The '''Gradient Patternpattern Analysisanalysis''' ( '''GPA ''') <ref (\cite{name=rosa2000 }>Rosa, \cite{rosa03}) R.R., basicallyPontes, consistsJ., inChristov, 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 geometrical bilateral [[symmetry breaking ]] of an ensemble of asymmetricsymmetric vectors regularly distributed in a square lattice. Usually, the lattice of vectors represent the first-order [[gradient ]] of a scalar field, here an $''M \times x M $'' square amplitude [[matrix (mathematics)|matrix ]]. An important property of the gradient representation is the following: A given $''M \times x M $'' matrix where all amplitudes are different results ain 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. ▼== Introduction ==
▲The ''Gradient Pattern Analysis'' (GPA) (\cite{rosa2000},\cite{rosa03}) basically consists in 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 $M\times M$ square amplitude matrix. An important property of the gradient representation is the following: A given $M\times M$ matrix where all amplitudes are different results a
$M\times M$ gradient lattice containing $N_{V}=M^{2}$ asymmetric vectors. As each vector can be characterized by its norm and phase, variations in the $M^{2}$ amplitudes can modify the respective $M^{2}$ gradient pattern.
The original ideas on GPA was introduced by Rosa, Sharma and Valdivia, 1999 \cite{Rosa99}. Usually GPA is applied for spatio-temporal pattern analysis in physics and environmental sciences operating on time-series and images. ▼
▲The original ideasconcept onof 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 ), \cite{ Rosa99{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.
== ContentsCalculation ==
By connecting all vectors using a [[Delaunay triangulation ]] criteriumcriterion it is possible to characterize gradient asymetriesasymmetries computing the so-called ''gradient asymmetry coefficient'', that has been defined as: ▼[[Calculation]]
<math>G_A=\frac{ |N_C-N_V |}{N_V} </math>, ▼
[[Relation to other methods]]
where $N_V<math> N_{V} > 0 $</math> is the total number of asymmetric vectors , and $N_C$<math>N_{C}</math> is the number of Delaunay connections among them . and the property <math>N_{C} > N_{V}</math> ▼[[References]]
is valid for any gradient square lattice.
[[External Links]]
As the asymmetry coefficient is very sensitive to small changes in the phase and modulus of each gradient vector, it can distinguish complex variability patterns (bilateral asymmetry) even when they are very similar but consist of a very fine structural difference. NotNote that, unlike most of the statistical tools, the GPA does not rely on the statistical properties of the data but ▼
== Calculation ==
▲By connecting all vectors using a Delaunay triangulation criterium it is possible to characterize gradient asymetries computing the so-called ''gradient asymmetry coefficient'', that has been defined as:
\begin{equation}
▲G_A=\frac{|N_C-N_V|}{N_V},
\end{equation}
▲where $N_V>0$ is the total number of asymmetric vectors and $N_C$ is the number of Delaunay connections among them.
▲As the asymmetry coefficient is very sensitive to small changes in the phase and modulus of each gradient vector, it can distinguish complex variability patterns even when they are very similar but consist of a very fine structural difference. Not that, unlike most of the statistical tools, the GPA does not rely on the statistical properties of the data but
depends solely on the local symmetry properties of the correspondent gradient pattern.
For a complex extended pattern (matrix of amplitudes of a spatio-temporal pattern) composed by locally asymmetric fluctuations, $<math>G_{A}</math> is nonzero, defining different classes of irregular fluctuation patterns (1/f noise, chaotic, reactive-diffusive, etc.).
G_{A}$ 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 formfrom the gradient lattice.<ref \cite{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 [[Wave turbulence|weak wave turbulence]] from X-ray images of [http://solar.physics.montana.edu/canfield/papers/EAA.2023.pdf 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.
== Relation to other methods ==
When GPA is conjugated with [[wavelet analysis]], then the method is called ''Gradient Spectralspectral Analysisanalysis'' (GSA), usually applied to short time series analysis.<ref \cite{name=rosa08})>Rosa, R.R. et al., ''Advances in Space Research'' '''42''', 844 (2008), {{doi|10.1016/j.asr.2007.08.015}}.</ref>
== References ==
\bibitem{rosa99} R. R. Rosa, A. S. Sharma and J. Valdivia, {\em Int. J. Mod. Phys. C \/} {\bf 10}, 147 (1999).
\bibitem{rosa2000} R. R. Rosa, J. Pontes, C. I. Christov, F. M. Ramos, C. Rodrigues Neto, E. L. Rempel, D. Walgraef, {\em Physica A \/} {\bf 283}, 156 (2000).
\bibitem{assireu2002} A. T. Assireu, R. R. Rosa, N. L. Vijaykumar, J. A. Lorenzetti, E. L. Rempel, F. M. Ramos, L. D. Abreu Sá, M. J. A. Bolzan, A. Zanandrea, {\em Physica D \/} {\bf 168} 397 (2002).
\bibitem{rosa03} R. R. Rosa, M. R. Campos, F. M. Ramos, N. L. Vijaykumar, S. Fujiwara, T. Sato, {\em Braz. Jour. Phys. \/} {\bf 33}, 605 (2003).
\bibitem{baroni06} M. P. M. A. Baroni, R. R. Rosa, A. Ferreira da Silva, I. Pepe, L. S. Roman, F. M. Ramos, R. Ahuja, C. Persson, E. Veje, {\em Microelectronics Journal \/} {\bf 37}, 290 (2006).
\bibitem{rosa07} R. R. Rosa, M. P. M. A. Baroni, G. T. Zaniboni, A. Ferreira da Silva, L. S. Roman, J. Pontes and M. J. A. Bolzan, {\em Physica A \/} {\bf 386}, 666 (2007).
\bibitem{rosa08} R.R.Rosa et al., {\em Advances in Space Research} {\bf 42}, 844 (2008), doi:10.1016/j.asr.2007.08.015.
<references/>
== External Links ==
[[Category:Geometric algorithms]]
Lab for Computing and Applied Mathematics
[[Category:Signal processing]]
(MATLAB code for Gradient Spectral Analysis).
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