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}}

===Aspect ratio===

'''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.

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

===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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The circularity of a circle is 1<!-- The number 1. NOT "...is one of them" -->, and much less than one for a [[starfish]] footprint. The reciprocal of the circularity equation is also used, such that ''f''_circ varies from one for a circle to infinity.

===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>

===Compactness shape factor===

The ''[[Compactness measure of a shape | compactness]] shape factor'' is a function of the polar second moment ''i_n'' of a particle and a circle of equal area ''A''. <ref name="Exner"/>

:<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===

The [[waviness]] shape factor of the perimeter is a function of the convex portion ''P''_cvx of the perimeter to the total. <ref name="Exner"/>

===Waviness shape factor===

The waviness shape factor of the perimeter is a function of the convex portion ''P''_cvx of the perimeter to the total. <ref name="Exner"/>

:<math>f_\text{wav} = \frac{P_\text{cvx}}{P}</math>

Some properties of metals and ceramics, such as [[fracture toughness]], have been linked to grain shapes.<ref>P.F. Becher, et al., "Microstructural Design of Silicon Nitride with Improved Fracture Toughness: I, Effects of Grain Shape and Size," ''J. American Ceramic Soc.'', Vol 81, Issue 11, P 2821-2830, Nov 1998.</ref> <ref>T. Huang, et al., "Anisotropic Grain Growth and Microstructural Evolution of Dense Mullite above 1550°C," ''J. American Ceramic Soc.'', Vol 83, Issue 1, P 204-10, Jan 2000.</ref>

==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 [[Coastlinecoastline paradox(see |the [[coastline paradox]]). 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.

[[Category:GeometryImage processing]]

[[Category:Microscopy]]