A common shape measure is compactness, computed by using the perimeter P and area A of a segmented region with

which quantifies how close an object is to the smoothest shape, the circle. The value of this unitless metric is minimal (4n) for a perfect circle because it is the shape that encloses a given area with the shortest perimeter. For spatially quantized circles, C can be slightly higher than 4re, such as in Fig. 1a where C = 13.6. The value of compactness increases with increasing

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FIGURE 1 Three binary regions for which compactness is (a) (b) 15.4, and (c) 27.6.

FIGURE 1 Three binary regions for which compactness is (a) (b) 15.4, and (c) 27.6.

13.6, shape complexity; for example, the region in Fig. 1b has C — 15.4. Due to this property, visual shape roughness perception often may have a good correlation with C; however, this common metric is not always a robust estimator of shape complexity. Although the elongated region in Fig. 1c is not perceived as rougher than that in Fig. 1b, it has a compactness of C — 27.6. Compactness should be used cautiously, by considering that it is simply a measure of similarity to a circle. Its advantages are computational simplicity as well as translation, rotation, and scale invariance within limits introduced by sampling and segmentation. A normalized variant C' — 1 — 4^/C ranging between zero and one also is commonly used. The values of C for benign and malignant calcifications in mammograms are illustrated in Figs. 2 and 3. Compactness has been used, for example, for quantifying calcifications [1] and breast tumors [2-4].

FIGURE 3 Distributions of normalized compactness C ' defined in Section 1.1, the radial distance metric f21 defined in section 1.3, and normalized Fourier descriptor FF defined in section 1.5, for 64 benign and 79 malignant microcalcifications in mammograms. C ' (a) and (d), f21 (b) and (e), and FF (c) and (f ) for benign and malignant microcalcifications respectively. Courtesy of L. Shen, Array Systems Computing Inc. and R. Rangayyan, University of Calgary.

FIGURE 3 Distributions of normalized compactness C ' defined in Section 1.1, the radial distance metric f21 defined in section 1.3, and normalized Fourier descriptor FF defined in section 1.5, for 64 benign and 79 malignant microcalcifications in mammograms. C ' (a) and (d), f21 (b) and (e), and FF (c) and (f ) for benign and malignant microcalcifications respectively. Courtesy of L. Shen, Array Systems Computing Inc. and R. Rangayyan, University of Calgary.

The concept of moments used to analyze statistical distributions can also be used to represent the spatial distribution of values in a 2D function [5]. Moments of a digital M by N image f (i, j) are given by

'Pq-EE^/ (i' i) P' 9 = 0' l 2' 3 ••• (2) i = 0 ; = 0

where p + q is the moment order in two dimensions. Note that moments can be computed for binary images as well as grayscale images. In binary images, moments quantify strictly the shape of the segmented region; in contrast, moments applied to gray scale images include information regarding the intensity distribution in addition to shape. Moments constitute an infinite set of transform coefficients from which /(¿, j) can be uniquely recovered. The finite number of moments used in practice do not retain all the image information, but they can provide an effective set of shape descriptors and can contribute to classification.

Translation invariance can be obtained by using central moments

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