05/13/2008

Morphological Parameters for Particles - Illustrations

These are some morphological parameters (statistics or 'stats') that Lispix can calculate and display.

Some of these parameters, such as length and width, are basic, exact, well defined, and determined directly from the convex hull, which is quickly calculated from the particle boundary.

• Length (maximum caliper diameter, or maximum Feret diameter – for all orientations).
• Width (minimum caliper diameter, or minimum Feret diameter – for all orientations).
• Minimum inscribed circle (another useful width measurement, the maximum of the Euclidian Distance Map).
• Bounding Rectangle (dependent on orientation, so not as useful.)

### Convex hull and Maximum Inscribed Circle

To display using the Blob Tool:  draw / convex hulls

The Convex Hull, green, is calculated from the particle outline using a fast and exact algorithm1.
Maximum Inscribed Circle, red,calculated from maximum of the Euclidian Distance Map2 of the mask.

Bounding rectangle, yellow.  The bounding rectangle, yellow, is calculated from the particle outline.
Note that it depends on the particle orientation.

Convex Hull references and algorithms.

### Maximum Caliper Diameter

To display using the Blob Tool:  draw / maximum caliper diameters.

Particle length (maximum caliper diameter) is the distance between the most separated points of the convex hull, and thus of the outline.

Minimum Caliper Diameter

To display using the Blob Tool:  draw / minimumcaliper diameters.

Particle width (minimum caliper diameter), yellow line, is calculated using the convex hull3,4. This is the minimum separation, for any particle orientation, for parallel lines that just touch each side of the particle.

The line of support, tan line, a segment of the convex hull, is one of the parallel lines, i.e., one of the ‘sides’ of the ‘caliper’

### References

1. Melkman, A.A., “Online Construction of the Convex-Hull of a simple Polyline”, Information Processing Letters 25 (1987) 11-12.
2. P. E. Danielsson, Comp. Graph Image Proc 14 (1980) 227-248.
3. Preparata, F.P. and Shamos, M.I., COMPUTATIONAL GEOMETRY, An Introduction. Springer-Verlag, N.Y., 1985, p. 174.
4. Hormoz Prizadeh, Rotating Calipers, http://cgm.cs.mcgill.ca/~orm/width.html