Shape factor (image analysis and microscopy): Difference between revisions

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'''Shape factors''' are [[Dimensionlessdimensionless quantity|dimensionless quantities]] used in [[image analysis]] and [[Microscope|microscopy]] that numerically describe the shape of a particle, independent of its size. Shape factors are calculated from measured [[dimension]]s, such as [[diameter]], [[Chord (geometry)|chord]] lengths, [[area]], [[perimeter]], [[centroid]], [[Moment (mathematics)|moments]], etc. The dimensions of the particles are usually [[Measure (mathematics)|measured]] from two-dimensional [[Cross section (geometry)|cross-sections]] or [[Orthographic projection|projections]], as in a microscope field, but shape factors also apply to three-dimensional objects. The particles could be the grains in a [[Metallography|metallurgical]] or [[Ceramography|ceramic microstructure]], or the microorganisms in a [[Microbiological culture|culture]], for example. The dimensionless quantities often represent the degree of [[Deviation (statistics)|deviation]] from an ideal shape, such as a [[Roundness (object)|circle]], sphere or equilateral [[polyhedron]].<ref>L. Wojnar & K.J. Kurzydłowski, et al., ''Practical Guide to Image Analysis'', [[ASM International (society)|ASM International]], 2000, p 157-160, {{ISBN |0-87170-688-1}}.</ref> Shape factors are often ''normalized'', that is, the value ranges from zero to one. A shape factor equal to one usually represents an ideal case or maximum symmetry, such as a circle, sphere, square or cube.

{{seealso|Shape factor}}

'''Shape factors''' are [[Dimensionless quantity|dimensionless quantities]] used in [[image analysis]] and [[Microscope|microscopy]] that numerically describe the shape of a particle, independent of its size. Shape factors are calculated from measured [[dimension]]s, such as [[diameter]], [[Chord (geometry)|chord]] lengths, [[area]], [[perimeter]], [[centroid]], [[Moment (mathematics)|moments]], etc. The dimensions of the particles are usually [[Measure (mathematics)|measured]] from two-dimensional [[Cross section (geometry)|cross-sections]] or [[Orthographic projection|projections]], as in a microscope field, but shape factors also apply to three-dimensional objects. The particles could be the grains in a [[Metallography|metallurgical]] or [[Ceramography|ceramic microstructure]], or the microorganisms in a [[Microbiological culture|culture]], for example. The dimensionless quantities often represent the degree of [[Deviation (statistics)|deviation]] from an ideal shape, such as a [[Roundness|circle]], sphere or equilateral [[polyhedron]].<ref>L. Wojnar & K.J. Kurzydłowski, et al., ''Practical Guide to Image Analysis'', [[ASM International]], 2000, p 157-160, ISBN 0-87170-688-1.</ref> Shape factors are often ''normalized'', that is, the value ranges from zero to one. A shape factor equal to one usually represents an ideal case or maximum symmetry, such as a circle, sphere, square or cube.

==Aspect ratio==

*The most common shape factor is the [[aspect ratio]], a function of the largest diameter and the smallest diameter [[Orthogonality|orthogonal]] to it:

:<math>A_R = \frac{d_\min}{d_\max}</math>

The normalized aspect ratio varies from approaching zero for a very elongated particle, such as a grain in a cold-worked metal, to near unity for an [[wikt:equiaxed|equiaxed]] grain. The reciprocal of the right side of the above equation is also used, such that the AR varies from one to approaching infinity.

==Circularity==

Another very common shape factor is the circularity (or [[isoperimetric quotient]]), a function of the perimeter ''P'' and the area ''A'':

:<math>f_\text{circ} = \frac {4 \pi A} {P^2}</math>

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==Elongation shape factor==

The less-common elongation shape factor is defined as the square root of the ratio of the two [[Moment of inertia|second moments]] ''i_n'' of the particle around its principal axes.<ref name="Exner">H.E. Exner & H.P. Hougardy, ''Quantitative Image Analysis of Microstructures'', DGM Informationsgesellschaft mbH, 1988, p 33-39, {{ISBN |3-88355-132-5}}.</ref>

:<math>f_\text{elong} = \sqrt{\frac{i_2}{i_1}}</math>

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:<math>f_\text{comp} = \frac{A^2}{2 \pi \sqrt{{i_1}^2 + {i_2}^2}}</math>

The ''f''_comp of a circle is one, and much less than one for the cross-section of an [[I-beam]].

==Waviness shape factor==

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==An application of shape factors==

[[Greenland]], the largest island in the world, has an area of 2,166,086&nbsp;km²; a coastline (perimeter) of 39,330&nbsp;km; a north-southnorth–south length of 2670&nbsp;km; and an east-westeast–west length of 1290&nbsp;km. The aspect ratio of Greenland is

:<math>A_R = \frac{1290}{2670} = 0.483</math>

The circularity of Greenland is

:<math>f_\text{circ} = \frac {4 \pi (2166086)} {39330^2} = 0.0176.</math>

The aspect ratio is agreeable with an eyeball-estimate on a globe. Such an estimate on a typical flat map, using the [[Mercator projection]], would be less accurate due to the distortiondistorted ofscale at high- [[latitude]] [[Orthographic projection (cartography)|projections]]s. The circularity is deceptively low, due to the [[fjord]]s that give Greenland a very jagged coastline (see the [[Coastlinecoastline paradox|coastline]]). A low value of circularity does not necessarily indicate a lack of symmetry!, Andand shape factors are not limited to microscopic objects!.

== References ==

{{reflist}}

== Further reading ==

* J.C. RustRuss & R.T. Dehoff, ''Practical Stereology'', 2nd Ed., Kluwer Academic, 2000.

* E.E. Underwood, ''Quantitative Stereology'', Addison-Wesley Publishing Co., 1970.

* G.F. VanderVoort, ''Metallography: Principles and Practice'', ASM International, 1984.