3D moment invariants

Bob Fisher

Moments are a measure of the spatial distribution of 'mass' of a shape. In the case of binary digital datasets, this is the distribution of pixels (in 2D) and voxels (in 3D) of a shape.

It is possible to compute moment invariants of 3D point distributions (i.e. voxels) that are invariant to translation and rotation, in the same manner as 2D moment invariants.

Let $\rho(x,y,z)$ be a local continuous density function. For example, this can be 1 inside voxels belonging to an object and 0 in free space.

Let $(\bar{x}. \bar{y}, \bar{z})$ be the centroid of the object. Using the centroid in the moment calculation below gives translation invariance.

Define the $\mu_{pqr}$ central moment as:

$$\mu_{pqr} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} ... ...bar{x})^p (y - \bar{y})^q (z - \bar{z})^r \rho(x,y,z) dx dy dz$$

Approximating this formula to digital voxel space is straightforward, using summation in place of integration.

With this, we can now define three second order moment invariants:

$$J_1 = \mu_{200} + \mu_{020} + \mu_{002}$$

$$J_2 = \mu_{200} \mu_{020} + \mu_{200} \mu_{002} + \mu_{020} \mu_{002} - \mu_{110}^2 - \mu_{101}^2 - \mu_{011}^2$$

$$J_3 = \mu_{200} \mu_{020} \mu_{002} + 2 \mu_{110} \mu_{101}... ...2} \mu_{110}^2 - \mu_{020} \mu_{101}^2 - \mu_{200} \mu_{011}^2$$

that are invariant to translation and rotation.

Fully closed shapes can have scale normalization applied to the centralized moments, in the same manner as 2D moment invariants, i.e. use:

$$\mu_{pqr} \leftarrow \frac{\mu_{pqr}}{(Volume)^{(p+q+r+3)/3}}$$

The invariants can also be used to describe points on surfaces in a translation and rotationally invariant manner. Here, the center of a sphere of a fixed radius is placed at the surface point. The portion of the object within the sphere around that point is used for computing the density function, with 1 for points belonging to the object and 0 for free space.

F. A. Sadjadi and E. L Hall. Three-dimensional moment invariants. IEEE Trans. on Pattern Analysis and Machine Intelligence. 2(2), pp 127-136, March 1980.