We now describe a fuzzy region growing algorithm to obtain mass regions in mam-mograms. An adaptive similarity criterion is used for region growing, with the mean and the standard deviation of the pixels in the region being grown as control parameters. The region is represented by a fuzzy set to preserve the transition information around boundary regions.

The algorithm starts with a seed pixel or region that lies inside the ROI and spreads by adding to the region 8-connected pixels that have similar properties.

Fig. 6. Original version of a 1,024 X 1,024-pixel portion of a mammogram with a benign mass. Pixel size = 50 ^m. Reproduced with permission from D. Guliato, R. M. Rangayyan, W. A. Carnielli, J. A. Zuffo, and J. E. L. Desautels, "Segmentation of breast tumors in mam-mograms using fuzzy sets", Journal of Electronic Imaging, 12(3): 369-378, 2003. SPIE and IS&T.

Fig. 6. Original version of a 1,024 X 1,024-pixel portion of a mammogram with a benign mass. Pixel size = 50 ^m. Reproduced with permission from D. Guliato, R. M. Rangayyan, W. A. Carnielli, J. A. Zuffo, and J. E. L. Desautels, "Segmentation of breast tumors in mam-mograms using fuzzy sets", Journal of Electronic Imaging, 12(3): 369-378, 2003. SPIE and IS&T.

Fig. 8. The white line shows the contour extracted by the closed-contour detection method for the mammogram in Fig. 6. The black line represents the boundary drawn by a radiologist (shown for comparison). Reproduced with permission from D. Guliato, R. M. Rangayyan, W. A. Carnielli, J. A. Zuffo, and J. E. L. Desautels, "Segmentation of breast tumors in mammograms using fuzzy sets", Journal of Electronic Imaging, 12(3): 369-378, 2003. © SPIE and IS&T.

Fig. 8. The white line shows the contour extracted by the closed-contour detection method for the mammogram in Fig. 6. The black line represents the boundary drawn by a radiologist (shown for comparison). Reproduced with permission from D. Guliato, R. M. Rangayyan, W. A. Carnielli, J. A. Zuffo, and J. E. L. Desautels, "Segmentation of breast tumors in mammograms using fuzzy sets", Journal of Electronic Imaging, 12(3): 369-378, 2003. © SPIE and IS&T.

The homogeneity of the region is evaluated by calculating the mean (p), standard deviation (a), and the coefficient of variation CV = a/p. CV gives a measure of inhomogeneity of the region, and allows one to compare different regions.

Let Apmax, ACVmx, and 3 be the control parameters for region growing. Apmax specifies the maximum allowed difference between the value of the pixel being analyzed and the mean of the subregion already grown. ACVmax indicates the desired degree of homogeneity between two subregions. 3 defines the opening of the membership function.

Let p be the next pixel to be analyzed and I(p) be the value of p. Let p and a be the mean and the standard deviation of the region already grown. The segmentation algorithm is executed in two steps:

If this condition is not satisfied, then the pixel is labelled as rejected. If the condition is satisfied, p is temporarily added to the subregion and ^new and anew are calculated.

If the condition is satisfied, then p must definitely be added to the subregion labelled as accepted, and p and a must be updated, i.e. p = pnew and a = anew.

If the condition is not satisfied, p is added to the subregion with the label accepted with restriction, and ^ and a are not modified.

Comparison 2 given above analyzes the distortion (in terms of CV) that the pixel p can produce if added to the subregion. At the beginning of the procedure, the region includes all the pixels in the seed region, and the standard deviation is set to zero. While the standard deviation of the region being grown is zero, a special procedure is executed for Comparison 2: |a/^ — anew/^,new| < 2ACVmax. The parameter ACVmax works as a filter that avoids the possibility that the mean and standard deviation measures suffer undesirable modification during the region growing procedure. Furthermore, the algorithm processes pixels in expanding concentric squares around the seed region, evaluating each pixel only once; these steps provide stability to the algorithm.

We now define a fuzzy membership function that maps the pixel values of the region resulting from the procedure described above to the unit interval [0,1] based upon the mean of the region. Pixels that are close to the mean will have a high membership degree, and, in the opposite case, a low membership degree. The desirable characteristics of the membership function are:

• the membership degree of the seed pixel or seed region must be 1;

• the membership degree of a pixel labelled as rejected must be 0;

• the membership function must be as independent of the seed pixel or region as possible;

• the membership degree must represent the proximity between a pixel labelled as accepted or accepted with restriction and the mean of the resulting region;

• the function must be symmetric with respect to the difference between the mean and the pixel value; and

• the function must decrease monotonically from 1 to 0.

The membership function r used in the present work is illustrated in Fig. 9, where a = \meanseedjregion + standard-deviationseedjregion — fj,\ and b = AyU,max; in this expression, ^ is the mean of the region obtained by the preceding region growing procedure. The value of a pixel p is mapped to the fuzzy membership degree r(p) as follows:

if |1 (p) — a then r(p) = 1 else if |1 (p) — ^ > b then r(p) = 0 else r(p) = 1/{1 + £[|/(p) — M|]}.

The method has been tested on several synthetic images with various levels of noise. Figure 10 illustrates three representative results of the method with a synthetic image and different seed pixels. The results do not differ significantly, indicating the low effect of noise on the method.

The fuzzy region provided by the method for the malignant tumor case shown in Fig. 2 is illustrated in Fig. 11. Figure 12 shows the fuzzy region obtained for the membership degree

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