N

where imax imax

is the mean of the distribution.

If the normalized third moment is used, m, m23/2 '

a unit-free measure is obtained. This normalization to the 3/2 power of m2 essentially removes the effect of the variance (the spread of the histogram) from the measure, while preserving the skewness information. This facilitates comparisons of different distributions and distributions of different variability.

To understand how the measure of skewness defined in Eqs. (5) and (6) would work on a mammogram, consider that, a priori, the distribution hi associated with a region of fibro-glandular tissue should tend toward higher gray levels (i.e., whiter). The tail of values below the mean (i<i) yields large negative contributions (i —i)3, which would result in a negative skewness (S<0). Similarly, a region of fatty tissue should yield a distribution that tends toward lower gray levels, so that a positive skewness (S>0) would be measured.

The skewness measurement is more sensitive to local tissue composition when it is calculated for individual small regions of the breast with a local measure of (ii) than when a single global measurement is used. A regional characterization can be obtained by dividing the projection of the breast into nonoverlapping square areas. For each region, the skewness can be calculated according to Eq. (6) and averaged over all regions to obtain a single number, Sr, for the image.

A nondense breast should have a more positive measure of Sr as the majority of local regions reflect predominately fatty tissue (each with more positive skewness). As the breast becomes more fibroglandular, a more negative Sr should be measured because a greater proportion of regions have more negative skewness. We investigated the effect of the region size on the measure of Sr and found that with decreasing region size a stronger trend between the measure and mam-mographic density was obtained, although the range of the measure also decreased [19]. A region size of 24 x 24 pixels (3.12 mm x 3.12 mm) was chosen as a compromise between regions small enough to represent local tissue patterns and large enough to provide a statistically meaningful result. In a study of 60 mammograms chosen to span the full range of the SCC measure, the regional skewness measurement ranged between — 0.27 and 0.19. The correlation between Sr and SCC was very good (Spearman — 0.88).

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