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=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 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.

:<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, a function of the perimeter ''P'' and the area ''A'':

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

The circularity of a circle is one, 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 to infinity.

===Circularity===

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

===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{elongcirc} = \sqrt{\frac {i_24 \pi A} {i_1}P^2}</math>

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.

===CompactnessElongation 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_{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]].

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{wavelong} = \sqrt{\frac{P_{cvx}i_2}{Pi_1}}</math>

===Waviness shape factor===

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

:<math>f_{wav} = \frac{P_{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>

===ElongationCompactness 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 of the perimeter ''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

==An Application of Shape Factors==

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

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

The circularity of Greenland is

:<math>f_{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 flat map would be less accurate due to the distortion of high-[[latitude]] [[Orthographic projection (cartography) | projections]]. The circularity is deceptively low, due to the [[fjord]]s that give Greenland a very jagged [[Coastline paradox | coastline]]. A low value of circularity does not necessarily indicate a lack of symmetry! And shape factors are not limited to microscopic objects!

The circularity of Greenland is

== References ==

:<math>f_\text{circ} = \frac {4 \pi A(2166086)} {P39330^2} = 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 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:Image processing]]

[[Category:Microscopy]]