The centralized GMs of some simple geometric shapes were shown in
Figure 1a to illustrate that low-order GMs were directly associated with perceivable global properties of 2-D images. For binary images (
f(
x, y) = 0 or 1) in
Figure 1a, the 0th-order GM is the number of ink pixels, which can be perceived as the general lightness or darkness of a letter. In
Figure 1a, patterns with more strokes have a larger
m 0,0 than those with fewer strokes and appear darker if individual strokes cannot be distinguished. Pure 2nd-order moments (moments that are 0-order in one direction)
m 2,0 and
m 0,2 are dispersion of ink distribution. For the vertical bar in
Figure 1a (height/width = 5/1),
m 2,0 <
m 0,2 (2.872 vs. 14.431). For the horizontal bar,
m 2,0 >
m 2,0 (14.431 vs. 2.872). In fact, when only GMs up to the second order are considered, the original image is completely equivalent to a constant irradiance ellipse, whose size, orientation, aspect ratio, and center are completely specified by the GMs (Teague,
1980). Pure 3rd-order moments
m3,0 and
m0,3 represent the skewness of ink distributions on the
x- and
y-directions. For the letter “
E,”
m3,0 has a positive value (0.18), but
m0,3 is zero. Perceptually, the letter appears darker on the left side (skewed to the left), but appears symmetric in the vertical direction. The distribution of letter “
L” is heavily skewed to the left in the horizontal direction and to the bottom in the vertical direction (
m3,0 = 0.85,
m0,3 = −0.85). Notice that the skewness is both directional specific (
m2,0 vs.
m0,2) and positional specific (positive vs. negative). Pure 4th-order moments
m4,0 and
m0,4 specify whether ink distributions on the
x- and
y-directions are more peaked or flat-topped than a Gaussian. In letter “
H”, the distribution along
x-axis is the lowest in the middle, due to the two vertical strokes, and thus
m4,0 has a large negative number (−1.72). The distribution of letter “
T”, on the other hand, has a strong peak in the middle, and
m4,0 has a positive number (0.046). Mixed moments indicate the clustering of ink pixels around oblique axes. For example,
m2,2 is large for letter “
X”, but small for Chinese character □ (1.60 vs. 0.81).