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its moments will be

which can be used to quantify shape independently from location and size.

In applications where the shape of a segmented region must be quantified in a manner that is insensitive to its orientation, rotation invariant metrics are used. The scale invariant central moments given in Eq. (8) can be combined further to obtain translation, scale, and rotation invariant descriptors .

=(i?3o + ^l2)2 + (i?2l + ^3 )2 (l2) = (i?3o - 3^l2)(^l2 + ^o) (^l2 + ^3o)2 - 3(^2l + ^

+ (3^2l - ^3)(^2l + ^o3) 3(^l2 + ^3o)2 - (ihl + %3)2

This definition yields = = 0. Central moments can be obtained in terms of noncentral moments. Some examples are:

= (iho - ^2) (^l2 + ^3o)2 - (i?2l + ^o3)2 + 4^ll(^l2 + ^3o)(^2l + ^3)

If an image is scaled up by a coefficient 5 larger than 1, so that the scaled image is

One way of achieving scale invariant quantification is to scale images first to a standard size by using a scale coefficient commensurate with the size of the object. The area of the object is given by ^00, and the object could be conceptually reduced to unit area by scaling down each axis with ^^00, that is /Vj"00). This transformation is equivalent to defining the scale invariant central moment

+ (3^12 - ^30)(^21 + ^03) |_3(^12 + ^30)2 - (ihi + ^03)2

The first row of Fig. 4 shows three shapes with increasing roughness. The shapes are scaled by a factor of two in the second row and rotated 60° counterclockwise in the third row of Fig. 4. The values of for these nine shapes are

"0.16766 0.17150 0.1760T 0.16796 0.17189 0.17618 0.16792 0.17185 0.17616

where each value is shown in the position of the corresponding shape in Fig. 4. Each column corresponds to one shape and exhibits a noteworthy level of invariance. A monotonous increase in the value of is also observed with increasing shape roughness. The other invariant descriptors also have similar trends; for example the values of for shapes of Fig. 4 are:

The orientation of an object, defined as the direction along FIGURE 4 Top row: three shapes with increasing roughness. Middle row: same shapes scaled up by two. Bottom row: shapes rotated counterclockwise by 60°.

is obtained by tracing all N pixels of the boundary. To achieve scale invariance, the normalized radial distance sequence r(n) is obtained by normalizing d(n) with the maximal distance. The sequence r(n) is analyzed further to extract shape metrics such as the entropy

where hk is the K-bin probability histogram that represents the distribution of r(n), as well as statistical moments

FIGURE 4 Top row: three shapes with increasing roughness. Middle row: same shapes scaled up by two. Bottom row: shapes rotated counterclockwise by 60°.

which the object is most elongated can be obtained with the angle 6: