Particle Analyser Outputs

As stated, I am using the particle analyser function in BoneJ. I am fitting particles with best-fit ellipsoids and I am interested in how these ellipsoids are oriented in space. One of the options allows me to keep track of the unit vectors of this ellipsoid. I am confused as to how they are displayed in results.

The documentation says that the corresponding output is a list of six vectors the following way: V00 - V22: elements of the 3×3 rotation matrix of the best-fit ellipsoid. I do not know what a 3×3 rotation matrix is. I suspect that this set of vectors that describe the Major axis, intermediate axis, and minor axis respectively, however this isn’t clear from the documentation.

Here’s some info on rotation matrices from Wolfram and from Wikipedia. You can think of this matrix as a set of 3 orthogonal column vectors, each a unit vector (i.e. with length = 1) and each pointing in the direction of one of the 3 ellipsoid axes. So, it is telling you how your best fit ellipsoid is rotated away from the coordinate frame of your image stack (i.e. away from the x, y, and z axes of your image). The rotation matrix is sorted so that the leftmost column (V0) relates to the largest eigenvalue, which relates to the shortest axis.

I see, so just to be clear, the First column vector is the the unit vector of the smallest axis, the second relates to the intermediate axis, and the third column vector relates to the largest axis. So if the largest axis was oriented along the y-axis, the third column vector would be <0 ,1 ,0>. I appreciate the help.

Yes, that’s exactly correct.

Related to this, some of the outputs I am getting look like this

Clearly, the volume of the ellipsoid is not remotely close to the calculated volume and enclosed volume. Do you know the reason for this discrepancy, and if so, which one is more accurate?

It’s possible that none of them are very accurate, if your resolution is low in relation to feature size. What’s your pixel spacing, and how many pixels are representing this particle?

The ellipsoid’s volume is only an accurate reflection of the fitted shape’s volume if the fitted shape is at least approximately ellipsoidal. If it’s something else, the ellipsoid dimensions are likely to be way off. You can check this out by cropping down and isolating this particle, then looking at it with the 3D visualisation.

The enclosed volume is measured from the surface mesh. If the particle is close in size to pixel spacing and you have applied a bit of smoothing or subsampling during mesh creation, it’s likely that your enclosed volume is completely wrong. Vol. is just a count of pixels in the particle multiplied by the x, y & z pixel spacing, so is probably the least inaccurate / best estimate in this case.