles, E. Kahn, R. Di Paola, “A fractal approach to the segmentation of microcalcifications in digital mammograms,” Med. Phys. 22, 381–390 (1995).
[CrossRef] [PubMed]

1994 (7)

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

A. F. Laine, S. Schuler, J. Fan, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
[CrossRef] [PubMed]

C. E. Priebe, J. L. Solka, R. A. Lorey, G. W. Rogers, W. L. Poston, M. Kallergi, W. Qian, L. P. Clarke, R. A. Clark, “The application of fractal analysis to mammographic tissue classification,” Cancer Lett. 77, 183–189 (1994).
[CrossRef] [PubMed]

P. Caligiuri, M. L. Giger, M. Favus, “Multifractal radiographic analysis of osteoporosis,” Med. Phys. 21, 503–508 (1994).
[CrossRef] [PubMed]

W. Qian, L. P. Clarke, H.-D. Li, R. Clark, M. L. Silbiger, “Digital mammography: M-channel quadrature mirror filters (QMF’s) for microcalcification extraction,” Comput. Med. Imag. Graph. 18, 301–314 (1994).
[CrossRef]

Q. Huang, J. R. Lorch, R. C. Dubes, “Can the fractal dimension of images be measured?” Pattern Recogn. 27, 339–349 (1994).
[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]

1993 (1)

K. Doi, M. L. Giger, R. M. Nishikawa, K. R. Hoffmann, H. MacMahon, R. A. Schmidt, K.-G. Chura, “Digital radiography,” Acta Radiol. 34, 426–439 (1993).
[CrossRef] [PubMed]

1992 (1)

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

1990 (1)

1989 (1)

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

1988 (1)

P. Kube, A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern. Anal. Mach. Intell. 10, 704–707 (1988).
[CrossRef]

1972 (1)

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

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]

Beckers, A. L. D.

J. F. Veenland, J. L. Grashuis, F. Van der Meer, A. L. D. Beckers, E. S. Gelsema, “Estimation of fractal dimension in radiographs,” Med. Phys. 23, 585–594 (1996).
[CrossRef] [PubMed]

Benali, H.

F. Lefebvre, H. Benali, R. Gilles, E. Kahn, R. Di Paola, “A fractal approach to the segmentation of microcalcifications in digital mammograms,” Med. Phys. 22, 381–390 (1995).
[CrossRef] [PubMed]

Bialek, W.

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

Bremond, A.

L. Valatx, I. E. Magnin, A. Bremond, “Automatic microcalcification and opacities detection in digitized mammograms using a multiscale approach,” in Digital Mammography, A. G. Gale, S. M. Astley, D. R. Dance, A. Y. Cairns, eds. (Elsevier, Amsterdam, 1994).

Buckingham, M. J.

M. J. Buckingham, Noise in Electronic Devices and Systems (Wiley, New York, 1983), pp. 152–175.

Caligiuri, P.

P. Caligiuri, M. L. Giger, M. Favus, “Multifractal radiographic analysis of osteoporosis,” Med. Phys. 21, 503–508 (1994).
[CrossRef] [PubMed]

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]

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 the IEEE International Conference on Acoustics, Speech, and Signal Processing ’93 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1993), Vol. IV, pp. 93–96.

K. C. Chou, S. Golden, A. S. Willsky, “Modeling and estimation of multiscale stochastic processes,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing ’91 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1991), Vol. 3D, pp. 1709–1712.

Chura, K.-G.

K. Doi, M. L. Giger, R. M. Nishikawa, K. R. Hoffmann, H. MacMahon, R. A. Schmidt, K.-G. Chura, “Digital radiography,” Acta Radiol. 34, 426–439 (1993).
[CrossRef] [PubMed]

Clark, R.

W. Qian, L. P. Clarke, H.-D. Li, R. Clark, M. L. Silbiger, “Digital mammography: M-channel quadrature mirror filters (QMF’s) for microcalcification extraction,” Comput. Med. Imag. Graph. 18, 301–314 (1994).
[CrossRef]

Clark, R. A.

W. Qian, M. Kallergi, L. P. Clarke, H.-D. Li, P. Venugopal, D. Song, R. A. Clark, “Tree structured wavelet segmentation of microcalcifications in digital mammography,” Med. Phys. 22, 1247–1254 (1995).
[CrossRef] [PubMed]

C. E. Priebe, J. L. Solka, R. A. Lorey, G. W. Rogers, W. L. Poston, M. Kallergi, W. Qian, L. P. Clarke, R. A. Clark, “The application of fractal analysis to mammographic tissue classification,” Cancer Lett. 77, 183–189 (1994).
[CrossRef] [PubMed]

Clarke, L. P.

J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar, L. P. Clarke, “Multiresolution probability analysis of gray-scaled images,” J. Opt. Soc. Am. A 15, 1048–1058 (1998).
[CrossRef]

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]

W. Qian, M. Kallergi, L. P. Clarke, H.-D. Li, P. Venugopal, D. Song, R. A. Clark, “Tree structured wavelet segmentation of microcalcifications in digital mammography,” Med. Phys. 22, 1247–1254 (1995).
[CrossRef] [PubMed]

W. Qian, L. P. Clarke, H.-D. Li, R. Clark, M. L. Silbiger, “Digital mammography: M-channel quadrature mirror filters (QMF’s) for microcalcification extraction,” Comput. Med. Imag. Graph. 18, 301–314 (1994).
[CrossRef]

C. E. Priebe, J. L. Solka, R. A. Lorey, G. W. Rogers, W. L. Poston, M. Kallergi, W. Qian, L. P. Clarke, R. A. Clark, “The application of fractal analysis to mammographic tissue classification,” Cancer Lett. 77, 183–189 (1994).
[CrossRef] [PubMed]

J. J. Heine, S. R. Deans, R. P. Velthuizen, L. P. Clarke, “On the statistical nature of mammograms,” Med. Phys.25, (to be published).

Cullers, D. K.

J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar, L. P. Clarke, “Multiresolution probability analysis of gray-scaled images,” J. Opt. Soc. Am. A 15, 1048–1058 (1998).
[CrossRef]

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 probability analysis of gray-scaled images,” J. Opt. Soc. Am. A 15, 1048–1058 (1998).
[CrossRef]

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, S. R. Deans, R. P. Velthuizen, L. P. Clarke, “On the statistical nature of mammograms,” Med. Phys.25, (to be published).

Devore, R. A.

R. A. Devore, B. Lucier, Z. Yang, “Feature extraction in digital mammography,” in Wavelets in Medicine and Biology, A. Aldroubi, M. Unser, eds. (CRC Press, Boca Raton, Fla., 1996), pp. 145–161.

Di Paola, R.

F. Lefebvre, H. Benali, R. Gilles, E. Kahn, R. Di Paola, “A fractal approach to the segmentation of microcalcifications in digital mammograms,” Med. Phys. 22, 381–390 (1995).
[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 the IEEE International Conference on Acoustics, Speech, and Signal Processing ’93 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1993), Vol. IV, pp. 93–96.

Doi, K.

K. Doi, M. L. Giger, R. M. Nishikawa, K. R. Hoffmann, H. MacMahon, R. A. Schmidt, K.-G. Chura, “Digital radiography,” Acta Radiol. 34, 426–439 (1993).
[CrossRef] [PubMed]

Dubes, R. C.

Q. Huang, J. R. Lorch, R. C. Dubes, “Can the fractal dimension of images be measured?” Pattern Recogn. 27, 339–349 (1994).
[CrossRef]

Fan, J.

A. F. Laine, S. Schuler, J. Fan, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
[CrossRef] [PubMed]

J. Fan, A. Laine, “Multiscale contrast enhancement and denoising in digital radiographs,” in Wavelets in Medicine and Biology, A. Aldroubi, M. Unser, eds. (CRC Press, Boca Raton, Fla., 1996), pp. 163–189.

Favus, M.

P. Caligiuri, M. L. Giger, M. Favus, “Multifractal radiographic analysis of osteoporosis,” Med. Phys. 21, 503–508 (1994).
[CrossRef] [PubMed]

Gelsema, E. S.

J. F. Veenland, J. L. Grashuis, F. Van der Meer, A. L. D. Beckers, E. S. Gelsema, “Estimation of fractal dimension in radiographs,” Med. Phys. 23, 585–594 (1996).
[CrossRef] [PubMed]

Giger, M. L.

P. Caligiuri, M. L. Giger, M. Favus, “Multifractal radiographic analysis of osteoporosis,” Med. Phys. 21, 503–508 (1994).
[CrossRef] [PubMed]

K. Doi, M. L. Giger, R. M. Nishikawa, K. R. Hoffmann, H. MacMahon, R. A. Schmidt, K.-G. Chura, “Digital radiography,” Acta Radiol. 34, 426–439 (1993).
[CrossRef] [PubMed]

Gilles, R.

F. Lefebvre, H. Benali, R. Gilles, E. Kahn, R. Di Paola, “A fractal approach to the segmentation of microcalcifications in digital mammograms,” Med. Phys. 22, 381–390 (1995).
[CrossRef] [PubMed]

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 (1953/1954).

Golden, S.

K. C. Chou, S. Golden, A. S. Willsky, “Modeling and estimation of multiscale stochastic processes,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing ’91 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1991), Vol. 3D, pp. 1709–1712.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, Boston, 1994), Eqs. 3.737 1, 3.771 2, and 3.754 2.

Grashuis, J. L.

J. F. Veenland, J. L. Grashuis, F. Van der Meer, A. L. D. Beckers, E. S. Gelsema, “Estimation of fractal dimension in radiographs,” Med. Phys. 23, 585–594 (1996).
[CrossRef] [PubMed]

Hann, H. I.

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

Heine, J. J.

J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar, L. P. Clarke, “Multiresolution probability analysis of gray-scaled images,” J. Opt. Soc. Am. A 15, 1048–1058 (1998).
[CrossRef]

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, S. R. Deans, R. P. Velthuizen, L. P. Clarke, “On the statistical nature of mammograms,” Med. Phys.25, (to be published).

Hoffmann, K. R.

K. Doi, M. L. Giger, R. M. Nishikawa, K. R. Hoffmann, H. MacMahon, R. A. Schmidt, K.-G. Chura, “Digital radiography,” Acta Radiol. 34, 426–439 (1993).
[CrossRef] [PubMed]

Huang, Q.

Q. Huang, J. R. Lorch, R. C. Dubes, “Can the fractal dimension of images be measured?” Pattern Recogn. 27, 339–349 (1994).
[CrossRef]

Huda, W.

A. F. Laine, S. Schuler, J. Fan, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
[CrossRef] [PubMed]

Judy, P. F.

P. F. Judy, “Detection of clusters of simulated calcifications in lumpy noise backgrounds,” in Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 39–46 (1996).
[CrossRef]

Kahn, E.

F. Lefebvre, H. Benali, R. Gilles, E. Kahn, R. Di Paola, “A fractal approach to the segmentation of microcalcifications in digital mammograms,” Med. Phys. 22, 381–390 (1995).
[CrossRef] [PubMed]

Kallergi, M.

W. Qian, M. Kallergi, L. P. Clarke, H.-D. Li, P. Venugopal, D. Song, R. A. Clark, “Tree structured wavelet segmentation of microcalcifications in digital mammography,” Med. Phys. 22, 1247–1254 (1995).
[CrossRef] [PubMed]

C. E. Priebe, J. L. Solka, R. A. Lorey, G. W. Rogers, W. L. Poston, M. Kallergi, W. Qian, L. P. Clarke, R. A. Clark, “The application of fractal analysis to mammographic tissue classification,” Cancer Lett. 77, 183–189 (1994).
[CrossRef] [PubMed]

Kube, P.

P. Kube, A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern. Anal. Mach. Intell. 10, 704–707 (1988).
[CrossRef]

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]

Laine, A.

J. Fan, A. Laine, “Multiscale contrast enhancement and denoising in digital radiographs,” in Wavelets in Medicine and Biology, A. Aldroubi, M. Unser, eds. (CRC Press, Boca Raton, Fla., 1996), pp. 163–189.

Laine, A. F.

A. F. Laine, S. Schuler, J. Fan, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
[CrossRef] [PubMed]

Lefebvre, F.

F. Lefebvre, H. Benali, R. Gilles, E. Kahn, R. Di Paola, “A fractal approach to the segmentation of microcalcifications in digital mammograms,” Med. Phys. 22, 381–390 (1995).
[CrossRef] [PubMed]

Li, H.-D.

W. Qian, M. Kallergi, L. P. Clarke, H.-D. Li, P. Venugopal, D. Song, R. A. Clark, “Tree structured wavelet segmentation of microcalcifications in digital mammography,” Med. Phys. 22, 1247–1254 (1995).
[CrossRef] [PubMed]

W. Qian, L. P. Clarke, H.-D. Li, R. Clark, M. L. Silbiger, “Digital mammography: M-channel quadrature mirror filters (QMF’s) for microcalcification extraction,” Comput. Med. Imag. Graph. 18, 301–314 (1994).
[CrossRef]

Lorch, J. R.

Q. Huang, J. R. Lorch, R. C. Dubes, “Can the fractal dimension of images be measured?” Pattern Recogn. 27, 339–349 (1994).
[CrossRef]

Lorey, R. A.

C. E. Priebe, J. L. Solka, R. A. Lorey, G. W. Rogers, W. L. Poston, M. Kallergi, W. Qian, L. P. Clarke, R. A. Clark, “The application of fractal analysis to mammographic tissue classification,” Cancer Lett. 77, 183–189 (1994).
[CrossRef] [PubMed]

Lucier, B.

R. A. Devore, B. Lucier, Z. Yang, “Feature extraction in digital mammography,” in Wavelets in Medicine and Biology, A. Aldroubi, M. Unser, eds. (CRC Press, Boca Raton, Fla., 1996), pp. 145–161.

MacMahon, H.

K. Doi, M. L. Giger, R. M. Nishikawa, K. R. Hoffmann, H. MacMahon, R. A. Schmidt, K.-G. Chura, “Digital radiography,” Acta Radiol. 34, 426–439 (1993).
[CrossRef] [PubMed]

Magnin, I. E.

L. Valatx, I. E. Magnin, A. Bremond, “Automatic microcalcification and opacities detection in digitized mammograms using a multiscale approach,” in Digital Mammography, A. G. Gale, S. M. Astley, D. R. Dance, A. Y. Cairns, eds. (Elsevier, Amsterdam, 1994).

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]

Nishikawa, R. M.

K. Doi, M. L. Giger, R. M. Nishikawa, K. R. Hoffmann, H. MacMahon, R. A. Schmidt, K.-G. Chura, “Digital radiography,” Acta Radiol. 34, 426–439 (1993).
[CrossRef] [PubMed]

Peli, T.

Pentland, A.

P. Kube, A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern. Anal. Mach. Intell. 10, 704–707 (1988).
[CrossRef]

Poston, W. L.

C. E. Priebe, J. L. Solka, R. A. Lorey, G. W. Rogers, W. L. Poston, M. Kallergi, W. Qian, L. P. Clarke, R. A. Clark, “The application of fractal analysis to mammographic tissue classification,” Cancer Lett. 77, 183–189 (1994).
[CrossRef] [PubMed]

Priebe, C. E.

C. E. Priebe, J. L. Solka, R. A. Lorey, G. W. Rogers, W. L. Poston, M. Kallergi, W. Qian, L. P. Clarke, R. A. Clark, “The application of fractal analysis to mammographic tissue classification,” Cancer Lett. 77, 183–189 (1994).
[CrossRef] [PubMed]

Qian, W.

W. Qian, M. Kallergi, L. P. Clarke, H.-D. Li, P. Venugopal, D. Song, R. A. Clark, “Tree structured wavelet segmentation of microcalcifications in digital mammography,” Med. Phys. 22, 1247–1254 (1995).
[CrossRef] [PubMed]

W. Qian, L. P. Clarke, H.-D. Li, R. Clark, M. L. Silbiger, “Digital mammography: M-channel quadrature mirror filters (QMF’s) for microcalcification extraction,” Comput. Med. Imag. Graph. 18, 301–314 (1994).
[CrossRef]

C. E. Priebe, J. L. Solka, R. A. Lorey, G. W. Rogers, W. L. Poston, M. Kallergi, W. Qian, L. P. Clarke, R. A. Clark, “The application of fractal analysis to mammographic tissue classification,” Cancer Lett. 77, 183–189 (1994).
[CrossRef] [PubMed]

Rogers, G. W.

C. E. Priebe, J. L. Solka, R. A. Lorey, G. W. Rogers, W. L. Poston, M. Kallergi, W. Qian, L. P. Clarke, R. A. Clark, “The application of fractal analysis to mammographic tissue classification,” Cancer Lett. 77, 183–189 (1994).
[CrossRef] [PubMed]

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]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, Boston, 1994), Eqs. 3.737 1, 3.771 2, and 3.754 2.

Schmidt, R. A.

K. Doi, M. L. Giger, R. M. Nishikawa, K. R. Hoffmann, H. MacMahon, R. A. Schmidt, K.-G. Chura, “Digital radiography,” Acta Radiol. 34, 426–439 (1993).
[CrossRef] [PubMed]

Schroeder, M.

M. Schroeder, Fractals, Chaos, Power Laws (Freeman, New York, 1991).

Schuler, S.

A. F. Laine, S. Schuler, J. Fan, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
[CrossRef] [PubMed]

Silbiger, M. L.

W. Qian, L. P. Clarke, H.-D. Li, R. Clark, M. L. Silbiger, “Digital mammography: M-channel quadrature mirror filters (QMF’s) for microcalcification extraction,” Comput. Med. Imag. Graph. 18, 301–314 (1994).
[CrossRef]

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]

Solka, J. L.

C. E. Priebe, J. L. Solka, R. A. Lorey, G. W. Rogers, W. L. Poston, M. Kallergi, W. Qian, L. P. Clarke, R. A. Clark, “The application of fractal analysis to mammographic tissue classification,” Cancer Lett. 77, 183–189 (1994).
[CrossRef] [PubMed]

Song, D.

W. Qian, M. Kallergi, L. P. Clarke, H.-D. Li, P. Venugopal, D. Song, R. A. Clark, “Tree structured wavelet segmentation of microcalcifications in digital mammography,” Med. Phys. 22, 1247–1254 (1995).
[CrossRef] [PubMed]

Stauduhar, R.

J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar, L. P. Clarke, “Multiresolution probability analysis of gray-scaled images,” J. Opt. Soc. Am. A 15, 1048–1058 (1998).
[CrossRef]

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]

Strickland, R. N.

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

Valatx, L.

L. Valatx, I. E. Magnin, A. Bremond, “Automatic microcalcification and opacities detection in digitized mammograms using a multiscale approach,” in Digital Mammography, A. G. Gale, S. M. Astley, D. R. Dance, A. Y. Cairns, eds. (Elsevier, Amsterdam, 1994).

Van der Meer, F.

J. F. Veenland, J. L. Grashuis, F. Van der Meer, A. L. D. Beckers, E. S. Gelsema, “Estimation of fractal dimension in radiographs,” Med. Phys. 23, 585–594 (1996).
[CrossRef] [PubMed]

Veenland, J. F.

J. F. Veenland, J. L. Grashuis, F. Van der Meer, A. L. D. Beckers, E. S. Gelsema, “Estimation of fractal dimension in radiographs,” Med. Phys. 23, 585–594 (1996).
[CrossRef] [PubMed]

Velanovich, V.

V. Velanovich, “Fractal analysis of mammographic lesions: a feasibility study quantifying the difference between benign and malignant masses,” Am. J. Med. Sci. 311, 211–214 (1996).
[CrossRef] [PubMed]

Velthuizen, R. P.

J. J. Heine, S. R. Deans, R. P. Velthuizen, L. P. Clarke, “On the statistical nature of mammograms,” Med. Phys.25, (to be published).

Venugopal, P.

W. Qian, M. Kallergi, L. P. Clarke, H.-D. Li, P. Venugopal, D. Song, R. A. Clark, “Tree structured wavelet segmentation of microcalcifications in digital mammography,” Med. Phys. 22, 1247–1254 (1995).
[CrossRef] [PubMed]

Willsky, A. S.

K. C. Chou, S. Golden, A. S. Willsky, “Modeling and estimation of multiscale stochastic processes,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing ’91 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1991), Vol. 3D, pp. 1709–1712.

Wornell, G.

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

Yang, Z.

R. A. Devore, B. Lucier, Z. Yang, “Feature extraction in digital mammography,” in Wavelets in Medicine and Biology, A. Aldroubi, M. Unser, eds. (CRC Press, Boca Raton, Fla., 1996), pp. 145–161.

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 (1953/1954).

Acta Radiol. (1)

K. Doi, M. L. Giger, R. M. Nishikawa, K. R. Hoffmann, H. MacMahon, R. A. Schmidt, K.-G. Chura, “Digital radiography,” Acta Radiol. 34, 426–439 (1993).
[CrossRef] [PubMed]

Am. J. Med. Sci. (1)

V. Velanovich, “Fractal analysis of mammographic lesions: a feasibility study quantifying the difference between benign and malignant masses,” Am. J. Med. Sci. 311, 211–214 (1996).
[CrossRef] [PubMed]

Cancer Lett. (1)

C. E. Priebe, J. L. Solka, R. A. Lorey, G. W. Rogers, W. L. Poston, M. Kallergi, W. Qian, L. P. Clarke, R. A. Clark, “The application of fractal analysis to mammographic tissue classification,” Cancer Lett. 77, 183–189 (1994).
[CrossRef] [PubMed]

Comput. Med. Imag. Graph. (1)

W. Qian, L. P. Clarke, H.-D. Li, R. Clark, M. L. Silbiger, “Digital mammography: M-channel quadrature mirror filters (QMF’s) for microcalcification extraction,” Comput. Med. Imag. Graph. 18, 301–314 (1994).
[CrossRef]

IEEE Trans. Comput. (1)

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. (1)

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 (3)

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]

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

A. F. Laine, S. Schuler, J. Fan, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
[CrossRef] [PubMed]

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

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]

P. Kube, A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern. Anal. Mach. Intell. 10, 704–707 (1988).
[CrossRef]

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

IRE Trans. Inf. Theory (1)

B. Gold, G. O. Young, “The response of linear systems to non-Gaussian noise,” IRE Trans. Inf. Theory PGIT 2–4, 63–67 (1953/1954).

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

Med. Phys. (4)

W. Qian, M. Kallergi, L. P. Clarke, H.-D. Li, P. Venugopal, D. Song, R. A. Clark, “Tree structured wavelet segmentation of microcalcifications in digital mammography,” Med. Phys. 22, 1247–1254 (1995).
[CrossRef] [PubMed]

J. F. Veenland, J. L. Grashuis, F. Van der Meer, A. L. D. Beckers, E. S. Gelsema, “Estimation of fractal dimension in radiographs,” Med. Phys. 23, 585–594 (1996).
[CrossRef] [PubMed]

P. Caligiuri, M. L. Giger, M. Favus, “Multifractal radiographic analysis of osteoporosis,” Med. Phys. 21, 503–508 (1994).
[CrossRef] [PubMed]

F. Lefebvre, H. Benali, R. Gilles, E. Kahn, R. Di Paola, “A fractal approach to the segmentation of microcalcifications in digital mammograms,” Med. Phys. 22, 381–390 (1995).
[CrossRef] [PubMed]

Pattern Recogn. (1)

Q. Huang, J. R. Lorch, R. C. Dubes, “Can the fractal dimension of images be measured?” Pattern Recogn. 27, 339–349 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

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

Other (12)

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

M. Schroeder, Fractals, Chaos, Power Laws (Freeman, New York, 1991).

M. J. Buckingham, Noise in Electronic Devices and Systems (Wiley, New York, 1983), pp. 152–175.

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

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, Boston, 1994), Eqs. 3.737 1, 3.771 2, and 3.754 2.

K. C. Chou, S. Golden, A. S. Willsky, “Modeling and estimation of multiscale stochastic processes,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing ’91 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1991), Vol. 3D, 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 the IEEE International Conference on Acoustics, Speech, and Signal Processing ’93 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1993), Vol. IV, pp. 93–96.

J. J. Heine, S. R. Deans, R. P. Velthuizen, L. P. Clarke, “On the statistical nature of mammograms,” Med. Phys.25, (to be published).

J. Fan, A. Laine, “Multiscale contrast enhancement and denoising in digital radiographs,” in Wavelets in Medicine and Biology, A. Aldroubi, M. Unser, eds. (CRC Press, Boca Raton, Fla., 1996), pp. 163–189.

L. Valatx, I. E. Magnin, A. Bremond, “Automatic microcalcification and opacities detection in digitized mammograms using a multiscale approach,” in Digital Mammography, A. G. Gale, S. M. Astley, D. R. Dance, A. Y. Cairns, eds. (Elsevier, Amsterdam, 1994).

P. F. Judy, “Detection of clusters of simulated calcifications in lumpy noise backgrounds,” in Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 39–46 (1996).
[CrossRef]

R. A. Devore, B. Lucier, Z. Yang, “Feature extraction in digital mammography,” in Wavelets in Medicine and Biology, A. Aldroubi, M. Unser, eds. (CRC Press, Boca Raton, Fla., 1996), pp. 145–161.

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Figures (5)

Fig. 1
Fig. 1

Raw f0 images, represented by a 256×256-pixel ROI, and associated empirical histograms for cases 1, 2, and 3 (top to bottom). Note the nonparametric form of the histograms and the lumpy image appearance.

Fig. 2
Fig. 2

f1 images for cases 1, 2, and 3 (top to bottom), represented by a 256×256-pixel ROI, and associated empirical histograms. Note that the images and the associated histograms are approximately identical to those shown in Fig. 1 for each case.

Fig. 3
Fig. 3

Case 1: detail image representation with associated empirical histograms (solid curves) and theoretical pdf’s (crosses) for the d1 to d4 images (a)–(d). The images are represented with a 256×256-pixel ROI. The empirical histograms are to be compared with the theoretical estimates. Points have been skipped on the theoretical plots to avoid confusion. This represents a case where all the theoretical pdf’s correspond to N=2 for all four expansion components.

Fig. 4
Fig. 4

Case 2: detail image representation with associated empirical histograms (solid curves) and theoretical pdf’s (crosses) for the d1 to d4 images (top to bottom). The images are represented with a 256×256-pixel ROI. The empirical histograms are to be compared with the theoretical estimates. Points have been skipped on the theoretical plots to avoid confusion. This represents a case where the theoretical pdf corresponds to N=4 for the d1 component and N=3 for the d2d4 expansion components.

Fig. 5
Fig. 5

Case 3: detail image representation with associated empirical histograms (solid curves) and theoretical pdf’s (crosses) for the d1 through d4 images (top to bottom). The images are represented with a 256×256-pixel ROI. The empirical histograms are to be compared with the theoretical estimates. Points have been skipped on the theoretical plots to avoid confusion. This represents a case where the theoretical pdf’s correspond to N=4 for all four expansion components.

Equations (16)

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

f0=d1+d2+d3++dJ+fJ,
f0=d1+f1
s0=s1*p1.
s1(x, τ2)1+τ ddx1-τ ddxs0(x)=1-τ2 d2dx2s0(x)=s0(x)-τ2s0(x),
s0(x)[s0(x)-τ2s0(x)]*p(x).
S0(ω)=[S0(ω)-(iτω)2S0(ω)]P(ω),
P(ω)=11+τ2ω2.
Dop(τ2)xm=xm-τ2m(m-1)xm-2,
Dopxm=xm-τ2m(m-1)xm-2.
s0(x, τ2)=Dopτ2NNs0(x)*p(x)=1-τ2Nd2dx2Ns0(x)*p(x).
P(ω)=11+τ2ω2NN.
p(x)=k exp(-k|x|)22N-1(N-1)!l=0N-1 (2N-l-2)!2lkl|x|ll!(N-l-1)!,
P(ω)=limN 11+τ2ω2NN=exp(-τ2ω2).
[p(x)]N=14πτ2 exp-x24τ2.
p(x)=β-ν+1π3/22|x|ν cos(πν)Γν+12k-ν(β|x|),
p(x)=βπk0(β|x|).

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