On Wavelet Transform General Modulus Maxima Metric for Singularity Classification in Mammograms


Continuous wavelet transform is employed to detect singularities in 2-D signals by tracking modulus maxima along maxima lines and particularly applied to microcalcification detection in mammograms. The microcalcifications are modeled as smoothed positive impulse functions. Other target property detection can be performed by adjusting its mathematical model. In this application, the general modulus maximum and its scale of each singular point are detected and statistically analyzed locally in its neighborhood. The diagnosed microcalcification cluster results are compared with health tissue results, showing that general modulus maxima can serve as a suspicious spot detection tool with the detection performance no significantly sensitive to the breast tissue background properties. Performed fractal analysis of selected singularities supports the statistical findings. It is important to select the suitable computation parameters-thresholds of magnitude, argument and frequency range-in accordance to mathematical description of the target property as well as spatial and numerical resolution of the analyzed signal. The tests are performed on a set of images with empirically selected parameters for 200 μm/pixel spatial and 8 bits/pixel numerical resolution, appropriate for detection of the suspicious spots in a mammogram. The results show that the magnitude of a singularity general maximum can play a significant role in the detection of microcalcification, while zooming into a cluster in image finer spatial resolution both magnitude of general maximum and the spatial distribution of the selected set of singularities may lead to the breast abnormality characterization.

Share and Cite:

T. Bujanovic and I. Abdel-Qader, "On Wavelet Transform General Modulus Maxima Metric for Singularity Classification in Mammograms," Open Journal of Medical Imaging, Vol. 3 No. 1, 2013, pp. 17-30. doi: 10.4236/ojmi.2013.31004.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] S. Mallat and W. L. Hwang, “Singularity Detection and Processing with Wavelets,” IEEE Transactions on Information Theory, Vol. 38, No. 2, 1992, pp. 617-643. doi:10.1109/18.119727
[2] S. Mallat and S. Zhong, “Characterization of Signals from Multiscale Edges,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 14, No. 7, 1992, pp. 710-732. doi:10.1109/34.142909
[3] S. Mallat, “A Wavelet Tour of Signal Processing,” Academic Press, Waltham, 1999.
[4] Y. Meyer, “Un Contre-Exemple a la Conjecture de Marr et Celle de S. Mallat,” Preprint, 1991.
[5] E. B. Arneodo and J. F. Muzy, “The Thermodynamics of Fractals Revisited with Wavelets,” Physica A, Vol. 213, 1995, pp. 232-275.
[6] A. Arneodo, B. Audit, E. Bacry, S. Manneville, J. F. Muzy and S. G. Roux, “Thermodynamics of Fractal Signals Based on Wavelet Analysis: Application to Fully Developed Turbulence Data and DNA Sequences,” Physica A, Vol. 254, 1998, pp. 24-45. doi:10.1016/S0378-4371(98)00002-8
[7] N. Decoster, “Analyse Multifractale d’Images de Surfaces Rigueuses à l’aide da la Transformation en Ondolettes,” Ph.D. Thesis, L’Université de Bordeaux I, 1999. http://decoster.free.fr/phd
[8] A. Arneodo, N. Decoster and S. G. Roux, “A WaveletBased Method for Multifractal Image Analysis. I. Methodology and Test Applications on Isotropic and Anisotropic Random Rough Surfaces,” The European Physical Journal B, Vol. 15, 2000, pp. 567-600.
[9] S. Roux, J. F. Muzzy and A Arneodo, “Detecting Vorticity Filaments Using Wavelet Analysis: About the Statistical Contribution of Vorticity Filaments to Intermittency in Swirling Turbulent Flows,” The European Physical Journal B, Condensed Matter Physics, Vol. 8, No. 22, 1999, pp. 301-322.
[10] J. J. Heine and P. Malhotra, “Mammographic Tissue, Breast Cancer Risk, Serial Image Analysis, and Digital Mammography. Part 1. Tissue and Related Risk Factors,” Academic Radiology, Vol. 9, No. 3, 2002, pp. 298-316. doi:10.1016/S1076-6332(03)80373-2
[11] J. J. Heine and P. Malhotra, “Mammographic Tissue, Breast Cancer Risk, Serial Image Analysis, and Digital Mammography. Part 2. Serial Breast Tissue Change and Related Temporal Influences,” Academic Radiology, Vol. 9, No. 3, 2002, pp. 317-335. doi:10.1016/S1076-6332(03)80374-4
[12] M. Kallergi, “Computer-Aided Diagnosis of Mammographic Microcalcification Clusters,” Medical Physics, Vol. 31, No. 2, 2004, pp. 314-326.
[13] J. J. Heine, S. R. Deans, D. K. Cullers, R. Stauduhar and L. P. Clarke, “Multiresolution Statistical Analysis of HighResolution Digital Mammograms,” IEEE Transaction on Medical Imaging, Vol. 16, No. 5, 1997, pp. 503-515. doi:10.1109/42.640740
[14] T. C. Wang and N. B. Karayiannis, “Detection of Microcalcifications in Digital Mammograms Using Wavelets,” IEEE Transactions on Medical Imaging, Vol. 17, No. 4, 1998, pp. 498-509. doi:10.1109/42.730395
[15] W. Zhang, H. Yoshida, R. M. Nishikawa and K. Doi, “Optimally Weighted Wavelet Transform Based on Supervised Training for Detection of Microcalcifications in Digital Mammograms,” Medical Physics, Vol. 25, No. 6, 1998, pp. 949-956. doi:10.1118/1.598273
[16] R. N. Strickland and H. I. Hahn, “Wavelet Transforms for Detecting Microcalcifications in Mammograms,” IEEE Transactions on Medical Imaging, Vol. 15, No. 2, 1996, pp. 218-229. doi:10.1109/42.491423
[17] F. J. Gilbert, S. M. Astley, M. A. McGee, M. G. Gillan, C. R. Boggis, P. M. Griffiths and S. W. Duffy, “Single Reading with Computer-Aided Detection and Double Reading of Screening Mammograms in the United Kingdom National Breast Screening Program,” Radiology, Vol. 241, No. 1, 2006, pp. 47-53. doi:10.1148/radiol.2411051092
[18] P. Taylor, J. Champness, R. Given-Wilson, K. Johnston, and H. Potts, “Impact of Computer-Aided Detection Prompts on the Sensitivity and Specificity of Screening Mammography,” Health Technology Assessment, Vol. 9, No. 3, 2005, pp. 1-58.
[19] J. J. Fenton, S. H. Taplin, P. A. Carney, L. Abraham, E. A. Sickles, C. D’Orsi, E. A. Berns, G. Cutter, R. E. Hendrick, W. E. Barlow and J. G. Elmore, “Influence of ComputerAided Detection on Performance of Screening Mammography,” The New England Journal of Medicine, Vol. 356, 2007, pp. 1399-409. doi:10.1056/NEJMoa066099
[20] L. E. Philpotts, “Can Computer-Aided Detection Be Detrimental to Mammographic Interpretation?” Radiology, Vol. 253, No. 1, 2009, pp. 17-22.
[21] J. Tang, R. M. Rangayyan, J. Xu, I. E. Naqa and Y. Yang, “Computer-Aided Detection and Diagnosis of Breast Cancer with Mammography: Recent Advances,” IEEE Transactions on Information Technology in Biomedicine, Vol. 13, No. 2, 2009, pp. 236-251. doi:10.1109/TITB.2008.2009441
[22] J. Suckling, “The Mammographic Image Analysis Society Digital Mammogram Database,” Exerpta Medica International Congress, Vol. 1069, 1994, pp. 375-378. http://peipa.essex.ac.uk/info/mias.html
[23] B. Mandelbrot, “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,” Science, New Series, Vol. 156, No. 3775, 1967, pp. 636-638.
[24] T. Netsch and H. O. Peitgen, “Scale-Space Signatures for the Detection of Clustered Microcalcifications in Digital Mammograms,” IEEE Transactions on Medical Imaging, Vol. 18, No. 9, 1999, pp. 774-786. doi:10.1109/42.802755

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.