====== Details about 3D ellipsoid fitting ======

The [[plugin:analysis:3d_analysis:start|3D ellipsoid fitting plugin]] will fit a 3D ellipsoid to objects in a labelled image. 

The ellipsoid axes and radii are computed using 3D moments : 

  * sx<sup>2</sup> = ∑(x-Cx)<sup>2</sup>/Vobj
  * sy<sup>2</sup> = ∑(y-Cy)<sup>2</sup>/Vobj
  * sz<sup>2</sup> = ∑(z-Cz)<sup>2</sup>/Vobj
  * sxy = ∑(x-Cx)(y-Cy)/Vobj
  * sxz = ∑(x-Cx)(z-Cz)/Vobj
  * syz= ∑(y-Cy)(z-Cz)/Vobj

The three eigen values and vectors are then computed from this matrix : 
| sx<sup>2</sup> | sxy | sxz |
| sxy | sy<sup>2</sup> | syz |
| sxz | syz | sz<sup>2</sup> |

The three radii are then : √(5*eigen value)


The different results are :

  * Cx,Cy,Cz : **centre** of the object (in pixel coordinate)
  * Vx, Vy, Vz : coordinate of the **main elongation** axis
  * R1, R2, R3 : the tree radii of the fitted ellipsoid
  * XY, XZ, YZ : **angle** between the main axis of the ellipsoid and the 3 planes
  * Vobj : **volume** of the object
  * Vell : volume of the fitted ellipsoid
  * Vbb : volume of the bounding box around the object
  * Vbbo (experimental) : volume of the bounding box oriented in the direction of the main axis
  * The two **poles** of the object as the two voxels used to compute Feret diameter, and the two poles of the fitting ellipsoid.

{{:tutorial:plugins:ellipsoid.png|Object and fitted ellipsoid with main elongation axis}}

Object and fitted ellipsoid with main elongation axis

{{:tutorial:plugins:ellipsoid1.png|Bounding box and oriented bounding box}}

Bounding box and oriented bounding box