Open Journal of Medical Imaging, 2013, 3, 17-30 http://dx.doi.org/10.4236/ojmi.2013.31004 Published Online March 2013 (http://www.scirp.org/journal/ojmi) On Wavelet Transform General Modulus Maxima Metric for Singularity Classification in Mammograms Tomislav Bujanovic 1 , Ikhlas Abdel-Qader 2 1 Department of Electrical Engineering & Computer Science, Syracuse University, Syracuse, USA 2 Department of Electrical & Computer Engineering, Western Michigan University, Kalamazoo, USA Email: tbujanov@syr.edu, ikhlas.abdelqader@wmich.edu Received January 15, 2013; revised February 19, 2013; accepted February 28, 2013 ABSTRACT 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 prop- erty 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 resolu- tion both magnitude of general maximum and the spatial distribution of the selected set of singularities may lead to the breast abnormality characterization. Keywords: Continuous Wavelet Transform; Fractal Dimension; General Modulus Maximum; Microcalcification; Singularity; Smoothed Impulse Function 1. Introduction Wavelet transform modulus maxima method was devel- oped for detection and characterization of signal singu- larities by Mallat and his collaborators [1-3]. Their me- thod detects signal singularities by tracking the wavelet coefficients magnitude maximum over scale. They proved that, if a wavelet function is derivative of a Gaussian, wavelet transform modulus maxima must propagate to- wards finer scales. Although the representation by dis- crete wavelet maxima is not complete since several sig- nals may exhibit the same wavelet maxima [4], Mallat’s numerical experiments have shown that it is possible to reconstruct signals with a relatively small mean square error (smaller than 1%) [3]. Arneodo and his team focused on how to recognize a sharp signal transition by tracking its behavior over scale. Specifically algorithms based on continuous wavelet transform modulus maxima method are able to detect singular points in a discrete 2-D and 3-D signals and supported by fractal analysis to give the metrics for the local signal regularity [5-9]. Arneodo’s team developed fractal based algorithm [8] supported by modulus maxi- ma method to analyze turbulent 2-D and 3-D signals. Microcalcifications in breast are residual calcium de- posits that originate not only from completely normal processes, but also abnormal ones. Shape, morphology, and spatial distribution of individual microcalcifications are some of the features detectable in X-ray mammo- grams that suggest benign or malignant breast abnormal- ity. Researchers have made few breast tissue classifica- tions and have recognized more than twenty of those conventional features [10-12]. One acceptable simplifi- cation is to describe microcalcifications as ellipsoids of diameters between 0.05 mm and 1 mm. For early cancer detection, calcifications with spatial extent less than 0.5 mm are most important for clinical diagnosis. Particu- larly, this corresponds to calcifications roughly in the order of 0.1 mm in diameter. Also, microcalcifications appearing in clusters may suggest malignancy while in- dividual occurrences are of low clinical significance [13]. Copyright © 2013 SciRes. OJMI