Moment-Based Shape Features

19.3.1 Defi nitions

In this section we present a systematic approach to object shape descrip- tion. We fi rst defi ne moments for gray value and binary images and then show how to extract useful shape parameters from this approach. We will discuss Fourier descriptors in a similar manner in Section 19.4.

We used moments in Section 3.2.2 to describe the probability den- sity function for gray values. Here we extend this description to two dimensions and defi ne the moments of the gray value function g( x ) of an object as

 

where

µ_{p, q} = ∫ (x₁ − x₁)p(x₂ − x₂)q g( x )d²x,                                       (19.1)

x_i = ∫ x_ig( x )d²x  ∫ g( x )d²x.                                      (19.2)

 

=

The integration includes the area of the object. Instead of the gray value, we may use more generally any pixel-based feature to compute object moments. The vector x (x₁, x₂) is called the center of mass of the object by analogy to classical mechanics. Think of g( x ) as the density ρ ( x ) of the object; then the zero-order moment µ_{0, 0} becomes the total mass of the object.

All the moments defi ned in Eq. (19.1) are related to the center of mass. Therefore they are often denoted as central moments. Central moments are translation invariant and thus are useful features for describing the shape of objects.

For discrete binary images, the moment calculation reduces to

µ_{p, q} =.(x₁ − x₁)p(x₂ − x₂)q.                                        (19.3)

The summation includes all pixels belonging to the object. For the de- scription of object shape we may use moments based on either binary, gray scale or feature images. Moments based on gray scale or feature images refl ect not only the geometrical shape of an object but also the distribution of features within the object. As such, they are generally diff erent from moments based on binary images.

19.3 Moment-Based Shape Features                                             501

 

Figure 19.5: Principal axes of the inertia tensor of an object for rotation around the center of mass.

 

Scale-Invariant Moments

=

Often it is necessary to use shape parameters that do not depend on the size of the object. This is always required if objects observed from diff erent distances must be compared. Moments can be normalized in the following way to obtain scale-invariant shape parameters. If we scale an object g( x ) by a factor of α, g'( x ) g( x /α ), its moments are scaled by

µp', q = α p+q+2 µp, q.

We can then normalize the moments with the zero-order moment, µ_{0, 0}, to gain scale-invariant moments

µ(p+q+2)/2

µ¯ =   µ p, q         .

0, 0

 

+ =

Because the zero-order moment of a binary object gives the area of the object (Eq. (19.3)), the normalized moments are scaled by the area of the object. Second-order moments (p q 2), for example, are scaled with the square of the area.

 

Moment Tensor

Shape analysis beyond area measurements starts with the second-order moments. The zero-order moment just gives the area or “total mass” of a binary or gray value object, respectively. The fi rst-order central moments are zero by defi nition.

The analogy to mechanics is again helpful to understand the meaning of the second-order moments µ_{2, 0}, µ_{0, 2}, and µ_{1, 1}. They contain terms in which the gray value function, i. e., the density of the object, is multiplied

502                                                             19 Shape Presentation and Analysis

 

Σ J =

by squared distances from the center of mass. Exactly the same terms are also included in the inertia tensor that was discussed in Section 13.3.7 (see Eqs. (13.23) and (13.24)). The three second-order moments form the components of the inertia tensor for rotation of the object around its center of mass:

µ2, 0  − µ1, 1

− µ1, 1         µ0, 2

Σ .                                 (19.4)

Because of this analogy, we can transfer all the results from Section 13.3 to shape description with second-order moments. The orientation of the object is defi ned as the angle between the x axis and the axis around which the object can be rotated with minimum inertia. This is the eigen- vector of the minimal eigenvalue. The object is most elongated in this direction (Fig. 19.5). According to Eq. (13.12), this angle is given by

φ = 1 arctan  2µ 1, 1              .                                (19.5)

2          µ2, 0 − µ0, 2

As a measure for the eccentricity ε, we can use what we have defi ned as a coherence measure for local orientation Eq. (13.15):

=

(µ2, 0 − µ0, 2)2 + 4µ2

ε

(µ2, 0 + µ0, 2)2

1, 1.                                (19.6)

The eccentricity ranges from 0 to 1. It is zero for a circular object and one for a line-shaped object. Thus, it is a better-defi ned quantity than circularity with its odd range (Section 19.5.3).

+

Shape description by second-order moments in the moment tensor essentially models the object as an ellipse. The combination of the three second-order moments into a tensor nicely results in two rotation-in- variant terms, the trace of the tensor, or µ_{2, 0} µ_{0, 2}, which gives the ra- dial distribution of features in the object, and the eccentricity Eq. (19.6), which measures the roundness, and one term which measures the ori- entation of the object. Moments allow for a complete shape description [148]. The shape description becomes more detailed the more higher- order moments are used.

 

Fourier Descriptors

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