Inertia3dData =	greyInertiaMsr3d (inImg)
Inertia3dData =	greyInertiaMsr3d (inImg,inMaskImg)

Detailed Description

Computes the global inertia based on the grey levels of the input 3D image.

This algorithm computes the second order central moment for the entire 3D image, ponderated by the pixels intensities, thanks to the inertia matrix $ M $ .

$M = \begin{bmatrix} m_{xx} & m_{xy} & m_{xz} \\ m_{yx} & m_{yy} & m_{yz} \\ m_{zx} & m_{zy} & m_{zz} \end{bmatrix}$

Whose components are computed as follows :

$\begin{array}{r c l} m_{xx} & = & \sum_{\lbrace x, y, z \rbrace \in \Omega}{ \left(x - \bar{x} \right)^2 I(x, y, z)} \\ m_{xy} = m_{yx} & = & \sum_{\lbrace x, y, z \rbrace \in \Omega}{ \left(x - \bar{x} \right) \left(y - \bar{y} \right) I(x, y, z)} \\ m_{xz} = m_{zx} & = & \sum_{\lbrace x, y, z \rbrace \in \Omega}{ \left(x - \bar{x} \right) \left(z - \bar{z} \right) I(x, y, z)} \\ m_{yy} & = & \sum_{\lbrace x, y, z \rbrace \in \Omega}{ \left(y - \bar{y} \right)^2 I(x, y, z)}\\ m_{yz} = m_{zy} & = & \sum_{\lbrace x, y, z \rbrace \in \Omega}{ \left(y - \bar{y} \right) \left(z - \bar{z} \right) I(x, y, z)}\\ m_{zz} & = & \sum_{\lbrace x, y, z \rbrace \in \Omega}{ \left(z - \bar{z} \right)^2 I(x, y, z)} \end{array}$

Where $ I(x, y, z) $ is the image intensity at the position $ (x, y, z) $ , $\bar{x}$ , $\bar{y}$ and $\bar{z}$ are the grey barycenter coordinates and $\Omega$ is the set of pixel coordinates in the image.

From these composents, the following descriptors are extracted :

the minimum, intermediate and maximum eigen values of ( $\lambda_{Min}$ , $\lambda_{Inter}$ and $\lambda_{Max}$ ),
the Euler orientation angles $\alpha$ , $\beta$ and $\chi$ ,
the spherical orientation angles $\theta$ and $\phi$ .