Gradient pattern analysis: Difference between revisions

Revision as of 05:30, 24 October 2010

Revision as of 04:53, 27 December 2010

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'''Gradient pattern analysis''' ('''GPA''')<ref name=rosa2000>Rosa, R.R., Pontes, J., Christov, C.I., Ramos, F.M., Rodrigues Neto, C., Rempel, E.L., Walgraef, D. ''Physica A'' '''283''', 156 (2000).</ref> is a geometric computing method for characterizing [[symmetry]] breaking of an ensemble of asymmetric vectors regularly distributed in a square lattice. Usually, the lattice of vectors represent the first-order [[gradient]] of a scalar field, here an ''M x M'' square amplitude [[matrix (mathematics)]]. An important property of the gradient representation is the following: A given ''M x M'' matrix where all amplitudes are different results in an ''M x M'' gradient lattice containing <math>N_{V} = M^2</math> asymmetric vectors. As each vector can be characterized by its norm and phase, variations in the <math>M^2</math> amplitudes can modify the respective <math>M^2</math> gradient pattern.

The original concept of GPA was introduced by Rosa, Sharma and Valdivia in 1999.<ref name=Rosa99>Rosa, R.R.; Sharma, A.S.and Valdivia, J.A. ''Int. J. Mod. Phys. C'', '''10''', 147 (1999), [[doi:10.1142/S0129183199000103]].</ref>. Usually GPA is applied for spatio-temporal pattern analysis in physics and environmental sciences operating on time-series and digital images.

== Calculation ==

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For a complex extended pattern (matrix of amplitudes of a spatio-temporal pattern) composed by locally asymmetric fluctuations, <math>G_{A}</math> is nonzero, defining different classes of irregular fluctuation patterns (1/f noise, chaotic, reactive-diffusive, etc.).

Besides <math>G_{A}</math> other measurements (called ''gradient moments'') can be calculated from the gradient lattice.<ref name=rosa03>Rosa, R.R.; Campos, M.R.; Ramos, F.M.; Vijaykumar, N.L.; Fujiwara, S.; Sato, T. ''Braz. J. Phys.'' '''33''', 605 (2003).</ref>. Considering the sets of local norms and phases as discrete compact groups, spatially distributed in a square lattice, the gradient moments have the basic property of being globally invariant (for rotation and modulation).

The primary research on gradient lattices applied to characterize [[weak turbulence]] from X-ray images of [[solar active regions]] was developed in the Department of Astronomy at [[University of Maryland, College Park]], USA. A key line of research on GPA's algorithms and applications has been developed at Lab for Computing and Applied Mathematics (LAC) at National Institute for Space Research ([[INPE]]) in Brazil.

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<references/>

{{DEFAULTSORT:Gradient Pattern Analysis}}

[[Category:Geometry]]

[[Category:Applied mathematics]]