Module: Moments Of Inertia ()

Description:

For an introduction, see section Analysis.

This module computes some global characteristics of the shape.

The first order moments define the centroid or center of mass. Formulas are given for binary images but the Moments of Inertia module works with all image types. For other cases, just replace the value by . They are defined as:



and in the discrete case as:



is the area and a point in the object.

The second order moments are defined in the continuous case as:







The second order moments are defined in the discrete case as:







The orientation is defined as the direction of its major inertia axis. It is given as the eigenvector of the largest eigenvalue of the inertia matrix:



This is a good measure of the orientation for simple, roughly convex objects. The result falls between and

The eccentricity is defined as:



and are the inertia matrix eigenvalues.

A disk or a cross has a null eccentricity as . The eccentricity increases with the difference between the eigenvalues, and thus measures the elongation of the object. It also indicates in some cases a privileged direction, corresponding to a large eigenvalue and a small one, i.e.: has a high value, though two orthogonal privileged directions mean two large eigenvalues and a smaller difference .

See also: Area, Euler Number, Fractal Dimension, Variance-Covariance matrix, Volume Fraction.

Connections:

Input Image [required]
The image to be analyzed. Supported types include: grayscale/color image (Uniform Scalar Field/Uniform Color Field), binary (Uniform Label Field with 2 labels) and label (Uniform Label Field) images.

Ports:

Interpretation

This port specifies whether the input will be interpreted as a 3D volume or a stack of 2D images for processing.